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11-03-09 BOT Agenda PacketThe Regular Meeting of the Board of Trustees will begin immediately following the conclusion of the Board of Trustees Workshop but not prior to the posted start time. Page 1 of 4 WESTLAKE ACADEMY Mission / Vision Statement Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who become well-balanced, responsible global citizens. BOARD OF TRUSTEES SPECIAL CALLED MEETING AGENDA November 3, 2009 WESTLAKE TOWN HALL 3 VILLAGE CIRCLE – 2ND FLOOR COUNCIL CHAMBERS/MUNICIPAL COURT ROOM Workshop Session 5:00 p.m. Regular Session 7:00 p.m. 1. CALL TO ORDER Workshop Session 2. REVIEW OF CONSENT AGENDA ITEMS FROM NOVEMBER 3, 2009, TRUSTEES REGULAR MEETING AGENDA. Page 2 of 4 (concepts and principles that underlie good governance) Board Assessment and Development 3. PRESENTATION AND DISCUSSION OF DRAFT BOARD OF TRUSTEES’ POLICY CONCERNING THE GRADUATION POLICY FOR WESTLAKE ACADEMY. 4. PRESENTATION AND DISCUSSION OF THE ACADEMY STAFF RETREAT TO THE RON CLARK FACILITY. 5. PRESENTATION AND DISCUSSION OF A DRAFT BOARD OF TRUSTEES POLICY CONCERNING THE PRESENTING OF INFORMATION/REPORTS FROM STAFF. 6. PRESENTATION AND DISCUSSION OF A DRAFT BOARD OF TRUSTEES POLICY CONCERNING THE ADHERENCE TO THE WESTLAKE ACADEMY CHARTER, TEXAS EDUCATION AGENCY AND INTERNATIONAL BACCALAUREATE GUIDELINES. (ensuring that operationally inappropriate things are not occurring.) Is everything occurring appropriately? 7. PRESENTATION AND DISCUSSION OF A DRAFT BOARD OF TRUSTEES’ POLICY RELATIVE TO CAPITAL PROJECTS AND FACILITY MODIFICATION/IMPROVEMENT PROJECTS AT THE WESTLAKE ACADEMY CAMPUS. (concepts and principles that underlie good governance) How well are the Students doing? 8. PRESENTATION AND DISCUSSION REGARDING THE WESTLAKE ACADEMY - SENIOR CLASS OF 2010 GRADUATION AND ACTIVITIES ALONG WITH GRADE LEVEL PLANNING AND COMMUNICATION. 9. PRESENTATION AND DISCUSSION BY WESTLAKE ACADEMY HEAD OF PRIMARY AND PRIMARY YEARS PROGRAMME COORDINATOR REGARDING DEVELOPMENTAL CONTINUA, AND, IN PARTICULAR, THE INTRODUCTION OF THE MATHEMATICS DEVELOPMENTAL CONTINUUM. 10. FUTURE AGENDA ITEMS: Any Board member may request at a workshop and / or Board meeting, under “Future Agenda Item Requests”, an agenda item for a future Board meeting. The Board member making the request will contact the CEO with the requested item and the CEO will list it on the agenda. At the meeting, the requesting Board member will explain the item, the need for Board discussion of the item, the item’s relationship to the Board’s strategic priorities, and the amount of estimated staff time necessary to prepare for Board discussion. If the requesting Board member receives a second, the CEO will place the item on the Board agenda calendar allowing for adequate time for staff preparation on the agenda item. Page 3 of 4 - None 11. ADJOURNMENT Regular Session 1. CALL TO ORDER 2. PLEDGE OF ALLEGIANCE 3. CITIZENS' PRESENTATIONS: This is an opportunity for citizens to address the Board on any matter whether or not it is posted on the agenda. The Board cannot by law take action nor have any discussion or deliberations on any presentation made to the Board at this time concerning an item not listed on the agenda. Any item presented may be noticed on a future agenda for deliberation or action. 4. CONSENT AGENDA: All items listed below are considered routine by the Board of Trustees and will be enacted with one motion. There will be no separate discussion of items unless a Board member or citizen so requests, in which event the item will be removed from the general order of business and considered in its normal sequence. a. Review and approve minutes of the School Board of Trustees of the workshop and regular meeting held on October 5, 2009. b. Consider a Resolution adopting homework, assessment and grade reporting, retention and transfer course credit, graduation policies and amending the responsibility for personnel decisions policy. 5. BOARD CALENDAR - Westlake Baja at Vaquero November 16, 2009, 3:30 p.m. - Barnes & Noble Book Fair November 18 -20, 2009 - Grandparents Day November 20, 2009 - Westlake Tree Lighting December 3, 2009, 6:30 p.m. - Board of Trustees Meeting December 7, 2009 - Employee Recognition Banquet December 9, 2009, TBD - WA Parents Shopping Day Out December 19, 2009 (Benefiting Project 2010) - Career Day Page 4 of 4 December TBD, 2009 - Westlake Academy Gallery Night event February 26, 2010 - Bandana Bonanza May 8, 2010 - Westlake Academy Graduation May 23, 2010 6. FUTURE AGENDA ITEMS: Any Board member may request at a workshop and / or Board meeting, under “Future Agenda Item Requests”, an agenda item for a future Board meeting. The Board member making the request will contact the CEO with the requested item and the CEO will list it on the agenda. At the meeting, the requesting Board member will explain the item, the need for Board discussion of the item, the item’s relationship to the Board’s strategic priorities, and the amount of estimated staff time necessary to prepare for Board discussion. If the requesting Board member receives a second, the CEO will place the item on the Board agenda calendar allowing for adequate time for staff preparation on the agenda item. - None 7. ADJOURNMENT ANY ITEM ON THIS POSTED AGENDA COULD BE DISCUSSED IN EXECUTIVE SESSION AS LONG AS IT IS WITHIN ONE OF THE PERMITTED CATEGORIES UNDER SECTIONS 551.071 THROUGH 551.076 AND SECTION 551.087 OF THE TEXAS GOVERNMENT CODE. I certify that the above notice was posted at the Town Hall of the Town of Westlake, 3 Village Circle, on Thursday, October 29, 2009, by 5:00 p.m. under the Open Meetings Act, Chapter 551 of the Texas Government Code. CERTIFICATION _____________________________________ Kelly Edwards, Town Secretary If you plan to attend this public meeting and have a disability that requires special needs, please advise the Town Secretary 48 hours in advance at 817-490-5710 and reasonable accommodations will be made to assist you. CONSENT AGENDA: All items listed below are considered routine by the Board of Trustees and will be enacted with one motion. There will be no separate discussion of items unless a Board member or citizen so requests, in which event the item will be removed from the general order of business and considered in its normal sequence. a. Review and approve minutes of the School Board of Trustees of the workshop and regular meeting held on October 5, 2009. b. Consider a Resolution adopting homework, assessment and grade reporting, retention and transfer course credit, graduation policies and amending the responsibility for personnel decisions policy. Westlake Academy Item # 2 – Review of Consent Agenda Items 1 Memo Westlake Academy To: Honorable President and Members of the Board of Trustees From: Tom Brymer, Town Manager/CEO Westlake Academy Amanda DeGan, Court Administrator Subject: Workshop Meeting of November 3, 2009 Date: October 28, 2009 Presentation and discussion of Draft Board of Trustees’ policy concerning the Graduation policy for Westlake Academy. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who become well-balanced, responsible citizens. WESTLAKE ACADEMY MISSION/ VISION STATEMENT Academic Achievement WESTLAKE ACADEMY VALUES PYP, MYP, DP (IB Continuum) Caring Environment Fiscal Stewardship Communication/Transparency Engaged Stakeholders Maximizing Each Child’s Potential BACKGROUND (including policy implications and options): Staff is presenting for Board discussion and feedback a policy for the Board’s Policy Handbook concerning the Graduation of students from Westlake Academy. The policy was presented to the Board at the previous Trustees meeting and has been revised and resubmitted for review. Contained within the prior draft, Staff presented a proposal to establish a Westlake Academy High School Diploma with specific and higher credit hour standards than those required by the state of Texas. The Board advised that the establishment of these standards was an appropriate course of action and directed Staff to resubmit the proposal with the necessary changes to implement the issuance of this type of diploma. The question arose as to the applicability of these standards to our current graduating class and what actions would be necessary for those students who might not meet the higher standards. For instance, what would be the plan for issuing a high school diploma to students in the current graduating class (2010) who qualified for the Distinguished High School Diploma from the State through the completion of 26 credit hours rather than the 29 required by Westlake Academy? 2 Our education legal counsel, Janet Bubert, spoke with representatives at TEA who advised that Westlake Academy could establish higher standards than the state minimum required graduation credit hours. However, we would need to provide ample notice to the current students of the additional/higher credit standards. The current graduating class of 2010 would need to receive notification of the policy standards and be allowed the opportunity to attend summer school or other approved alternative educational programs to attain the credit hours in order for them to be denied the Distinguished High School Diploma they would otherwise be eligible to receive upon completion of their senior year. In keeping with this recommendation from counsel, Staff has added a statement designating a policy effective date for the graduation class of 2011. FUNDING : N/A at this time RECOMMENDATION : Review the draft policies and provide Staff feedback and direction on the content of these draft policies, as well as whether the Board wishes to consider them for adoption at a future meeting. ATTACHMENTS : Draft policy on: Graduation 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name : Graduation Policy Policy Category : Student Achievement Policy Goal : Defining the commencement participation requirements; establishing Valedictorian, Salutatorian and Honor Graduate policy; Policy Description : Eligibility for Commencement – Westlake Academy seniors who have satisfied all Academy requirements for graduation are eligible for participation in the graduation ceremonies. For students who do not meet the requirements, an exception may be made in the event that a senior lacks one (1) credit towards the graduation requirements. The student must have a plan approved by the Head of Secondary to complete the credit prior to the beginning of the next school year. No other exceptions to participation in the graduation ceremony will be granted. Only students who have completed all the requirements of Westlake Academy will receive a diploma. Those requirements are as follows: • Completion of a minimum of 29 credit hours according to the Westlake Academy Program of Studies • Not assigned to a Disciplinary Alternative Education Placement (DEAP) setting during his/her senior year at the Academy or any other educational facility 2 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : As reflective of the Academy’s rigorous and enriching IB curriculum and desire to foster college readiness, students who graduate from Westlake Academy will complete a program of study, which includes higher academic standards and additional credit hours than those established through the guidelines of the State of Texas. Students who receive a Westlake Academy high school diploma will have completed a minimum of 29 hours of course work, which reflects those higher levels of academic achievement. The graduation requirements detailing the necessary courses are incorporated into the policy by addendum A. The provision for the Westlake Academy Diploma will become effective for the students in the graduating class of 2011. Approved Courses Included in Formal Grade Point Average (GPA) - All courses taken at the secondary level at Westlake Academy which contain specific objectives, determined by the Texas Education Agency (TEA) and are not modified in content will be included when calculating grade point average. The following general criteria govern the calculation of the grade point average: • GPA will be calculated by the Counselor and rounded up to the nearest whole number • GPA will be determined using semester grades earned in state approved courses • Grades earned in courses completed at other fully accredited school(s) and transferred to the Academy will be included in determining grade point average Course criteria must be consistent with the standards established at Westlake Academy. Correspondence courses, Credit-by-Exam, Dual Credit courses, courses modified in content, Pass/Fail courses, Local Credit courses not required for graduation, Advanced Placement Exams, International Baccalaureate Exams and grades earned in credit-bearing courses taken prior to Grade 9 will not be included in the calculation of the formal GPA, but will be reflected on the student’s Academic Achievement Record/Transcript. 3 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : Quality Points for IB Courses - Beginning in Grade 9, Westlake Academy will award ten (10) points to the final grade for any IB Diploma Program or Advanced Placement course and no less than five (5) points for any honors (pre-IB/AP) course, as determined by the school administration. Raw grades in the course must be a passing grade of 70% or better to receive quality points. Raw semester grades are shown on a student’s report card and Academic Achievement Record (AAR)/Transcript. Indicators for quality points will be included on the Report Card and the AAR/Transcript. Quality points will be included in GPA calculation and this weighted GPA will appear on the student’s transcript. Grades earned in honors (pre-IB/AP) courses and IB/AP courses taken at other accredited districts will receive quality points if the following conditions are met: • Quality points awarded from other schools will not exceed or be less than the numerical weight of quality points awarded at Westlake Academy • The previous school provides official documentation of advanced courses Academic Recognition - Westlake Academy will encourage and recognize academic achievement through the establishment of a class Valedictorian, Salutatorian and Honor Graduates. Only grades earned at Westlake Academy and those earned at schools approved by Westlake Academy will determine the student’s academic recognition. Calculating and Reporting GPA - A counselor will determine a student’s GPA calculation at regular intervals as determined necessary by the Head of Secondary. The intervals will be communicated through the use of the Parent/Student Handbook. 4 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : Rank in Class - Rank in class will only be reported for the purposes of: • Automatic admission of the top 10% of the class into a Texas public college or university • Consideration for scholarships Valedictorian and Salutatorian - Graduating seniors with the top two cumulative grade point averages (CGPA), as determined by the Head of Secondary or his/her designee, will be eligible to serve as the Valedictorian and Salutatorian, respectively. In the event of a tie for Valedictorian, the students will share the honor and no Salutatorian will be designated. All courses and corresponding numerical grades used to determine GPA must not be modified in content and must be identified by the state as regular, honors, and AP or IB courses in order to qualify for Valedictorian and Salutatorian. Valedictorian and Salutatorian eligibility requires attendance at Westlake Academy for six (6) consecutive semesters preceding graduation. Further, a student who is in violation of the Academy codes and requirements, Town code, or State codes, may be deemed ineligible by the Chief Executive Officer or his/her designee to represent Westlake Academy as the Valedictorian or Salutatorian. Honor Graduates - A graduating senior’s weighted cumulative grade point average (CGPA) rounded to 90% or above will be determined to be an Honor Graduate and will be duly recognized at commencement. All courses and corresponding numerical grades earned to determine GPA must not be modified in content and must be identified by the state as regular, honors, and AP or IB courses in order to qualify for Honor Graduate status. The Head of Secondary and his/her designee are granted the authority to include students whose CGPA is 90% or above at the end of the final semester. Operational guidelines consistent with the above policy directives will be detailed and published by Academy Staff in the Parent/Student Handbook on an annual basis. 5 Graduation Requirements Side-by-Side Comparison WA WA NHP TEA Actual Required Distinguished* English 4 4 4 4 Mathematics 5 4 4 4 Science 4 4 4 4 Social Studies 4 4 4 3.0 Gov’t 0.5 0.5 0.5 0.5 Economics 0.5 0.5 0.5 0.5 Foreign Language 5 3 3 3 Physical Education 1.5 1.5 1.5 1.5 Health 0.5 0.5 0.5 0.5 Speech 0.5 0.5 0.5 0.5 Technology 2.0 2.0 1.0 1.0 Fine Arts 2.0 1.0 2.0 1.0 Tok 2.0 2.0 1.0 n/a Electives 1.0 1.0 2.5 2.5 CAS/Extended Essay 1.0 0.5 n/a n/a (32.5) (29.0) (29.0) (26.0) *Source: TEA Website http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074f.html 1 Memo Westlake Academy To: Honorable President and Members of the Board of Trustees From: Tom Brymer, Town Manager/CEO Westlake Academy Mark Garcia, Head of Secondary Subject: Workshop Meeting of November 3, 2009 Date: October 29, 2009 Presentation and Discussion of a Report Concerning Recent Staff Training and Visit to the Ron Clark Academy in Atlanta, Georgia ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who become well-balanced, responsible citizens. WESTLAKE ACADEMY MISSION/ VISION STATEMENT  Academic Achievement WESTLAKE ACADEMY VALUES  PYP, MYP, DP (IB Continuum)  Caring Environment  Fiscal Stewardship  Communication/Transparency  Engaged Stakeholders  Maximizing Each Child’s Potential Recently a group of staff attended training funded by the Hudson Foundation at the Ron Clark Academy in Atlanta, Georgia. This Grades 5-8 private school in a low to moderate income Atlanta neighborhood is becoming a well-known, bench mark school for innovative teaching methods that focus on students’ critical thinking skills and setting high academic standards for all students regardless of academic proficiency and socio-economic level. Staff will report on their experiences and observations during this training as well as transferability of knowledge and practices gained to Westlake Academy. Attending from Westlake Academy were CEO Tom Brymer, Head of Secondary Mark Garcia, PYP Coordinator Terry Watson, Grade 7 Teacher Livia Miller, and Grade 9-10 Teacher Linday Lemons. BACKGROUND (including policy options and implications): FUNDING : N/A at this time RECOMMENDATION : Receive the staff report, ask questions and engage in dialogue with staff about transferability of this professional development experience to Westlake Academy for the betterment of our students. Also recommend that all section heads and additional staff attend this training in the futures, finances allowing. ATTACHMENTS: For information about the Ron Clark Academy go to: http://www.ronclarkacademy.com/ Memo Town of Westlake To: Honorable President and Members of the Board of Trustees From: Tom Brymer, Town Manager/CEO of Westlake Academy Amanda DeGan, Court Administrator Subject: Workshop Meeting of November 3, 2009 Date: October 26, 2009 Presentation and discussion of a Draft Board of Trustees Policy concerning the presenting of information/reports from Staff. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who well balanced, responsible citizens. WESTLAKE ACADEMY MISSION/VISION STATEMENT ATTACHMENTS: • DRAFT – Policy concerning Board Parameters for Staff Recommendations to the Board of Trustees • Excerpt from John Carver book on Governance – “Free Your Board and Staff Through Executive Limitations”, Board Leadership, Number 4, Nov.-Dec. 1992 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name Board of Trustees : Board Parameters for Staff Recommendations to the Policy Category : Governance Policy Goal/Ends Staff compilation of reports and recommendations : Board definition of executive limitations and boundaries for presented to the Board Policy Description In an effort to provide the CEO and Staff of Westlake Academy with the proper guidance necessary to operate the school in an effective manner and ensure the Board is provided with the most current and complete information possible when making decisions, the following policy statement regarding executive limitations is established: : Information, advice, reports, as well as recommendations and research presented to the Board of Trustees by the Staff will contain no significant gaps in regards to timeliness of the information, completeness of the research, or accuracy of the facts associated with the presentation. Adherence to this policy, will allow the Board to make well informed and appropriate decisions necessary to achieve the success of Westlake Academy and its students. Memo Town of Westlake To: Honorable President and Members of the Board of Trustees From: Tom Brymer, Town Manager/CEO of Westlake Academy Amanda DeGan, Court Administrator Subject: Workshop Meeting of November 3, 2009 Date: October 26, 2009 Presentation and discussion of a Draft Board of Trustees Policy concerning the adherence to the Westlake Academy Charter, Texas Education Agency and International Baccalaureate guidelines. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who well balanced, responsible citizens. WESTLAKE ACADEMY MISSION/VISION STATEMENT 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name Baccalaureate and Charter Guidelines : Adherence to the Texas Education Agency, International Policy Category : Governance Policy Goal/Ends activities and decisions related to the students at Westlake Academy : Establish Board policy governance parameters by which the education will be conducted Policy Description As Westlake Academy is an open-enrollment public charter school, which provides education to students utilizing the International Baccalaureate (IB) curriculum, the school is required to adhere to criteria and regulations promulgated by various authorizing agencies. : In an effort to provide the CEO and Staff of Westlake Academy with clear guidance from the Board of Trustees, as it relates to these multiple authorizing agencies, the following policy statement regarding operational guidelines is established: Any action in the activities or education of the students at Westlake Academy, which materially deviates from the International Baccalaureate curriculum, Texas Education Agency (TEA) guidelines for Charter schools and/or the Academy’s Charter passed and approved by the TEA (and as may be amended and approved by TEA) will be considered imprudent by the Board of Trustees. The statement is issued to indicate the importance the Board places in the mission/vision of Westlake Academy to educate students utilizing the IB methods and organizational goals specified in the Charter as approved by our authorizing entity, the Texas Education Agency. Memo Town of Westlake To: Honorable President and Members of the Board of Trustees From: Tom Brymer, Town Manager/CEO of Westlake Academy Amanda DeGan, Court Administrator Subject: Workshop Meeting of November 3, 2009 Date: October 26, 2009 Presentation and discussion of a Draft Board of Trustees’ Policy relative to capital projects and facility modification/improvement projects at the Westlake Academy campus. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who well balanced, responsible citizens. WESTLAKE ACADEMY MISSION/VISION STATEMENT 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name Projects : Campus Facility Renovation/Modification Projects and Capital Policy Category : Governance Policy Goal coordination for Westlake Academy campus facility additions, : Establishment of a process and central point of contact and renovations, modifications, and proposed capital projects Policy Description As Westlake Academy has transitioned into a K-12 educational facility, the need now exists to formally identify a specific individual/position for capital project submission requests as well as coordination of all campus facility renovation and modification projects. The CEO, or his/her designee, shall be the central point of contact for all such projects. : All projects involving additions or modifications to the existing campus will be submitted during the Academic Services Budget preparation process with a full description of the project, its budget, and any impact on on-going operating costs. All capital projects will be submitted in accordance with the capital budgeting process and the budget calendar on forms provided by the Town. All projects will be submitted through the CEO or his/her designate. All capital projects will be submitted in accordance with the Town’s current financial policies adopted with the Town budget. This policy applies to projects submitted by Staff and by Academy affiliate organizations. Memo Westlake Academy To: Honorable President and Members of the Board of Trustees From: Mark Garcia, Head of Secondary Subject: Workshop Meeting of November 3, 2009 Date: October 27, 2009 Presentation and discussion regarding the Westlake Academy - Senior Class of 2010 graduation and activities along with grade level planning and communication. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who become well- balanced, responsible citizens. WESTLAKE ACADEMY MISSION/ VISION STATEMENT Memo Westlake Academy To: Honorable President and Members of the Board of Trustees From: Jamie Schmitz, Head of Primary Subject: Workshop of November 3, 2009 Date: October 19th, 2009 Presentation and discussion by Westlake Academy Head of Primary and Primary Years Programme Coordinator regarding developmental continua, and, in particular, the introduction of the Mathematics Developmental Continuum. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who become well-balanced, responsible citizens. WESTLAKE ACADEMY MISSION/ VISION STATEMENT X Academic Achievement WESTLAKE ACADEMY VALUES X PYP, MYP, DP (IB Continuum) X Caring Environment Fiscal Stewardship X Communication/Transparency X Engaged Stakeholders X Maximizing Each Child’s Potential A developmental continuum is comprised of particular subject’s expectations (what we and TEA would like each child to understand and be able to do), in this case for mathematics, arranged in developmental phases, with each phase building on the one before it. The advantage of using a developmental continuum is that, unlike a traditional approach, where teachers aim to have students master only the grade level subject expectations in an academic year, teachers will work with students at their developmental level, whether that is within a higher or lower phase than the traditional model prescribes. This allows teachers to facilitate student learning in accordance with every child’s individual level of understanding. In essence, this is an individualized developmental map for each student, which will be vital in moving him or her forward at his or her own developmental pace. BACKGROUND (policy implications) N/A FUNDING To implement the Language Developmental Continuum in November, 2009, with a view to developing and implementing developmental continua for science and social studies during the 2010/2011 academic year. RECOMMENDATION 1. Westlake Academy Primary Mathematics Developmental Continuum ATTACHMENTS: 2. Detailed mathematics developmental continuum strand (number) 3. Official International Baccalaureate documentation regarding developmental continua (mathematics), including sample developmental continuum MATH LEARNING CONTINUUM FOR NUMBER IB/TEKS ALIGNMENT PHASE 1 PHASE 2 PHASE 3 PHASE 4 PHASE 5 PHASE 6 PHASE 7 PHASE 8 Connect number names and numerals to the quantities they represent (through 20) using verbal and symbolic descriptions Read and write whole numbers up to hundreds Read and write whole numbers up to hundreds or beyond Use place value to read, write, and describe the value of whole numbers through 999,999 Use place value to read, write, and describe the value of whole numbers through 999,999,999 Use place value to read, write, compare and order whole numbers through 999,999,999,999 Compare and order non-negative rational numbers Compare and order integers and positive rational numbers Use sets of concrete objects to represent quantities given in verbal or written form (through 20) Compare and order whole numbers up to 99 (less than, greater than, or equal to) using sets of concrete objects and pictorial models Use place value to read, write, and describe the value of whole numbers to 999 Use place value to compare and order whole numbers through 9,999 Use place value to read, write, compare and order decimals involving tenths and hundredths, including money, using concrete objects and pictorial models Use place value to read, write, compare and order decimals through thousandths Generate equivalent forms of rational numbers including whole numbers, fractions and decimals Convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator Name the ordinal positions in a sequence such as first, second, third, etc. Create sets of tens and ones using concrete objects to describe, compare, and order whole numbers Use place value to compare and order whole numbers to 999 and record comparisons using numbers and symbols Determine the value of a collection of coins and bills Use concrete objects and pictorial models to generate equivalent fractions Convert improper fractions to mixed numbers and vice versa Use integers to represent real-life situations Represent squares and square roots using geometric models Share a whole by separating it into two equal parts Use words and numbers to describe the values of individual coins such as penny, nickel, dime and quarter and their relationships Use concrete models to represent and name fractional parts of a whole object or a set of objects (with denominators of 12 or less) Construct concrete models of fractions Model fraction quantities greater than one using concrete objects and pictorial models Simplify fractions Write prime factorizations using exponents Represent multiplication and division situations involving fractions and decimals with models, including concrete objects, pictures, words, and numbers Explain why a given part is half the whole Share a whole by separating it into equal parts and use appropriate language to describe the parts such as three out of four equal parts Use concrete models to determine if a fractional part of a whole is closer to 0, ½, or 1 Compare fractional parts of whole objects or sets of objects in a problem situation using concrete models Compare and order fractions using concrete objects and pictorial models Use models to relate decimals to fractions Identify factors of a positive integer, common factors, and the greatest common factor of a set of positive integers Use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals Model and create addition and subtraction problems in real situations with concrete objects Use appropriate language to describe part of a set such as three out of the eight crayons are red Recall and apply basic addition and subtraction facts (to 18) Use fraction names and symbols to describe parts of whole objects or sets of objects Relate decimals to fractions that name tenths and hundredths using concrete objects and pictorial models Generate equivalent fractions Identify multiples of a positive integer and common multiples and the least common multiple of a set of positive integers Use models, such as concrete objects, pictorial models and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms Solve problems with guidance that incorporate processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solutions for reasonableness Model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences Model addition and subtraction of two-digit numbers with objects, pictures, words and numbers Construct concrete models of equivalent fractions Use addition and subtraction to solve problems involving whole numbers Compare fractions Use addition and subtraction to solve problems involving fractions and decimals Use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes and student- teacher ratio Select or develop an appropriate problem-solving strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem Learn and apply basic addition facts (sums up to 18) using concrete models Determine the value of a collection of coins up to one dollar Model addition and subtraction using pictures, words and numbers Add and subtract decimals to the hundredths place using concrete objects and pictorial models Use addition and subtraction to solve problems involving whole numbers and decimals Use multiplication and division of whole numbers to solve problems Simplify numerical expressions involving order of operations and exponents Use tools such as real objects, manipulatives and technology to solve problems Use a problem-solving model, with guidance as needed, that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness Describe how the cent symbol, dollar symbol, and the decimal point are used to name the value of a collection of coins Select addition or subtraction and use the operation to solve problems involving whole numbers through 999 Model factors and products using arrays and area models Use multiplication to solve problems involving whole numbers Estimate and round to approximate reasonable results Select and use appropriate operations to solve problems and justify the selections Communicate mathematical ideas using objects, words, pictures, numbers, and technology Use tools such as real objects, manipulatives, and technology to solve problems Model, create and describe multiplication situations in which equivalent sets of concrete objects are joined or separated Learn and apply multiplication facts through 12s using concrete models and objects Represent multiplication and division situations in picture, word, and number form Use division to solve problems involving whole numbers (two-digit divisors and three-digit dividends), including interpreting the remainder within a given context Use order of operations to simplify whole number expressions in problem solving situations Solve problems that incorporate the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness Relate everyday language to mathematical language and symbols Explain and record observations using objects, words, pictures, numbers, and technology Select addition or subtraction to solve problems using two-digit numbers, whether or not regrouping is necessary Solve and record multiplication problems (up to two digit times one digit) Recall and apply multiplication facts through 12 x 12 Identify common factors of a set of whole numbers Use ratios to describe proportional situations Use logical reasoning to justify thinking Use logical reasoning Use fractions in real-life situations Select or develop an appropriate problem- solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem Use models to solve division problems and use number sentences to record the solutions Use multiplication to solve problems (two digits times two digits without technology) Estimate solutions to determine reasonable results using strategies such as rounding Represent ratios and percents with concrete models, fractions and decimals Use tools to solve problems Use logical reasoning Use logical reasoning to justify thinking Round whole numbers to the nearest ten or hundred to estimate reasonable results Use division to solve problems (one-digit divisors and three-digit dividends without technology) Explain and record observations Use ratios to make predictions in proportional situations Use strategies, including rounding, to estimate solutions to addition and subtraction problems Round whole numbers to the nearest ten, hundred, or thousand to estimate reasonable results Use fractions and decimals interchangeably Simplify fractions Develop strategies for memorizing addition, subtraction, multiplication and division number facts Use strategies including rounding and compatible numbers to estimate solutions to multiplication and division problems Add and subtract fractions with related denominators Solve problems that incorporate the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness Solve problems that incorporate the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness Solve problems that incorporate the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness Solve problems that incorporate the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness MATH LEARNING CONTINUUM FOR MEASUREMENT IB/TEKS ALIGNMENT PHASE 1 PHASE 2 PHASE 3 PHASE 4 PHASE 5 PHASE 6 PHASE 7 PHASE 8 Compare and order two or three concrete objects according to length (longer/shorter than, or the same) Estimate and measure length, capacity, and weight of objects using non-standard units Estimate and measure objects using standard units of measurement: length, mass, capacity, money and temperature Estimate and measure using standard units of measurement: length and perimeter Estimate and measure using standard units of measurement: perimeter, area, volume and weight/mass Develop and describe formulas for finding perimeter, area and volume Estimate measurements (including circumference) and evaluate reasonableness of results Estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes Compare the areas of two flat surfaces of 2D figures (covers more, covers less, or covers the same) Compare and order two or more concrete objects according to length, area, capacity, weight/mass, and temperature Read and write the time to the hour, half hour and quarter hour Use concrete and pictorial models of square units to determine the area of a 2D surface Perform simple conversions between different units of length, between different units of capacity and between different units of weight within the customary measurement system Carry out simple unit conversions within a system of measurement (metric or customary) Select and use appropriate units, tools, or formulas to measure and to solve problems involving length (including perimeter), area, time, temperature, volume and weight Connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders Compare two containers according to capacity (holds more, holds less, or holds the same) Describe the relationship between the size of the unit and the number of units needed in a measurement Estimate and compare lengths of time: second, minute, hour, day, week, and month Identify concrete models that approximate standard units of weight/mass and use them to measure weight/mass Use concrete models of standard cubic units to measure volume Solve problems involving changes in temperature Measure and construct angles in degrees using a protractor Estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders Compare two objects according to weight/mass (heavier than, lighter than or equal to) Recognize temperatures such as a hot day or a cold day Use standard units of measurement to solve problems in real-life situations involving length, mass, capacity, money and temperature Identify concrete models that approximate standard units for capacity and use them to measure capacity Estimate volume in cubic units Solve problems involving elapsed time Convert measures within the same measurement system (customary and metric) based on relationships between units Select and use appropriate units of measurement and tools to solve problems in real-life situations Compare situations or objects according to relative temperature (hotter/colder than, or the same as) Describe time on an analog and digital clock using hours and half hours Use measures of time to assist with problem solving in real-life situations Use concrete models that approximate cubic units to determine the volume of a given container or other 3D geometric figure Explain the difference between weight and mass Select and use appropriate units of measurement and tools to solve problems in real-life situations Use decimal and fraction notation in measurement, for example, 3.2 cm, 1.47 kg, 1 ½ miles Use decimal and fractional notation in measurement Compare events according to duration, such as more time than or less time than Use non-standard units of measurement to solve problems in real-life situation involving length, mass, and capacity Use a thermometer to measure temperature Use a thermometer to measure temperature and changes in temperature (in degrees Fahrenheit and Celsius) Use timetables and schedules in real-life situations Determine and justify the level of accuracy required to solve real-life problems involving measurement Determine and justify the level of accuracy required to solve real-life problems involving measurement Sequence events (up to three) Use measures of time to assist with problem solving in real- life situations Tell and write time shown on analog and digital clocks Use tools such as a clock with gears or a stopwatch to solve problems involving elapsed time Determine times worldwide Use timetables and schedules in real-life situations Read a calendar using days, weeks, and months Use standard units of measurement to solve problems in real-life situations involving perimeter, area and volume Use standard units of measurement to solve problems in real-life situations involving perimeter, area, volume and weight/mass Determine times worldwide Use non-standard units of measurement to solve problems in real-life situations involving length, mass and capacity Select appropriate tools and units of measurement Select appropriate tools and units of measurement Use timelines in units of inquiry and other real-life situations Use timelines in units of inquiry and other real-life situations MATH LEARNING CONTINUUM FOR DATA HANDLING IB/TEKS ALIGNMENT PHASE 1 PHASE 2 PHASE 3 PHASE 4 PHASE 5 PHASE 6 PHASE 7 PHASE 8 Sort and label real objects by attributes Collect, sort and represent data in different types of graphs Collect and represent data in different types of graphs, including picture graphs and bar-type graphs Collect, organize, record, display and interpret data using simple graphs where each picture or cell might represent more than one piece of data Collect, display and interpret data using bar graphs Collect, display and interpret data using graphs Identify, describe and explain the range, mode, median and mean in a set of data Select and use an appropriate representation for presenting and displaying relationships among collected data Construct graphs using real objects or pictures Represent the relationship between objects in sets using tree and Venn diagrams Use data to describe events as more likely or less likely, such as drawing a certain color crayon from a bag of seven red crayons and three green crayons Interpret information from picture graphs and bar graphs Use concrete objects or pictures to make generalizations about determining all possible combinations of a given set of data or of objects in a problem situation Identify, describe and explain the range, mode, median and mean in a set of data Set up a spreadsheet using simple formulas to manipulate data and create graphs Make inferences and convincing arguments based on an analysis of given or collected data Construct and use graphs of real objects or pictures to answer questions Express the chance of an event happening using words or phrases (impossible, less likely, maybe, most likely, certain) Represent the relationship between objects in sets using tree and Venn diagrams Use data to describe events as more likely than, less likely than, or equally likely as Design a survey and systematically collect, organize and display data in bar graphs Use fractions to describe the results of an experiment Express probabilities using scale (0-1) or percent (0% - 100%) Describe a set of data using mean, median, mode and range Create pictographs and tally marks Collect, display and interpret data for the purpose of answering questions and drawing conclusions Draw conclusions and answer questions based on picture graphs and bar-type graphs Design a survey and systematically collect, organize and display data in picture graphs and bar graphs Express probability using simple fractions Use experimental results to make predictions Construct sample spaces using lists and tree diagrams Choose among mean, median, mode, or range to describe a set of data and justify the choice for a particular situation Describe real objects and events by attributes Create a pictograph and sample bar graph of real objects and interpret data by comparing quantities (more, fewer, less than, greater than) Create a pictograph and sample bar graph of real objects and interpret data by comparing quantities Select appropriate graph form(s) to display data Solve problems by collecting, organizing, displaying and interpreting sets of data List all possible outcomes of a probability experiment, such as tossing a coin Find the probabilities of a simple event and its complement and describe the relationship between the two Find the probability of independent events Use tree and Venn diagrams to explore relationships between data Identify and describe chance in daily events Use tables of related number pairs to make line graphs Select and use an appropriate representation for presenting and displaying different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot Create and manipulate an electronic database for own purposes Identify and describe chance in daily events (impossible, less likely, maybe, most likely, certain) Design a survey and systematically collect, record, organize and display the data in a bar graph, circle graph, line graph Sketch circle graphs to display data Design a survey and systematically collect, record, organize and display the data in a bar graph, circle graph, line graph, stem and leaf plot and justify the selection Use probability to determine mathematically fair and unfair games and to explain possible outcomes Solve problems by collecting, organizing, displaying and interpreting data Design a survey and systematically collect, record, organize and display the data in a bar graph, circle graph, line graph Determine the theoretical probability of an event and explain why it might differ from experimental probability MATH LEARNING CONTINUUM FOR PATTERN AND FUNCTION IB/TEKS ALIGNMENT PHASE 1 PHASE 2 PHASE 3 PHASE 4 PHASE 5 PHASE 6 PHASE 7 PHASE 8 Describe patterns in various ways, for example, using words, drawings, symbols, materials, actions, numbers Represent patterns in a variety of ways, for example, using words, drawings, symbols, materials, actions, numbers Find patterns in numbers, such as in a 100s chart Describe the rule for a pattern in a variety of ways Describe the rule for a pattern in a variety of ways Describe the relationship between sets of data in graphic organizers, such as lists, tables, charts, and diagrams Represent the rule of a pattern by using a function Estimate and find solutions to application problems involving percent Identify, extend and create patterns of sounds, physical movement and concrete objects Describe number patterns, for example, odd and even numbers, skip counting (by twos, fives and tens) Use patterns in place value to compare and order whole numbers through 999 Represent rules for patterns using words, symbols and tables Identify a sequence of operations relating one set of numbers to another set Identify prime and composite numbers using concrete objects, pictorial models, and patterns in factor pairs Analyze pattern and function using words, tables and graphs, and, when possible, symbolic rules Estimate and find solutions to application problems involving proportional relationships such as similarity, scaling, unit costs, and related measurement units Use patterns to predict what comes next, including cause and effect relationships Use patterns to develop strategies to solve basic addition and basic subtraction problems Use patterns and relationships to develop strategies to remember basic addition and subtraction facts (fact families) Identify and extend whole- number and geometric patterns to make predictions and solve problems Represent rules for patterns using words, symbols and tables Select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations Formulate equations from problem situations described by linear relationships Generate formulas involving unit conversions within the same system (customary and metric), perimeter, area, circumference, volume, and scaling Count by ones to 100 Extend and create patterns in numbers, for example, odd and even numbers, skip counting Generate a list of paired numbers based on a real-life situation, such as number of tricycles related to number of wheels Identify patterns in multiplication facts using concrete objects, pictorial models or technology Use patterns and relationships to develop strategies to remember basic multiplication and division facts Select appropriate methods to analyze patterns and identify rules Use tables of data to generate formulas representing relationships involving perimeter, area, volume of a rectangular prism, etc. Use words and symbols to describe the relationship between the terms in an arithmetic sequence (with a constant rate of change) and their positions in the sequence Identify, extend and create patterns Use number patterns to represent and understand real- life situations and to make predictions Use number patterns to represent and understand real-life situations, solve problems and make predictions Identify patterns in related multiplication and division sentences (fact families) Use patterns to multiply by 10 and 100 Use number patterns to make predictions and solve problems Use tables and symbols to represent and describe proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change), perimeter, and area Use concrete and pictorial models to solve equations and use symbols to record the actions Use patterns to make predictions Use the properties and relationships of addition and subtraction to solve problems Generate a table of paired numbers based on a real-life situation, such as insects and legs Describe the relationship between two sets of related data, such as ordered pairs in a table Select appropriate methods to analyze patterns and identify rules Formulate problem situations when given a simple equation and formulate an equation when given a problem situation Identify and describe patterns in a table of related number pairs based on a meaningful problem and extend the table Select appropriate methods for representing patterns, for example, using words, symbols and tables Use functions to solve problems Select appropriate methods to analyze patterns and identify rules Select appropriate methods for representing patterns, for example, using words, symbols and tables Use number patterns to make predictions and solve problems Use letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes Use functions to solve problems Use letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes MATH LEARNING CONTINUUM FOR SHAPE AND SPACE IB/TEKS ALIGNMENT PHASE 1 PHASE 2 PHASE 3 PHASE 4 PHASE 5 PHASE 6 PHASE 7 PHASE 8 Sort, describe and compare 3D shapes based on attributes using informal language Sort, describe and identify 2D and 3D shapes Sort, describe, and compare 2D and 3D shapes Identify congruent 2D figures Sort, describe and model regular and irregular polygons Identify essential attributes including parallel, perpendicular and congruent parts of 2D and 3D geometric figures Use angle measurements to classify angles as acute, obtuse or right Use angle measurements to classify pairs of angles as complementary or supplementary Describe position and direction, for example, inside, outside, above, below, next to, behind, in front of, up, down Represent ideas about the real world using geometric vocabulary and symbols, for example, through oral description, drawing, modeling, labeling Describe attributes (the number of vertices, faces, edges, sides) of 2D and 3D geometric figures such as circles, polygons, spheres, cones, cylinders, prisms, and pyramids Create 2D figures with lines of symmetry using concrete models and technology Describe and model congruency and similarity in 2D shapes Sketch the results of translations, rotations and reflections on a coordinate grid Identify relationships involving angles in triangles and quadrilaterals Use properties to classify triangles and quadrilaterals Sort a variety of objects, including 2D and 3D geometric figures, according to their attributes and describe how the objects are sorted Use concrete models to combine 2D geometric figures to make new geometric figures Cut 2D geometric figures apart and identify new figures formed Identify lines of symmetry in 2D geometric figures Identify and describe right, acute, and obtuse angles Locate and name points on a coordinate grid using ordered pairs of whole numbers Describe the relationship between radius, diameter, and circumference of a circle Use properties to classify 3D figures, including pyramids, cones, prisms and cylinders Describe and compare the attributes of real-life objects such as balls, boxes, cans and cones or models of 3D geometric figures or models of 3D geometric figures Analyze and use information about 3D shapes to describe and work with 2D shapes Use whole numbers to locate and name points on a number line Locate and name points on a number line using whole numbers and fractions, including halves and fourths Identify and describe parallel and intersecting (including perpendicular) lines using concrete objects and pictorial models Use geometric vocabulary when describing shape and space in mathematical situations and beyond Locate and name points on a coordinate plane using ordered pairs of non-negative rational numbers Use critical attributes to define similarity Describe, identify, and compare circles, triangles, rectangles, and squares (a special type of rectangle) Analyze and use information about 3D shapes to describe and work with 2D shapes Analyze and describe 2D and 3D shapes using geometrical vocabulary Use essential attributes to define 2D and 3D geometric figures Use 2D representations of 3D objects to visualize and solve problems, for example using drawings or models Create and model how a 2D net converts into a 3D shape and vice versa Locate and name points on a coordinate plane using ordered pairs of integers Explore and describe the paths, regions and boundaries of immediate environment (inside, outside, above, below) and their position (next to, behind, in front of, up, down) Identify, describe and model congruency in 2D shapes Demonstrate translations, reflections and rotations using concrete models Apply knowledge of transformations to problem- solving situations Explore the use of geometric ideas and relationships to solve problems in other areas of mathematics Graph reflections across the horizontal or vertical axis and graph translations on a coordinate plane Recognize lines of symmetry in the environment Use reflections to verify that a shape has symmetry Use geometric vocabulary when describing shape and space in mathematical situations and beyond Sketch 3D figures when given the top, side, and front views Recognize that a line can be used to represent numbers and fractions and their properties and relationships Locate and name points on a number line using whole numbers, fractions, such as halves and fourths, and decimals, such as tenths Use scale (ratios) to enlarge and reduce shapes Make a net (2D model) of the surface area of a 3D figure Apply knowledge of transformations to problem- solving situations Apply the language and notation of bearing to describe direction and position Use geometric concepts and properties to solve problems in fields such as art and architecture Use 2D representations of 3D objects to visualize and solve problems Apply the language and notation of bearing to describe direction and position Use 2D representations of 3D objects to visualize and solve problems Learning Continuum for Number IB/TEKS Alignment Phase 1 Conceptual Understandings • Numbers are a naming system. • Numbers can be used in many ways for different purposes in the real world. • Numbers are connected to each other through a variety of relationships. • Making connections between our experiences with number can help us to develop number sense. Learning Outcomes When constructing meaning, learners: • Understand one-to-one correspondence and language such as more than, same number as, or two less than to describe relative sizes of sets of concrete objects • Understand that, for a set of objects, the number name of the last object counted describes the quantity of the whole set • Understand that numbers can be constructed in multiple ways, for example, by adding or subtracting • Understand conservation of number • Understand the relative magnitude of whole numbers • Recognize groups of zero to five objects without counting • Understand whole-part relationships • Use the language of mathematics to compare quantities, for example, more, less, first, second • Identify mathematics in everyday situations When transferring meaning into symbols, learners: • Connect number names and numerals to the quantities they represent (through 20) using verbal and symbolic descriptions • Use sets of concrete objects to represent quantities given in verbal or written form (through 20) • Name the ordinal positions in a sequence such as first, second, third, etc. • Share a whole by separating it into two equal parts • Explain why a given part is half the whole • Model and create addition and subtraction problems in real situations with concrete objects • Solve problems with guidance that incorporates processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solutions for reasonableness • Select or develop an appropriate problem-solving strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem • Use tools such as real objects, manipulatives and technology to solve problems • Communicate mathematical ideas using objects, words, pictures, numbers, and technology • Relate everyday language to mathematical language and symbols When applying with understanding, learners: • Count to determine the number of objects in a set • Use number words and numerals to represent quantities in real-life situations • Apply mathematics to solve problems connected to everyday experiences and activities in and outside of school • Use logical reasoning Learning Continuum for Number IB/TEKS Alignment Phase 2 Conceptual Understandings • Numbers can be used in many ways for different purposes in the real world. • Numbers are connected to each other through a variety of relationships. • The base 10 place value system is used to represent numbers and number relationships. • Fractions are ways of representing whole-part relationships. • The operations of addition and subtraction are related to each other and are used to process information to solve problems. • Numbers operations can be modeled in a variety of ways. Learning Outcomes When constructing meaning, learners: • Use whole numbers to describe and compare quantities • Model simple fraction relationships • Model addition and subtraction of whole numbers When transferring meaning into symbols, learners: • Read and write whole numbers up to hundreds • Compare and order whole numbers up to 99 (less than, greater than, or equal to) using sets of concrete objects and pictorial models • Create sets of tens and ones using concrete objects to describe, compare, and order whole numbers • Identify individual coins by name and describe their relationships • Share a whole by separating it into equal parts and use appropriate language to describe the parts such as three out of four equal parts • Use appropriate language to describe part of a set such as three out of the eight crayons are red • Model and create addition and subtraction problem situations with concrete objects and write corresponding number sentences • Learn and apply basic addition facts (sums up to 18) using concrete models • Use a problem-solving model, with guidance as needed, that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness • Use tools such as real objects, manipulatives, and technology to solve problems • Explain and record observations using objects, words, pictures, numbers, and technology When applying with understanding, learners: • Use whole numbers up to 99 in real-life situations • Use fractions in real-life situations • Recognizes and solves problems in addition and subtraction situations • Use strategies to evaluate the reasonableness of answers • Communicates mathematic concepts using informal language • Use logical reasoning to make sense of the world • Reason and support their thinking using objects, words, pictures, numbers and technology Learning Continuum for Number IB/TEKS Alignment Phase 3 Conceptual Understandings • The base 10 place value system is used to represent numbers and number relationships. • Fractions are ways of representing whole-part relationships. • The operations of addition, subtraction, multiplication and division are related to each other and are used to process information to solve problems. • Number operations can be modeled in a variety of ways. • There are many mental methods that can be applied for exact and approximate computations. Learning Outcomes When constructing meaning, learners: • Understand how place value is used to represent whole numbers • Describe how fractions are used to name parts of whole objects or sets of objects • Add and subtract whole numbers to solve problems • Model multiplication and division • Model numbers to hundreds or beyond When transferring meaning into symbols, learners: • Read and write whole numbers up to hundreds or beyond • Use place value to read, write, and describe the value of whole numbers to 999 • Use place value to compare and order whole numbers to 999 and record comparisons using numbers and symbols • Use concrete models to represent and name fractional parts of a whole object or a set of objects (with denominators of 12 or less) • Use concrete models to determine if a fractional part of a whole is closer to 0, ½, or 1 • Recall and apply basic addition and subtraction facts (to 18) • Model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers • Determine the value of a collection of coins up to one dollar • Describe how the cent symbol, dollar symbol, and the decimal point are used to name the value of a collection of coins • Model, create and describe multiplication situations in which equivalent sets of concrete objects are joined or separated When applying with understanding, learners: • Select addition or subtraction to solve problems using two-digit numbers, whether or not regrouping is necessary • Use whole numbers up to hundreds or beyond in real-life situations • Identify mathematics in everyday situations • Solve problems with guidance that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness • Select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem • Use tools such as real objects, manipulatives, and technology to solve problems • Explain and record observations using objects, words, pictures numbers and technology • Relate informal language to mathematical language and symbols • Use logical reasoning to justify thinking Learning Continuum for Number IB/TEKS Alignment Phase 4 Conceptual Understandings • The base 10 place value system is used to represent numbers and number relationships. • Fractions are ways of representing whole-part relationships. • The operations of addition, subtraction, multiplication and division are related to each other and are used to process information to solve problems. • Number operations can be modeled in a variety of ways. Learning Outcomes When constructing meaning, learners: • Model numbers to thousands or beyond using place value • Estimate sums and differences • Model equivalent fractions • Model decimal fractions to hundredths • Use the language of multiplication for example factor and product • Model multiplication and division of whole numbers When transferring meaning into symbols, learners: • Use place value to read, write and describe the value of whole numbers through 999,999 • Use place value to compare and order whole numbers through 9,999 • Determine the value of a collection of coins and bills • Construct concrete models of fractions • Compare fractional parts of whole objects or sets of objects in a problem situation using concrete models • Use fraction names and symbols to describe frational parts of whole objects or sets of objects • Construct concrete models of equivalent fractions • Model addition and subtraction using pictures, words, and numbers • Select addition or subtraction and use the operation to solve problems involving whole numbers through 999 • Learn and apply multiplications facts through 12s using concrete models and objects • Solve and record multiplication problems (up to two digits times one digit • Use models to solve division problems and use number sentences to record the solutions • Round whole numbers to the nearest ten or hundred to approximate reasonable results in problem situations • Use strategies including rounding and compatible numbers to estimate solutions to addition and subtraction problems When applying with understanding, learners: • Read, write, compare and order whole numbers up to thousands or beyond • Develop strategies for memorizing addition, subtraction, multiplication and division number facts • Read, write and compare fractions • Solve problems that incorporate the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness • Select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem • Use tools such as real objects, manipulatives, and technology to solve problems • Explain and record observations using objects, words, pictures numbers and technology • Relate informal language to mathematical language and symbols • Use logical reasoning to justify thinking • Make generalizations from patterns or sets of examples • Explain and record observations using objects, words, pictures, numbers and technology • Relate informal language to mathematical language and symbols • Identify mathematics in everyday situations Learning Continuum for Number IB/TEKS Alignment Phase 5 Conceptual Understandings • The base 10 place value system can be extended to represent numbers and number relationships. • Fractions and decimals are ways of representing whole-part relationships. • The operations of addition, subtraction, multiplication and division are related to each other and are used to process information to solve problems. Learning Outcomes When constructing meaning, learners: • Model numbers to hundred thousands or beyond • Model equivalent fractions • Use the language of fractions • Model decimal fractions to hundredths or beyond • Model multiplication and division of whole numbers use the language of multiplication and division When transferring meaning into symbols, learners: • Use place value to read, write, compare, and order whole numbers through 999,999,999 • Use place value to read, write, compare and order decimals involving tenths, and hundredths, including money, using concrete objects and pictorial models • Use concrete objects and pictorial models to generalte equivalent fractions • Model fraction quantities greater than one using concrete objects and pictorial models • Compare and order fractions using concrete objects and pictorial models • Relate decimals to fractions that name tenths and hundredths using concrete objects and pictorial models • Use addition and subtraction to solve problems involving whole numbers • Add and subtract decimals to the hundredths place using concrete objects and pictorial models • Model factors and products using arrays and area models • Represent multiplication and division situations in picture, word, and nuamber formrecall and apply multiplication facts through 12 x 12 • Use multiplication to solve problems (two digits times two digits without technology) • Use division to solve problems (one-digit divisors and three-digit dividends without technology • Round whole numbers to the nearest ten, hundred, or thousand to approximate reasonable results in problem situations • Use strategies including rounding and compatible numbers to estimate solutions to multiplication and division problems • Identify mathematics in everyday situations • Solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness • Select or develop an appropriate problem-solving plan or strategy • Use tools to solve problems • Explain and record observations • Make generalizations from patterns or sets of examples • Use logical reasoning to justify an answer When applying with understanding, learners: • Use whole numbers up to hundred thousands or beyond in real-life situations • Use fast recall of multiplication and division number facts in real-life situations • Use decimal fractions in real-life situations • Use mental and written strategies for multiplication and division in real-life situations • Select an efficient method for solving a problem, for example, mental estimation, mental or written strategies, or by using a calculator • Use strategies to evaluate the reasonableness of answers • Estimate sum, difference, product and quotient in real-life situations Learning Continuum for Number IB/TEKS Alignment Phase 7 Conceptual Understandings • The base 10 place value system extends infinitely in two directions. • Fractions, decimals and percentages are ways of representing whole-part relationships. • For fractional and decimal computation, the ideas developed for whole-number computation can apply. • Ratios are a comparison of two numbers or quantities. Learning Outcomes When constructing meaning, learners: • Model numbers to millions or beyond using the base 10 place value system • Model ratios • Model integers in appropriate contexts • Model exponents • Model improper fractions and mixed numbers • Simplify fractions using manipulatives • Model decimals to thousandths or beyond • Model percentages • Understand the relationship between fractions, decimals and percentages • Model addition, subtractions, multiplication and division of fractions • Model addition, subtraction, multiplication and division of decimals When transferring meaning into symbols, learners: • Compare and order non-negative rational numbers • Generate equivalent forms of rational numbers including whole numbers, fractions and decimals • Use integers to represent real-life situations • Write prime factorizations using exponents • Identify factors of a positive integer, common factors, and the greatest common factor of a set of positive integers • Identify multiples of a positive integer and common multiples and the least common multiple of a set of positive integers • Use addition and subtraction to solve problems involving fractions and decimals • Use multiplication and division of whole numbers to solve problems • Estimate and round to approximate reasonable results • Use order of operations to simplify whole number expressions in problem solving situations • Use ratios to describe proportional situations • Represent ratios and percents with concrete models, fractions and decimals • Use ratios to make predictions in proportional situations • Identify and apply mathematics to everyday experiences, to activities in and outside of school, etc. • Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan and evaluating the solution for reasonableness • Select from a variety of tools and techniques to solve problems • Select an appropriate problem-solving strategy • Communicate mathematical ideas • Use logical reasoning to support answers When applying with understanding, learners: • Use whole numbers up to millions or beyond in real-life situations • Use ratios in real-life situations • Use integers in real-life situations • Convert improper fractions to mixed numbers and vice versa in real-life situations • Simplify fractions in computation answers • Use fractions, decimals and percentages interchangeably in real-life situations • Select and use an appropriate sequence of operations to solve word problems • Select an efficient method of solving a problem • Use strategies to evaluate the reasonableness of answers • Use mental and written strategies of adding, subtracting, multiplying and dividing fractions and decimals in real-life situations • Estimate and make approximations in real-life situations involving fractions, decimals and percentages Learning Continuum for Number IB/TEKS Alignment Phase 8 Conceptual Understandings • Fractions, decimals and percentages are ways of representing whole-part relationships and can be used interchangeably. • For fractional and decimal computation, the ideas developed for whole-number computation can apply. • Mathematics can be used to solve problems connected to everyday experiences, investigations in other disciplines and activities in and outside of school Learning Outcomes When constructing meaning, learners: • Understand the relationship between fractions, decimals and percentages • Model addition, subtractions, multiplication and division of fractions • Model addition, subtraction, multiplication and division of decimals When transferring meaning into symbols, learners: • Compare and order integers and positive rational numbers • Convert between fractions, decimals, whole numbers, and percents mentally, on paper, or with a calculator • Represent squares and square roots using geometric models • Represent multiplication and division situations involving fractions and decimals with models, including concrete objects, pictures, words, and numbers • Use addition, subtraction, multiplication and division to solve problems involving fractions and decimals • Use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms • Use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes and student-teacher ratio • Simplify numerical expressions involving order of operations and exponents • Select and use appropriate operations to solve problems and justify the selections • Determine the reasonableness of a solution to a problem • Identify and apply mathematics to everyday experiences, to activities in and outside of school, etc. • Use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan and evaluating the solution for reasonableness • Select from a variety of tools and techniques to solve problems • Select an appropriate problem-solving strategy • Communicate mathematical ideas • Use logical reasoning to support answers • Evaluate the effectiveness of different representations to communicate ideas When applying with understanding, learners: • Use ratios in real-life situations • Use integers in real-life situations • Use fractions, decimals and percentages interchangeably in real-life situations • Select and use an appropriate sequence of operations to solve word problems • Select an efficient method of solving a problem • Use strategies to evaluate the reasonableness of answers • Use mental and written strategies of adding, subtracting, multiplying and dividing fractions and decimals in real-life situations • Estimate and make approximations in real-life situations involving fractions, decimals and percentages Mathematics scope and sequence Primary Years Programme Mathematics scope and sequence Primary Years Programme PYP110Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire Published February 2009 International Baccalaureate Peterson House, Malthouse Avenue, Cardiff Gate Cardiff, Wales GB CF23 8GL United Kingdom Phone: +44 29 2054 7777 Fax: +44 29 2054 7778 Website: http://www.ibo.org © International Baccalaureate Organization 2009 The International Baccalaureate (IB) offers three high quality and challenging educational programmes for a worldwide community of schools, aiming to create a better, more peaceful world. 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Phone: +44 29 2054 7746 Fax: +44 29 2054 7779 Email: sales@ibo.org Primary Years Programme Mathematics scope and sequence IB mission statement The International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help to create a better and more peaceful world through intercultural understanding and respect. To this end the organization works with schools, governments and international organizations to develop challenging programmes of international education and rigorous assessment. These programmes encourage students across the world to become active, compassionate and lifelong learners who understand that other people, with their differences, can also be right. IB learner profile The aim of all IB programmes is to develop internationally minded people who, recognizing their common humanity and shared guardianship of the planet, help to create a better and more peaceful world. IB learners strive to be: Inquirers They develop their natural curiosity. They acquire the skills necessary to conduct inquiry and research and show independence in learning. They actively enjoy learning and this love of learning will be sustained throughout their lives. Knowledgeable They explore concepts, ideas and issues that have local and global significance. In so doing, they acquire in-depth knowledge and develop understanding across a broad and balanced range of disciplines. Thinkers They exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems, and make reasoned, ethical decisions. Communicators They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication. They work effectively and willingly in collaboration with others. Principled They act with integrity and honesty, with a strong sense of fairness, justice and respect for the dignity of the individual, groups and communities. They take responsibility for their own actions and the consequences that accompany them. Open-minded They understand and appreciate their own cultures and personal histories, and are open to the perspectives, values and traditions of other individuals and communities. They are accustomed to seeking and evaluating a range of points of view, and are willing to grow from the experience. Caring They show empathy, compassion and respect towards the needs and feelings of others. They have a personal commitment to service, and act to make a positive difference to the lives of others and to the environment. Risk-takers They approach unfamiliar situations and uncertainty with courage and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are brave and articulate in defending their beliefs. Balanced They understand the importance of intellectual, physical and emotional balance to achieve personal well-being for themselves and others. Reflective They give thoughtful consideration to their own learning and experience. They are able to assess and understand their strengths and limitations in order to support their learning and personal development. © International Baccalaureate Organization 2007 Mathematics scope and sequence Contents Introduction to the PYP mathematics scope and sequence 1 What the PYP believes about learning mathematics 1 Mathematics in a transdisciplinary programme 2 The structure of the PYP mathematics scope and sequence 3 How to use the PYP mathematics scope and sequence 4 Viewing a unit of inquiry through the lens of mathematics 5 Learning continuums 6 Data handling 6 Measurement 10 Shape and space 14 Pattern and function 18 Number 21 Samples 26 Mathematics scope and sequence 1 Introduction to the PYP mathematics scope and sequence The information in this scope and sequence document should be read in conjunction with the mathematics subject annex in Making the PYP happen: A curriculum framework for international primary education (2007). What the PYP believes about learning mathematics The power of mathematics for describing and analysing the world around us is such that it has become a highly effective tool for solving problems. It is also recognized that students can appreciate the intrinsic fascination of mathematics and explore the world through its unique perceptions. In the same way that students describe themselves as “authors” or “artists”, a school’s programme should also provide students with the opportunity to see themselves as “mathematicians”, where they enjoy and are enthusiastic when exploring and learning about mathematics. In the IB Primary Years Programme (PYP), mathematics is also viewed as a vehicle to support inquiry, providing a global language through which we make sense of the world around us. It is intended that students become competent users of the language of mathematics, and can begin to use it as a way of thinking, as opposed to seeing it as a series of facts and equations to be memorized. How children learn mathematics It is important that learners acquire mathematical understanding by constructing their own meaning through ever-increasing levels of abstraction, starting with exploring their own personal experiences, understandings and knowledge. Additionally, it is fundamental to the philosophy of the PYP that, since it is to be used in real-life situations, mathematics needs to be taught in relevant, realistic contexts, rather than by attempting to impart a fixed body of knowledge directly to students. How children learn mathematics can be described using the following stages (see figure 1). Applying with understanding Transferring meaning Constructing meaning Figure 1 How children learn mathematics Introduction to the PYP mathematics scope and sequence Mathematics scope and sequence2 Constructing meaning about mathematics Learners construct meaning based on their previous experiences and understanding, and by reflecting upon their interactions with objects and ideas. Therefore, involving learners in an active learning process, where they are provided with possibilities to interact with manipulatives and to engage in conversations with others, is paramount to this stage of learning mathematics. When making sense of new ideas all learners either interpret these ideas to conform to their present understanding or they generate a new understanding that accounts for what they perceive to be occurring. This construct will continue to evolve as learners experience new situations and ideas, have an opportunity to reflect on their understandings and make connections about their learning. Transferring meaning into symbols Only when learners have constructed their ideas about a mathematical concept should they attempt to transfer this understanding into symbols. Symbolic notation can take the form of pictures, diagrams, modelling with concrete objects and mathematical notation. Learners should be given the opportunity to describe their understanding using their own method of symbolic notation, then learning to transfer them into conventional mathematical notation. Applying with understanding Applying with understanding can be viewed as the learners demonstrating and acting on their understanding. Through authentic activities, learners should independently select and use appropriate symbolic notation to process and record their thinking. These authentic activities should include a range of practical hands-on problem-solving activities and realistic situations that provide the opportunity to demonstrate mathematical thinking through presented or recorded formats. In this way, learners are able to apply their understanding of mathematical concepts as well as utilize mathematical skills and knowledge. As they work through these stages of learning, students and teachers use certain processes of mathematical reasoning. They use patterns and relationships to analyse the problem situations upon which they are working.• They make and evaluate their own and each other’s ideas.• They use models, facts, properties and relationships to explain their thinking.• They justify their answers and the processes by which they arrive at solutions.• In this way, students validate the meaning they construct from their experiences with mathematical situations. By explaining their ideas, theories and results, both orally and in writing, they invite constructive feedback and also lay out alternative models of thinking for the class. Consequently, all benefit from this interactive process. Mathematics in a transdisciplinary programme Wherever possible, mathematics should be taught through the relevant, realistic context of the units of inquiry. The direct teaching of mathematics in a unit of inquiry may not always be feasible but, where appropriate, prior learning or follow-up activities may be useful to help students make connections between the different aspects of the curriculum. Students also need opportunities to identify and reflect on “big ideas” within and between the different strands of mathematics, the programme of inquiry and other subject areas. Links to the transdisciplinary themes should be explicitly made, whether or not the mathematics is being taught within the programme of inquiry. A developing understanding of these links will contribute to the students’ understanding of mathematics in the world and to their understanding of the transdisciplinary Introduction to the PYP mathematics scope and sequence Mathematics scope and sequence 3 theme. The role of inquiry in mathematics is important, regardless of whether it is being taught inside or outside the programme of inquiry. However, it should also be recognized that there are occasions when it is preferable for students to be given a series of strategies for learning mathematical skills in order to progress in their mathematical understanding rather than struggling to proceed. The structure of the PYP mathematics scope and sequence This scope and sequence aims to provide information for the whole school community of the learning that is going on in the subject area of mathematics. It has been designed in recognition that learning mathematics is a developmental process and that the phases a learner passes through are not always linear or age related. For this reason the content is presented in continuums for each of the five strands of mathematics—data handling, measurement, shape and space, pattern and function, and number. For each of the strands there is a strand description and a set of overall expectations. The overall expectations provide a summary of the understandings and subsequent learning being developed for each phase within a strand. The content of each continuum has been organized into four phases of development, with each phase building upon and complementing the previous phase. The continuums make explicit the conceptual understandings that need to be developed at each phase. Evidence of these understandings is described in the behaviours or learning outcomes associated with each phase and these learning outcomes relate specifically to mathematical concepts, knowledge and skills. The learning outcomes have been written to reflect the stages a learner goes through when developing conceptual understanding in mathematics—constructing meaning, transferring meaning into symbols and applying with understanding (see figure 1). To begin with, the learning outcomes identified in the constructing meaning stage strongly emphasize the need for students to develop understanding of mathematical concepts in order to provide them with a secure base for further learning. In the planning process, teachers will need to discuss the ways in which students may demonstrate this understanding. The amount of time and experiences dedicated to this stage of learning will vary from student to student. The learning outcomes in the transferring meaning into symbols stage are more obviously demonstrable and observable. The expectation for students working in this stage is that they have demonstrated understanding of the underlying concepts before being asked to transfer this meaning into symbols. It is acknowledged that, in some strands, symbolic representation will form part of the constructing meaning stage. For example, it is difficult to imagine how a student could construct meaning about the way in which information is expressed as organized and structured data without having the opportunity to collect and represent this data in graphs. In this type of example, perhaps the difference between the two stages is that in the transferring meaning into symbols stage the student will be able to demonstrate increased independence with decreasing amounts of teacher prompting required for them to make connections. Another difference could be that a student’s own symbolic representation may be extended to include more conventional methods of symbolic representation. In the final stage, a number of learning outcomes have been developed to reflect the kind of actions and behaviours that students might demonstrate when applying with understanding. It is important to note that other forms of application might be in evidence in classrooms where there are authentic opportunities for students to make spontaneous connections between the learning that is going on in mathematics and other areas of the curriculum and daily life. When a continuum for a particular strand is observed as a whole, it is clear how the conceptual understandings and the associated learning outcomes develop in complexity as they are viewed across the phases. In each of the phases, there is also a vertical progression where most learning outcomes identified in the constructing meaning stage of the phase are often described as outcomes relating to the transferring Introduction to the PYP mathematics scope and sequence Mathematics scope and sequence4 meaning into symbols and applying with understanding stages of the same phase. However, on some occasions, a mathematical concept is introduced in one phase but students are not expected to apply the concept until a later phase. This is a deliberate decision aimed at providing students with adequate time and opportunities for the ongoing development of understanding of particular concepts. Each of the continuums contains a notes section which provides extra information to clarify certain learning outcomes and to support planning, teaching and learning of particular concepts. How to use the PYP mathematics scope and sequence In the course of reviewing the PYP mathematics scope and sequence, a decision was made to provide schools with a view of how students construct meaning about mathematics concepts in a more developmental way rather than in fixed age bands. Teachers will need to be given time to discuss this introduction and accompanying continuums and how they can be used to inform planning, teaching and assessing of mathematics in the school. The following points should also be considered in this discussion process. It is acknowledged that there are earlier and later phases that have not been described in these • continuums. Each learner is a unique individual with different life experiences and no two learning pathways are • the same. Learners within the same age group will have different proficiency levels and needs; therefore, teachers • should consider a range of phases when planning mathematics learning experiences for a class. Learners are likely to display understanding and skills from more than one of the phases at a time. • Consequently, it is recognized that teachers will interpret this scope and sequence according to the needs of their students and their particular teaching situations. The continuums are not prescriptive tools that assume a learner must attain all the outcomes of a • particular phase before moving on to the next phase, nor that the learner should be in the same phase for each strand. Each teacher needs to identify the extent to which these factors affect the learner. Plotting a mathematical profile for each student is a complex process for PYP teachers. Prior knowledge should therefore never be assumed before embarking on the presentation or introduction of a mathematical concept. Schools may decide to use and adapt the PYP scope and sequences according to their needs. For example, a school may decide to frame their mathematics scope and sequence document around the conceptual understandings outlined in the PYP document, but develop other aspects (for example outcomes, indicators, benchmarks, standards) differently. Alternatively, they may decide to incorporate the continuums from the PYP documents into their existing school documents. Schools need to be mindful of practice C1.23 in the IB Programme standards and practices (2005) that states, “If the school adapts, or develops, its own scope and sequence documents for each PYP subject area, the level of overall expectation regarding student achievement expressed in these documents at least matches that expressed in the PYP scope and sequence documents”. To arrive at such a judgment, and given that the overall expectations in the PYP mathematics scope and sequence are presented as broad generalities, it is recommended that the entire document be read and considered. Introduction to the PYP mathematics scope and sequence Mathematics scope and sequence 5 Viewing a unit of inquiry through the lens of mathematics The following diagram shows a sample process for viewing a unit of inquiry through the lens of mathematics. This has been developed as an example of how teachers can identify the mathematical concepts, skills and knowledge required to successfully engage in the units of inquiry. Note: It is important that the integrity of a central idea and ensuing inquiry is not jeopardized by a subject- specific focus too early in the collaborative planning process. Once an inquiry has been planned through to identification of learning experiences, it would be appropriate to consider the following process. Figure 2 Sample processes for viewing a unit of inquiry through the lens of mathematics Will mathematics inform this unit? Do aspects of the transdisciplinary theme initially stand out as being mathematics related? Will mathematical knowledge, concepts and skills be needed to understand the central idea? Will mathematical knowledge, concepts and skills be needed to develop the lines of inquiry within the unit? What mathematical knowledge, concepts and skills will the students need to be able to engage with and/or inquire into the following? (Refer to mathematics scope and sequence documents.) Central idea• Lines of inquiry• Assessment tasks• Teacher questions, student questions• Learning experiences• In your planning team, list the knowledge, concepts and skills. What prior knowledge, concepts and skills do the students have that can be utilized and built upon? Which stages of understanding are the learners in the class working at—constructing meaning, transferring meaning into symbols, or applying with understanding? How will we know what they have learned? Identify opportunities for assessment. Decide which aspects can be learned: within the unit of inquiry (learning through mathematics)• as subject-specific, prior to being used and applied in the context of the inquiry (inquiry into • mathematics). Mathematics scope and sequence6 Learning continuums Data handling Data handling allows us to make a summary of what we know about the world and to make inferences about what we do not know. Data can be collected, organized, represented and summarized in a variety of ways to highlight • similarities, differences and trends; the chosen format should illustrate the information without bias or distortion. Probability can be expressed qualitatively by using terms such as “unlikely”, “certain” or “impossible”. It • can be expressed quantitatively on a numerical scale. Overall expectations Phase 1 Learners will develop an understanding of how the collection and organization of information helps to make sense of the world. They will sort, describe and label objects by attributes and represent information in graphs including pictographs and tally marks. The learners will discuss chance in daily events. Phase 2 Learners will understand how information can be expressed as organized and structured data and that this can occur in a range of ways. They will collect and represent data in different types of graphs, interpreting the resulting information for the purpose of answering questions. The learners will develop an understanding that some events in daily life are more likely to happen than others and they will identify and describe likelihood using appropriate vocabulary. Phase 3 Learners will continue to collect, organize, display and analyse data, developing an understanding of how different graphs highlight different aspects of data more efficiently. They will understand that scale can represent different quantities in graphs and that mode can be used to summarize a set of data. The learners will make the connection that probability is based on experimental events and can be expressed numerically. Phase 4 Learners will collect, organize and display data for the purposes of valid interpretation and communication. They will be able to use the mode, median, mean and range to summarize a set of data. They will create and manipulate an electronic database for their own purposes, including setting up spreadsheets and using simple formulas to create graphs. Learners will understand that probability can be expressed on a scale (0–1 or 0%–100%) and that the probability of an event can be predicted theoretically. Learning continuums Mathematics scope and sequence 7 Le a r n i n g c o n t i n u u m f o r d a t a h a n d l i n g Ph a s e 1 Ph a s e 2 Ph a s e 3 Ph a s e 4 Co n c e p t u a l u n d e r s t a n d i n g s We c o l l e c t i n f o r m a t i o n t o m a k e s e n s e o f th e w o r l d a r o u n d u s . Or g a n i z i n g o b j e c t s a n d e v e n t s h e l p s u s t o so l v e p r o b l e m s . Ev e n t s i n d a i l y l i f e i n v o l v e c h a n c e . Co n c e p t u a l u n d e r s t a n d i n g s In f o r m a t i o n c a n b e e x p r e s s e d a s or g a n i z e d a n d s t r u c t u r e d d a t a . Ob j e c t s a n d e v e n t s c a n b e o r g a n i z e d i n di f f e r e n t w a y s . So m e e v e n t s i n d a i l y l i f e a r e m o r e l i k e l y t o ha p p e n t h a n o t h e r s . Co n c e p t u a l u n d e r s t a n d i n g s Da t a c a n b e c o l l e c t e d , o r g a n i z e d , di s p l a y e d a n d a n a l y s e d i n d i f f e r e n t w a y s . Di f f e r e n t g r a p h f o r m s h i g h l i g h t d i f f e r e n t as p e c t s o f d a t a m o r e e f f i c i e n t l y . Pr o b a b i l i t y c a n b e b a s e d o n e x p e r i m e n t a l ev e n t s i n d a i l y l i f e . Pr o b a b i l i t y c a n b e e x p r e s s e d i n n u m e r i c a l no t a t i o n s . Co n c e p t u a l u n d e r s t a n d i n g s Da t a c a n b e p r e s e n t e d e f f e c t i v e l y f o r v a l i d in t e r p r e t a t i o n a n d c o m m u n i c a t i o n . Ra n g e , m o d e , m e d i a n a n d m e a n c a n b e us e d t o a n a l y s e s t a t i s t i c a l d a t a . Pr o b a b i l i t y c a n b e r e p r e s e n t e d o n a s c a l e be t w e e n 0 – 1 o r 0 % – 1 0 0 % . Th e p r o b a b i l i t y o f a n e v e n t c a n b e pr e d i c t e d t h e o r e t i c a l l y . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t s e t s c a n b e o r g a n i z e d • by d i f f e r e n t a t t r i b u t e s un d e r s t a n d t h a t i n f o r m a t i o n a b o u t • th e m s e l v e s a n d t h e i r s u r r o u n d i n g s ca n b e o b t a i n e d i n d i f f e r e n t w a y s di s c u s s c h a n c e i n d a i l y e v e n t s • (i m p o s s i b l e , m a y b e , c e r t a i n ) . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t s e t s c a n b e o r g a n i z e d • by o n e o r m o r e a t t r i b u t e s un d e r s t a n d t h a t i n f o r m a t i o n a b o u t • th e m s e l v e s a n d t h e i r s u r r o u n d i n g s ca n b e c o l l e c t e d a n d r e c o r d e d i n di f f e r e n t w a y s un d e r s t a n d t h e c o n c e p t o f c h a n c e i n • da i l y e v e n t s ( i m p o s s i b l e , l e s s l i k e l y , ma y b e , m o s t l i k e l y , c e r t a i n ) . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t d a t a c a n b e • co l l e c t e d , d i s p l a y e d a n d i n t e r p r e t e d us i n g s i m p l e g r a p h s , f o r e x a m p l e , b a r gr a p h s , l i n e g r a p h s un d e r s t a n d t h a t s c a l e c a n r e p r e s e n t • di f f e r e n t q u a n t i t i e s i n g r a p h s un d e r s t a n d t h a t t h e m o d e c a n b e • us e d t o s u m m a r i z e a s e t o f d a t a un d e r s t a n d t h a t o n e o f t h e p u r p o s e s • of a d a t a b a s e i s t o a n s w e r q u e s t i o n s an d s o l v e p r o b l e m s un d e r s t a n d t h a t p r o b a b i l i t y i s b a s e d • on e x p e r i m e n t a l e v e n t s . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t d i f f e r e n t t y p e s o f • gr a p h s h a v e s p e c i a l p u r p o s e s un d e r s t a n d t h a t t h e m o d e , m e d i a n , • me a n a n d r a n g e c a n s u m m a r i z e a s e t of d a t a un d e r s t a n d t h a t p r o b a b i l i t y c a n b e • ex p r e s s e d i n s c a l e ( 0 – 1 ) o r p e r c e n t (0 % – 1 0 0 % ) un d e r s t a n d t h e d i f f e r e n c e b e t w e e n • ex p e r i m e n t a l a n d t h e o r e t i c a l pr o b a b i l i t y . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : re p r e s e n t i n f o r m a t i o n t h r o u g h • pi c t o g r a p h s a n d t a l l y m a r k s so r t a n d l a b e l r e a l o b j e c t s b y • at t r i b u t e s . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : co l l e c t a n d r e p r e s e n t d a t a i n d i f f e r e n t • ty p e s o f g r a p h s , f o r e x a m p l e , t a l l y ma r k s , b a r g r a p h s Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : co l l e c t , d i s p l a y a n d i n t e r p r e t d a t a • us i n g s i m p l e g r a p h s , f o r e x a m p l e , b a r gr a p h s , l i n e g r a p h s id e n t i f y , r e a d a n d i n t e r p r e t r a n g e a n d • sc a l e o n g r a p h s Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : co l l e c t , d i s p l a y a n d i n t e r p r e t d a t a i n • ci r c l e g r a p h s ( p i e c h a r t s ) a n d l i n e g r a p h s id e n t i f y , d e s c r i b e a n d e x p l a i n t h e • ra n g e , m o d e , m e d i a n a n d m e a n i n a se t o f d a t a Learning continuums Mathematics scope and sequence8 re p r e s e n t t h e r e l a t i o n s h i p b e t w e e n • ob j e c t s i n s e t s u s i n g t r e e , V e n n a n d Ca r r o l l d i a g r a m s ex p r e s s t h e c h a n c e o f a n e v e n t • ha p p e n i n g u s i n g w o r d s o r p h r a s e s (i m p o s s i b l e , l e s s l i k e l y , m a y b e , m o s t li k e l y , c e r t a i n ) . id e n t i f y t h e m o d e o f a s e t o f d a t a • us e t r e e d i a g r a m s t o e x p r e s s • pr o b a b i l i t y u s i n g s i m p l e f r a c t i o n s . se t u p a s p r e a d s h e e t u s i n g s i m p l e • fo r m u l a s t o m a n i p u l a t e d a t a a n d t o cr e a t e g r a p h s ex p r e s s p r o b a b i l i t i e s u s i n g s c a l e ( 0 – 1 ) • or p e r c e n t ( 0 % – 1 0 0 % ) . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : cr e a t e p i c t o g r a p h s a n d t a l l y m a r k s • cr e a t e l i v i n g g r a p h s u s i n g r e a l o b j e c t s • an d p e o p l e * de s c r i b e r e a l o b j e c t s a n d e v e n t s b y • at t r i b u t e s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : co l l e c t , d i s p l a y a n d i n t e r p r e t d a t a f o r • th e p u r p o s e o f a n s w e r i n g q u e s t i o n s cr e a t e a p i c t o g r a p h a n d s a m p l e b a r • gr a p h o f r e a l o b j e c t s a n d i n t e r p r e t da t a b y c o m p a r i n g q u a n t i t i e s ( f o r ex a m p l e , m o r e , f e w e r , l e s s t h a n , gr e a t e r t h a n ) us e t r e e , V e n n a n d C a r r o l l d i a g r a m s t o • ex p l o r e r e l a t i o n s h i p s b e t w e e n d a t a id e n t i f y a n d d e s c r i b e c h a n c e i n d a i l y • ev e n t s ( i m p o s s i b l e , l e s s l i k e l y , m a y b e , mo s t l i k e l y , c e r t a i n ) . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : de s i g n a s u r v e y a n d s y s t e m a t i c a l l y • co l l e c t , o r g a n i z e a n d d i s p l a y d a t a i n pi c t o g r a p h s a n d b a r g r a p h s se l e c t a p p r o p r i a t e g r a p h f o r m ( s ) t o • di s p l a y d a t a in t e r p r e t r a n g e a n d s c a l e o n g r a p h s • us e p r o b a b i l i t y t o d e t e r m i n e • ma t h e m a t i c a l l y f a i r a n d u n f a i r g a m e s an d t o e x p l a i n p o s s i b l e o u t c o m e s ex p r e s s p r o b a b i l i t y u s i n g s i m p l e • fr a c t i o n s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : de s i g n a s u r v e y a n d s y s t e m a t i c a l l y • co l l e c t , r e c o r d , o r g a n i z e a n d d i s p l a y th e d a t a i n a b a r g r a p h , c i r c l e g r a p h , li n e g r a p h id e n t i f y , d e s c r i b e a n d e x p l a i n t h e • ra n g e , m o d e , m e d i a n a n d m e a n i n a se t o f d a t a cr e a t e a n d m a n i p u l a t e a n e l e c t r o n i c • da t a b a s e f o r t h e i r o w n p u r p o s e s de t e r m i n e t h e t h e o r e t i c a l p r o b a b i l i t y • of a n e v e n t a n d e x p l a i n w h y i t m i g h t di f f e r f r o m e x p e r i m e n t a l p r o b a b i l i t y . No t e s Un i t s o f i n q u i r y w i l l b e r i c h i n op p o r t u n i t i e s f o r c o l l e c t i n g a n d or g a n i z i n g i n f o r m a t i o n . I t m a y b e u s e f u l fo r t h e t e a c h e r t o p r o v i d e s c a f f o l d s , s u c h as q u e s t i o n s f o r e x p l o r a t i o n , a n d t h e mo d e l l i n g o f g r a p h s a n d d i a g r a m s . *L i v i n g g r a p h s r e f e r t o d a t a t h a t i s or g a n i z e d b y p h y s i c a l l y m o v i n g a n d ar r a n g i n g s t u d e n t s o r a c t u a l m a t e r i a l s in s u c h a w a y a s t o s h o w a n d c o m p a r e qu a n t i t i e s . No t e s An i n c r e a s i n g n u m b e r o f c o m p u t e r a n d we b - b a s e d a p p l i c a t i o n s a r e a v a i l a b l e t h a t en a b l e l e a r n e r s t o m a n i p u l a t e d a t a i n or d e r t o c r e a t e g r a p h s . St u d e n t s s h o u l d h a v e a l o t o f e x p e r i e n c e of o r g a n i z i n g d a t a i n a v a r i e t y o f w a y s , an d o f t a l k i n g a b o u t t h e a d v a n t a g e s a n d di s a d v a n t a g e s o f e a c h . I n t e r p r e t a t i o n s o f da t a s h o u l d i n c l u d e t h e i n f o r m a t i o n t h a t ca n n o t b e c o n c l u d e d a s w e l l a s t h a t w h i c h ca n . I t i s i m p o r t a n t t o r e m e m b e r t h a t th e c h o s e n f o r m a t s h o u l d i l l u s t r a t e t h e in f o r m a t i o n w i t h o u t b i a s . No t e s Us i n g d a t a t h a t h a s b e e n c o l l e c t e d a n d sa v e d i s a s i m p l e w a y t o b e g i n d i s c u s s i n g th e m o d e . A f u r t h e r e x t e n s i o n o f m o d e i s to f o r m u l a t e t h e o r i e s a b o u t w h y a c e r t a i n ch o i c e i s t h e m o d e . St u d e n t s s h o u l d h a v e t h e o p p o r t u n i t y to u s e d a t a b a s e s , i d e a l l y , t h o s e c r e a t e d us i n g d a t a c o l l e c t e d b y t h e s t u d e n t s t h e n en t e r e d i n t o a d a t a b a s e b y t h e t e a c h e r o r to g e t h e r . No t e s A d a t a b a s e i s a c o l l e c t i o n o f d a t a , w h e r e th e d a t a c a n b e d i s p l a y e d i n m a n y f o r m s . Th e d a t a c a n b e c h a n g e d a t a n y t i m e . A sp r e a d s h e e t i s a t y p e o f d a t a b a s e w h e r e in f o r m a t i o n i s s e t o u t i n a t a b l e . U s i n g a c o m m o n s e t o f d a t a i s a g o o d w a y f o r st u d e n t s t o s t a r t t o s e t u p t h e i r o w n da t a b a s e s . A u n i t o f i n q u i r y w o u l d b e an e x c e l l e n t s o u r c e o f c o m m o n d a t a f o r st u d e n t p r a c t i c e . Learning continuums Mathematics scope and sequence 9 Ve r y y o u n g c h i l d r e n v i e w t h e w o r l d a s a pl a c e o f p o s s i b i l i t i e s . T h e t e a c h e r s h o u l d tr y t o i n t r o d u c e p r a c t i c a l e x a m p l e s a n d sh o u l d u s e a p p r o p r i a t e v o c a b u l a r y . Di s c u s s i o n s a b o u t c h a n c e i n d a i l y e v e n t s sh o u l d b e r e l e v a n t t o t h e c o n t e x t o f t h e le a r n e r s . Si t u a t i o n s t h a t c o m e u p n a t u r a l l y i n th e c l a s s r o o m , o f t e n t h r o u g h l i t e r a t u r e , pr e s e n t o p p o r t u n i t i e s f o r d i s c u s s i n g pr o b a b i l i t y . D i s c u s s i o n s n e e d t o t a k e pl a c e i n w h i c h s t u d e n t s c a n s h a r e t h e i r se n s e o f l i k e l i h o o d i n t e r m s t h a t a r e u s e f u l to t h e m . Si t u a t i o n s t h a t c o m e u p n a t u r a l l y i n t h e cl a s s r o o m o r f o r m p a r t o f t h e u n i t s o f in q u i r y p r e s e n t o p p o r t u n i t i e s f o r s t u d e n t s to f u r t h e r d e v e l o p t h e i r u n d e r s t a n d i n g o f st a t i s t i c s a n d p r o b a b i l i t y c o n c e p t s . Te c h n o l o g y g i v e s u s t h e o p t i o n o f cr e a t i n g a g r a p h a t t h e p r e s s o f a k e y . Be i n g a b l e t o g e n e r a t e d i f f e r e n t t y p e s of g r a p h s a l l o w s l e a r n e r s t o e x p l o r e a n d ap p r e c i a t e t h e a t t r i b u t e s o f e a c h t y p e o f gr a p h a n d i t s e f f i c a c y i n d i s p l a y i n g t h e da t a . Te c h n o l o g y a l s o g i v e s u s t h e p o s s i b i l i t y of r a p i d l y r e p l i c a t i n g r a n d o m e v e n t s . Co m p u t e r a n d w e b - b a s e d a p p l i c a t i o n s ca n b e u s e d t o t o s s c o i n s , r o l l d i c e , a n d ta b u l a t e a n d g r a p h t h e r e s u l t s . Learning continuums Mathematics scope and sequence10 Measurement To measure is to attach a number to a quantity using a chosen unit. Since the attributes being measured are continuous, ways must be found to deal with quantities that fall between numbers. It is important to know how accurate a measurement needs to be or can ever be. Overall expectations Phase 1 Learners will develop an understanding of how measurement involves the comparison of objects and the ordering and sequencing of events. They will be able to identify, compare and describe attributes of real objects as well as describe and sequence familiar events in their daily routine. Phase 2 Learners will understand that standard units allow us to have a common language to measure and describe objects and events, and that while estimation is a strategy that can be applied for approximate measurements, particular tools allow us to measure and describe attributes of objects and events with more accuracy. Learners will develop these understandings in relation to measurement involving length, mass, capacity, money, temperature and time. Phase 3 Learners will continue to use standard units to measure objects, in particular developing their understanding of measuring perimeter, area and volume. They will select and use appropriate tools and units of measurement, and will be able to describe measures that fall between two numbers on a scale. The learners will be given the opportunity to construct meaning about the concept of an angle as a measure of rotation. Phase 4 Learners will understand that a range of procedures exists to measure different attributes of objects and events, for example, the use of formulas for finding area, perimeter and volume. They will be able to decide on the level of accuracy required for measuring and using decimal and fraction notation when precise measurements are necessary. To demonstrate their understanding of angles as a measure of rotation, the learners will be able to measure and construct angles. Learning continuums Mathematics scope and sequence 11 Le a r n i n g c o n t i n u u m f o r m e a s u r e m e n t Ph a s e 1 Ph a s e 2 Ph a s e 3 Ph a s e 4 Co n c e p t u a l u n d e r s t a n d i n g s Me a s u r e m e n t i n v o l v e s c o m p a r i n g o b j e c t s an d e v e n t s . Ob j e c t s h a v e a t t r i b u t e s t h a t c a n b e me a s u r e d u s i n g n o n - s t a n d a r d u n i t s . Ev e n t s c a n b e o r d e r e d a n d s e q u e n c e d . Co n c e p t u a l u n d e r s t a n d i n g s St a n d a r d u n i t s a l l o w u s t o h a v e a c o m m o n la n g u a g e t o i d e n t i f y , c o m p a r e , o r d e r a n d se q u e n c e o b j e c t s a n d e v e n t s . We u s e t o o l s t o m e a s u r e t h e a t t r i b u t e s o f ob j e c t s a n d e v e n t s . Es t i m a t i o n a l l o w s u s t o m e a s u r e w i t h di f f e r e n t l e v e l s o f a c c u r a c y . Co n c e p t u a l u n d e r s t a n d i n g s Ob j e c t s a n d e v e n t s h a v e a t t r i b u t e s t h a t ca n b e m e a s u r e d u s i n g a p p r o p r i a t e t o o l s . Re l a t i o n s h i p s e x i s t b e t w e e n s t a n d a r d un i t s t h a t m e a s u r e t h e s a m e a t t r i b u t e s . Co n c e p t u a l u n d e r s t a n d i n g s Ac c u r a c y o f m e a s u r e m e n t s d e p e n d s o n th e s i t u a t i o n a n d t h e p r e c i s i o n o f t h e t o o l . Co n v e r s i o n o f u n i t s a n d m e a s u r e m e n t s al l o w s u s t o m a k e s e n s e o f t h e w o r l d w e li v e i n . A r a n g e o f p r o c e d u r e s e x i s t s t o m e a s u r e di f f e r e n t a t t r i b u t e s o f o b j e c t s a n d e v e n t s . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t a t t r i b u t e s o f r e a l • ob j e c t s c a n b e c o m p a r e d a n d de s c r i b e d , f o r e x a m p l e , l o n g e r , sh o r t e r , h e a v i e r , e m p t y , f u l l , h o t t e r , co l d e r un d e r s t a n d t h a t e v e n t s i n d a i l y • ro u t i n e s c a n b e d e s c r i b e d a n d se q u e n c e d , f o r e x a m p l e , b e f o r e , a f t e r , be d t i m e , s t o r y t i m e , t o d a y , t o m o r r o w . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h e u s e o f s t a n d a r d u n i t s • to m e a s u r e , f o r e x a m p l e , l e n g t h , m a s s , mo n e y , t i m e , t e m p e r a t u r e un d e r s t a n d t h a t t o o l s c a n b e u s e d t o • me a s u r e un d e r s t a n d t h a t c a l e n d a r s c a n b e • us e d t o d e t e r m i n e t h e d a t e , a n d t o id e n t i f y a n d s e q u e n c e d a y s o f t h e we e k a n d m o n t h s o f t h e y e a r un d e r s t a n d t h a t t i m e i s m e a s u r e d • us i n g u n i v e r s a l u n i t s o f m e a s u r e , f o r ex a m p l e , y e a r s , m o n t h s , d a y s , h o u r s , mi n u t e s a n d s e c o n d s . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h e u s e o f s t a n d a r d • un i t s t o m e a s u r e p e r i m e t e r , a r e a a n d vo l u m e un d e r s t a n d t h a t m e a s u r e s c a n f a l l • be t w e e n n u m b e r s o n a m e a s u r e m e n t sc a l e , f o r e x a m p l e , 3 ½ kg , b e t w e e n 4 cm a n d 5 cm un d e r s t a n d r e l a t i o n s h i p s b e t w e e n • un i t s , f o r e x a m p l e , m e t r e s , ce n t i m e t r e s a n d m i l l i m e t r e s un d e r s t a n d a n a n g l e a s a m e a s u r e o f • ro t a t i o n . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d p r o c e d u r e s f o r f i n d i n g • ar e a , p e r i m e t e r a n d v o l u m e un d e r s t a n d t h e r e l a t i o n s h i p s b e t w e e n • ar e a a n d p e r i m e t e r , b e t w e e n a r e a a n d vo l u m e , a n d b e t w e e n v o l u m e a n d ca p a c i t y un d e r s t a n d u n i t c o n v e r s i o n s w i t h i n • me a s u r e m e n t s y s t e m s ( m e t r i c o r cu s t o m a r y ) . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : id e n t i f y , c o m p a r e a n d d e s c r i b e • at t r i b u t e s o f r e a l o b j e c t s , f o r e x a m p l e , lo n g e r , s h o r t e r , h e a v i e r , e m p t y , f u l l , ho t t e r , c o l d e r Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : es t i m a t e a n d m e a s u r e o b j e c t s u s i n g • st a n d a r d u n i t s o f m e a s u r e m e n t : le n g t h , m a s s , c a p a c i t y , m o n e y a n d te m p e r a t u r e Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : es t i m a t e a n d m e a s u r e u s i n g s t a n d a r d • un i t s o f m e a s u r e m e n t : p e r i m e t e r , a r e a an d v o l u m e Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : de v e l o p a n d d e s c r i b e f o r m u l a s f o r • fi n d i n g p e r i m e t e r , a r e a a n d v o l u m e us e d e c i m a l a n d f r a c t i o n n o t a t i o n i n • me a s u r e m e n t , f o r e x a m p l e , 3 . 2 cm , 1. 4 7 kg , 1 ½ mi l e s Learning continuums Mathematics scope and sequence12 co m p a r e t h e l e n g t h , m a s s a n d • ca p a c i t y o f o b j e c t s u s i n g n o n - st a n d a r d u n i t s id e n t i f y , d e s c r i b e a n d s e q u e n c e • ev e n t s i n t h e i r d a i l y r o u t i n e , f o r ex a m p l e , b e f o r e , a f t e r , b e d t i m e , st o r y t i m e , t o d a y , t o m o r r o w . re a d a n d w r i t e t h e t i m e t o t h e h o u r , • ha l f h o u r a n d q u a r t e r h o u r es t i m a t e a n d c o m p a r e l e n g t h s o f t i m e : • se c o n d , m i n u t e , h o u r , d a y , w e e k a n d mo n t h . de s c r i b e m e a s u r e s t h a t f a l l b e t w e e n • nu m b e r s o n a s c a l e re a d a n d w r i t e d i g i t a l a n d a n a l o g u e • ti m e o n 1 2 - h o u r a n d 2 4 - h o u r c l o c k s . re a d a n d i n t e r p r e t s c a l e s o n a r a n g e o f • me a s u r i n g i n s t r u m e n t s me a s u r e a n d c o n s t r u c t a n g l e s i n • de g r e e s u s i n g a p r o t r a c t o r ca r r y o u t s i m p l e u n i t c o n v e r s i o n s • wi t h i n a s y s t e m o f m e a s u r e m e n t (m e t r i c o r c u s t o m a r y ) . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : de s c r i b e o b s e r v a t i o n s a b o u t e v e n t s • an d o b j e c t s i n r e a l - l i f e s i t u a t i o n s us e n o n - s t a n d a r d u n i t s o f • me a s u r e m e n t t o s o l v e p r o b l e m s i n re a l - l i f e s i t u a t i o n s i n v o l v i n g l e n g t h , ma s s a n d c a p a c i t y . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : us e s t a n d a r d u n i t s o f m e a s u r e m e n t t o • so l v e p r o b l e m s i n r e a l - l i f e s i t u a t i o n s in v o l v i n g l e n g t h , m a s s , c a p a c i t y , mo n e y a n d t e m p e r a t u r e us e m e a s u r e s o f t i m e t o a s s i s t w i t h • pr o b l e m s o l v i n g i n r e a l - l i f e s i t u a t i o n s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : us e s t a n d a r d u n i t s o f m e a s u r e m e n t t o • so l v e p r o b l e m s i n r e a l - l i f e s i t u a t i o n s in v o l v i n g p e r i m e t e r , a r e a a n d v o l u m e se l e c t a p p r o p r i a t e t o o l s a n d u n i t s o f • me a s u r e m e n t us e t i m e l i n e s i n u n i t s o f i n q u i r y a n d • ot h e r r e a l - l i f e s i t u a t i o n s . Wh e n • ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : se l e c t a n d u s e a p p r o p r i a t e u n i t s • of m e a s u r e m e n t a n d t o o l s t o s o l v e pr o b l e m s i n r e a l - l i f e s i t u a t i o n s de t e r m i n e a n d j u s t i f y t h e l e v e l o f • ac c u r a c y r e q u i r e d t o s o l v e r e a l - l i f e pr o b l e m s i n v o l v i n g m e a s u r e m e n t us e d e c i m a l a n d f r a c t i o n a l n o t a t i o n • in m e a s u r e m e n t , f o r e x a m p l e , 3 . 2 cm , 1. 4 7 kg , 1 ½ mi l e s us e t i m e t a b l e s a n d s c h e d u l e s ( 1 2 - • ho u r a n d 2 4 - h o u r c l o c k s ) i n r e a l - l i f e si t u a t i o n s de t e r m i n e t i m e s w o r l d w i d e . • Learning continuums Mathematics scope and sequence 13 No t e s Le a r n e r s n e e d m a n y o p p o r t u n i t i e s t o ex p e r i e n c e a n d q u a n t i f y m e a s u r e m e n t in a d i r e c t k i n e s t h e t i c m a n n e r . T h e y w i l l de v e l o p u n d e r s t a n d i n g o f m e a s u r e m e n t by u s i n g m a n i p u l a t i v e s a n d m a t e r i a l s fr o m t h e i r i m m e d i a t e e n v i r o n m e n t , f o r ex a m p l e , c o n t a i n e r s o f d i f f e r e n t s i z e s , sa n d , w a t e r , b e a d s , c o r k s a n d b e a n s . No t e s Us i n g m a t e r i a l s f r o m t h e i r i m m e d i a t e en v i r o n m e n t , l e a r n e r s c a n i n v e s t i g a t e ho w u n i t s a r e u s e d f o r m e a s u r e m e n t an d h o w m e a s u r e m e n t s v a r y d e p e n d i n g on t h e u n i t t h a t i s u s e d . L e a r n e r s w i l l re f i n e t h e i r e s t i m a t i o n a n d m e a s u r e m e n t sk i l l s b y b a s i n g e s t i m a t i o n s o n p r i o r kn o w l e d g e , m e a s u r i n g t h e o b j e c t a n d co m p a r i n g a c t u a l m e a s u r e m e n t s w i t h th e i r e s t i m a t i o n s . No t e s In o r d e r t o u s e m e a s u r e m e n t m o r e au t h e n t i c a l l y , l e a r n e r s s h o u l d h a v e t h e op p o r t u n i t y t o m e a s u r e r e a l o b j e c t s i n re a l s i t u a t i o n s . T h e u n i t s o f i n q u i r y c a n of t e n p r o v i d e t h e s e r e a l i s t i c c o n t e x t s . A w i d e r a n g e o f m e a s u r i n g t o o l s s h o u l d be a v a i l a b l e t o t h e s t u d e n t s , f o r e x a m p l e , ru l e r s , t r u n d l e w h e e l s , t a p e m e a s u r e s , ba t h r o o m s c a l e s , k i t c h e n s c a l e s , ti m e r s , a n a l o g u e c l o c k s , d i g i t a l c l o c k s , st o p w a t c h e s a n d c a l e n d a r s . T h e r e a r e a n in c r e a s i n g n u m b e r o f c o m p u t e r a n d w e b - ba s e d a p p l i c a t i o n s a v a i l a b l e f o r s t u d e n t s to u s e i n a u t h e n t i c c o n t e x t s . Pl e a s e n o t e t h a t o u t c o m e s r e l a t i n g t o an g l e s a l s o a p p e a r i n t h e s h a p e a n d s p a c e st r a n d . No t e s Le a r n e r s g e n e r a l i z e t h e i r m e a s u r i n g ex p e r i e n c e s a s t h e y d e v i s e p r o c e d u r e s an d f o r m u l a s f o r w o r k i n g o u t p e r i m e t e r , ar e a a n d v o l u m e . Wh i l e t h e e m p h a s i s f o r u n d e r s t a n d i n g i s on m e a s u r e m e n t s y s t e m s c o m m o n l y u s e d in t h e l e a r n e r ’ s w o r l d , i t i s w o r t h w h i l e be i n g a w a r e o f t h e e x i s t e n c e o f o t h e r sy s t e m s a n d h o w c o n v e r s i o n s b e t w e e n sy s t e m s h e l p u s t o m a k e s e n s e o f t h e m . Learning continuums Mathematics scope and sequence14 Shape and space The regions, paths and boundaries of natural space can be described by shape. An understanding of the interrelationships of shape allows us to interpret, understand and appreciate our two-dimensional (2D) and three-dimensional (3D) world. Overall expectations Phase 1 Learners will understand that shapes have characteristics that can be described and compared. They will understand and use common language to describe paths, regions and boundaries of their immediate environment. Phase 2 Learners will continue to work with 2D and 3D shapes, developing the understanding that shapes are classified and named according to their properties. They will understand that examples of symmetry and transformations can be found in their immediate environment. Learners will interpret, create and use simple directions and specific vocabulary to describe paths, regions, positions and boundaries of their immediate environment. Phase 3 Learners will sort, describe and model regular and irregular polygons, developing an understanding of their properties. They will be able to describe and model congruency and similarity in 2D shapes. Learners will continue to develop their understanding of symmetry, in particular reflective and rotational symmetry. They will understand how geometric shapes and associated vocabulary are useful for representing and describing objects and events in real-world situations. Phase 4 Learners will understand the properties of regular and irregular polyhedra. They will understand the properties of 2D shapes and understand that 2D representations of 3D objects can be used to visualize and solve problems in the real world, for example, through the use of drawing and modelling. Learners will develop their understanding of the use of scale (ratio) to enlarge and reduce shapes. They will apply the language and notation of bearing to describe direction and position. Learning continuums Mathematics scope and sequence 15 Le a r n i n g c o n t i n u u m f o r s h a p e a n d s p a c e Ph a s e 1 Ph a s e 2 Ph a s e 3 Ph a s e 4 Co n c e p t u a l u n d e r s t a n d i n g s Sh a p e s c a n b e d e s c r i b e d a n d o r g a n i z e d ac c o r d i n g t o t h e i r p r o p e r t i e s . Ob j e c t s i n o u r i m m e d i a t e e n v i r o n m e n t ha v e a p o s i t i o n i n s p a c e t h a t c a n b e de s c r i b e d a c c o r d i n g t o a p o i n t o f re f e r e n c e . Co n c e p t u a l u n d e r s t a n d i n g s Sh a p e s a r e c l a s s i f i e d a n d n a m e d ac c o r d i n g t o t h e i r p r o p e r t i e s . So m e s h a p e s a r e m a d e u p o f p a r t s t h a t re p e a t i n s o m e w a y . Sp e c i f i c v o c a b u l a r y c a n b e u s e d t o de s c r i b e a n o b j e c t ’ s p o s i t i o n i n s p a c e . Co n c e p t u a l u n d e r s t a n d i n g s Ch a n g i n g t h e p o s i t i o n o f a s h a p e d o e s no t a l t e r i t s p r o p e r t i e s . Sh a p e s c a n b e t r a n s f o r m e d i n d i f f e r e n t wa y s . Ge o m e t r i c s h a p e s a n d v o c a b u l a r y a r e us e f u l f o r r e p r e s e n t i n g a n d d e s c r i b i n g ob j e c t s a n d e v e n t s i n r e a l - w o r l d si t u a t i o n s . Co n c e p t u a l u n d e r s t a n d i n g s Ma n i p u l a t i o n o f s h a p e a n d s p a c e t a k e s pl a c e f o r a p a r t i c u l a r p u r p o s e . Co n s o l i d a t i n g w h a t w e k n o w o f ge o m e t r i c c o n c e p t s a l l o w s u s t o m a k e se n s e o f a n d i n t e r a c t w i t h o u r w o r l d . Ge o m e t r i c t o o l s a n d m e t h o d s c a n b e us e d t o s o l v e p r o b l e m s r e l a t i n g t o s h a p e an d s p a c e . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t 2 D a n d 3 D s h a p e s • ha v e c h a r a c t e r i s t i c s t h a t c a n b e de s c r i b e d a n d c o m p a r e d un d e r s t a n d t h a t c o m m o n l a n g u a g e • ca n b e u s e d t o d e s c r i b e p o s i t i o n a n d di r e c t i o n , f o r e x a m p l e , i n s i d e , o u t s i d e , ab o v e , b e l o w , n e x t t o , b e h i n d , i n f r o n t of , u p , d o w n . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t t h e r e a r e • re l a t i o n s h i p s a m o n g a n d b e t w e e n 2 D an d 3 D s h a p e s un d e r s t a n d t h a t 2 D a n d 3 D s h a p e s • ca n b e c r e a t e d b y p u t t i n g t o g e t h e r an d / o r t a k i n g a p a r t o t h e r s h a p e s un d e r s t a n d t h a t e x a m p l e s o f • sy m m e t r y a n d t r a n s f o r m a t i o n s c a n b e fo u n d i n t h e i r i m m e d i a t e e n v i r o n m e n t un d e r s t a n d t h a t g e o m e t r i c s h a p e s • ar e u s e f u l f o r r e p r e s e n t i n g r e a l - w o r l d si t u a t i o n s un d e r s t a n d t h a t d i r e c t i o n s c a n b e • us e d t o d e s c r i b e p a t h w a y s , r e g i o n s , po s i t i o n s a n d b o u n d a r i e s o f t h e i r im m e d i a t e e n v i r o n m e n t . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h e c o m m o n l a n g u a g e • us e d t o d e s c r i b e s h a p e s un d e r s t a n d t h e p r o p e r t i e s o f r e g u l a r • an d i r r e g u l a r p o l y g o n s un d e r s t a n d c o n g r u e n t o r s i m i l a r • sh a p e s un d e r s t a n d t h a t l i n e s a n d a x e s o f • re f l e c t i v e a n d r o t a t i o n a l s y m m e t r y as s i s t w i t h t h e c o n s t r u c t i o n o f s h a p e s un d e r s t a n d a n a n g l e a s a m e a s u r e o f • ro t a t i o n un d e r s t a n d t h a t d i r e c t i o n s f o r • lo c a t i o n c a n b e r e p r e s e n t e d b y co o r d i n a t e s o n a g r i d un d e r s t a n d t h a t v i s u a l i z a t i o n o f s h a p e • an d s p a c e i s a s t r a t e g y f o r s o l v i n g pr o b l e m s . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h e c o m m o n l a n g u a g e • us e d t o d e s c r i b e s h a p e s un d e r s t a n d t h e p r o p e r t i e s o f r e g u l a r • an d i r r e g u l a r p o l y h e d r a un d e r s t a n d t h e p r o p e r t i e s o f c i r c l e s • un d e r s t a n d h o w s c a l e ( r a t i o s ) i s u s e d • to e n l a r g e a n d r e d u c e s h a p e s un d e r s t a n d s y s t e m s f o r d e s c r i b i n g • po s i t i o n a n d d i r e c t i o n un d e r s t a n d t h a t 2 D r e p r e s e n t a t i o n s • of 3 D o b j e c t s c a n b e u s e d t o v i s u a l i z e an d s o l v e p r o b l e m s un d e r s t a n d t h a t g e o m e t r i c i d e a s • an d r e l a t i o n s h i p s c a n b e u s e d t o so l v e p r o b l e m s i n o t h e r a r e a s o f ma t h e m a t i c s a n d i n r e a l l i f e . Learning continuums Mathematics scope and sequence16 Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : so r t , d e s c r i b e a n d c o m p a r e 3 D s h a p e s • de s c r i b e p o s i t i o n a n d d i r e c t i o n , f o r • ex a m p l e , i n s i d e , o u t s i d e , a b o v e , be l o w , n e x t t o , b e h i n d , i n f r o n t o f , u p , do w n . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : so r t , d e s c r i b e a n d l a b e l 2 D a n d 3 D • sh a p e s an a l y s e a n d d e s c r i b e t h e r e l a t i o n s h i p s • be t w e e n 2 D a n d 3 D s h a p e s cr e a t e a n d d e s c r i b e s y m m e t r i c a l a n d • te s s e l l a t i n g p a t t e r n s id e n t i f y l i n e s o f r e f l e c t i v e s y m m e t r y • re p r e s e n t i d e a s a b o u t t h e r e a l w o r l d • us i n g g e o m e t r i c v o c a b u l a r y a n d sy m b o l s , f o r e x a m p l e , t h r o u g h o r a l de s c r i p t i o n , d r a w i n g , m o d e l l i n g , la b e l l i n g in t e r p r e t a n d c r e a t e s i m p l e d i r e c t i o n s , • de s c r i b i n g p a t h s , r e g i o n s , p o s i t i o n s an d b o u n d a r i e s o f t h e i r i m m e d i a t e en v i r o n m e n t . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : so r t , d e s c r i b e a n d m o d e l r e g u l a r a n d • ir r e g u l a r p o l y g o n s de s c r i b e a n d m o d e l c o n g r u e n c y a n d • si m i l a r i t y i n 2 D s h a p e s an a l y s e a n g l e s b y c o m p a r i n g a n d • de s c r i b i n g r o t a t i o n s : w h o l e t u r n ; h a l f tu r n ; q u a r t e r t u r n ; n o r t h , s o u t h , e a s t an d w e s t o n a c o m p a s s lo c a t e f e a t u r e s o n a g r i d u s i n g • co o r d i n a t e s de s c r i b e a n d / o r r e p r e s e n t m e n t a l • im a g e s o f o b j e c t s , p a t t e r n s , a n d pa t h s . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : an a l y s e , d e s c r i b e , c l a s s i f y a n d v i s u a l i z e • 2D ( i n c l u d i n g c i r c l e s , t r i a n g l e s a n d qu a d r i l a t e r a l s ) a n d 3 D s h a p e s , u s i n g ge o m e t r i c v o c a b u l a r y de s c r i b e l i n e s a n d a n g l e s u s i n g • ge o m e t r i c v o c a b u l a r y id e n t i f y a n d u s e s c a l e ( r a t i o s ) t o • en l a r g e a n d r e d u c e s h a p e s id e n t i f y a n d u s e t h e l a n g u a g e a n d • no t a t i o n o f b e a r i n g t o d e s c r i b e di r e c t i o n a n d p o s i t i o n cr e a t e a n d m o d e l h o w a 2 D n e t • co n v e r t s i n t o a 3 D s h a p e a n d v i c e ve r s a ex p l o r e t h e u s e o f g e o m e t r i c i d e a s • an d r e l a t i o n s h i p s t o s o l v e p r o b l e m s i n ot h e r a r e a s o f m a t h e m a t i c s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : ex p l o r e a n d d e s c r i b e t h e p a t h s , • re g i o n s a n d b o u n d a r i e s o f t h e i r im m e d i a t e e n v i r o n m e n t ( i n s i d e , ou t s i d e , a b o v e , b e l o w ) a n d t h e i r po s i t i o n ( n e x t t o , b e h i n d , i n f r o n t o f , up , d o w n ) . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : an a l y s e a n d u s e w h a t t h e y k n o w • ab o u t 3 D s h a p e s t o d e s c r i b e a n d w o r k wi t h 2 D s h a p e s re c o g n i z e a n d e x p l a i n s i m p l e • sy m m e t r i c a l d e s i g n s i n t h e en v i r o n m e n t ap p l y k n o w l e d g e o f s y m m e t r y t o • pr o b l e m - s o l v i n g s i t u a t i o n s in t e r p r e t a n d u s e s i m p l e d i r e c t i o n s , • de s c r i b i n g p a t h s , r e g i o n s , p o s i t i o n s an d b o u n d a r i e s o f t h e i r i m m e d i a t e en v i r o n m e n t . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : an a l y s e a n d d e s c r i b e 2 D a n d 3 D • sh a p e s , i n c l u d i n g r e g u l a r a n d ir r e g u l a r p o l y g o n s , u s i n g g e o m e t r i c a l vo c a b u l a r y id e n t i f y , d e s c r i b e a n d m o d e l • co n g r u e n c y a n d s i m i l a r i t y i n 2 D sh a p e s re c o g n i z e a n d e x p l a i n s y m m e t r i c a l • pa t t e r n s , i n c l u d i n g t e s s e l l a t i o n , i n t h e en v i r o n m e n t ap p l y k n o w l e d g e o f t r a n s f o r m a t i o n s • to p r o b l e m - s o l v i n g s i t u a t i o n s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : us e g e o m e t r i c v o c a b u l a r y w h e n • de s c r i b i n g s h a p e a n d s p a c e i n ma t h e m a t i c a l s i t u a t i o n s a n d b e y o n d us e s c a l e ( r a t i o s ) t o e n l a r g e a n d • re d u c e s h a p e s ap p l y t h e l a n g u a g e a n d n o t a t i o n o f • be a r i n g t o d e s c r i b e d i r e c t i o n a n d po s i t i o n us e 2 D r e p r e s e n t a t i o n s o f 3 D o b j e c t s • to v i s u a l i z e a n d s o l v e p r o b l e m s , f o r ex a m p l e u s i n g d r a w i n g s o r m o d e l s . Learning continuums Mathematics scope and sequence 17 No t e s Le a r n e r s n e e d m a n y o p p o r t u n i t i e s t o ex p e r i e n c e s h a p e a n d s p a c e i n a d i r e c t ki n e s t h e t i c m a n n e r , f o r e x a m p l e , t h r o u g h pl a y , c o n s t r u c t i o n a n d m o v e m e n t . Th e m a n i p u l a t i v e s t h a t t h e y i n t e r a c t w i t h sh o u l d i n c l u d e a r a n g e o f 3 D s h a p e s , i n pa r t i c u l a r t h e r e a l - l i f e o b j e c t s w i t h w h i c h ch i l d r e n a r e f a m i l i a r . 2 D s h a p e s ( p l a n e sh a p e s ) a r e a m o r e a b s t r a c t c o n c e p t b u t ca n b e u n d e r s t o o d a s f a c e s o f 3 D s h a p e s . No t e s Le a r n e r s n e e d t o u n d e r s t a n d t h e pr o p e r t i e s o f 2 D a n d 3 D s h a p e s be f o r e t h e m a t h e m a t i c a l v o c a b u l a r y as s o c i a t e d w i t h s h a p e s m a k e s s e n s e to t h e m . T h r o u g h c r e a t i n g a n d ma n i p u l a t i n g s h a p e s , l e a r n e r s a l i g n t h e i r na t u r a l v o c a b u l a r y w i t h m o r e f o r m a l ma t h e m a t i c a l v o c a b u l a r y a n d b e g i n t o ap p r e c i a t e t h e n e e d f o r t h i s p r e c i s i o n . No t e s Co m p u t e r a n d w e b - b a s e d a p p l i c a t i o n s ca n b e u s e d t o e x p l o r e s h a p e a n d s p a c e co n c e p t s s u c h a s s y m m e t r y , a n g l e s a n d co o r d i n a t e s . Th e u n i t s o f i n q u i r y c a n p r o v i d e a u t h e n t i c co n t e x t s f o r d e v e l o p i n g u n d e r s t a n d i n g of c o n c e p t s r e l a t i n g t o l o c a t i o n a n d di r e c t i o n s . No t e s To o l s s u c h a s c o m p a s s e s a n d p r o t r a c t o r s ar e c o m m o n l y u s e d t o s o l v e p r o b l e m s i n re a l - l i f e s i t u a t i o n s . H o w e v e r , c a r e s h o u l d be t a k e n t o e n s u r e t h a t s t u d e n t s h a v e a st r o n g u n d e r s t a n d i n g o f t h e c o n c e p t s em b e d d e d i n t h e p r o b l e m t o e n s u r e me a n i n g f u l e n g a g e m e n t w i t h t h e t o o l s an d f u l l u n d e r s t a n d i n g o f t h e s o l u t i o n . Learning continuums Mathematics scope and sequence18 Pattern and function To identify pattern is to begin to understand how mathematics applies to the world in which we live. The repetitive features of patterns can be identified and described as generalized rules called “functions”. This builds a foundation for the later study of algebra. Overall expectations Phase 1 Learners will understand that patterns and sequences occur in everyday situations. They will be able to identify, describe, extend and create patterns in various ways. Phase 2 Learners will understand that whole numbers exhibit patterns and relationships that can be observed and described, and that the patterns can be represented using numbers and other symbols. As a result, learners will understand the inverse relationship between addition and subtraction, and the associative and commutative properties of addition. They will be able to use their understanding of pattern to represent and make sense of real-life situations and, where appropriate, to solve problems involving addition and subtraction. Phase 3 Learners will analyse patterns and identify rules for patterns, developing the understanding that functions describe the relationship or rules that uniquely associate members of one set with members of another set. They will understand the inverse relationship between multiplication and division, and the associative and commutative properties of multiplication. They will be able to use their understanding of pattern and function to represent and make sense of real-life situations and, where appropriate, to solve problems involving the four operations. Phase 4 Learners will understand that patterns can be represented, analysed and generalized using algebraic expressions, equations or functions. They will use words, tables, graphs and, where possible, symbolic rules to analyse and represent patterns. They will develop an understanding of exponential notation as a way to express repeated products, and of the inverse relationship that exists between exponents and roots. The students will continue to use their understanding of pattern and function to represent and make sense of real-life situations and to solve problems involving the four operations. Learning continuums Mathematics scope and sequence 19 Le a r n i n g c o n t i n u u m f o r p a t t e r n a n d f u n c t i o n Ph a s e 1 Ph a s e 2 Ph a s e 3 Ph a s e 4 Co n c e p t u a l u n d e r s t a n d i n g s Pa t t e r n s a n d s e q u e n c e s o c c u r i n e v e r y d a y si t u a t i o n s . Pa t t e r n s r e p e a t a n d g r o w . Co n c e p t u a l u n d e r s t a n d i n g s Wh o l e n u m b e r s e x h i b i t p a t t e r n s a n d re l a t i o n s h i p s t h a t c a n b e o b s e r v e d a n d de s c r i b e d . Pa t t e r n s c a n b e r e p r e s e n t e d u s i n g nu m b e r s a n d o t h e r s y m b o l s . Co n c e p t u a l u n d e r s t a n d i n g s Fu n c t i o n s a r e r e l a t i o n s h i p s o r r u l e s t h a t un i q u e l y a s s o c i a t e m e m b e r s o f o n e s e t wi t h m e m b e r s o f a n o t h e r s e t . By a n a l y s i n g p a t t e r n s a n d i d e n t i f y i n g ru l e s f o r p a t t e r n s i t i s p o s s i b l e t o m a k e pr e d i c t i o n s . Co n c e p t u a l u n d e r s t a n d i n g s Pa t t e r n s c a n o f t e n b e g e n e r a l i z e d u s i n g al g e b r a i c e x p r e s s i o n s , e q u a t i o n s o r fu n c t i o n s . Ex p o n e n t i a l n o t a t i o n i s a p o w e r f u l w a y to e x p r e s s r e p e a t e d p r o d u c t s o f t h e s a m e nu m b e r . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t p a t t e r n s c a n b e • fo u n d i n e v e r y d a y s i t u a t i o n s , f o r ex a m p l e , s o u n d s , a c t i o n s , o b j e c t s , na t u r e . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t p a t t e r n s c a n b e • fo u n d i n n u m b e r s , f o r e x a m p l e , o d d an d e v e n n u m b e r s , s k i p c o u n t i n g un d e r s t a n d t h e i n v e r s e r e l a t i o n s h i p • be t w e e n a d d i t i o n a n d s u b t r a c t i o n un d e r s t a n d t h e a s s o c i a t i v e a n d • co m m u t a t i v e p r o p e r t i e s o f a d d i t i o n . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t p a t t e r n s c a n b e • an a l y s e d a n d r u l e s i d e n t i f i e d un d e r s t a n d t h a t m u l t i p l i c a t i o n i s • re p e a t e d a d d i t i o n a n d t h a t d i v i s i o n i s re p e a t e d s u b t r a c t i o n un d e r s t a n d t h e i n v e r s e r e l a t i o n s h i p • be t w e e n m u l t i p l i c a t i o n a n d d i v i s i o n un d e r s t a n d t h e a s s o c i a t i v e • an d c o m m u t a t i v e p r o p e r t i e s o f mu l t i p l i c a t i o n . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d t h a t p a t t e r n s c a n b e • ge n e r a l i z e d b y a r u l e un d e r s t a n d e x p o n e n t s a s r e p e a t e d • mu l t i p l i c a t i o n un d e r s t a n d t h e i n v e r s e r e l a t i o n s h i p • be t w e e n e x p o n e n t s a n d r o o t s un d e r s t a n d t h a t p a t t e r n s c a n b e • re p r e s e n t e d , a n a l y s e d a n d g e n e r a l i z e d us i n g t a b l e s , g r a p h s , w o r d s , a n d , wh e n p o s s i b l e , s y m b o l i c r u l e s . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : de s c r i b e p a t t e r n s i n v a r i o u s w a y s , • fo r e x a m p l e , u s i n g w o r d s , d r a w i n g s , sy m b o l s , m a t e r i a l s , a c t i o n s , n u m b e r s . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : re p r e s e n t p a t t e r n s i n a v a r i e t y o f w a y s , • fo r e x a m p l e , u s i n g w o r d s , d r a w i n g s , sy m b o l s , m a t e r i a l s , a c t i o n s , n u m b e r s de s c r i b e n u m b e r p a t t e r n s , f o r • ex a m p l e , o d d a n d e v e n n u m b e r s , s k i p co u n t i n g . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : de s c r i b e t h e r u l e f o r a p a t t e r n i n a • va r i e t y o f w a y s re p r e s e n t r u l e s f o r p a t t e r n s u s i n g • wo r d s , s y m b o l s a n d t a b l e s id e n t i f y a s e q u e n c e o f o p e r a t i o n s • re l a t i n g o n e s e t o f n u m b e r s t o a n o t h e r se t . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : re p r e s e n t t h e r u l e o f a p a t t e r n b y • us i n g a f u n c t i o n an a l y s e p a t t e r n a n d f u n c t i o n u s i n g • wo r d s , t a b l e s a n d g r a p h s , a n d , w h e n po s s i b l e , s y m b o l i c r u l e s . Learning continuums Mathematics scope and sequence20 Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : ex t e n d a n d c r e a t e p a t t e r n s . • Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : ex t e n d a n d c r e a t e p a t t e r n s i n • nu m b e r s , f o r e x a m p l e , o d d a n d e v e n nu m b e r s , s k i p c o u n t i n g us e n u m b e r p a t t e r n s t o r e p r e s e n t a n d • un d e r s t a n d r e a l - l i f e s i t u a t i o n s us e t h e p r o p e r t i e s a n d r e l a t i o n s h i p s • of a d d i t i o n a n d s u b t r a c t i o n t o s o l v e pr o b l e m s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : se l e c t a p p r o p r i a t e m e t h o d s f o r • re p r e s e n t i n g p a t t e r n s , f o r e x a m p l e us i n g w o r d s , s y m b o l s a n d t a b l e s us e n u m b e r p a t t e r n s t o m a k e • pr e d i c t i o n s a n d s o l v e p r o b l e m s us e t h e p r o p e r t i e s a n d r e l a t i o n s h i p s o f • th e f o u r o p e r a t i o n s t o s o l v e p r o b l e m s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : se l e c t a p p r o p r i a t e m e t h o d s t o a n a l y s e • pa t t e r n s a n d i d e n t i f y r u l e s us e f u n c t i o n s t o s o l v e p r o b l e m s . • No t e s Th e w o r l d i s f i l l e d w i t h p a t t e r n a n d t h e r e wi l l b e m a n y o p p o r t u n i t i e s f o r l e a r n e r s to m a k e t h i s c o n n e c t i o n a c r o s s t h e cu r r i c u l u m . A r a n g e o f m a n i p u l a t i v e s c a n b e u s e d t o ex p l o r e p a t t e r n s i n c l u d i n g p a t t e r n b l o c k s , at t r i b u t e b l o c k s , c o l o u r t i l e s , c a l c u l a t o r s , nu m b e r c h a r t s , b e a n s a n d b u t t o n s . No t e s St u d e n t s w i l l a p p l y t h e i r u n d e r s t a n d i n g of p a t t e r n t o t h e n u m b e r s t h e y a l r e a d y kn o w . T h e p a t t e r n s t h e y f i n d w i l l h e l p t o de e p e n t h e i r u n d e r s t a n d i n g o f a r a n g e o f nu m b e r c o n c e p t s . Fo u r - f u n c t i o n c a l c u l a t o r s c a n b e u s e d t o ex p l o r e n u m b e r p a t t e r n s . No t e s Pa t t e r n s a r e c e n t r a l t o t h e u n d e r s t a n d i n g of a l l c o n c e p t s i n m a t h e m a t i c s . T h e y a r e th e b a s i s o f h o w o u r n u m b e r s y s t e m i s or g a n i z e d . S e a r c h i n g f o r , a n d i d e n t i f y i n g , pa t t e r n s h e l p s u s t o s e e r e l a t i o n s h i p s , ma k e g e n e r a l i z a t i o n s , a n d i s a p o w e r f u l st r a t e g y f o r p r o b l e m s o l v i n g . F u n c t i o n s de v e l o p f r o m t h e s t u d y o f p a t t e r n s a n d ma k e i t p o s s i b l e t o p r e d i c t i n m a t h e m a t i c s pr o b l e m s . No t e s Al g e b r a i s a m a t h e m a t i c a l l a n g u a g e us i n g n u m b e r s a n d s y m b o l s t o e x p r e s s re l a t i o n s h i p s . W h e n t h e s a m e r e l a t i o n s h i p wo r k s w i t h a n y n u m b e r , a l g e b r a u s e s le t t e r s t o r e p r e s e n t t h e g e n e r a l i z a t i o n . Le t t e r s c a n b e u s e d t o r e p r e s e n t t h e qu a n t i t y . Learning continuums Mathematics scope and sequence 21 Number Our number system is a language for describing quantities and the relationships between quantities. For example, the value attributed to a digit depends on its place within a base system. Numbers are used to interpret information, make decisions and solve problems. For example, the operations of addition, subtraction, multiplication and division are related to one another and are used to process information in order to solve problems. The degree of precision needed in calculating depends on how the result will be used. Overall expectations Phase 1 Learners will understand that numbers are used for many different purposes in the real world. They will develop an understanding of one-to-one correspondence and conservation of number, and be able to count and use number words and numerals to represent quantities. Phase 2 Learners will develop their understanding of the base 10 place value system and will model, read, write, estimate, compare and order numbers to hundreds or beyond. They will have automatic recall of addition and subtraction facts and be able to model addition and subtraction of whole numbers using the appropriate mathematical language to describe their mental and written strategies. Learners will have an understanding of fractions as representations of whole-part relationships and will be able to model fractions and use fraction names in real-life situations. Phase 3 Learners will develop the understanding that fractions and decimals are ways of representing whole-part relationships and will demonstrate this understanding by modelling equivalent fractions and decimal fractions to hundredths or beyond. They will be able to model, read, write, compare and order fractions, and use them in real-life situations. Learners will have automatic recall of addition, subtraction, multiplication and division facts. They will select, use and describe a range of strategies to solve problems involving addition, subtraction, multiplication and division, using estimation strategies to check the reasonableness of their answers. Phase 4 Learners will understand that the base 10 place value system extends infinitely in two directions and will be able to model, compare, read, write and order numbers to millions or beyond, as well as model integers. They will develop an understanding of ratios. They will understand that fractions, decimals and percentages are ways of representing whole-part relationships and will work towards modelling, comparing, reading, writing, ordering and converting fractions, decimals and percentages. They will use mental and written strategies to solve problems involving whole numbers, fractions and decimals in real-life situations, using a range of strategies to evaluate reasonableness of answers. Learning continuums Mathematics scope and sequence22 Le a r n i n g c o n t i n u u m f o r n u m b e r Ph a s e 1 Ph a s e 2 Ph a s e 3 Ph a s e 4 Co n c e p t u a l u n d e r s t a n d i n g s Nu m b e r s a r e a n a m i n g s y s t e m . Nu m b e r s c a n b e u s e d i n m a n y w a y s f o r di f f e r e n t p u r p o s e s i n t h e r e a l w o r l d . Nu m b e r s a r e c o n n e c t e d t o e a c h o t h e r th r o u g h a v a r i e t y o f r e l a t i o n s h i p s . Ma k i n g c o n n e c t i o n s b e t w e e n o u r ex p e r i e n c e s w i t h n u m b e r c a n h e l p u s t o de v e l o p n u m b e r s e n s e . Co n c e p t u a l u n d e r s t a n d i n g s Th e b a s e 1 0 p l a c e v a l u e s y s t e m i s u s e d to r e p r e s e n t n u m b e r s a n d n u m b e r re l a t i o n s h i p s . Fr a c t i o n s a r e w a y s o f r e p r e s e n t i n g w h o l e - pa r t r e l a t i o n s h i p s . Th e o p e r a t i o n s o f a d d i t i o n , s u b t r a c t i o n , mu l t i p l i c a t i o n a n d d i v i s i o n a r e r e l a t e d to e a c h o t h e r a n d a r e u s e d t o p r o c e s s in f o r m a t i o n t o s o l v e p r o b l e m s . Nu m b e r o p e r a t i o n s c a n b e m o d e l l e d i n a va r i e t y o f w a y s . Th e r e a r e m a n y m e n t a l m e t h o d s t h a t c a n be a p p l i e d f o r e x a c t a n d a p p r o x i m a t e co m p u t a t i o n s . Co n c e p t u a l u n d e r s t a n d i n g s Th e b a s e 1 0 p l a c e v a l u e s y s t e m c a n b e ex t e n d e d t o r e p r e s e n t m a g n i t u d e . Fr a c t i o n s a n d d e c i m a l s a r e w a y s o f re p r e s e n t i n g w h o l e - p a r t r e l a t i o n s h i p s . Th e o p e r a t i o n s o f a d d i t i o n , s u b t r a c t i o n , mu l t i p l i c a t i o n a n d d i v i s i o n a r e r e l a t e d to e a c h o t h e r a n d a r e u s e d t o p r o c e s s in f o r m a t i o n t o s o l v e p r o b l e m s . Ev e n c o m p l e x o p e r a t i o n s c a n b e mo d e l l e d i n a v a r i e t y o f w a y s , f o r ex a m p l e , a n a l g o r i t h m i s a w a y t o re p r e s e n t a n o p e r a t i o n . Co n c e p t u a l u n d e r s t a n d i n g s Th e b a s e 1 0 p l a c e v a l u e s y s t e m e x t e n d s in f i n i t e l y i n t w o d i r e c t i o n s . Fr a c t i o n s , d e c i m a l f r a c t i o n s a n d pe r c e n t a g e s a r e w a y s o f r e p r e s e n t i n g wh o l e - p a r t r e l a t i o n s h i p s . Fo r f r a c t i o n a l a n d d e c i m a l c o m p u t a t i o n , th e i d e a s d e v e l o p e d f o r w h o l e - n u m b e r co m p u t a t i o n c a n a p p l y . Ra t i o s a r e a c o m p a r i s o n o f t w o n u m b e r s or q u a n t i t i e s . Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : un d e r s t a n d o n e - t o - o n e • co r r e s p o n d e n c e un d e r s t a n d t h a t , f o r a s e t o f o b j e c t s , • th e n u m b e r n a m e o f t h e l a s t o b j e c t co u n t e d d e s c r i b e s t h e q u a n t i t y o f t h e wh o l e s e t un d e r s t a n d t h a t n u m b e r s c a n • be c o n s t r u c t e d i n m u l t i p l e w a y s , fo r e x a m p l e , b y c o m b i n i n g a n d pa r t i t i o n i n g un d e r s t a n d c o n s e r v a t i o n o f n u m b e r * • un d e r s t a n d t h e r e l a t i v e m a g n i t u d e o f • wh o l e n u m b e r s re c o g n i z e g r o u p s o f z e r o t o f i v e • ob j e c t s w i t h o u t c o u n t i n g ( s u b i t i z i n g ) Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : mo d e l n u m b e r s t o h u n d r e d s o r • be y o n d u s i n g t h e b a s e 1 0 p l a c e v a l u e sy s t e m * * es t i m a t e q u a n t i t i e s t o 1 0 0 o r b e y o n d • mo d e l s i m p l e f r a c t i o n r e l a t i o n s h i p s • us e t h e l a n g u a g e o f a d d i t i o n a n d • su b t r a c t i o n , f o r e x a m p l e , a d d , t a k e aw a y , p l u s , m i n u s , s u m , d i f f e r e n c e mo d e l a d d i t i o n a n d s u b t r a c t i o n o f • wh o l e n u m b e r s de v e l o p s t r a t e g i e s f o r m e m o r i z i n g • ad d i t i o n a n d s u b t r a c t i o n n u m b e r fa c t s es t i m a t e s u m s a n d d i f f e r e n c e s • Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : mo d e l n u m b e r s t o t h o u s a n d s o r • be y o n d u s i n g t h e b a s e 1 0 p l a c e v a l u e sy s t e m mo d e l e q u i v a l e n t f r a c t i o n s • us e t h e l a n g u a g e o f f r a c t i o n s , f o r • ex a m p l e , n u m e r a t o r , d e n o m i n a t o r mo d e l d e c i m a l f r a c t i o n s t o • hu n d r e d t h s o r b e y o n d mo d e l m u l t i p l i c a t i o n a n d d i v i s i o n o f • wh o l e n u m b e r s us e t h e l a n g u a g e o f m u l t i p l i c a t i o n • an d d i v i s i o n , f o r e x a m p l e , f a c t o r , mu l t i p l e , p r o d u c t , q u o t i e n t , p r i m e nu m b e r s , c o m p o s i t e n u m b e r Le a r n i n g o u t c o m e s Wh e n co n s t r u c t i n g m e a n i n g l e a r n e r s : mo d e l n u m b e r s t o m i l l i o n s o r b e y o n d • us i n g t h e b a s e 1 0 p l a c e v a l u e s y s t e m mo d e l r a t i o s • mo d e l i n t e g e r s i n a p p r o p r i a t e • co n t e x t s mo d e l e x p o n e n t s a n d s q u a r e r o o t s • mo d e l i m p r o p e r f r a c t i o n s a n d m i x e d • nu m b e r s si m p l i f y f r a c t i o n s u s i n g m a n i p u l a t i v e s • mo d e l d e c i m a l f r a c t i o n s t o • th o u s a n d t h s o r b e y o n d mo d e l p e r c e n t a g e s • un d e r s t a n d t h e r e l a t i o n s h i p b e t w e e n • fr a c t i o n s , d e c i m a l s a n d p e r c e n t a g e s Learning continuums Mathematics scope and sequence 23 un d e r s t a n d w h o l e - p a r t r e l a t i o n s h i p s • us e t h e l a n g u a g e o f m a t h e m a t i c s • to c o m p a r e q u a n t i t i e s , f o r e x a m p l e , mo r e , l e s s , f i r s t , s e c o n d . un d e r s t a n d s i t u a t i o n s t h a t i n v o l v e • mu l t i p l i c a t i o n a n d d i v i s i o n mo d e l a d d i t i o n a n d s u b t r a c t i o n o f • fr a c t i o n s w i t h t h e s a m e d e n o m i n a t o r . mo d e l a d d i t i o n a n d s u b t r a c t i o n • of f r a c t i o n s w i t h r e l a t e d de n o m i n a t o r s * * * mo d e l a d d i t i o n a n d s u b t r a c t i o n o f • de c i m a l s . mo d e l a d d i t i o n , s u b t r a c t i o n , • mu l t i p l i c a t i o n a n d d i v i s i o n o f f r a c t i o n s mo d e l a d d i t i o n , s u b t r a c t i o n , • mu l t i p l i c a t i o n a n d d i v i s i o n o f de c i m a l s . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : co n n e c t n u m b e r n a m e s a n d • nu m e r a l s t o t h e q u a n t i t i e s t h e y re p r e s e n t . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : re a d a n d w r i t e w h o l e n u m b e r s u p t o • hu n d r e d s o r b e y o n d re a d , w r i t e , c o m p a r e a n d o r d e r • ca r d i n a l a n d o r d i n a l n u m b e r s de s c r i b e m e n t a l a n d w r i t t e n • st r a t e g i e s f o r a d d i n g a n d s u b t r a c t i n g tw o - d i g i t n u m b e r s . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : re a d , w r i t e , c o m p a r e a n d o r d e r w h o l e • nu m b e r s u p t o t h o u s a n d s o r b e y o n d de v e l o p s t r a t e g i e s f o r m e m o r i z i n g • ad d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n an d d i v i s i o n n u m b e r f a c t s re a d , w r i t e , c o m p a r e a n d o r d e r • fr a c t i o n s re a d a n d w r i t e e q u i v a l e n t f r a c t i o n s • re a d , w r i t e , c o m p a r e a n d o r d e r • fr a c t i o n s t o h u n d r e d t h s o r b e y o n d de s c r i b e m e n t a l a n d w r i t t e n • st r a t e g i e s f o r m u l t i p l i c a t i o n a n d di v i s i o n . Wh e n tr a n s f e r r i n g m e a n i n g i n t o sy m b o l s l e a r n e r s : re a d , w r i t e , c o m p a r e a n d o r d e r w h o l e • nu m b e r s u p t o m i l l i o n s o r b e y o n d re a d a n d w r i t e r a t i o s • re a d a n d w r i t e i n t e g e r s i n a p p r o p r i a t e • co n t e x t s re a d a n d w r i t e e x p o n e n t s a n d s q u a r e • ro o t s co n v e r t i m p r o p e r f r a c t i o n s t o m i x e d • nu m b e r s a n d v i c e v e r s a si m p l i f y f r a c t i o n s i n m e n t a l a n d • wr i t t e n f o r m re a d , w r i t e , c o m p a r e a n d o r d e r • de c i m a l f r a c t i o n s t o t h o u s a n d t h s o r be y o n d re a d , w r i t e , c o m p a r e a n d o r d e r • pe r c e n t a g e s co n v e r t b e t w e e n f r a c t i o n s , d e c i m a l s • an d p e r c e n t a g e s . Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : co u n t t o d e t e r m i n e t h e n u m b e r o f • ob j e c t s i n a s e t us e n u m b e r w o r d s a n d n u m e r a l s • to r e p r e s e n t q u a n t i t i e s i n r e a l - l i f e si t u a t i o n s Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : us e w h o l e n u m b e r s u p t o h u n d r e d s o r • be y o n d i n r e a l - l i f e s i t u a t i o n s us e c a r d i n a l a n d o r d i n a l n u m b e r s i n • re a l - l i f e s i t u a t i o n s Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : us e w h o l e n u m b e r s u p t o t h o u s a n d s • or b e y o n d i n r e a l - l i f e s i t u a t i o n s us e f a s t r e c a l l o f m u l t i p l i c a t i o n a n d • di v i s i o n n u m b e r f a c t s i n r e a l - l i f e si t u a t i o n s Wh e n ap p l y i n g w i t h u n d e r s t a n d i n g le a r n e r s : us e w h o l e n u m b e r s u p t o m i l l i o n s o r • be y o n d i n r e a l - l i f e s i t u a t i o n s us e r a t i o s i n r e a l - l i f e s i t u a t i o n s • us e i n t e g e r s i n r e a l - l i f e s i t u a t i o n s • Learning continuums Mathematics scope and sequence24 us e t h e l a n g u a g e o f m a t h e m a t i c s • to c o m p a r e q u a n t i t i e s i n r e a l - l i f e si t u a t i o n s , f o r e x a m p l e , m o r e , l e s s , fi r s t , s e c o n d su b i t i z e i n r e a l - l i f e s i t u a t i o n s • us e s i m p l e f r a c t i o n n a m e s i n r e a l - l i f e • si t u a t i o n s . us e f a s t r e c a l l o f a d d i t i o n a n d • su b t r a c t i o n n u m b e r f a c t s i n r e a l - l i f e si t u a t i o n s us e f r a c t i o n s i n r e a l - l i f e s i t u a t i o n s • us e m e n t a l a n d w r i t t e n s t r a t e g i e s f o r • ad d i t i o n a n d s u b t r a c t i o n o f t w o - di g i t n u m b e r s o r b e y o n d i n r e a l - l i f e si t u a t i o n s se l e c t a n a p p r o p r i a t e m e t h o d f o r • so l v i n g a p r o b l e m , f o r e x a m p l e , me n t a l e s t i m a t i o n , m e n t a l o r w r i t t e n st r a t e g i e s , o r b y u s i n g a c a l c u l a t o r us e s t r a t e g i e s t o e v a l u a t e t h e • re a s o n a b l e n e s s o f a n s w e r s . us e d e c i m a l f r a c t i o n s i n r e a l - l i f e • si t u a t i o n s us e m e n t a l a n d w r i t t e n s t r a t e g i e s f o r • mu l t i p l i c a t i o n a n d d i v i s i o n i n r e a l - l i f e si t u a t i o n s se l e c t a n e f f i c i e n t m e t h o d f o r • so l v i n g a p r o b l e m , f o r e x a m p l e , me n t a l e s t i m a t i o n , m e n t a l o r w r i t t e n st r a t e g i e s , o r b y u s i n g a c a l c u l a t o r us e s t r a t e g i e s t o e v a l u a t e t h e • re a s o n a b l e n e s s o f a n s w e r s ad d a n d s u b t r a c t f r a c t i o n s w i t h • re l a t e d d e n o m i n a t o r s i n r e a l - l i f e si t u a t i o n s ad d a n d s u b t r a c t d e c i m a l s i n r e a l - l i f e • si t u a t i o n s , i n c l u d i n g m o n e y es t i m a t e s u m , d i f f e r e n c e , p r o d u c t • an d q u o t i e n t i n r e a l - l i f e s i t u a t i o n s , in c l u d i n g f r a c t i o n s a n d d e c i m a l s . co n v e r t i m p r o p e r f r a c t i o n s t o m i x e d • nu m b e r s a n d v i c e v e r s a i n r e a l - l i f e si t u a t i o n s si m p l i f y f r a c t i o n s i n c o m p u t a t i o n • an s w e r s us e f r a c t i o n s , d e c i m a l s a n d • pe r c e n t a g e s i n t e r c h a n g e a b l y i n r e a l - li f e s i t u a t i o n s se l e c t a n d u s e a n a p p r o p r i a t e • se q u e n c e o f o p e r a t i o n s t o s o l v e w o r d pr o b l e m s se l e c t a n e f f i c i e n t m e t h o d f o r s o l v i n g • a p r o b l e m : m e n t a l e s t i m a t i o n , m e n t a l co m p u t a t i o n , w r i t t e n a l g o r i t h m s , b y us i n g a c a l c u l a t o r us e s t r a t e g i e s t o e v a l u a t e t h e • re a s o n a b l e n e s s o f a n s w e r s us e m e n t a l a n d w r i t t e n s t r a t e g i e s f o r • ad d i n g , s u b t r a c t i n g , m u l t i p l y i n g a n d di v i d i n g f r a c t i o n s a n d d e c i m a l s i n re a l - l i f e s i t u a t i o n s es t i m a t e a n d m a k e a p p r o x i m a t i o n s i n • re a l - l i f e s i t u a t i o n s i n v o l v i n g f r a c t i o n s , de c i m a l s a n d p e r c e n t a g e s . Learning continuums Mathematics scope and sequence 25 No t e s *T o c o n s e r v e , i n m a t h e m a t i c a l t e r m s , me a n s t h e a m o u n t s t a y s t h e s a m e re g a r d l e s s o f t h e a r r a n g e m e n t . Le a r n e r s w h o h a v e b e e n e n c o u r a g e d to s e l e c t t h e i r o w n a p p a r a t u s a n d me t h o d s , a n d w h o b e c o m e a c c u s t o m e d to d i s c u s s i n g a n d q u e s t i o n i n g t h e i r wo r k , w i l l h a v e c o n f i d e n c e i n l o o k i n g f o r al t e r n a t i v e a p p r o a c h e s w h e n a n i n i t i a l at t e m p t i s u n s u c c e s s f u l . Es t i m a t i o n i s a s k i l l t h a t w i l l d e v e l o p wi t h e x p e r i e n c e a n d w i l l h e l p c h i l d r e n ga i n a “ f e e l ” f o r n u m b e r s . C h i l d r e n m u s t be g i v e n t h e o p p o r t u n i t y t o c h e c k t h e i r es t i m a t e s s o t h a t t h e y a r e a b l e t o f u r t h e r re f i n e a n d i m p r o v e t h e i r e s t i m a t i o n s k i l l s . Th e r e a r e m a n y o p p o r t u n i t i e s i n t h e u n i t s of i n q u i r y a n d d u r i n g t h e s c h o o l d a y f o r st u d e n t s t o p r a c t i s e a n d a p p l y n u m b e r co n c e p t s a u t h e n t i c a l l y . No t e s ** M o d e l l i n g i n v o l v e s u s i n g c o n c r e t e ma t e r i a l s t o r e p r e s e n t n u m b e r s o r nu m b e r o p e r a t i o n s , f o r e x a m p l e , t h e u s e of p a t t e r n b l o c k s o r f r a c t i o n p i e c e s t o re p r e s e n t f r a c t i o n s a n d t h e u s e o f b a s e 1 0 bl o c k s t o r e p r e s e n t n u m b e r o p e r a t i o n s . St u d e n t s n e e d t o u s e n u m b e r s i n ma n y s i t u a t i o n s i n o r d e r t o a p p l y t h e i r un d e r s t a n d i n g t o n e w s i t u a t i o n s . I n ad d i t i o n t o t h e u n i t s o f i n q u i r y , c h i l d r e n ’ s li t e r a t u r e a l s o p r o v i d e s r i c h o p p o r t u n i t i e s fo r d e v e l o p i n g n u m b e r c o n c e p t s . To b e u s e f u l , a d d i t i o n a n d s u b t r a c t i o n fa c t s n e e d t o b e r e c a l l e d a u t o m a t i c a l l y . Re s e a r c h c l e a r l y i n d i c a t e s t h a t t h e r e a r e mo r e e f f e c t i v e w a y s t o d o t h i s t h a n “ d r i l l an d p r a c t i c e ” . A b o v e a l l , i t h e l p s t o h a v e st r a t e g i e s f o r w o r k i n g t h e m o u t . C o u n t i n g on , u s i n g d o u b l e s a n d u s i n g 1 0 s a r e g o o d st r a t e g i e s , a l t h o u g h l e a r n e r s f r e q u e n t l y in v e n t m e t h o d s t h a t w o r k e q u a l l y w e l l f o r th e m s e l v e s . Di f f i c u l t i e s w i t h f r a c t i o n s c a n a r i s e w h e n fr a c t i o n a l n o t a t i o n i s i n t r o d u c e d b e f o r e st u d e n t s h a v e f u l l y c o n s t r u c t e d m e a n i n g ab o u t f r a c t i o n c o n c e p t s . No t e s Mo d e l l i n g u s i n g m a n i p u l a t i v e s p r o v i d e s a v a l u a b l e s c a f f o l d f o r c o n s t r u c t i n g me a n i n g a b o u t m a t h e m a t i c a l co n c e p t s . T h e r e s h o u l d b e r e g u l a r op p o r t u n i t i e s f o r l e a r n e r s t o w o r k wi t h a r a n g e o f m a n i p u l a t i v e s a n d t o di s c u s s a n d n e g o t i a t e t h e i r d e v e l o p i n g un d e r s t a n d i n g s w i t h o t h e r s . ** * E x a m p l e s o f r e l a t e d d e n o m i n a t o r s in c l u d e h a l v e s , q u a r t e r s ( f o u r t h s ) a n d ei g h t h s . T h e s e c a n b e m o d e l l e d e a s i l y b y fo l d i n g s t r i p s o r s q u a r e s o f p a p e r . Th e i n t e r p r e t a t i o n a n d m e a n i n g o f re m a i n d e r s c a n c a u s e d i f f i c u l t y f o r so m e l e a r n e r s . T h i s i s e s p e c i a l l y t r u e i f ca l c u l a t o r s a r e b e i n g u s e d . F o r e x a m p l e , 67 ÷ 4 = 16 . 7 5 . T h i s c a n a l s o b e s h o w n as 1 6 ¾ o r 1 6 r3 . L e a r n e r s n e e d p r a c t i c e in p r o d u c i n g a p p r o p r i a t e a n s w e r s w h e n us i n g r e m a i n d e r s . F o r e x a m p l e , f o r a sc h o o l t r i p w i t h 2 5 s t u d e n t s , o n l y b u s e s th a t c a r r y 2 0 s t u d e n t s a r e a v a i l a b l e . A re m a i n d e r c o u l d n o t b e l e f t b e h i n d , s o an o t h e r b u s w o u l d b e r e q u i r e d ! Ca l c u l a t o r s k i l l s m u s t n o t b e i g n o r e d . Al l a n s w e r s s h o u l d b e c h e c k e d f o r t h e i r re a s o n a b l e n e s s . By r e f l e c t i n g o n a n d r e c o r d i n g t h e i r fi n d i n g s i n m a t h e m a t i c s l e a r n i n g l o g s , st u d e n t s b e g i n t o n o t i c e p a t t e r n s i n t h e nu m b e r s t h a t w i l l f u r t h e r d e v e l o p t h e i r un d e r s t a n d i n g . No t e s It i s n o t p r a c t i c a l t o c o n t i n u e t o d e v e l o p an d u s e b a s e 1 0 m a t e r i a l s b e y o n d 1 , 0 0 0 . Le a r n e r s s h o u l d h a v e l i t t l e d i f f i c u l t y i n ex t e n d i n g t h e p l a c e v a l u e s y s t e m o n c e th e y h a v e u n d e r s t o o d t h e g r o u p i n g pa t t e r n u p t o 1 , 0 0 0 . T h e r e a r e a n u m b e r of w e b s i t e s w h e r e v i r t u a l m a n i p u l a t i v e s ca n b e u t i l i z e d f o r w o r k i n g w i t h l a r g e r nu m b e r s . Es t i m a t i o n p l a y s a k e y r o l e i n c h e c k i n g th e f e a s i b i l i t y o f a n s w e r s . T h e m e t h o d of m u l t i p l y i n g n u m b e r s a n d i g n o r i n g th e d e c i m a l p o i n t , t h e n a d j u s t i n g t h e an s w e r b y c o u n t i n g d e c i m a l p l a c e s , d o e s no t g i v e t h e l e a r n e r a n u n d e r s t a n d i n g o f wh y i t i s d o n e . A p p l i c a t i o n o f p l a c e v a l u e kn o w l e d g e m u s t p r e c e d e t h i s a p p l i c a t i o n of p a t t e r n . Me a s u r e m e n t i s a n e x c e l l e n t w a y o f ex p l o r i n g t h e u s e o f f r a c t i o n s a n d de c i m a l s a n d t h e i r i n t e r c h a n g e . St u d e n t s s h o u l d b e g i v e n m a n y op p o r t u n i t i e s t o d i s c o v e r t h e l i n k be t w e e n f r a c t i o n s a n d d i v i s i o n . A t h o r o u g h u n d e r s t a n d i n g o f mu l t i p l i c a t i o n , f a c t o r s a n d l a r g e nu m b e r s i s r e q u i r e d b e f o r e w o r k i n g w i t h ex p o n e n t s . Mathematics scope and sequence26 Samples Several examples of how schools are using the planner to facilitate mathematics inquiries have been developed and trialled by IB World Schools offering the PYP. These examples are included in the HTML version of the mathematics scope and sequence on the online curriculum centre. The IB is interested in receiving planners that have been developed for mathematics inquiries or for units of inquiry where mathematics concepts are strongly evident. Please send planners to pyp@ibo.org for possible inclusion on this site. FUTURE AGENDA ITEMS: Any Council member may request at a workshop and / or Council meeting, under “Future Agenda Item Requests”, an agenda item for a future Council meeting. The Council Member making the request will contact the Town Manager with the requested item and the Town Manager will list it on the agenda. At the meeting, the requesting Council Member will explain the item, the need for Council discussion of the item, the item’s relationship to the Council’s strategic priorities, and the amount of estimated staff time necessary to prepare for Council discussion. If the requesting Council Member receives a second, the Town Manager will place the item on the Council agenda calendar allowing for adequate time for staff preparation on the agenda item. - None Town of Westlake Item #10 - Future Workshop Agenda Items Westlake Academy Item # 11 – Workshop Adjournment Back up material has not been provided for this item. Westlake Academy Item # 2 – Pledge of Allegiance Texas Pledge: "Honor the Texas flag; I pledge allegiance to thee, Texas, one state under God, one and indivisible." Westlake Academy Item # 3 – Citizens’ Presentations This is an opportunity for citizens to address the Board on any matter whether or not it is posted on the agenda. The Board cannot by law take action nor have any discussion or deliberations on any presentation made to the Board at this time concerning an item not listed on the agenda. The Board will receive the information, ask staff to review the matter, or an item may be noticed on a future agenda for deliberation or action. CONSENT AGENDA: All items listed below are considered routine by the Board of Trustees and will be enacted with one motion. There will be no separate discussion of items unless a Board member or citizen so requests, in which event the item will be removed from the general order of business and considered in its normal sequence. a. Review and approve minutes of the School Board of Trustees of the workshop and regular meeting held on October 5, 2009. b. Consider a Resolution adopting homework, assessment and grade reporting, retention and transfer course credit, graduation policies and amending the responsibility for personnel decisions policy. Town of Westlake Item # 4 - Consent Agenda Items BOT Minutes October 5, 2009 Page 1 of 5 WESTLAKE ACADEMY BOARD OF TRUSTEES WORKSHOP & REGULAR MEETINGS October 5, 2009 PRESENT: President Laura Wheat and Trustees Tim Brittan, Larry Corson, Carol Langdon, and Rebecca Rollins. Rick Rennhack arrived at 4:42 p.m. ABSENT: None OTHERS PRESENT: Chief Executive Officer Thomas Brymer, Town Attorney Stan Lowery, School Board Attorney, Janet Bubert, Secretary Kelly Edwards, Finance Director Debbie Piper, Assistant to the Town Manager Ginger Awtry, Facilities and Recreation Director Troy Meyer, Head of Secondary Mark Garcia, Head of Primary Jamie Schmitz, Administrative Coordinator Darcy McFarlane, Director of Human Resources and Administrative Services Todd Wood, Management Intern Ben Nibarger, and Municipal Court Administrator Amanda DeGan. 1. CALL TO ORDER WORKSHOP SESSION President Wheat called the workshop meeting to order at 4:33 p.m. 2. REVIEW OF CONSENT AGENDA ITEMS FROM OCTOBER 5, 2009, TRUSTEES REGULAR MEETING AGENDA. There were no questions. 3. REVIEW AND DISCUSS STRATEGIC PLANNING PROCESS CHART. CEO Brymer presented the Strategic Planning process for Westlake Academy. BOT Minutes October 5, 2009 Page 2 of 5 4. PRESENTATION AND DISCUSSION OF A REPORT ON MUNICIPAL CHARTER SCHOOLS. Chief Executive Officer Brymer introduced Management Intern Ben Nibarger to review the first portion of presentation with the Board. This presentation compares Westlake Academy to other Charter Schools. The presentation reviewed key findings and overviewed the population, budgets and state funding per student using Charter Schools in Florida. None of the schools in Florida offer the IB programme. Discussion ensued regarding open enrollment, the boundaries matching ISD boundaries, and debt service of the bonds. Chief Executive Officer Brymer presented Option Evaluation reviewing the organizational structures of the academy. Discussion then ensued on the options as presented and the best option for the academy. CEO Brymer recommends Option 2 with single point of touch. 5. FUTURE AGENDA ITEMS There were no items submitted for consideration. 6. ADJOURNMENT There being no further business before the Board, President Wheat adjourned the workshop at 6:33 p.m. Regular Session 1. CALL TO ORDER President Wheat called the Regular meeting to order at 6:56 p.m. with all Trustees present. 2. PLEDGE OF ALLEGIANCE President Wheat led the pledge of allegiance to the United States and Texas flags. BOT Minutes October 5, 2009 Page 3 of 5 3. CITIZENS' PRESENTATIONS Ann Dunlap spoke in favor of weighting the MYP classes to provide the students equal opportunity to compete with other students applying college applications. Ken Gordon encouraged the board to not only consider the comparisons of the academy to other charter school but to all schools, to consider the accountability issues with Option 2 and if that is a good structure and asked the board to consider selecting a professional with academic experience to manage the school. 4. CONSENT AGENDA a. Review and approve minutes of the School Board of Trustees of the workshop and regular meeting held on September 21, 2009. MOTION: Trustee Corson made a motion to approve the consent agenda. Trustee Brittan seconded the motion. The motion carried by a vote of 6-0 5. PRESENTATION BY WESTLAKE ACADEMY HEAD OF PRIMARY REGARDING THE WESTLAKE P.R.O.U.D. CAMPAIGN. Jamie Schmitz explained the P.R.O.U.D. campaign and how it works with the IB learner profile and support the K-12 continuum. The program was launched on September 25th with a pep rally. They are working on how to document certain behaviors and other avenues to promote P.R.O.U.D. 6. CONSIDER A RESOLUTION ADOPTING HOMEWORK, ASSESSMENT AND GRADE REPORTING, RETENTION AND TRANSFER COURSE CREDIT AND GRADUATION POLICIES. Head of Secondary Mark Garcia provided a presentation providing an overview of the policies and answered Trustee questions. Trustee Brittan left the meeting at 7:28 p.m. MOTION: No Action taken. 7. BOARD CALENDAR - Monster Mash o October 24, 2009 - Pancake Breakfast o October 10, 2009 (8-11 am WA Dining Hall) - Board of Trustees Meeting o November 2, 2009 BOT Minutes October 5, 2009 Page 4 of 5 - Westlake Baja at Vaquero o November 16, 2009, 3:30 p.m. - Westlake Tree Lighting o December 3, 2009, TBD - Board of Trustees Meeting December 7, 2009 - WA Parents Shopping Day Out o December 19, 2009 (Benefiting Project 2010) - Career Day December TBD, 2009 - Westlake Academy Gallery Night event o February 26, 2010 - Bandana Bonanza o May 8, 2010 - Westlake Academy Graduation o May 23, 2010 President Wheat recessed the meeting at 8:01 p.m. President Wheat reconvened the meeting at 9:27 p.m. 8. FUTURE AGENDA ITEMS There were no items submitted for consideration. 9. EXECUTIVE SESSION The Council will conduct a closed session pursuant to Texas Government Code, annotated, Chapter 551, Subchapter D for the following: a. Section 551.071(2) – Consultation with School Attorney on a matter in which the duty of the attorney to the governmental body under the Texas Disciplinary Rules of Professional Conduct of the State Bar of Texas clearly conflicts with this chapter: Student Transfers. The Board of Trustees convened into Executive Session at 9:27 p.m. 10. RECONVENE MEETING The Board of Trustees reconvened into Regular Session at 10:11 p.m. 11. TAKE ANY NECESSARY ACTION FROM EXECUTIVE SESSION, IF NECESSARY MOTION: No action taken BOT Minutes October 5, 2009 Page 5 of 5 12. ADJOURNMENT There being no further business before the Board, President Wheat asked for a motion to adjourn. MOTION: Trustee Rennhack made a motion to adjourn the meeting. Trustee Langdon seconded the motion. The motion carried by a vote of 6-0. President Wheat adjourned the meeting at 10:11 p.m. APPROVED BY THE BOARD OF TRUSTEES ON NOVEMBER 2, 2009. ____________________________ Laura Wheat, Board President ATTEST: ______________________________ Kelly Edwards, Board Secretary Memo Town of Westlake To: Honorable President and Members of the Board of Trustees From: Tom Brymer, CEO Subject: Regular Meeting of November 3, 2009 Date: October 28, 2009 Consider a Resolution adopting homework, assessment and grade reporting, retention and transfer course credit, graduation policies and amending the responsibility for personnel decisions policy. ITEM Westlake Academy is a nurturing, community owned International Baccalaureate Charter School whose mission is to achieve academic excellence and to develop life-long learners who well balanced, responsible citizens. WESTLAKE ACADEMY MISSION/VISION STATEMENT Academic Achievement WESTLAKE ACADEMY VALUES PYP, MYP, DP (IB Continuum) Caring Environment Fiscal Stewardship Communication/Transparency Engaged Stakeholders Maximizing Each Child’s Potential BACKGROUND The Board has adopted a series of policies intended to direct and guide the Staff in the education of the students at Westlake Academy. To further support those efforts, the above mentioned topics are recommended by Staff for adoption. Guidelines consistent with the Board policy directives will be published in the Parent/Student Handbook. These policies were reviewed by the Board at the September 21, 2009, workshop. Not applicable at this time. FUNDING Adoption of the Resolution of policy proposals and amendments for Westlake Academy. RECOMMENDATION ATTACHMENTS: Resolution 09-18 adopting Board policies related to the following topics: • Student Achievement: o Homework Policy o Assessment and Grade Reporting Policy – Secondary o Retention and Transfer Course Credit Policy o Graduation Policy And, amending the policy regarding Responsibility for Personnel Decisions to be titled along with appropriate language added pursuant to this amendment: o Responsibility for Personnel Decisions and Setting Parameters for the CEO Regarding Establishment of Managerial Reporting and Organizational Structure 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name : Homework Policy Policy Category : Student Achievement Policy Goal : Creation of an effective and supportive learning environment Policy Description : Westlake Academy is a school with a desired outcome of high academic achievement for each student. In support of this outcome, homework is considered as an integral component of a student’s school career and serves the vital purpose of assisting the student in their pursuit of knowledge. Homework is viewed as: • An avenue to increase and enhance student learning • Providing additional drill, practice and study • A support vehicle for teaching and developing personal responsibility • Informing parents of the subject matter and classroom lessons of their child Staff will develop uniform homework requirement guidelines for all grades and place the information in the Parent/Student Handbook and on the Academy’s web site on an annual basis. Allowances and contingency plans will be developed for excused absences along with consequences for unexcused absences and the submittal of late assignments. In addition, schedules for student/teacher tutorials will be included in the homework requirement guidelines. 2 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : Parents are encouraged to assist the Academy’s instruction program by providing the student with the necessary tools and time to complete the homework requirements. Failure to adhere to the homework requirements will subject the student to the possibility of receiving poor/failing grades on assignments. Continued abuse of the homework requirement guidelines will subject the student to the progressive disciplinary policy of the Academy. 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name Secondary – Middle Years Program and Diploma Program : Assessment and Grade Reporting Policy Policy Category : Student Achievement Policy Goal : Definition, uniformity, and transparency in the Academy’s Secondary (MYP and DP) assessment and grading policy Policy Description : Uniform assessments and grade reporting standards are a significant component of the success of the Secondary (MYP and DP) student’s at Westlake Academy. As such, the following purposes of these standards are necessary to accurately reflect the academic achievement of the student: • Ability of the assessment / grading system to report actual classroom performance on exams, homework assignments, projects, etc. • Allow the parent to easily identify and gauge their child’s performance • Consistency across grade level(s) of assessment application and grade reporting • Assist colleges in determining the achievements of the students • Demonstration of college readiness • Identification of any subject area where a student may need additional assistance to reach educational goals 2 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) In consideration of the need to establish comprehensive grade reporting and assessment policies for the student body of Westlake Academy, the following numerical grades will be utilized by the Staff to indicate academic performance on report cards. : A = 100 - 90 points B = 89 - 80 points C = 79 - 70 points F = 69 points and below I = Incomplete for extenuating circumstances (to be determined by the Head(s) of Section) Staff will administer a variety of assessments to measure student knowledge and progress throughout the year. The frequency and type of assessment utilized will be program specific, - i.e. Primary, Middle or Diploma Program. In the Secondary School (Middle Years and Diploma Program), a student shall not be required to complete more than two major assessments on the same day. In addition, SENIORS in the Diploma Program sitting for the IB exams, may be eligible for final exam exemption based on absence criteria Report Cards - Westlake Academy will send home four (4) report cards and two (2) interval progress reports each year. The report cards will be sent home at the end of each quarter. The interval progress reports will be issued within the quarter grading period. Parent Teacher Meetings – Staff will conduct two (2) parent/teacher conferences per year − one (1) each semester. This will allow parents to meet with teachers to briefly discuss their child’s progress. Additional concerns will be addressed at individually scheduled conferences available at any time during the school year. 3 Letter Grades for Citizenship - Staff will assign a letter grade for student citizenship using the following criteria: E = Excellent Observes the rules all of the time S = Satisfactory Observes the rules most of the time N = Needs Improvement Observes rules infrequently/inconsistently U = Unsatisfactory Almost never observes the rules Elements of the IB Learner Profile will be to be used to assess citizenship grades. • Practices respect • Uses self-control • Talks at appropriate times • Obeys school rules • Works well with others • Uses acceptable language • Follows Uniform Code • Follows Honor Code Guidelines consistent with these policies will be developed by Staff and published in the Parent/Student Handbook. 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name : Retention and Transfer Course Credit Policy Policy Category : Student Achievement Policy Goal : Describe criteria/circumstances for student retention based on unsatisfactory academic achievement and develop guidelines for acceptance of transfer course credits Policy Description : In order to ensure academic achievement and the mastery of the necessary skills to advance to each subsequent grade level, the Staff of Westlake Academy will develop specific Retention Guidelines for the students, which will be published in the Parent/Student Handbook on an annual basis. Legally mandated and internally required assessments will be administered to assist the teachers in determining if a student has attained the necessary educational understanding to advance to the next grade level. The assessments may include, but are not limited to, grade/subject specific examinations, essays, homework assignments, group and individual projects, and state mandated assessment tests. Summer School/Alternative Educational Programs - Students, who do not meet the necessary requirements for advancement, must complete a comparable course in summer school or through an alternative education program. A student who fails three (3) or more core classes may not advance to the next grade level. 2 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued): Students who do not satisfactorily complete a state required course for high school graduation must complete the course through a summer or alternative educational program approved in advance by the Head of Secondary or his/her designee prior to being considered for graduation from Westlake Academy. Students who fail a core course must retake the class during the summer session immediately following the academic year in which the class was failed. If not offered at Westlake Academy, it is the parent/guardian’s responsibility to locate a summer school or alternative educational program within their home district for remediation that is acceptable for credit and approved by the Head of Secondary or his/her designee. Westlake Academy reserves the right to accept or deny summer school credit. Prior to attending summer school or any alternative educational program, parents and students should obtain written approval from the Head of Secondary or his/her designee to ensure that the credit has been approved and is acceptable by the Academy. Evaluation of Credits for Transfer Students – The Academy accepts credits from other schools accredited by the State of Texas. Courses will be evaluated by a counselor to determine if the course meets the requirements for graduation from a Texas high school, as well as from Westlake Academy. All transfer grades earned in accredited schools will be converted to Westlake Academy grading scale and course designation consistent with the Academy’s standards. In order to receive credit for completed work, students entering Westlake Academy from a non-accredited school or from home schooling must take credit-by-exam (CBE) tests approved in advance by Westlake Academy. Students are responsible for any applicable fees associated with these tests and must achieve acceptable test scores as determined by the Head(s) of Section. Texas Assessment of Knowledge and Skills Testing and Remediation – The Board of Trustees and the Staff of Westlake Academy are committed to student success on the Texas Assessment of Knowledge and Skills (TAKS) and takes seriously its obligation to provide students the support and assistance they need to succeed. 3 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued): All students at TAKS-testing grade levels, (Grades 3-11) will be administered regular benchmark testing to ensure they are making appropriate progress through the required TAKS curriculum. Students who are not making adequate progress will receive remedial assistance. In addition, students who have failed a TAKS test in the previous year will be required to attend remedial classes/tutorials. The schedule(s) will be developed by Staff and communicated to the parent/guardian of the students needing additional assistance. Operational guidelines consistent with the above policy directives will be detailed and published as necessary by Academy Staff in the Parent/Student Handbook on an annual basis. 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09- : Date Board Adopted: Date Board Amended: N/A Effective Date : Policy Name : Graduation Policy Policy Category : Student Achievement Policy Goal : Defining the commencement participation requirements; establishing Valedictorian, Salutatorian and Honor Graduate policy; Policy Description : Eligibility for Commencement – Westlake Academy seniors who have satisfied all Academy requirements for graduation are eligible for participation in the graduation ceremonies. For students who do not meet the requirements, an exception may be made in the event that a senior lacks one (1) credit towards the graduation requirements. The student must have a plan approved by the Head of Secondary to complete the credit prior to the beginning of the next school year. No other exceptions to participation in the graduation ceremony will be granted. Only students who have completed all the requirements of Westlake Academy will receive a diploma. Those requirements are as follows: • Completion of a minimum of 29 credit hours according to the Westlake Academy Program of Studies • Not assigned to a Disciplinary Alternative Education Placement (DEAP) setting during his/her senior year at the Academy or any other educational facility 2 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : As reflective of the Academy’s rigorous and enriching IB curriculum and desire to foster college readiness, students who graduate from Westlake Academy will complete a program of study, which includes higher academic standards and additional credit hours than those established through the guidelines of the State of Texas. Students who receive a Westlake Academy high school diploma will have completed a minimum of 29 hours of course work, which reflects those higher levels of academic achievement. The graduation requirements detailing the necessary courses are incorporated into the policy by addendum A. The provision for the Westlake Academy Diploma will become effective for the students in the graduating class of 2011. Approved Courses Included in Formal Grade Point Average (GPA) - All courses taken at the secondary level at Westlake Academy which contain specific objectives, determined by the Texas Education Agency (TEA) and are not modified in content will be included when calculating grade point average. The following general criteria govern the calculation of the grade point average: • GPA will be calculated by the Counselor and rounded up to the nearest whole number • GPA will be determined using semester grades earned in state approved courses • Grades earned in courses completed at other fully accredited school(s) and transferred to the Academy will be included in determining grade point average Course criteria must be consistent with the standards established at Westlake Academy. Correspondence courses, Credit-by-Exam, Dual Credit courses, courses modified in content, Pass/Fail courses, Local Credit courses not required for graduation, Advanced Placement Exams, International Baccalaureate Exams and grades earned in credit-bearing courses taken prior to Grade 9 will not be included in the calculation of the formal GPA, but will be reflected on the student’s Academic Achievement Record/Transcript. 3 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : Quality Points for IB Courses - Beginning in Grade 9, Westlake Academy will award ten (10) points to the final grade for any IB Diploma Program or Advanced Placement course and no less than five (5) points for any honors (pre-IB/AP) course, as determined by the school administration. Raw grades in the course must be a passing grade of 70% or better to receive quality points. Raw semester grades are shown on a student’s report card and Academic Achievement Record (AAR)/Transcript. Indicators for quality points will be included on the Report Card and the AAR/Transcript. Quality points will be included in GPA calculation and this weighted GPA will appear on the student’s transcript. Grades earned in honors (pre-IB/AP) courses and IB/AP courses taken at other accredited districts will receive quality points if the following conditions are met: • Quality points awarded from other schools will not exceed or be less than the numerical weight of quality points awarded at Westlake Academy • The previous school provides official documentation of advanced courses Academic Recognition - Westlake Academy will encourage and recognize academic achievement through the establishment of a class Valedictorian, Salutatorian and Honor Graduates. Only grades earned at Westlake Academy and those earned at schools approved by Westlake Academy will determine the student’s academic recognition. Calculating and Reporting GPA - A counselor will determine a student’s GPA calculation at regular intervals as determined necessary by the Head of Secondary. The intervals will be communicated through the use of the Parent/Student Handbook. 4 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09 - (continued) : Rank in Class - Rank in class will only be reported for the purposes of: • Automatic admission of the top 10% of the class into a Texas public college or university • Consideration for scholarships Valedictorian and Salutatorian - Graduating seniors with the top two cumulative grade point averages (CGPA), as determined by the Head of Secondary or his/her designee, will be eligible to serve as the Valedictorian and Salutatorian, respectively. In the event of a tie for Valedictorian, the students will share the honor and no Salutatorian will be designated. All courses and corresponding numerical grades used to determine GPA must not be modified in content and must be identified by the state as regular, honors, and AP or IB courses in order to qualify for Valedictorian and Salutatorian. Valedictorian and Salutatorian eligibility requires attendance at Westlake Academy for six (6) consecutive semesters preceding graduation. Further, a student who is in violation of the Academy codes and requirements, Town code, or State codes, may be deemed ineligible by the Chief Executive Officer or his/her designee to represent Westlake Academy as the Valedictorian or Salutatorian. Honor Graduates - A graduating senior’s weighted cumulative grade point average (CGPA) rounded to 90% or above will be determined to be an Honor Graduate and will be duly recognized at commencement. All courses and corresponding numerical grades earned to determine GPA must not be modified in content and must be identified by the state as regular, honors, and AP or IB courses in order to qualify for Honor Graduate status. The Head of Secondary and his/her designee are granted the authority to include students whose CGPA is 90% or above at the end of the final semester. Operational guidelines consistent with the above policy directives will be detailed and published by Academy Staff in the Parent/Student Handbook on an annual basis. 5 Graduation Requirements Side-by-Side Comparison WA WA NHP TEA Actual Required Distinguished* English 4 4 4 4 Mathematics 5 4 4 4 Science 4 4 4 4 Social Studies 4 4 4 3.0 Gov’t 0.5 0.5 0.5 0.5 Economics 0.5 0.5 0.5 0.5 Foreign Language 5 3 3 3 Physical Education 1.5 1.5 1.5 1.5 Health 0.5 0.5 0.5 0.5 Speech 0.5 0.5 0.5 0.5 Technology 2.0 2.0 1.0 1.0 Fine Arts 2.0 1.0 2.0 1.0 Tok 2.0 2.0 1.0 n/a Electives 1.0 1.0 2.5 2.5 CAS/Extended Essay 1.0 0.5 n/a n/a (32.5) (29.0) (29.0) (26.0) *Source: TEA Website http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074f.html 1 TOWN OF WESTLAKE WESTLAKE ACADEMY BOARD OF TRUSTEES POLICY Policy No. 09-09 : Date Board Adopted : February 9, 2009 Date Board Amended : Effective Date : Policy Name : Responsibility for Personnel Decisions and Setting Parameters for the CEO Regarding Establishment of Managerial Reporting and Organizational Structure Policy Category : Faculty Attraction and Retention Policy Goal : Efficient and Effective Recruitment/Accountability/Retention of Staff Policy Description : Administrative regulations governing Texas Charter Schools allow the Board of Trustees to delegate to its Chief Executive Officer (CEO) responsibility for personnel decisions. As such, it is the purpose of this Board policy to delegate this function to the CEO as follows: a.) The CEO or his/her designate is delegated all authority to offer employment, terminate employment, evaluate, promote, demote, appoint, and employ all Westlake Academy employees, except the Superintendent/Head of School. b.) The CEO or his/her designate shall define the qualifications (excepting those that may be established by the Board), duties and responsibilities of all Academy positions and shall ensure that job descriptions are current and accessible to all employees and supervisors. c.) All compensation will be in accordance with Board approved teacher salary scales and establishment of new positions shall be approved by the Board. d.) The Board shall have input into the process utilized for the selection of section head positions. 2 e.) The CEO shall be responsible to the Board for establishing (and modifying when necessary) a managerial reporting and organizational structure, pursuant to applicable Board budget approvals, that meets the following criteria as well as facilitates and enhances: • student achievement in alignment with the IB Learner profile • communication with all Academy stakeholder groups • timely staff decision making • staff accountability • employee empowerment and teamwork • attainment of Board policies and adopted strategic plan desired outcomes • staff professional development and effectiveness • stewardship regarding the effective and efficient use of available resources • clarity of reporting structure, lines of staff authority, and staff roles Resolution 09-18 Page 1 of 2 WESTLAKE ACADEMY RESOLUTION NO. 09-18 ESTABLISHING AND AMENDING BOARD POLICIES INTENDED TO DIRECT AND GUIDE THE DAILY OPERATIONS AND EDUCATIONAL OUTCOMES OF THE STUDENTS AT WESTLAKE ACADEMY, IN COMPLIANCE WITH STATE AND FEDERAL LAWS. WHEREAS, the Board of Trustees desires to establish policies to assist the staff and students in attaining their educational goals at Westlake Academy; and WHEREAS, the Board of Trustees desires to comply with State and Federal laws and regulations as it pertains to education and school requirements; and WHEREAS, the Board of Trustees finds that the passage of this Resolution is in the best interest of the Academy; NOW, THEREFORE, BE IT RESOLVED BY THE BOARD OF TRUSTEES OF WESTLAKE ACADEMY: SECTION 1: That, all matters stated in the Recitals hereinabove are found to be true and correct and are incorporated herein by reference as if copied in their entirety SECTION 2: That the Board of Trustees hereby adopts the following Board policies attached as Exhibit “A”,: Homework Assessment and Grade Reporting Retention and Transfer Course Credits Graduation SECTION 3: That the Board of Trustees hereby amends the following Board policy attached as Exhibit “B”,: Responsibility for Personnel Decisions and Setting Parameters for the CEO Regarding Establishment of Managerial Reporting and Organizational Structure SECTION 4: That this Resolution shall become effective upon the date of its passage. Resolution 09-18 Page 2 of 2 PASSED AND APPROVED ON THIS 2nd DAY OF NOVEMBER, 2009. ______________________________________ ____________________________________ Laura Wheat, President Larry Corson, Trustee _____________________________________ ____________________________________ Tim Brittan, Trustee Rebecca Rollins, Trustee _______________________________________ ____________________________________ Carol Langdon, Trustee Rick Rennhack, Trustee ATTEST: _________________________________ ___________________________________ Kelly Edwards, Secretary Thomas Brymer, Chief Executive Officer APPROVED AS TO FORM: __________________________________ Janet Bubert, School Attorney BOARD CALENDAR - Westlake Baja at Vaquero November 16, 2009, 3:30 p.m. - Barnes & Noble Bookfair November 18 -20, 2009 - Grandparents Day November 20, 2009 - Westlake Tree Lighting December 3, 2009, 6:30 p.m. - Board of Trustees Meeting December 7, 2009 - Employee Recognition Banquet December 9, 2009, TBD - WA Parents Shopping Day Out December 19, 2009 (Benefiting Project 2010) - Career Day December TBD, 2009 - Westlake Academy Gallery Night event February 26, 2010 - Bandana Bonanza May 8, 2010 - Westlake Academy Graduation May 23, 2010 Westlake Academy Item #5 Board Calendar FUTURE AGENDA ITEMS: Any Board member may request at a workshop and or Board meeting, under “Future Agenda Item Requests”, an agenda item for a future Board meeting. The Board member making the request will contact the CEO with the requested item and the CEO will list it on the agenda. At the meeting, the requesting Board member will explain the item, the need for Board discussion of the item, the item’s relationship to the Board’s strategic priorities, and the amount of estimated staff time necessary to prepare for Board discussion. If the requesting Board member receives a second, the CEO will place the item on the Board agenda calendar allowing for adequate time for staff preparation on the agenda item. - None Westlake Academy Item # 6 – Future Agenda Items Westlake Academy Item # 7 – Adjournment Back up material has not been provided for this item.