SFUSD Mathematics Core Curriculum Development Project€¦ · · 2014-12-13SFUSD Mathematics Core...

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SFUSD Mathematics Core Curriculum, Grade 7, 7.5: Algebraic Thinking, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Grade 7

7.5 Algebraic Thinking

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task CPM CCC2 4.3.1 (4 pages) 1 per pair Colored pencils Algebra tiles

5 Lesson Series 1 CPM CCC2 4.3.2 (4 pages) Distributing and Factoring Area (Georgia) (2 pages) Generating Equivalent Expressions (Engage NY) (2 pages) CPM CCC2 4.3.3 (4 pages)

1 per pair 1 per student 1 per student 1 per pair

Colored pencils Algebra tiles Graph paper Materials for Opening Exercise Resource, 3 copies per class on card stock Optional: Resource Page 4.3.3, 1 class set on colored card stock

2 Apprentice Task Interpreting Expressions Card Set (4 pages) HW: Interpreting Expressions

1 per pair 1 per student

5 Lesson Series 2 Algebra Magic (Georgia) (2 pages) CPM CCC2 6.2.1 (3 pages) CPM CCC2 6.2.2 (4 pages) CPM CCC2 6.2.3 (4 pages) CPM CCC2 6.2.3 Resource Page

1 per student 1 per pair 1 per pair 1 per pair 1 per student

Resource Page 6.2.1, one class set copied on cardstock Calculators Algebra Tiles Colored pencils

2 Expert Task Express Yourself (2 pages) Card Set (3 pages) Express Yourself (revisited) (2 pages)

1 per student 1 per pair 1 per student

Glue stick and large poster paper Scissors Markers Sticky notes

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5 Lesson Series 3 Glencoe 8.6 Reading Guide & Skills Practice CPM CCC2 6.1.1 (3 pages) CPM CCC2 6.1.1B Resource Page CPM CCC2 6.1.2 (3 pages) CPM CCC2 6.1.2 Resource Page CPM CCC2 6.1.3 (3 pages) CPM CCC2 6.2.4 problems 6-94 & 6-95 (2 pages)

1 per student 1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per pair

Large paper (butcher or 11x14 copy paper) for posters Resource 6.1.1A, 1 class set copied onto cardstock Algebra tiles Markers Sticky dots Graph poster paper (1 piece)

1 Milestone Task Pool Party (2 pages) 1 per student

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Unit Overview

Big Idea

Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value.

Unit Objectives

● Use properties of operations to generate equivalent expressions. ● Solve real-life and mathematical problems using numerical and algebraic expressions and equations. ● Identify variables and write equations and inequalities to solve word problems.

Unit Description

Students begin by applying properties of operations in order to add, subtract, factor, expand, and rewrite linear equations. They then solve multi-step real life and mathematical problems using numerical and algebraic expressions and equations. Finally they write equations and inequalities to solve word problems.

CCSS-M Content Standards

Use properties of operations to generate equivalent expressions. 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

Solve real-life and mathematical problems using numerical and algebraic expressions and equations. 7.EE.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. 7.EE.4. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

In first grade students learn about properties of operations for addition and subtraction. In 3rd grade the understanding of properties of operations extends to multiplication and division. In sixth grade students write, read, and evaluate expressions in which letters stand for numbers. They solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q.

Students use properties of operations to generate equivalent expressions. They also use this concept of equality to solve equations using whole numbers, fractions and decimals. Students extend their understanding of equations to include inequalities and how their meanings are different from equations.

In eighth grade students will apply the properties of integer exponents to generate equivalent numerical expressions. They will analyze and solve pairs of simultaneous linear equations. Students will also define, evaluate, and compare functions, and use functions to model relationships between quantities.

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Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 Day 5 Days 2 Day 5 Days 2 Days 5 Days 1 Day

Total Days: 21

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Entry Task How Can I Simplify? (CPM CCC2 4.3.1)

Apprentice Task Interpreting Algebraic Expressions

Expert Task Steps to Solving Equations

Milestone Task Pool Party

CCSS-M Standards

7.EE.1 7.EE.1 7.EE.2 7.EE.3 7.EE.4a 7.EE.1 7.EE.2, 7.EE.3, 7.EE.4

Brief Description of Task

Students use algebra tiles to build area models of polynomials and write expressions for those areas. They use the tiles to help them visualize and combine like terms, and to write multiple equivalent expressions for the same area.

Students are given twenty-four cards (six sets of four cards each) that include six algebraic expressions, six verbal sentences, six x/y tables, and six area models. Students match the correct algebraic expression, verbal expression, area model, and table.

Students match algebraic equations to verbal expressions. Then they sort cards into corresponding word problems, algebraic equations, and solution steps.

Students extend a pattern of square pools surrounded by a border. Students generate an expression to let them find the border of any pool of side n. They then look at two expressions generated by two “students” and use their knowledge of rewriting expressions in order to determine if the students’ expressions are correct or not.

Source CPM Core Connections Course 2 Adapted from MARS FAL MARS FAL Adapted from NCSM Great Tasks for Mathematics “How Do You Visualize It?”

Lesson Series 1 Lesson Series 2 Lesson Series 3

CCSS-M Standards

7.EE.1 7.EE.2 7.EE.3, 7.EE.4a 7.EE.4b

Brief Description of Lessons

Students apply properties of operations as strategies to simplify, add, subtract, factor, and expand linear expressions with rational coefficients.

Students expand the skills that they learned in the first Lesson Series to solve multi-step linear equations. They continue to use algebra tiles to model equations and the step-by-step methods they use to solve them.

Students practice writing and solving one-and-two step inequalities. Students also develop skills to write expressions, equations, and inequalities in order to solve word problems.

Sources

CPM Core Connections Course 2 EngageNY

CPM Core Connections Course 2 CPM Core Connections Course 2 EngageNY Glencoe

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Entry Task

How Can I Simplify? (CPM CCC2 4.3.1)

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Find area of algebra tiles. ● Write expressions to represent area and perimeter of a group of

algebra tiles in multiple ways. CCSS-M Standards Addressed: 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” Potential Misconceptions Students may try to get an actual measure for length of “x”. Note: If you do not have Algebra Tiles, create them using the Algebra Tiles Resource Page. You need 3 “x squared”, 5 “x”, and 18 “1” minimum per group (all positive).

See CPM teacher guide for detailed teaching suggestions. Launch: To find out what the students already know, students will explore the shapes of the three main algebra tile shapes determining what the area of each is. Use the algebra tiles to represent algebraic expressions. Have students work in pairs or small groups. During: Students will determine the length of the sides of the tiles by comparing the pieces to each other. They are told that the small square is a “one” or a “unit tile”. They will understand that the unknown length will be known as “x”. Students will write multiple algebraic expressions representing the areas for a group of tiles. They will do this with several different tile groupings. Next they will write expressions for the perimeters of a group of tiles. Closure/Extension: Finally the students will understand that terms are the parts of the expressions that are separated by addition and subtraction. Like terms with the same exponents raised to the same power, can be combined to simplify an expression.

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How Can I Simplify? (CPM CCC2 4.3.1)

How will students do this?

Focus Standards for Mathematical Practice: SMP 5: Use appropriate tools strategically. SMP 1: Make sense of problems and persevere in solving them. Structures for Student Learning: Academic Language Support:

Vocabulary: Algebraic Expression, Coefficient, Term, Like Term, Variable

Sentence frames: The area of these tiles is___(algebraic expression)__. I know this because__________. The perimeter of these tiles is __(algebraic expression)__. I know this because________.

Differentiation Strategies: Students working in pairs. Participation Structures (group, partners, individual, other): students will work in partners and also join up with another pair before whole group debriefing.

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Lesson Series #1

Lesson Series Overview: Students apply properties of operations as strategies to simplify, add, subtract, factor, and expand linear expressions with rational coefficients. CCSS-M Standards Addressed: 7.EE.1, 7.EE.2 Time: 5 days

Lesson Overview – Day 1 Resources

Description of Lesson: Students simplify algebraic expressions by combining like terms and using the Distributive Property. Notes:

CPM Core Connections Course 2, Lesson 4.3.2 Problems 4-85 through 4-88 Suggested HW: Choose from Review & Preview


Description of Lesson: Students continue to simplify algebraic expressions by combining like terms and using the Distributive Property. Notes:

CPM Core Connections Course 2, Lesson 4.3.2 Problems 4-89 through 4-92 Suggested HW: Choose from Review & Preview


Description of Lesson: Students apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. Notes: At end of this class, students can cut up triangles and quadrilaterals from the Materials for Opening Exercise for tomorrow.

Distributing and Factoring Area (Georgia) For projector: Resource: Vocabulary List

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Description of Lesson: This exercise requires students to represent unknown quantities with variable symbols and reason mathematically with those symbols to represent another unknown value. Notes: You will need to have 3 copies of Triangle and Quadrilateral Resource (2 pages) printed on cardstock and divided into envelopes, 1 envelope of triangles (4) and quadrilaterals (2) per pair

Generating Equivalent Expressions, Opening Exercise (EngageNY) Problems a through g Resource: Materials for Opening Exercise


Description of Lesson: Students simplify algebraic expressions by combining like terms and using the Distributive Property. Includes concepts of simplifying with negatives and zero. Notes:

CPM Core Connections Course 2, Lesson 4.3.3 Optional: Resource Page 4.3.3, class set copied onto colored cardstock Suggested HW: Choose from Review & Preview

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Apprentice Task

Interpreting Algebraic Expressions



Math Objectives: ● Understand the distributive law of multiplication over addition. ● Recognize equivalent expressions.

CCSS-M Standards Addressed: 7.EE.1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

7.EE.2. Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” Potential Misconceptions

● Students may work problems left to right, showing no understanding of order of operations.

● Students may construct/expand parenthesis incorrectly.

Launch: Lead the class in a discussion around the “Matching Expressions and Words” projector slide. -One pair of words/expression does not match up. This is deliberate. If students match these, have them work through the problem to explore errors. Students can also work to create an accurate matching expression/phrase for the non-matched. During: Students work in pairs to match up sets of four cards (one expression, one verbal phrase, one table, and one area model). -Three cards (two phrases and one table) are blank and will have to be filled in by students. -E4 and E5 are equivalent and either one can be represented by A2 or A6. Closure/Extension: Students explain to class how they decided that different representations of an expression went together. Exit slip: Write another problem on the board, such as 3(x+5) on the board. Have students copy and write a phrase and area model that matches the expression, and write an equivalent expression. Homework: Interpreting Expressions Handout

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Interpreting Algebraic Expressions


Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure Structures for Student Learning: Academic Language Support:

Vocabulary: Expression, Equivalent, Table, Variable, Model, Expand, Distributive,

Sentence frames: “Another way to write this expression is __________________________________________________.”

“I/we think _________________and ________________ are/are not equivalent because_________________.”

Differentiation Strategies: Physical algebra tiles may be used to model expressions, expansions, or areas. Calculator may be used for completing table. Participation Structures (group, partners, individual, other): Do now is done individually and then whole-class as a discussion. Matching activity is done collaboratively as pairs, then discussed as a whole group. Exit slips and homework are done individually.

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Lesson Series #2

Lesson Series Overview: In the second series of lessons, students expand the skills that they learned in the first series to solve multi-step linear equations. They continue to use algebra tiles to model equations and the step-by-step methods they use to solve them. Students will verbally explain their reasons for choosing the steps they did to each other. By the Expert Task, they are able to write an algebraic equation for a given word problem and solve it. CCSS-M Standards Addressed: 7.EE.3 Solve real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically. 7.EE.4a Solve word problems leading to equations of the form px + q = r or p(x + q) = r, where p, q, and r are specific rational numbers. Time: 4 days


Description of Lesson: Using math magic tricks, students substitute numbers into a verbal expression. Then, they write an expression and rewrite it to demonstrate the operation being performed in the trick. Notes: The first exercise (Variables: Algebra Magic) should be done as a teacher-led activity, then students can work individually or in pairs to complete the Georgia handout.

Algebra magic – Teaching Guidelines (The Futures Channel) Use algebra tiles (x tiles and 1 tiles) in place of the recommended materials (discs, etc.) Algebra Magic (Georgia)


Description of Lesson: Students apply strategies for simplifying expressions to determine if one expression is greater, and begin finding solutions when expressions are equal. Notes:

CPM CCC2 Lesson 6.2.1 Resource Page 6.2.1, one class set copied on cardstock Algebra tiles Suggested HW: Choose from Review & Preview

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Description of Lesson: Students develop an understanding of checking solutions while continuing to develop their understanding of solving equations and the Distributive Property. Notes:

CPM CCC2 Lesson 6.2.2 Suggested HW: Choose from Review & Preview

Lesson Overview – Days 4-5 Resources

Description of Lesson: Students use formal notation while simplifying expressions and solving equations. They compare arithmetic and algebraic methods for solving problems. Notes: Algebra tiles and mats are not required for this lesson, but should be available for students who would benefit from a tangible model.

CPM CCC2 Lesson 6.2.3 Resource Page 6.2.2: Recording Work Suggested HW: Choose from Review & Preview

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Expert Task

Steps to Solving Equations



Math Objectives: ● Generate equivalent expressions. ● Solve real-life and mathematical problems using numerical and

algebraic expressions and equations. CCSS-M Standards Addressed: 7.EE.1, 7.EE.2, 7.EE.3, 7.EE.4a Potential Misconceptions

● Students may think there is only one way to write an equation/expression.

● Students may expect one correct answer from each set. ● Students may follow the wrong order of operations. ● Student may confuse area and perimeter.

Note Slides can also be made into a Nearpod lesson for iPad.

Launch: Do Now: “Express Yourself” Handout In addition to, or in lieu of, above activity, hand each student a copy of the word problem cards and have them individually write an equation for each situation. They will be solving these later in groups, so they do not need to do every one, or solve them. During: Students work in groups to match word problems to equation cards. They will need a new set of word problem cards and a set of equation cards (one per group) that students will cut out and match up. Discuss as a class which equations were matched with which problem. Look for different groups of students with different equations for the same situation. Have them explain their reasoning and determine which equation is correct. Pass out the solution steps cards and a large poster paper. Students put aside E5 and E6 and the matching story problems. Students divide their poster paper into quarters. Students put one equation/word problem in each quarter, then add the appropriate solution steps in the correct order. Then they write the process used in each step (“divide each side by two”) in between each card. Closure/Extension: Lesson can end with a gallery walk where students use sticky notes to make meaningful comments on each other’s posters. Homework: “Express Yourself (revisited)”

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Steps to Solving Equations


Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 4. Model with mathematics. Structures for Student Learning: Academic Language Support:

Vocabulary: Area, Perimeter, Expression, Equation, Represents, Consecutive Sentence frames: “The variable represents _______________________.” “The equation _________________ represents this story because _____________________________.” “The next step to solving this is ___________________________________________.”

Differentiation Strategies: -Area and perimeter problems can be presented first with just numbers to review the procedure for finding them, then repeat the process with variable measures. -For the individual equation writing activity, students may write a solution strategy if they cannot come up with an equation at first. Participation Structures (group, partners, individual, other): Do Now activity is done individually. Slide show activity is done as whole class. Equation writing activity is done individually. Equation matching activity is done in groups/pairs. Discussion on matching is done as whole class. Poster is done in groups.

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Lesson Series #3

Lesson Series Overview: Students practice writing and solving one- and two-step inequalities. Students also write expressions, equations, and inequalities in order to solve word problems. CCSS-M Standards Addressed: 7.EE.3, 7.EE.4.a, 7.EE.4.b. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Time: 5 days


Description of Lesson: Students understand the concept of inequalities, translate verbal expressions into inequalities and vice versa, and match common phrases associated with inequalities with the correct inequality. You may begin the day with a discussion on inequalities in the real world (see truck example). Lesson Reading Guide (from Chapter 8 Glencoe Resources) 8-6 Problem #3: Students work with a partner to choose correct symbol for each phrase, write a complete statement around that phrase, and name at least 3 values that fit their statement. Example: “Is at least”

● symbol would be “greater than or equal to” ● statement: “Must be at least 54” to ride” ● examples: you can ride if you are 54”, 60”, or 72” tall, but not if you are 53”

Students make a poster to show their work. Notes:

Do Now: truck example Glencoe Lesson 8.6 Reading Guide Suggested HW: Glencoe Lesson 8.6 Practice or Skills Practice

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Description of Lesson: Students use a mat and algebra tiles to model writing inequalities. They use a game format to model simplifying inequalities so that they can be compared. See CPM teacher guide for detailed teaching suggestions. Notes:

CPM CCC2 Lesson 6.1.1 Core problems: 6-1 through 6-4 Resource page 6.1.1A (Expression Comparison Mat), 1 class set copied onto card stock Resource page 6.1.1B Algebra tiles Suggested HW: Choose from Review & Preview


Description of Lesson: Students compare inequalities with variables (similar to day 2, but with variables). See CPM teacher guide for detailed teaching suggestions. Notes:

CPM CCC2 Lesson 6.1.2 Problems 6-12 to 6-15 Resource page 6.1.1A (Expression Comparison Mat) Resource page 6.1.2 Algebra tiles Suggested HW: Choose from Review & Preview


Description of Lesson: Students model and solve inequalities in one variable. See CPM teacher guide for detailed teaching suggestions. Notes: Mats and tiles are not required, but are suggested if students need visuals/manipulatives

CPM CCC2 Lesson 6.1.3 Core problems: 6-23 through 6-26 Resource page 6.1.1A (Expression Comparison Mat) Algebra tiles Poster graph paper (1 piece) Marker Sticky dots Suggested HW: Choose from Review & Preview

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Description of Lesson: Students participate in a math activity based on problem 6-94, and then apply strategies learned to analyze and solve word problems independently. Do not hand out problems 6-94 and 6-95 until after the following activity. Project or hand out the following situation: “While looking at the country of Jamaica’s results from the 2008 Beijing Olympics, Gemma noticed that the number of gold medals Jamaica received was twice the number of silver medals. She also realized that Jamaica received 1 fewer bronze medal than silver medals. Altogether, Jamaica received 11 medals.”

• Ask pairs of students to come up with a question based on the information. • Share out possible questions. • Students solve a question of their choice in pairs and make a poster. • Students do a “gallery walk” and leave comments on post-its on others’ posters.

Students finish the rest of 6-94 and 6-95 independently.

CPM CCC2 lesson 6.2.4 Problems: 6-94 and 6-95 Suggested HW: Choose from Review & Preview

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Milestone Task

Pool Party



Math Objectives: ● Write an expression to describe an area. ● Decide if two different expressions are equivalent or not.

CCSS-M Standards Addressed: 7.EE.1, 7.EE.2, 7.EE.3, 7.EE.4 Potential Misconceptions

● Students see multiples of four and think that the perimeter of pool 10 is 40.

● Students fail to apply the distributive property correctly in simplifying Elan’s expression.

● Students confuse “n” as the number of tiles in the edge of the entire square, not just the pool.

Launch: Do Now: Draw a rectangle with the dimensions length = 5 and width = x + 3 on the board. Have students write the area and perimeter as many ways as they can. Share out responses with a partner and then with the class. This can also be done as a Math Talk. During: Students complete Pool Party activity individually. Closure/Extension:

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Pool Party


Focus Standards for Mathematical Practice: Structures for Student Learning: Academic Language Support:

Vocabulary: Sequence, surrounded, visualize Sentence frames: “I think _______________________ is correct because ___________________________.”

Differentiation Strategies: Students may use tiles or graph paper to help with pool models. Students may use algebra tiles to model expressions. Students may use whatever strategy they prefer to find the 10th pool. Participation Structures (group, partners, individual, other): Individual

SFUSD Mathematics Core Curriculum Development Project€¦ · · 2014-12-13SFUSD Mathematics Core...

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Transcript of SFUSD Mathematics Core Curriculum Development Project€¦ · · 2014-12-13SFUSD Mathematics Core...