Roots and Radicals. Radicals (also called roots) are directly related to exponents.

# SFUSD Mathematics Core Curriculum Development Project · a notation for radicals in terms of...

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 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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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Algebra 1

A.4 Working with Expressions

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Equivalent Forms Individual Work (2 pages) 1 per student

4 Lesson Series 1 CPM CCA Lessons 3.2.2 and 3.2.3 (6 pages) Working with Parentheses (2 pages) CPM CCA 3.2.4 (4 pages)

1 per pair 1 per student 1 per pair

Algebra tiles

2 Apprentice Task Matching Equivalent Quadratic Expressions (2 pages) 1 per student

3 Lesson Series 2 CPM CCA Lesson 3.1.1 (2 pages) CPM CCA Lesson 3.1.2 (2 pages) CPM CCA Lesson 7.2.1 (2 pages)

1 per pair 1 per pair 1 per pair

Graphing calculators

1 Expert Task Exponents! Exponents! (2 pages) 1 per student

1 Lesson Series 3 Table cards

1 Milestone Task Working with Expressions (4 pages) 1 per student

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Unit Overview

Big Idea

Algebraic expressions and equations can be rearranged into other equivalent forms in which different mathematical structures are evident.

Unit Objectives

● Students will be able to manipulate expressions using rules of integer and fractional exponents. ● Students will be able to explain why a particular form of an equivalent expression is more desirable given the context and question. (This objective will

be introduced in this unit but mastered in A.5 with Quadratics.) ● Students will be able to perform operations--adding, subtracting and multiplying--on polynomials and other functions (rational, radical, linear, etc.) and

expressions in order to extract information and confirm relationships.

Unit Description

Students learn to recognize equivalent expressions through multiple representations (table, graph, context, algebra tiles) as they review operations with linear expressions from 8th grade. They use operations with polynomials to show equivalent forms of quadratic expressions—standard form and factored form—using Algebra Tiles. They connect to operations with polynomials to show how to move from factored form to standard form. Students review properties of integer exponents from eighth grade and are Introduced to fractional exponents and their connection to radicals.

CCSS-M Content Standards

The Real Number System Extend the properties of exponents to rational exponents. N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5. N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Use properties of rational and irrational numbers N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Seeing Structure in Expressions

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Interpret the structure of expressions A.SSE.1 Interpret expressions that represent a quantity in terms of its context. A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. ***The standard below will be previewed in our unit and more deeply addressed in A.5 and A.6. Interpreting Functions Analyze functions using different representations F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* F.IF.7a Graph linear and quadratic functions and show intercepts, maxima and minima.

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

8th Grade: 8.2: Roots and Exponents 1. Work with the square root and cube root symbols 2. Know properties of integer exponents 3. Estimates of very small and very large quantities

9th Grade A.4 Working with Expressions 1. Procedure: Use properties of operations to move between different polynomial representations and expressions 2. Concept: Recognize different forms of equivalent expressions and know when they are useful for a certain purpose, both for exponents and polynomials

11th Grade A.10 Polynomials and Rational Expressions 1. Construct polynomial functions satisfying given conditions 2. Simplify more complex rational expressions and expressions with radicals 3. Recognize different forms of rational functions

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are formative 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 4 days 2 days 3 days 1 day 1 day 1 day

Total: 13 days

Lesson Series 1

Lesson Series 2

Lesson Series 3

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Entry Task Equivalent Equations

Apprentice Task Matching Equivalent Quadratic

Expressions

Expert Task Exponents! Exponents!

Milestone Task Working with Expressions

CCSS-M Standards

A.SSE.2 A.SSE.1 A.APR.1 F.IF.7a (preview)

N.RN.1, N.RN.2 A.SSE.2

N.RN.1, N.RN.2 A.SSE.1, A.SSE.2 A.APR.1

Brief Description of Task

Graph different forms of linear equations and have students discuss how to manipulate different forms (3 or 4 students with different forms of the same linear equation per group). Explain why each form is important.

Add and subtract polynomials and multiply two binomials with algebra tiles; recognize equivalent forms of quadratic expressions.

Apply understanding of exponents and laws of exponents.

Write equivalent forms of expressions (fractional exponent, polynomial). Explain error analysis in equivalent forms of expressions.

Source SFUSD Teacher Created SFUSD Teacher Created; CPM CPM Algebra Connections SFUSD Teacher Created

Lesson Series 1 Lesson Series 2 Lesson Series 3

CCSS-M Standards

A.SSE.1 A.APR.1

A.SSE.1 F.IF.7E

N.RN.1, N.RN.2 A.SSE.1, A.SSE.2 A.APR.1 F.IF.7

Brief Description of Lessons

This lesson series includes istributing negatives, multiplying binomials, combining like terms, and building area models.

This lesson series includes working with exponential form (multiplying terms with a common base, power of a power, powers of natural numbers and variables, alone or combined).

This lesson series will review all unit topics–distribution of binomials, distribution of negative, exponents, square roots–in preparation for the Milestone Task.

Sources CPM Core Connections Algebra 3.2.2, 3.2.3, 3.2.4 SFUSD Teacher Created

CPM Core Connections Algebra 3.1.1, 3.1.2, 7.2.1

SFUSD Teacher Created

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Entry Task

Equivalent Equations

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to explain why a particular form of an equivalent

expression is more desirable given the context and question. CCSS-M Standards Addressed: A.SSE.2 Potential Misconceptions:

● Student can not apply operations to move from one form of a linear equation to another (for example from point slope to slope intercept).

● Students may not remember how to graph more complicated equations (point slope or other).

Launch: It is helpful to assign Team Jobs at the beginning of the task: Facilitator, Recorder/Reporter, Team Captain, and Resource Monitor. You could also use Group Grades. (These documents are in Resources.) Students will start with a review of graphing linear equations in different forms. Each student graphs his/her equation and labels important points on a graph. During: Students will work with their groups to identify important information (intercepts, slope) in each equation and on the graph. Students compare the graphs and should notice that although the equations are in different forms the graphs are identical. Each group categorizes all equations in their group and when that form is best to use. Each group shows mathematically how to move between the forms. Each group discusses the questions:

1. Why are different forms of the same equation useful? 2. What mathematics did you use to move from one equation to another? 3. What important points on the graph and in the equation might be useful

in different functions? (e.g., exponential, quadratic, etc. ) 4. What do you think we are going to do next?

Closure/Extension: Have the class share out about why different forms of the same equation are useful and predict where the class is going next.

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Equivalent Equations

How will students do this?

Focus Standards for Mathematical Practice: 7. Look for and make use of structure. (They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.) 1. Make sense of problems and persevere in solving them. (Older students might transform algebraic expressions. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs.)

Structures for Student Learning: Academic Language Support:

Vocabulary: linear equation, slope, x- and y-intercepts, standard form, slope intercept form, point slope form, compare Sentence frames: To move from standard form of the equation to slope intercept form, I used _______ mathematical operations to _______. Different forms of the same equation are useful because _______. _______ and _______ are important points on the graph and in the equation that might be useful in other types of functions.

Differentiation Strategies: ● You can assign students different equations when starting the lesson.

Participation Structures (group, partners, individual, other):

● Groups, with some individual work. ● Class discussion/share out at the end.

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Lesson Series #1

Lesson Series Overview: Review of combining like terms and distribution of a negative. Introduce students to multiplying binomials through algebra tiles and the area model. Introduce generic rectangle as a way of organizing terms when multiplying. CCSS-M Standards Addressed: A.SSE.1, A.APR.1 Time: 4 days

Lesson Overview - Day 1 and 2 Resources

Description of Lesson: Introduce algebra tiles as a way to represent area and (base)(height). Review the names of pieces and asks students to build a rectangle and label the base and height. Then, given base and height, have students find the area using the algebra tiles in the L-piece. This assumes that students have already worked with algebra tiles. If not, two or three days may be needed to introduce the pieces and find perimeter and area before building rectangles. Notes: As students multiply with algebra tiles, give them a generic rectangle to show the steps on paper and to see the distribution in a more abstract way.

CPM Core Connections Algebra Lessons 3.2.2 and 3.2.3, all problems

Lesson Overview - Day 3 and 4 Resources

Description of Lesson: Individual practice with distribution of 2 binomials. Remind students that

so they can eventually distribute quadratic expressions in vertex form. Connect this to building a square versus a rectangle with algebra tiles. Have students simplify expressions with binomial distribution and combining like terms. Review distributing the negative with all terms inside parentheses. Model this with the tiles and then have students practice. In the debrief, make sure to highlight the differences between multiplying binomials and adding or subtracting binomials.

Working with Parentheses CPM Core Connections Algebra Lesson 3.2.4, Problems 3-64 through 3-69. Problem 3-69 should be used as an extension, but is not entirely necessary for a full understanding of this unit. (If you have CPM Algebra Connections textbooks, Lesson 5.1.3 is a similar lesson.)

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Apprentice Task

Matching Equivalent Quadratic Expressions

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to use distribution and combining like terms to

move between different algebraic forms of a quadratic function. ● Students will be able to use the key elements of a parabola--y-

intercept, roots and vertex--to determine which form of a quadratic function is most appropriate.

● Students will be able to graph a parabola using the vertex, roots, and the y-intercept.

CCSS-M Standards Addressed: A.SSE.1, A.APR.1, F.IF.7a (preview) Potential Misconceptions

● Students may forget to distribute the negative with all terms. ● Students may only multiply two terms when multiplying two binomials.

Launch: Videos, words, and music are all used in different situations for certain purposes. For example, what would you use if you wanted to see how to do something? What about if you wanted to dance? And how about learning how to spell something or read something? The same is true for expressions in mathematics; different forms serve different purposes. Today, you will determine which form is the most appropriate to see certain elements of a graph. During: How do you know the expressions are equivalent? Which form shows you the integers you need for the roots, y-intercept, and vertex? Closure/Extension: Standard form shows you the y-intercept. Factored form shows you the roots. Vertex form show you the vertex.

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Matching Equivalent Quadratic Expression

How will students do this?

Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: x-intercepts (roots), y-intercept, vertex, distribute, combining like terms

Sentence frames: I can combine ____ and ____ because _______. The factored form shows the _______. The standard form shows the _______. The vertex form shows the _______.

Differentiation Strategies: • Instead of making a table into graphs, students can use a graphing calculator or a pre-make a graph. • Ask students to graph more than one graph with roots, y-intercept, and vertex. • Provide combining like terms notes for students who struggle with this skill.

Participation Structures (group, partners, individual, other): Give each person in a group of four a different form of a quadratic expression and ask them to match. At the first checkpoint, check that the matches are correct and do a shuffle quiz to see how students distributed. At the second checkpoint, ask students to defend why they chose a particular form for a certain element of the parabola.

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Lesson Series #2 Lesson Series Overview: Students work with the properties of exponents to become efficient at manipulating them. They begin with multiplying and dividing exponents in expanded form to develop and derive the shortcut of adding and subtracting exponents. They will then use powers of powers to develop the shortcut of multiplying the exponents. These ideas will be extended to negative exponents and then fractional exponents. CCSS-M Standards Addressed: N.RN.1, N.RN.2 Time: 3 days

Lesson Overview – Days 1-2 Resources

Description of Lesson: Students will review simplification of exponents, formalize laws of exponents, and understand exponent of 0 and –1. Notes: Refer to the notes in the CPM teacher pages.

CPM Core Connections Algebra Lesson 3.1.1, Problems 3-1 to 3-5 CPM Core Connections Algebra Lesson 3.1.2, Problems 3-13 to 3-16 Homework: Use additional problems from CPM

Lesson Overview – Day 3 Resources

Description of Lesson: Students will make sense and interpret fractional exponents. Notes: Students will need to know that = y means that y3 = x. Refer to the notes in the CPM teacher pages.

CPM Core Connections Algebra Lesson 7.2.1, Problems 7-82 and 7-83

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Expert Task Exponents! Exponents!

What will Students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will apply their understanding of powers, laws of exponents,

negative exponents, and fractional exponents to solve problems CCSS-M Standards Addressed: N.RN.1, N.RN.2, A.SSE.2 Potential Misconceptions:

Launch: Let students know they will be applying all their knowledge about exponents in this task. They will be working in pairs or groups because the discussions are a huge part of the learning. Students will be asked to:

● Come to consensus for each problem. (Therefore, they must stay together.)

● Have individual written explanations for each problem after they have come to consensus.

● Have each person be ready to explain orally to you. Only give each pair one Task sheet (or each group two sheets) to encourage discussion. During: Have some possible questions ready to ask groups or students to check for understanding. (e.g., Why did you do that? What does this… mean? Why does this choice …. not work) Answer only group/pair questions and address entire team (versus an individual student). Check in with each group before the end of the period. Ask random students in each team to explain their solution. Closure/Extension:

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Exponents! Exponents!

How will students do this?

Focus Standards for Mathematical Practice: 7. Look for and make use of structure. 1. Make sense of problems and persevere in solving them.

Structures for Student Learning: Academic Language Support:

Vocabulary: Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other):

Groups or pairs

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Lesson Series #3 Lesson Series Overview: Students will spend one day reviewing the somewhat disparate topics that are introduced in this unit (it’s very difficult to connect meaningfully) through a menu structure. Each station will review a particular topic, and solutions will be provided so students can check their work. CCSS-M Standards Addressed: All in the unit. Time: 1 day

Lesson Overview – Day 1 Resources

Description of Lesson: Each table will have a different type of problem to review from the unit. Tell students to start with the concept/skill that they still feel the most challenged by. Provide notes or student work at each table in the event that students get stuck or forget how to approach the problem. If students finish early and are correct, provide challenge problems that go above and beyond the required skills.

Menu Review

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Milestone Task Working with Expressions

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to manipulate expressions using rules of

integers and fractional exponents. ● Students will be able to perform operations--adding, subtracting and

multiplying--on polynomials and other functions (rational, radical, linear, etc.) and expressions to extract information and confirm relationships.

CCSS-M Standards Addressed: N.RN.1, N.RN.2, A.SSE.1, A.SSE.2, A.APR.1 Potential Misconceptions:

● Students may incorrectly multiply binomials. ● Students may incorrectly distribute a negative. ● Showing how radicals are equivalent to fractional exponents. ● Bases must be the same to simplify exponents. ● Mixing up Exponent rules.

Launch: Review operations with exponents, fractional exponents, and polynomials. During: This is an individual task. Closure/Extension: Follow up with common mistakes on Milestone Task the next day. You could also have students do error corrections on individual assignments after grading.

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SFUSD Mathematics Core Curriculum, Grade 9, Unit A.4: Working with Expressions, 2014–2015

Working with Expressions

How will students do this?

Focus Standards for Mathematical Practice: 7. Look for and make use of structure. 1. Make sense of problems and persevere in solving them

Structures for Student Learning: Academic Language Support:

Vocabulary: equivalent, expressions, explain

Sentence frames: These expressions are equivalent because _____________. The error in this step is _____________.

Differentiation Strategies: Modify questions as needed.

Participation Structures (group, partners, individual, other):

Individual