SFUSD Mathematics Core Curriculum Development Project · 2020-02-17 · 1 SFUSD Mathematics Core...

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SFUSD Mathematics Core Curriculum, Algebra 1, Unit A.6: Quadratic Equations, 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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Algebra 1

A.6 Quadratic Equations

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Rectiles (2 pages) 1 per student Algebra tiles (optional)

7 Lesson Series 1 CPM CCA Lesson 8.1.1 (2 pages) CPM CAA Lesson 8.1.2 (2 pages) CPM CAA Lesson 8.1.3 (2 pages) CPM CAA Lesson 8.1.4 (2 pages) CPM CAA Lesson 8.1.5 (2 pages) CPM CAA Lesson 8.2.2 (4 pages) CPM CAA Lesson 8.2.3

1 per pair 1 per pair 1 per pair 1 per pair 1 per pair 1 per pair 1 per pair

Algebra tiles

1 Apprentice Task Quadratic Functions Domino Cards (2 pages)

1 per student 1 per pair

Scissors

4 Lesson Series 2 CPM Algebra Connections, Lesson 10.2.4 CPM Algebra Connections, Lesson 10.3.1 (3 pages) CPM Algebra Connections, Lesson 10.3.2 (3 pages) CPM CCA Lesson 10.2.5 (6 pages)

1 per pair 1 per pair 1 per pair 1 per pair

Algebra tiles

2 Expert Task Victory Celebration (2 pages) 1 per pair Graph paper

4 Lesson Series 3 CPM CCA Lesson 9.1.2 (2 pages) CPM CCA Lesson 9.1.2 HW (2 pages) CPM CCA Lesson 9.1.3 (2 pages) CPM CCA Lesson 9.1.3 HW (2 pages) CPM CCA Lesson 9.1.4 (2 pages) CPM CCA Lesson 9.1.4 HW (2 pages) Graphs (2 pages)

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

Graphing calculators or software

1 Milestone Task CLA 2 Constructed Response Questions CLA 2 Performance Assessment – Falling Shoes

Provided by AAO Provided by AAO

Graph paper

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Unit Overview

Big Idea

Quadratic equations have equivalent forms (standard, factored, vertex) that reveal different properties of the relationship represented by the given equation. These properties, along with the quadratic formula, can be applied to solve problems and interpret them in context.

Unit Objectives

● Students will be able to convert between equivalent forms of the quadratic equation by factoring, completing the square, and using the quadratic formula.

● Students will be able to choose and produce the form of the quadratic equation that is best suited to solve a given problem or model a given situation or context.

Unit Description

In this unit, students will explore equivalent forms of quadratic equations. Students will convert between equivalent forms of quadratic equations by factoring, completing the square, and using the quadratic formula. In the first lesson series, students will use diamonds, generic rectangles and find the root(s) of a quadratic equation by factoring. In the second lesson series, students will: 1. Find the root(s) of a quadratic equation by factoring, completing the square, and using the quadratic formula, 2. Find the roots, vertex, maximum or minimum, axis symmetry, and produce a rough graph, and 3. Model a situation with a quadratic equation and interpret in context.

CCSS-M Content Standards

The Complex Number System

Perform arithmetic operations with complex numbers

N.CN.1 Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.

Use complex numbers in polynomial identities and equations

N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.

Seeing Structure in Expressions

Interpret the structure of expressions

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).

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Write expressions in equivalent forms to solve problems

A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines.

A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Arithmetic with Polynomials and Rational Expressions

Understand the relationship between zeros and factors of polynomials

A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Creating Equations

Create equations that describe numbers or relationships

A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Reasoning with Equations and Inequalities

Understand solving equations as a process of reasoning and explain the reasoning

A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve equations and inequalities in one variable

A.REI.4 Solve quadratic equations in one variable.

A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

A.REI.4b Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

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Solve systems of equations

A.REI.7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.

Interpreting Functions

Analyze functions using different representations

F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

Students previously identified and explained the parts of graphs of quadratic equations. Students learned that there are different representations of quadratic functions and that different forms of the quadratic equation reveal different information. Students learned basic mathematics needed to manipulate equations (exponents, negative numbers, square roots and radicals, solving equations, etc.). Students learned to multiply polynomials.

In algebra, students find the root(s) of a quadratic equation by factoring, completing the square, and the quadratic formula as well as discerning when to use each method. Students find the roots, vertex, maximum or minimum, and axis symmetry, and they produce a rough graph. Students model a situation with a quadratic equation and interpret in context. Students simplify radicals.

In Algebra 2, students will conduct more in-depth explorations of imaginary numbers and connection to roots of polynomials as well as being able to write equivalent rational expressions.

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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 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 7 Days 1 Day 4 Days 2 Days 4 Days 1 Day

Total Days: 20

Lesson Series 1

Lesson Series 2

Lesson Series 3

Entr

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Entry Task Rectiles

Apprentice Task Forming Quadratics

Expert Task Victory Celebration

Milestone Task Falling Shoes

CCSS-M Standards

A.SSE.3a, F.IF.8a

A.SSE.2, A.SSE.3 A.SSE.3b, A.CED.1, A.REI.2, A.REI.4

A.CED.1, A.REI.2, A.REI.4

Brief Description of Task

Students will use algebra tiles to find area and side lengths as a lead into factoring.

Given the standard form of an equation, students will produce the factored form and make a graph and produce the vertex form and make a graph.

Given the starting height and initial speed of a firework, students will find information about the situation, including a sketch, maximum height, time it takes the fireworks to reach the maximum height, etc.

Students use basic information about two thrown shoes to extract information about the situation (connecting graphs to equations, finding points of intersection, etc.).

Source Rectiles, Silicon Valley Mathematics Initiative - Course 1, Spring 2012 Performance Test

Forming Quadratics FAL, MARS/Shell 2012

IMP Year 2, p. 280, pp. 316–319 Teacher Created

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Lesson Series 1 Lesson Series 2 Lesson Series 3

CCSS-M Standards

A.SSE.2, A.SSE.3a A.APR.3 F.IF.8a

A.SSE.3b A.REI.4a F.IF.8a

A.CEDa A.REI.1, A.REI.2, A.REI.4b, A.REI.7 N.CN.1, N.CN.7

Brief Description of Lessons

Themes: Factoring A. Students will use the diamond method, algebra tiles, and generic rectangles to factor quadratic expressions in the form ax2 + bx + c. B. Students will factor completely expressions that have a common factor in each term. C. Students will learn relationships with factoring and the graph of quadratic equations in the form y = ax2 + bx + c, for y = 0.

● For a = 1 diamond problems work CPM-CCA 8-1 to 8-4, 8-6, 8-7,8-9, 8-10, 8-24.

● For a <> 1 diamond method with the generic rectangle 8-13 to 8-16.

● Any method may be used to factor the following: 8-24 to 8-27, 8-31.

● Factoring Completely: 8-37,8-38, 8-45 to 8-47

D. Zero Product Property ● 8.2.2 CPM 8-64 to 8-67

Themes: Completing the Square, Deriving the Quadratic Formula Students will look at equivalent quadratic equations and learn how to complete the square, both with and without algebra tiles and equation mats, then derive the quadratic formula by completing the square.

Themes: Solving Quadratic Equations With the Quadratic Formula Solving Quadratic Equations with the Quadratic Formula

Sources CPM Core Connections Algebra, Chapter 8 CPM Core Connections Algebra, Chapter 10 CPM Algebra Connections, Chapter 10

CPM Core Connections Algebra, Chapter 9 MARS (Graphs)

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Entry Task

Rectiles

Students will find the area of shapes with algebra tiles

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will be able to find area of polynomials in trinomial form. ● Students will be able to prepare for factoring trinomials.

CCSS-M Standards Addressed: A.SSE.3, F.IF.8a Potential Misconceptions:

● Area is length times width.

Launch: Make sure students understand the situation. During: Students perform the task, finding the area of the figure. Closure/Extension: Students create new shapes and find their areas. Note: The task uses tiles with lengths of x and y. You might consider relabeling the diagrams so the tiles have lengths of x and 1 to match the dimensions of algebra tiles.

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Rectiles

How will students do this?

Focus Standards for Mathematical Practice: 5. Use appropriate tools strategically. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: configuration, rectangular, width, length, area

Sentence frames:

Differentiation Strategies: Provide students with algebra tiles to make their own configurations and figure out the area. Participation Structures (group, partners, individual, other): Students should do this task with partners.

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Lesson Series #1 Lesson Series Overview: Students factor quadratic expressions, ax2 + x + c using manipulatives and visually enhanced algebraic methods. Emphasis will be on distinguishing methods based on a = 1 and a <> 1. Algebra tiles will be used to help with factoring and transition to transferring the visual information to a generic rectangle. Students connect that factoring leads to solving for the x-intercepts of a quadratic function. CCSS-M Standards Addressed: A.SSE.2, A.SSE.3a, A.APR.3, F.IF.8a Time: 7 days

Lesson Overview – Day 1 Resources

Description of Lesson: Introduction to factoring quadratics

Algebra Tiles CPM-CCA Lesson 8.1.1: Core problems 8-1 to 8-5 (If you have CPM Algebra Connections textbooks, Lesson 8.1.1 is a similar lesson.)


Description of Lesson: Factoring with generic rectangles

CPM-CCA Lesson 8.1.2: Core problems 8-13 to 8-16 (If you have CPM Algebra Connections textbooks, Lesson 8.1.2 is a similar lesson.)


Description of Lesson: Factoring with special cases



Description of Lesson: Factoring Completely


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Description of Lesson: Factoring Shortcuts

CPM-CCA Lesson 8.1.5: Core problems 8-45 to 8-47


Description of Lesson: Opening: Students will learn relationship between factoring and the graph of quadratic equations in the form y = ax2 + bx + c, for y = 0. Notes: Make sure students remember difference between x- and y- intercepts and vertex.

CPM-CCA Lesson 8.2.2: problems 8-64 to 8-68 Methods and Meanings p. 393 (If you have CPM Algebra Connections textbooks, Lesson 8.2.3 is a similar lesson.)


Description of Lesson: Opening: Students will be given quadratics equations in the form y = ax2 + bx + c. Students will factor and solve for x using the zero product property. They will also graph/sketch parabolas for those equations. Notes: You can also have students find the vertex if they have had experience with this already. It is good fore them to know that its x-value lies halfway between the x-intercepts.

CPM-CCA Lesson 8.2.2 and 8.2.3: problems 8-71, 8-73 can omit finding vertex, 8-77

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Apprentice Task Forming Quadratics

What will students do?


Math Objectives: ● Students will be able to convert the standard form of a quadratic

equation into factored form. ● Students will be able to convert the standard form of a quadratic

equation into vertex form. ● Students will be able to explain how these forms are different.

CCSS-M Standards Addressed: A.SSE.2, A.SSE.3 Potential Misconceptions:

● Students may connect numbers in equations with numbers on the graph incorrectly.

● Students may factor incorrectly (by just changing x2 to x and bx to b, for example)

● Students may connect negative coefficients to 3rd and 4th quadrant on the graph.

Launch: Have students work on “Quadratic Functions” handout. During: Give pairs of students each of the “Domino Cards” to cut out and match. Students convert quadratic equations in standard form to factored form and vertex form and match them to the graphs of the equations. Closure: Debrief the lesson, assign competence, and review common misconceptions (or call student to share answers). Extension: Have students group graphs/equations into similar groups and write observations (open-ended: graphs with negative a coefficient, graphs that are above/below x-axis, etc.).

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Forming Quadratics


Focus Standards for Mathematical Practice: 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning;


Vocabulary: quadratic equation, coefficient, constant term, standard form for quadratics, simplifying, exponent, factored form Sentence frames: The factors of this equation are ________________. If students get stuck: What should we do next? To clarify: How did you do that? First, we _________. Then, we ____________.

Differentiation Strategies: Remediation: Some students may not complete all problems, focusing instead on key problems. Review factoring integers with students. Review

diamonds with students. Challenge: Have students look for patterns among equations.

Participation Structures (group, partners, individual, other): Partners with help from rest of group. Option for Participation Quiz.

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Lesson Series #2 Lesson Series Overview: Day 1: Types of solutions and completing the square. Day 2: Generalize completing the square. Day 3: Derive the quadratic formula. CCSS-M Standards Addressed: A.SSE.3b, A.REI.4a, F.IF.8a Time: 4 days


Description of Lesson: Opening: Show students graphs of quadratic equations with different types of solutions. Discuss what it looks like to have two real roots, one real root, and zero real roots. Classwork: Students use algebra tiles and equation mats to connect equivalent quadratic equations and explore completing the square. Notes: Students need to know diamonds, algebra tiles, equation mat. You may want to practice/front load with warm-ups/Do Nows.

Opening: Show graphs of CPM Core Connections Algebra, Lesson 10.2.4, Problem 10-60, parts a, b, and d. Classwork: CPM Algebra Connections, Lesson 10.3.1, Problems 10-99 to 10-104


Description of Lesson: Students find ways to complete the square without using algebra tiles. Notes: You may want to give students homework to help prepare/front load for deriving the quadratic equation on Day 3 (equations, rational terms, square roots, squares).

Classwork/Practice: CPM Algebra Connections, Lesson 10.3.2, Problems 10-110 to 10-112 (Table from Standard Form to Perfect Square Form; looks more intermediate) Reference/guided practice: CPM Algebra Connections, Lesson 10.4.1, Methods and Meanings (Completing the Square), Problem 10-125

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Lesson Overview – Days 3 and 4 Resources

Description of Lesson: Students use Completing the Square to derive the quadratic formula. After practicing using the quadratic formula, they are introduced to imaginary numbers. Notes: Problem 10-74 can be redone as a graphic organizer for students to fill in.

Day 3: Derivation of the Quadratic Formula CPM Core Connections Algebra, Lesson 10.2.5, Problems 10-74 to 10-75, 10-81. Day 4: Imaginary Numbers CPM Core Connections Algebra, Lesson 10.2.5, Problems 10-76 to 10-80.

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Expert Task Victory Celebration



Math Objectives: ● Students manipulate a quadratic model of a real-life situation to find

the maximum and roots of the parabola. ● Students interpret the equivalent forms on a quadratic function and its

solutions in terms of the situation it represents. CCSS-M Standards Addressed: A.SSE.3b, A.CED.1, A.REI.2, A.REI.4 Potential Misconceptions:

● Students may think that the parabola represents the 2-dimensional trajectory of the rocket rather than the function of the rocket’s height over time.

Adapted from IMP Year 2 pp. 280, 317–319 Launch: Use the Three Read Protocol (described on pages 10-11 of the Math Teaching Toolkit) to introduce the situation described on the first page. During: Students work in groups to perform the tasks described on the second page. Provide graph paper. Closure/Extension: Have groups create posters and have a gallery walk for the class to write questions and comments.

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Victory Celebration


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 4. Model with mathematics.


Vocabulary: launch, base, maximum

Differentiation Strategies: Allow students to draw diagrams and graphs to help visualize the situation. Participation Structures (group, partners, individual, other): Students should work in groups of four.

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Lesson Series #3 Lesson Series Overview: Students start with the quadratic formula and use it to solve quadratic equations. They practice simplifying answers that are both rational and irrational. They solve a simple system with a quadratic equation and a linear equation. CCSS-M Standards Addressed: A.CED.a, A.REI.1, A.REI.2, A.REI.4b, A.REI.7, N.CN.1, N.CN.7 Time: 4 days


Description of Lesson: Students will solve quadratic equations with rational answers using the quadratic formula. Notes: If students need practice using graphing calculators, have them look at CPM Core Connections Algebra Lesson 9.1.1 (Problems 9-1, 9-2).

CPM Core Connections Algebra, Lesson 9.1.2, Problems 9-12 to 9-15 (9-15 uses graphing calculator), Methods and Meaning (The Quadratic Formula), Problems 9-17 to 9-19


Description of Lesson: Students look at more difficult applications of the quadratic formula. Notes: Do you teach how to simplify radicals? Show imaginary numbers and show what a complex solution looks like.

CPM Core Connections Algebra, Lesson 9.1.3, Problems 9-24 to 9-26, Methods and Meanings (Solving a Quadratic Equation), Problems 9-27 to 9-30, 9-33 (factoring polynomials) (If you have CPM Algebra Connections textbooks, Lesson 8.3.2 is a similar lesson.)


Description of Lesson: Students look at a variety of problems and choose the method that best suits the situation.

CPM Core Connections Algebra, Lesson 9.1.4 (p.429), Problems 9-34 to 9-37 (If you have CPM Algebra Connections textbooks, Lesson 8.3.3 is a similar lesson.)

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Description of Lesson: Students solve a system with a quadratic equation and linear equation.

Intersection of Quadratic and Linear Equation (“Graphs,” MAP 2007)

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Milestone Task

Falling Shoes



Math Objectives: ● Students will write quadratic equations based on a situation. ● Students will say how different parameters (height, starting velocity,

etc.) affect the graph of a quadratic equation. CCSS-M Standards Addressed: A.CED.1, A.REI.2, A.REI.4 Potential Misconceptions:

● Students may put coefficients in the wrong place.

Launch: Remind students of the work they’ve done earlier in this unit, especially in the Expert Task. You might also show a video of a thrown or falling object, or throw a shoe up in the air. During: In the constructed response portion, students graph a quadratic function, solve the related equation using at least two techniques (factoring and using the zero product property, completing the square, applying the quadratic formula), and explain the connection between the solutions and the graph. In the performance assessment, students write quadratic functions for two thrown shoes and determine their maximum height and when they will hit the ground. Closure/Extension: Students who need an extra challenge can solve the quadratic equation in the constructed response portion using a third technique, or they can write their own shoe problem.

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Falling Shoes


Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.


Vocabulary: parabola, vertex, line of symmetry, intercepts, velocity

Differentiation Strategies: Remediation: Students may struggle with finding the coefficients. Some students may be given the generic form of a quadratic equation and

prompted/helped to fill in the coefficients. Challenge: What happens if one of the shoes is thrown later? Students can write their own problems. What if shoes were thrown on the moon (where

the gravity is)? Participation Structures (group, partners, individual, other): Individual

SFUSD Mathematics Core Curriculum Development Project · 2020-02-17 · 1 SFUSD Mathematics Core...

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Transcript of SFUSD Mathematics Core Curriculum Development Project · 2020-02-17 · 1 SFUSD Mathematics Core...