Grades Algebra 1. Polynomial Arithmetic Equations and Identities Quadratics. By Henri Picciotto. 395 Main Street Rowley, MA

Transcription

1 Grades 7 10 ALGEBRA LAB GEAR Algebra 1 Polynomial Arithmetic Equations and Identities Quadratics Factoring Graphing Connections By Henri Picciotto 395 Main Street Rowley, MA

2 Contents Correlations to the Common Core Standards To the Teacher The Algebra Lab Gear vii viii xiii Lesson 2-1 Perimeter 20 Activity 2-1A Perimeter 22 Section 1 Meet the Algebra Lab Gear: A Model for Polynomials 1 Lesson 1-1 The Blocks and the Corner Piece 3 Activity 1-1A The Yellow Blocks and the Corner Piece 6 Activity 1-1B The Blue Blocks and the Corner Piece 7 Activity 1-1C The 3-D Blocks and the Corner Piece 8 Activity 1-1D Face to Face 9 Lesson 1-2 Dimension and Drawing 10 Activity 1-2A Sketch the Lab Gear 2-D Sketches 12 Activity 1-2B Sketch the Lab Gear 3-D Sketches 13 Activity 2-1B Perimeter Challenges 23 Lesson 2-2 Surface Area 24 Activity 2-2A Surface Area 26 Activity 2-2B Surface Area Using 3-D Blocks 27 Activity 2-2C Surface Area Puzzles Using 3-D Blocks 28 Lesson 2-3 Sequences 29 Activity 2-3A Perimeter Sequences 31 Activity 2-3B Surface Area Sequences 32 Section 3 Multiplying, Factoring, and Dividing Polynomials 33 Lesson 1-3 Modeling Polynomials 14 Activity 1-3A Naming Polynomials 17 Activity 1-3B Naming Polynomials Using the 3-D Blocks 18 Section 2 Using Polynomials: Perimeter and Surface Area 19 Lesson 3-1 The Geometry of Multiplication 35 Activity 3-1A Make a Rectangle 38 Activity 3-1B Make a Box 39 Activity 3-1C Make a Box Using 3-D Blocks 40 Activity 3-1D Multiplying in the Corner Piece 41 Didax, Inc. iii Contents

3 Activity 3-1E Multiplication Using the 3-D Blocks 42 Activity 3-5D Perfect Cubes 68 Lesson 3-2 The Distributive Property 43 Activity 3-2A Multiplying in a Table 46 Activity 3-2B Multiplication Table Puzzles 47 Activity 3-2C Finding the Dimensions 48 Lesson 3-3 Factoring 49 Activity 3-3A Trinomial Patterns 52 Activity 3-3B Factoring in a Table 53 Activity 3-3C Three Factors 54 Activity 3-3D Three Factors Using the 3-D Blocks 55 Lesson 3-4 Dividing by a Monomial 56 Activity 3-4A Dividing in the Corner Piece 59 Activity 3-4B Dividing in the Corner Piece Using the 3-D Blocks 60 Activity 3-4C Dividing in a Table 61 Lesson 3-5 Perfect Square Trinomials 62 Activity 3-5A Make a Square 65 Activity 3-5B Perfect Squares 66 Activity 3-5C Patterns in Making a Square 67 Lesson 3-6 Multiplying, Factoring, Dividing 69 Section 4 Linear Equations 71 Lesson 4-1 Modeling Minus 73 Activity 4-1A Naming Polynomials with Minus 78 Activity 4-1B Opposites 79 Lesson 4-2 Comparing Expressions 80 Activity 4-2A Which Is Greater? 82 Activity 4-2B Can You Tell? 83 Activity 4-2C Which x Makes the Expressions Equal? 84 Lesson 4-3 Solving Linear Equations 85 Activity 4-3A Solving Linear Equations 89 Activity 4-3B Doing the Same Thing to Both Sides 90 Activity 4-3C More Linear Equations 91 Activity 4-3D Solving Two Ways 92 Lesson 4-4 Systems of Linear Equations 93 Activity 4-4A Solving for One Variable 97 Activity 4-4B Solving a System of Equations 98 Algebra Lab Gear: Algebra 1 iv Didax, Inc.

4 Section 5 More on Polynomials and Linear Functions 99 Lesson 5-1 Dimension and Degree 101 Lesson 5-2 Evaluating Polynomials 103 Activity 5-2A Evaluating Polynomials 106 Activity 5-2B Always, Sometimes, Never 107 Lesson 5-3 The Graphing Connection: Solving Linear Equations 108 Lesson 5-4 The Graphing Connection: Lines 111 Activity 5-4A Lab Gear Graphs 113 Activity 5-4B Analyzing Linear Functions 114 Section 6 The Distributive Property and Quadratics 115 Activity 6-2D Multiplying Trinomials 126 Activity 6-2E More Minus Practice 127 Lesson 6-3 Identities 128 Activity 6-3A Using Identities 131 Activity 6-3B Factoring a Difference of Squares 132 Lesson 6-4 The Graphing Connection: Parabolas 133 Activity 6-4A Analyzing Parabolas 137 Activity 6-4B How Many Solutions 138 Lesson 6-5 Solving Some Quadratic Equations 139 Activity 6-5A The Zero Product Principle 142 Activity 6-5B Solving by Factoring 143 Activity 6-5C Solving by Equal Squares 144 Lesson 6-1 Simplifying Fractions 117 Activity 6-1A Simplifying Fractions 118 Lesson 6-6 Solving Any Quadratic Equation 145 Activity 6-6A Solving by Completing the Square 147 Lesson 6-2 Minus in the Corner Piece 119 Activity 6-2A Minus in the Corner Piece 123 Selected Answers 148 Workmat Master 154 Activity 6-2B Polynomial Multiplication 124 Activity 6-2C Multiplying Binomials 125 Didax, Inc. v Contents

5 Correlation to the Common Core Standards CCSS Standard Description Activities 5.OA.3 5.MD.5a 6.NS.6a 6.NS.7a 6.EE.2c Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as values, e.g., to represent the associative property of multiplication. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., ( 3) = 3, and that 0 is its own opposite. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. Write, read, and evaluate expressions in which letters stand for numbers. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). 2.3A, 2.3B 3.1C, 3.1E 4.1A, 4.1B 4.2A, 4.2B 6.EE.3 Apply the properties of operations to generate equivalent expressions. (Lesson) 3.6, 4.2C 6.EE.6 6.EE.9 6.G.4 7.G.6 8.EE.1 Use variables to represent numbers and write expressions when solving realworld or mathematical problems; understand that a variable can represent an unknown number, or depending on the purpose at hand, any number in a specified set. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Know and apply the properties of integer exponents to generate equivalent numerical expressions. 8.EE.7 Solve linear equations in one variable. 4.3A 8.EE.7a Solve linear equations in one variable. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 5.2A 1.1B, 1.1C 5.4A 2.2A 1.1A, 2.2B, 2.2C 3.5B, 3.5D 6.4B Algebra Lab Gear: Algebra 1 vi Didax, Inc.

6 CCSS Standard Description Activities 8.EE.7b 8.EE.8a 8.EE.8b 8.SP.3 Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of two linear equations in two variables algeraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 1.3A, 1.3B, 3.2A, 3.2B (Lesson) 5.3 A.SSE.2 Use the structure of an expression to identify ways to rewrite it. 6.3A, 6.3B A-APR.1 F-IF.8a Not correlated 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. 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. 4.4B 5.4B 3.1D, 3.4A, 3.4B, 6.2A E 6.5B, 6.5C, 6.6A 1.1D, 1.2A, 1.2B, 2.1A, 2.1B, 3.1A, 3.1B, 3.2C, 3.3A D, 3.5A, 3.5C Didax, Inc. vii Correlation to Standards

7 SECTION 3 Multiplying, Factoring, and Dividing Polynomials The Main Concepts This section covers multiplication of polynomials and several related topics. This is the most important use of Lab Gear. We start with Make a Rectangle and Make a Box, geometric puzzles for which all students can readily understand the instructions. These serve as an introduction to the geometric representation of multiplication in the corner piece, using the formulas for area of a rectangle and volume of a box. Once this foundation has been laid, we can proceed to the introduction of the distributive property and to several topics related to multiplication: factoring, division, squaring, cubing, and simplifying fractions. In a way, factoring is the key to this section, since the initial make a rectangle problems are in fact informal factoring problems. In addition to the pedagogic reason given above for starting that way, there is another more fundamental reason: a full understanding of multiplication and the distributive property is not possible without some conceptual grasp of factoring. Similarly, one cannot fully understand addition without understanding reversed problems of the type find two numbers that add up to 10 or what plus 1.41 equals 10? This is the principle of reversibility. On the other hand, do not get too concerned about having your students become fast and accurate trinomial factoring experts. While factoring is important on a conceptual level, it is becoming obsolete as a skill. Like other symbolic manipulations, it will be performed primarily by electronic means in the near future. When working with the Lab Gear in this section, we do not use minus. Beginning in Lesson 3.2, we also work in the multiplication table format, what students often call the box, using both + and. This is because the corner piece model of multiplication for a multiplication like (y l)(y 2), while mathematically correct, is somewhat tricky. On the other hand, the multiplication table format readily generalizes to minus. We tackle the modeling of minus in the corner piece in Section 6. Pacing and Selection There is a lot in this section, and you should not expect to do it all in a few days. Once you have decided how much of it you want to cover, plan to alternate between one week of concentrated effort on one lesson and a few activities from this book and a week of other work. Doing the work in this section will take time, but because your students will have a more solid understanding, you will need less drill than you may have assigned in the past on topics such as the distributive property and factoring. In any case, be sure to do Lessons 3.1 and 3.2 and their activities early in the year. Didax, Inc. 33 Section 3

8 SECTION 3 (continued) The following lessons and activities are essential: Lesson 3.1, Activities 3.1A and 3.1D Lesson 3.2, Activities 3.2A, and 3.2B Lesson 3.3, Activities 3.3A, 3.3B, and 3.3C Lesson 3.4, Activity 3.4C Lesson 3.5, Activities 3.5B and 3.5C The Transition to Symbolic Algebra The general progression within each topic area in this section is: 1. Build with the Lab Gear. 2. Sketch. 3. Use the multiplication table format. Students are often asked to sketch corner piece figures for the various types of problems in the activities. By encouraging them to use pictorial representations as a substitute for work with the blocks, you help them make the transition from the concrete environment of the Lab Gear to the multiplication table format and the world of symbols. About the Lessons Lesson 3.1, The Geometry of Multiplication. This lesson introduces the geometric model for multiplication, laying the groundwork for the whole section. Lesson 3.2, The Distributive Property. This lesson helps students generalize what they learn from Lab Gear multiplication, so that they can correctly multiply using symbols. Mathematically, the distributive property is the central structure underlying the entire section. Lesson 3.3, Factoring. This lesson focuses on the concept of factoring, with a particular emphasis on the inverse relationship with distributing. Lesson 3.4, Dividing by a Monomial. This lesson is about the distributive property of division over addition and subtraction. Lesson 3.5, Perfect Square Trinomials. This lesson introduces the identity for the square of a sum, which becomes important for completing the square in the context of quadratic equations and functions. Lesson 3.6, Multiplying, Factoring, Dividing. This lesson summarizes the whole section. Algebra Lab Gear: Algebra 1 34 Didax, Inc.

9 Lesson 3.1 The Geometry of Multiplication The purpose of this lesson is to model multiplication and factoring with the Lab Gear and to introduce the corresponding symbolic notation. Parentheses x(x + 6) = x 2 + 6x Ask the students to make a rectangle with x 2 and 6x, and then to measure it in the corner piece. Ask the students to write an equation in the form length width = area to describe the figure. Tell the students that beginners in algebra will often write this multiplication x x + 6 or x + 6 x. However, mathematicians agree that multiplication comes before addition, so that the first expression means multiply x by x, then add 6 and the second expression means multiply 6 by x, and add that to x. Build these two expressions with the Lab Gear on the screen. x x + 6 x + 6x Explain the correct use of parentheses to write the original multiplication: x(x + 6) or (x + 6)x. Make a Rectangle 2(2x + y) = 4x + 2y Ask students to make a rectangle with 4x + 2y, and to write a length width = area equation for the rectangle. Show that 2 is a common dimension of the subrectangles made by 4x and 2y. Say that 2 is called a common factor of each of the original two terms 4x and 2y. A factor of a monomial (or term) is a monomial that divides it evenly. 2 2x = 4x, so 2 is a factor of 4x, and 2 y = 2y, so 2 is a factor of 2y. If rectangles have a common dimension, their areas have a common factor. Didax, Inc. 35 Lesson 3.1

10 Lesson 3.1 (continued) Ask students to make rectangles with each of the following collections of blocks. In each case, they should write the length width = area equation, and identify the common factor. 3x y + 8 5x x + x 2 + xy 5x + 2x 2 (Answers: 3(x + 5), 4(y + 2), 5(x + 4), x(3 + x + y), x(5 + 2x)) Discuss the two alternate solutions for 2x 2 + 4x (common factor x, or common factor 2x) Make a Box 2 x (x + 2) Now show how the common factor 2 in 2x 2 + 4x leads to a box. If the three dimensions are measured separately, the box can be described by an equation in the form length width height = volume, 2 x (x + 2) = 2x 2 + 4x. Describe how the collection of blocks can be arranged into a box and measured in three more ways: 2(x 2 + 2x), x(2x + 4), and 2x(x + 2). These three arrangements show the common factors of 2, x, and 2x respectively. Geometrically, they represent the multiplication area of base times height equals volume. 2(x 2 + 2x) x(2x + 4) 2x(x + 2) It is difficult to demonstrate the above example on the screen, since it involves three-dimensional arrangements. You might have the students gather around a central desk or table for a demonstration with the student blocks. Or you can show the process to representatives of groups, and have them show their neighbors. Have students make boxes (or rectangles) for each of the following collections of blocks. In each case they should write one or more multiplications and identify the common factors. Encourage them to compare answers with one another, or to look for alternate solutions because each collection has many different solutions. (Do not worry at this stage about which version is the most factored. This is a level of sophistication most students are probably not ready for.) 12x x 2 + 6x 2xy + l0x These require 3-D blocks: x 3 + 2x 2 + x 2 y 2x 2 y + 4xy Algebra Lab Gear: Algebra 1 36 Didax, Inc.

11 Lesson 3.1 (continued) Note that when multiplying in the corner piece, the dimension of the product is the sum of the dimensions of the factors. It is not crucial for students to understand this fact, but many find it interesting. You may lead a discussion of it by showing that the x 3 block, for example, can be shown as the product of three x s, each of which is one-dimensional, or as the product of a one-dimensional x and a two-dimensional x 2. Some students may build it as x 2 x 2, but that is one dimension too many. Wrap up this lesson by asking students if they have found ways to look carefully at the terms of an expression and find the common factor and the multiplication without the Lab Gear. Even though not all the students may see this yet, it is important to begin this discussion because it lays the groundwork for the introduction of the distributive property in the next lesson. About the Activities Activity 3.1A, Make a Rectangle. These geometric puzzles provide practice in writing multiplication equations of the form length width = area. A wrap-up of the completed activity may include a discussion of the common factor in each exercise. Activity 3.1B, Make a Box. These geometric puzzles provide practice in writing multiplication equations of the form length width height = volume. Encourage students to build boxes rather than rectangles in this activity. Activity 3.1C, Make a Box Using the 3-D Blocks. If you have the 3-D blocks in your classroom, use this activity to reinforce the connection between geometric volume and algebraic multiplication equations. Activity 3.1D, Multiplying in the Corner Piece. In this activity, students build multiplication problems with the Lab Gear blocks and the corner piece, and then record the product. This work prepares students for the discussion in Lesson 3.2 about the distributive property and symbolic multiplication. Activity 3.1E, Multiplication Using the 3-D Blocks. This activity is similar to the preceding one, but requires the 3-D blocks. Didax, Inc. 37 Lesson 3.1

12 Activity 3.1A Make a Rectangle For each of the polynomials shown below: a. Build the polynomial and arrange the blocks into a rectangle. b. Sketch the rectangle in the corner piece. c. Write an equation of the form length width = area. 1. x 2 + 3x 2. 5x + 5y Equation: 3. 3y + 6x Equation: 4. 4x y Equation: 5. xy + x 2 + 2x Equation: 6. 4y + 8x + 20 Equation: Equation: Algebra Lab Gear: Algebra 1 38 Didax, Inc.

13 Activity 3.1B Make a Box For each of the polynomials shown below: a. Build the polynomial and arrange the blocks into a rectangle. b. Make a 3-D sketch of the box. c. Complete the equation volume = length width height. 1. 4y + 4 = 2. 2xy + 2y = 3. 6y + 18 = 4. 2xy + 2x + 2x 2 = 5. 4x 2 + 8x = = Find two different ways. Didax, Inc. 39 Algebra Lab Gear: Algebra 1

14 Activity 3.1C Make a Box Using 3-D Blocks For each of the polynomials shown below: a. Build the polynomial and arrange the blocks into a box. b. Make a 3-D sketch of the box. c. Complete the equation volume = length width height. 1. x 2 y + 4xy = 2. x 3 + 5x 2 = 3. xy 2 + x 2 y + xy + x 2 = 4. y 3 + x 2 y + 2xy 2 = 5. 2x 2 y + 2xy 2 = = = Find three different ways. Algebra Lab Gear: Algebra 1 40 Didax, Inc.