1 Grade Level: 7-9 (Set up for eighth grade) Time Span: 5 Day Unit Tools/Technology: Algebra Tiles, Overhead
2 Day 1 Day 2 Day 3 Day 4 Day 5 Introduction to Subtraction of Multiplication Division of Introduction to Algebra Tiles, And Addition And Subtraction of Polynomials Using Algebra Tiles Of Polynomials Using Algebra Tiles Polynomials Using Algebra Tiles Factoring Trinomials Using Algebra Tiles Polynomials Materials: Algebra Tiles, Overhead, Product Mat (can Draw) Resources: Algebra tile Investigations - McDougal Littell Algebra 1 Concepts and Skills - pg. 36-47 Algebra Tiles Workbook Learning Resources pg. 3-37
3 Day 1 (Introduction to Algebra Tiles and Addition of Polynomials) Goal: To help students understand the process of addition of Polynomials by the use of Algebra Tiles. Student Learning Objectives: Students will become acquainted with Algebra tiles Students will be able to create physical models representing polynomials using Algebra tiles and use these to perform operations on the polynomials Students will become familiar with the Zero pair Students will discover the necessary skills and procedure used to add polynomials with integer coefficients When students have completed this lesson they will be able to use physical models to add polynomials and also be able to add polynomials without the use of Algebra Tiles. Student Pre-Knowledge: Students have some knowledge and experience with Integers Students have worked with inverse and additive inverse Students can perform basic operations and have basic algebraic skills Materials: Overhead, Algebra Tiles, Chalk Board, Writing Materials NYS Standards Addressed:
4 Algebra Strand: (8.A.5) Use physical models to perform operations with polynomials (8.A.7) Add and Subtract polynomials (integer coefficients) Problem Solving Strand: (8.PS.1) Use a variety of strategies to understand new mathematical content and to develop more efficient methods (8.PS.4) Observe patterns and formulate generalizations Reasoning and Proof Strand: (8.RP.2) Use mathematical strategies to reach a conclusion Communication Strand: (8.CM.4) Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form (8.CM.5) Answer clarifying questions from others (if necessary) NCTM Standards: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Model and solve problems using various representations, such as graphs, tables, and equations Create and Use representations to organize, record and communicate mathematical ideas Warm-up (Anticipatory Set):
5 Introduce students to the algebra tiles Show students what each tile represents and why X 1 I.e. X, X, 1 1, first has an area of X^2, and the second has an area of X and the last one has and area of 1 Introduce the students to the idea of Blue tiles representing Positive and 1 1 1 1 Red tiles representing Negative, 1 1 X X X X X X, Review with students what happens if you add a number with its additive inverse or opposite, this will help them understand the zero pair rule when working with the algebra tiles Developmental Activities: I will start the lesson by supplying the students with examples of how to set up simple expressions using the tiles on the overhead - 4X^2 =
6-3X = - -2X = - X+2 = - X^2-4 = - X^2+8X = - -3X^2+2X = - 8X^2+6X+5 = - 2X^2-X+6 = Next I will put a few problems on the board or overhead and ask students to use their Algebra Tiles to model the polynomials, I will give the students some time and move about the room working with students. I will continue this practice until the students are comfortable with the process.
7-4x - 3X^2-3X^2-X-3-2X^2+5X+1 - -X^2+2X Next I will model a few problems on the board or overhead using the Algebra Tiles and ask the students to write on their paper the polynomial that the Algebra Tiles represent, this will ensure that the students are truly comfortable working with the tiles and can work both forwards and backwards When I feel that the students are comfortable and capable of moving on I will talk about the principle of the Zero Pair - I will tell students that when tiles of the exact same dimension and value are presented together yet one is positive and the other is negative, they cancel each other out and form a model of zero. (I m assuming that the students have had previous experience working with integers and understand the concept of additive inverse or opposite) - Using the algebra tiles I will show students that there is many ways to represent the Zero pair. = 0, = 0,, Etc After the students understand the zero pair I will introduce them to a simple addition problem and tell them that when adding polynomials, you
8 employ the Zero pair wherever its possible and then combine tiles that represent like terms - (X^2+3) + (X^2+1) = 2X^2+4: + = - (X^2+2X+3) + (2x^2+X-1) = 3X^2+3X+2: + = Pair) - (3X^2-5X+4) + (2X-6) = 3X^2-3X-2: (Note The Zero + = Pairs) - (X^2+9X-2) + (-2X^2-7X+3) = -X^2+2X+1: (Note the Zero
9 + = After the students have proven their ability to use the principles and techniques I have taught them I will try to get the student to notice any patterns that developed as we were solving the problems. Hopefully the students will see the result of adding the polynomials is simply adding the coefficients of like terms and combining for the answer. If the students are having a hard time coming up with the answer or rule for adding polynomial I will line up the equations on the board that they have already solved using their Algebra Tiles and when they see the answer below they might have a better understanding of the rule. Ex. X^2 + 2X + 3 + 2X^2 + 1X 1 3X^2 + 3X + 2 We will discuss the discovery as a class and talk about how this rule is used when adding polynomials We will solve some problems on the board using no tiles at all and observe how the rule we discovered easily works when adding polynomials Assessment:
10 I will check for understanding of the material presented by monitoring students progress and by asking questions and observing the students work as I move about the room I will ask specific questions that deal with the subject to be sure that students comprehend the material Students successful completion of practice problems Closure: For a closure of the lesson I will go over important information discovered by the class from the use of the Algebra Tiles. I will call upon students to restate the procedure to use when working on adding polynomials. I will review or restate that when adding polynomials you can simply perform the operation on like terms and combine to formulate the answer. At this point students should be comfortable adding polynomials without the use of the Tiles. After I have went through the review for the closure I will have the students create a problem of their own and group together with a partner to solve each others problem, after they have done this they are to explain the problem back to the person who created it and compare answers. This will ensure that the students understand the concept and can communicate their ideas and techniques to their peers.
11 Day 2 (Subtraction of Polynomials using Algebra Tiles) Goal: To help students understand the process of subtraction of Polynomials by the use of Algebra Tiles. Student Learning Objectives: Students will be able to create physical models representing polynomials using Algebra tiles and perform subtraction on the polynomials Students will use the Zero principle to model subtraction of polynomials Students will discover the necessary skills and procedures used to subtract polynomials with integer coefficients When students have completed this lesson they will be able to use physical models to perform subtraction of polynomials and also be able to subtract polynomials without the use of Algebra Tiles. Student Pre-Knowledge: Students have had previous experience with addition of polynomials using Algebra Tiles Students have been introduced to the zero pair Materials: Overhead, Algebra Tiles, Chalk Board, and Writing Materials NYS Standards Addressed: Algebra Strand: (8.A.5) Use physical models to perform operations with polynomials
12 (8.A.7) Add and Subtract polynomials (integer coefficients) Problem Solving Strand: (8.PS.1) Use a variety of strategies to understand new mathematical content and to develop more efficient methods (8.PS.4) Observe patterns and formulate generalizations Reasoning and Proof Strand: (8.RP.2) Use mathematical strategies to reach a conclusion Communication Strand: (8.CM.4) Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form (8.CM.5) Answer clarifying questions from others (if necessary) NCTM Standards: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Model and solve problems using various representations, such as graphs, tables, and equations Create and Use representations to organize, record and communicate mathematical ideas Warm-up (Anticipatory Set): Review the conclusion s of yesterday s lesson and remodel a few problems quick on the board
13 Discuss with the class how they think subtraction could be modeled using the tiles I will tell the students that when subtracting polynomials, you take away tiles that represent the subtrahend or bottom portion of the equation, and discuss with the class if they think that this would lead to a similar rule for subtraction Developmental Activities: I will start the lesson by supplying the students with examples of how to set up simple subtraction problems using the tiles on the overhead, noting that you have to take away the tiles that represent the subtrahend or bottom portion of the equation - 2X^2-5x + 3 - (X^2-2x + 1) X^2-3x + 2 = - 3X^2-7X + 4 - (X^2-2X + 2) 2X^2-5X + 2 =
14-4X^2-2X + 3 - (2X^2-1X + 2) 2X^2 X + 1 = Next I will explain that the take away method works well when there are tiles readily available to take away. When there aren t enough of the right tiles to take away, students can apply the zero principal and then subtract. In this problem there are no X tiles to take away, only X tiles. However, you can provide as many X tiles as you need by applying the zero principle. Since you need two X tiles, you can represent the problem like this - 2x^2-7X+4 - (X^2+2X+2) X^2-9X + 2 Zero Pair Zero Pair Now, you can take away the tiles that represent the subtrahend X^2 + 2X + 2 and count the remaining tiles to determine the difference.
15 Zero Pair Zero Pair = Once the students are confident with this technique I will have them solve some of the following equations, and again look for a pattern or rule that they can apply to use without the Algebra Tiles, again noticing that all they have to do is subtract the numbers or coefficients in front of the variables. The students should have an easier time understanding and coming up with this after already observing a similar rule with addition. If they cannot I will align the equation up so that the students can see the answer they got and this will help them see the rule or pattern we developed. - X^2-4X+6 - (X^2+1X+3) -5X - 3 Zero Pair =
16-2X^2+2X-5 - (-X^2+3X-2) 3X^2 X 3 Zero Pair Zero Pair = For Homework students will complete a worksheet involving the concepts learned over the past two days. Students will perform operations of addition and subtraction on polynomials, drawing their Algebra Tiles to help them solve their equations and using the zero pair when necessary. Students will verify their answers by using the rule they discovered to complete the problems without the Tiles Assessment: I will check for understanding of the material presented by monitoring students progress and by asking questions and observing the students work as I move about the room I will ask specific questions that deal with the subject to be sure that students comprehend the material Students successful completion of practice problems and the assigned homework Closure:
17 For a closure of the lesson I will go over important information discovered by the class from the use of the Algebra Tiles. I will restate the rules that we discovered as a class, which are when adding or subtracting polynomials with integer coefficients you perform the operation on like terms and combine to come up with the answer. At this point students should be comfortable working with polynomials without the use of the Tiles and can add and subtract them efficiently After I have went through the review for the closure I will have the students create a problem of their own involving subtraction and group together with a partner to solve each others problem, after they have done this they are to explain the problem back to the person who created it and compare answers. This will ensure that the students understand the concept and can communicate their ideas and techniques to their peers.
18 Day 3 (Introduction of Multiplying Polynomials When All terms are Positive) Goal: To help students understand the process of Multiplication of Polynomials and demonstrate the Distributive Property of Multiplication by the use of Algebra Tiles. Student Learning Objectives: Students will be able to create physical models representing polynomials using Algebra tiles and perform multiplication of these polynomials Students will discover the necessary skills and procedures used to multiply polynomials with integer coefficients When students have completed this lesson they will be able to use physical models to perform multiplication of polynomials and also be able to multiply polynomials without the use of Algebra Tiles. Student Pre-Knowledge: Students have had previous experience with addition and subtraction of polynomials using Algebra Tiles Students have worked with multiplication of integer numbers Students understand and can compute and area of a quadrilateral Materials: Overhead, Algebra Tiles, Chalk Board, Writing Materials, and Product Mat NYS Standards Addressed: Algebra Strand: (8.A.5) Use physical models to perform operations with polynomials (8.A.8) Multiply a binomial by monomial or a binomial (integer coefficients)
19 Problem Solving Strand: (8.PS.1) Use a variety of strategies to understand new mathematical content and to develop more efficient methods (8.PS.4) Observe patterns and formulate generalizations Reasoning and Proof Strand: (8.RP.2) Use mathematical strategies to reach a conclusion Communication Strand: (8.CM.4) Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form (8.CM.5) Answer clarifying questions from others (if necessary) NCTM Standards: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Model and solve problems using various representations, such as graphs, tables, and equations Create and Use representations to organize, record and communicate mathematical ideas Warm-up (Anticipatory Set): Review the conclusion of yesterday s lesson, and remodel a few problems quick on the board. Restate the procedures that we have discovered for addition and subtraction of polynomials
20 Go over yesterdays homework and collect, and clarify any questions that the students might have Introduce the students to the Product Mat, and hand out worksheet copies of these mats, or have students draw their own The Product map is an organizational tool that allows students to develop rectangles by their length and width. While the task can be accomplished without a Product Mat, students may have a more difficult time deciding which tiles are being used Tell students that the quantity represented by a rectangular array of tiles represents the area of the array. Using tiles to multiply polynomials builds on the basic concept of multiplication. The tile s dimensions and why, are known from the first lesson where I introduced them to the tiles. Thus the kids understand that each dimension is known from computing the area with known or given lengths for the tiles I will explain that when using the Product Mat students should put one expression on the top of the Product Mat and the other expression on the side of the Product Mat Developmental Activities: I will start the lesson by supplying the students with examples of how to set up multiplication problems and complete them, by using the tiles and
21 the mat on the overhead, noting that you place one expression on the top of the mat and the other expression on the side of the mat Dimensions X (X + 4) = X^2 + 4X = Total Area (Note how the array demonstrates the Distributive Property of Multiplication) To Multiply 2X by (X+7), the first dimension 2X is shown on the vertical axis of the Product Mat. The second dimension, X+7, is shown on the horizontal axis of the Product Mat. When the dimensions are multiplied, they form the area, which is represented inside the Product Mat. For example, the area of 2X (X+7) = 2X^2 + 14X - I will have students use their Product Mats to create an array of tiles that represent the dimensions. By counting X^2 tiles and the
22 X tiles in the array, you can see that 2X (X+7) = 2X^2 + 14X. Then after they have attempted this by their selves I will reveal the answer or complete picture Next I will create another example for the product of (X + 2)(X + 3), using the Product Mat on the overhead to form the rectangular array. X + 3 X + 2 ----> X (X + 3) ----> 2 (X + 3) - The rectangle formed is equal to (X^2 + 5X + 4). Note that the Distributive Property is used twice, which implies the foil method. Also, notice how the model illustrates traditional multiplication After I went over the examples I will have students form small groups of two or three to practice the concept and complete a few problems together. The students will use the tiles to model the operations. I will walk around and observe the student s techniques and answer or give hints to any questions presented. - (X + 1)(X + 4)
23 - (X + 2)(X + 5) - 3X (X + 2) - X (5X + 6) - (3X + 3)(2X + 1) After students have completed these problems I will have a couple of students come present them on the board and we will talk about them as a class. Once again I will inform the students to look for a generalization or a rule/pattern we could use to perform multiplication without the Tiles. Students may have a hard time coming up with the rule by themselves even though they can successfully complete the exercise, therefore I will go over the process and model how to do the problems without the Tiles, explaining the Distributive Property, this will help them understand what they are doing with the tiles and also help them with their homework. Also I will talk about F.O.I.L (Distributive Property twice) and what it represents and how this process is what we model in some of the problems - Model without Tiles first and then with Tiles and notice pattern: 2X (4X + 6) X (5X + 1) (X + 6)(3X + 1) (X + 9)(X + 7) For Homework students will complete a worksheet involving the concepts learned today. Students will complete multiplication problems drawing their Algebra Tiles to help them solve their equations. Students will verify
24 their answers by using the rule they discovered and we discussed to complete the problems without the Tiles Assessment: I will check for understanding of the material presented by monitoring students progress and by asking questions and observing the students work as I move about the room I will ask specific questions that deal with the subject to be sure that students comprehend the material and have the students restate in their own words what they have discovered Students successful completion of practice problems and the assigned homework Closure: For a closure of the lesson I will go over important information discovered by the class from the use of the Algebra Tiles. I will restate the rules that we discovered as a class, which are the Distributive Property and the F.O.I.L method. After I have went through the review for the closure I will have the students create a problem of their own involving any multiplication problem of their choice and group together with a partner to solve each others problem, after they have done this they are to explain the problem back to the person who created it and compare answers. This will ensure
25 that the students understand the concept and can communicate their ideas and techniques to their peers.
26 Day 4 (Introduction to Dividing Trinomials) Goal: To help students understand the process of dividing trinomials Student Learning Objectives: Students will be able to create physical models representing polynomials using Algebra tiles and perform division of these polynomials Students will discover the necessary skills and procedures used to divide polynomials with integer coefficients When students have completed this lesson they will be able to use physical models to perform division of polynomials and also be able to divide polynomials without the use of Algebra Tiles. Student Pre-Knowledge: Students have had previous experience with addition, subtraction and multiplication of polynomials using Algebra Tiles Students have worked with multiplication of integer numbers Students understand and can compute and area of a quadrilateral Materials: Overhead, Algebra Tiles, Chalk Board, Writing Materials, and Product Mat NYS Standards Addressed: Algebra Strand: (8.A.5) Use physical models to perform operations with polynomials
27 (8.A.9) Divide a polynomial by a monomial (integer coefficients) The degree of the denominator is less than or equal to the degree of the numerator for all variables Problem Solving Strand: (8.PS.1) Use a variety of strategies to understand new mathematical content and to develop more efficient methods (8.PS.4) Observe patterns and formulate generalizations (8.PS.9) Work backwards from a solution (8.PS.11) Work in collaboration with others to solve problems Reasoning and Proof Strand: (8.RP.2) Use mathematical strategies to reach a conclusion Communication Strand: (8.CM.4) Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form (8.CM.5) Answer clarifying questions from others (if necessary) (8.CM.7) Compare strategies used and solutions found by others in relation to their own work NCTM Standards: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Model and solve problems using various representations, such as graphs, tables, and equations
28 Create and Use representations to organize, record and communicate mathematical ideas Warm-up (Anticipatory Set): Review the conclusion of yesterday s lesson, and remodel a few problems quick on the board. Restate the procedures that we have discovered for multiplication of polynomials, talk about the distributive property and F.O.I.L method Go over yesterdays homework and collect, and clarify any questions that the students might have Hand out worksheet copies of Product mats, or have students draw their own Developmental Activities: I will start the lesson by supplying the students with examples of how to set up division problems and complete them by using the tiles and the mat on the overhead I will explain that in division, the area of the product rectangle and one of its dimensions are given, and you must determine the other dimension To model (X^2 + 6X + 8) (X + 4), you must arrange six X tiles and eight 1 tiles in a rectangular array that has (X + 4) as one of its dimensions
29 Arrange these tiles into a rectangular array that has (X + 4) as one of its dimensions.? X + 4 From the students experience with modeling multiplication, they saw that a rectangular array can be formed by placing the X^2 tile at the upper left corner of the array, the 1 tiles at the lower right and the X tiles at the lower left, Thus (X^2 + 6X + 8) (X + 4) = X + 2
30 X + 2 X + 4 I will then have the students use their Algebra Tiles to model the following: - (X^2+7X+12) (X + 3) = - (X^2 + 6X + 5) (X + 1) = After I give the students time to complete these problems I will demonstrate them on the board or overhead while the students follow along and guide me to the answer. Next I will continue the lesson by switching around the operation. I will ask the students to use their Algebra Tiles to help them determine the length and width if given the area. - Ex. Area = The students will arrange the tiles in their Product Mat so they can see the dimensions. The students will show that the length and width can be portrayed
31 Or X (X + 2) or (X + 2) X Next the students will do one of these problems on their own. I will display the following tiles and ask student to derive the length and width Area = 4X + X^2 + 3 Ans. = Dimensions = (X + 3)(X + 1) The students will work in small groups of two or three to solve and model some problems as I rotate around the room answering questions and aiding in problems. Students will be on the look out for a new pattern or rule they can use without having tiles to represent the problem - X^2 + 5X + 6-4X^2 + 4X - X^2 + 3X + 2 - X^2 + 5X + 4
32 For homework student will have to complete a worksheet involving finding the other dimension (length or width) of each rectangle and also problems where they use their Algebra Tiles and Product Mat to determine the length, width, and area of the rectangles formed by the tile groupings given a specific amount of tiles Assessment: I will check for understanding of the material presented by monitoring students progress and by asking questions and observing the students work as I move about the room I will ask specific questions that deal with the subject to be sure that students comprehend the material Students successful completion of practice problems and the assigned homework Closure: For a closure I will discuss and model with the class how the tiles help represent the procedure we are really doing when performing these operations with polynomials We will discuss a generalization of how to divide a polynomial by a monomial (integer coefficients) The degree of the denominator is less than or equal to the degree of the numerator for all variables I will answer any last minute questions about the homework or material covered
33 Day 5 (Introduction to factoring Trinomials) Goal: To investigate and help students recognize through the use of Algebra Tiles whether or not a trinomial of the form ax^2 + bx + c, where a, b, and c are positive integers can be factored, and if so to determine the factors. Student Learning Objectives: Students will be able to create physical models representing polynomials using Algebra tiles and factor these polynomials with the use of tiles Students will discover that not all trinomial with integer coefficients can be factored When students have completed this lesson they will be able to use physical models to factor polynomials and have a better understanding of a polynomial and its factors Student Pre-Knowledge: Students have had previous experience with addition, subtraction, multiplication, and division of polynomials using Algebra Tiles Students have worked with Product Mats Materials: Overhead, Algebra Tiles, Chalk Board, Writing Materials, and Product Mat NYS Standards Addressed: Algebra Strand: (8.A.5) Use physical models to perform operations with polynomials (8.A.11) Factor a trinomial in the form ax^2 + bx + c; a=1 and c having no more that three factors
34 Problem Solving Strand: (8.PS.1) Use a variety of strategies to understand new mathematical content and to develop more efficient methods (8.PS.4) Observe patterns and formulate generalizations (8.PS.11) Work in collaboration with others to solve problems Reasoning and Proof Strand: (8.RP.2) Use mathematical strategies to reach a conclusion Communication Strand: (8.CM.4) Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form (8.CM.5) Answer clarifying questions from others (if necessary) (8.CM.7) Compare strategies used and solutions found by others in relation to their own work NCTM Standards: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others Model and solve problems using various representations, such as graphs, tables, and equations Create and Use representations to organize, record and communicate mathematical ideas Warm-up (Anticipatory Set):
35 Review the conclusion of yesterday s lesson, and remodel a few problems quick on the board. Restate the procedures that we have discovered for division of polynomials Go over yesterdays homework and collect, and clarify any questions that the students might have Developmental Activities: I will start the lesson by modeling a few problems for them on the overhead and talk about what we were doing in yesterday s lesson (Since factoring with Algebra Tiles is similar to division with Algebra Tiles the students will not have a difficult time with this introduction to factoring) I will show students the following model X^2 + 5X + 4 - Next I will rearrange the Algebra Tiles so that a rectangle is formed. I will show the students that either one of the following models accomplishes this task
36 Next I will discuss the concept that X^2 + 5X + 4 can be factored as (X +4)(X + 1) or (X + 1)(X + 4) Next I will show the students the following model for X^2 + 2X + 2 And ask them to arrange these in the usual fashion with the X^2 in the top and with the X s in the bottom left or top right and the 1 s where they belong. I will ask the students for suggestions of ways to rearrange the tiles to form our rectangle as before. The students will find that there is no such arrangement and I will discuss the fact that not all trinomials of the form ax^2 + bx + c with integer coefficients can be factored. I will check for student s understanding by repeating the process for the following trinomials - 2X^2+5X+2
37-3X^2+4X+5 The students will work in small groups for the rest of class while trying to complete a work sheet of related problems using the tiles to determine the factors if possible and stating which one can t be factored, the students will finish the problems for homework Assessment: I will check for understanding of the material presented by monitoring students progress and by asking questions and observing the students work as I move about the room I will ask specific questions that deal with the subject to be sure that students comprehend the material Students successful completion of practice problems and the assigned homework worksheets Closure: For a closure I will discuss and model with the class how the tiles help represent the procedure we will be doing when we learn more about factoring trinomials We will talk about how factoring with the Algebra Tiles is similar to Division with Algebra Tiles and how this might help us when factoring trinomials
38 I will answer any last minute questions about the homework or material covered
39 Worksheet for Day 1 & 2 (Addition and Subtraction) Directions: Draw Models and find the correct answers to the following problems 1. 2X^2-2X + 6 9. -3X^2-2X + 4 + (-X^2 1X - 2) - (X^2 + X + 2) 2. X + 2 10. 4X - 3 + (-X - 2) - (X + 2) 3. 2X^2 + 4 11. 4X^2 + 6 + (2X^2 2) - (2X^2 + 2) 4. 3X^2-5X 1 12. 4X^2 3X - 6 + (2X^2 X + 4) - (-2X^2 + X + 4) 5. 3X^2-4X + 1 13. 6X^2-3X + 4 + (-2X^2 1X - 3) - (2X^2 + X + 3) 6. 4X^2-3X + 1 14. 2X^2-2X + 1 + (-3X^2 1X - 1) - (3X^2 + X + 1) 7. 3X^2 4 15. 3X + 4 + (-3X^2-2) - (-2X - 3) 8. 3X + 4 16. 5X^2-4 + (2X + 3) - (3X^2 + 2)
40 Worksheet for Day 3 (multiplication) Directions: Multiply the polynomials below by drawing a product map and creating a model for the problem. 1. X (X + 1) 9. (X + 2)(X + 4) 2. (X + 1)(X + 3) 10. X (2X + 5) 3. (X + 2)(X + 2) 11. (X + 1)(2X + 2) 4. (X + 2)(X + 1) 12. (2X + 3)(X + 2) 5. 3X (X + 1) 13. (3X + 1)(X + 1) 6. X (3X + 2) 14. (2X + 2)(X + 1) 7. X (X + 4) 15. (X + 3)(3X + 4) 8. (X + 1)(X + 5) 16. (2X + 5)(X + 2)
41 Worksheet for Day 4 (Division) Directions: Draw your algebra tiles and product mat and find the missing dimension (length or width) of the rectangle to divide these trinomials 1. (X^2 + 4X) (X + 4) 2. (X^2 + 6X + 5) (X + 1) 3. (X^2 + 6X + 8) (X + 2) 4. (2X^2 + 5X) X 5. (2X^2 + 4X + 2) (2X + 2) 6. (2X^2 + 7X + 6) (X + 2) 7. (3X^2 + 4X + 1) (X + 1) 8. (2X^2 + 4X + 2) (2X + 2) 9. (3X^2 + 13X + 12) (X + 3) 10. (2X^2 + 9X + 10) (X + 2)
42 Worksheet for Day 5 (factoring trinomials) Directions: Complete the following questions drawing tiles to help find the dimensions and factor if possible 1. Use Algebra Tiles to find the dimensions of the rectangle with an area of 4X^2 + 8X +3 2. The Area of a rectangle is given by A = X^2 +6X +5. The width of the rectangle is X+5. Use Algebra Tile to find the length 3. The area of a rectangle is given by A=2X^2 + 11X + 15. The length of the rectangle is X + 3. Use algebra tiles to find the width 4. Use Algebra Tiles to factor the trinomial X^2 + 8X +16 5. Use Algebra Tiles to factor the trinomial x^2 + 4X +8 6. Use Algebra Tiles to factor the trinomial 3X^2 + 8X + 4 7. Use Algebra Tiles to factor the trinomial, if possible: X^2 + 8X + 7 8. Use Algebra Tiles to factor the trinomial, if possible: X^2 +7X +12 9. Use Algebra Tiles to factor the trinomial, if possible: 2X^2 + 9X +4
43 Answer sheet for Day 1 & 2 (Addition and Subtraction) Directions: Draw Models and find the correct answers to the following problems - Compare student s answer to the correct answer and look to see if drawing is correct. 1. 2X^2-2X + 6 9. -3X^2-2X + 4 + (-X^2 1X - 2) - (X^2 + X + 2) X^2 3X + 4-4X 3 + 2 2. X + 2 10. 4X - 3 + (-X - 2) - (X + 2) 0 3X - 5 3. 2X^2 + 4 11. 4X^2 + 6 + (2X^2 2) - (2X^2 + 2) 4X^2 + 2 2X^2 + 4 4. 3X^2-5X 1 12. 4X^2 3X - 6 + (2X^2 X + 4) - (-2X^2 + X + 4) -X^2 6X + 3-2X^2 4X - 10 5. 3X^2-4X + 1 13. 6X^2-3X + 4 + (-2X^2 1X - 3) - (2X^2 + X + 3) -5X^2 5X + 2-8X^2 4X + 1 6. 4X^2-3X + 1 14. 2X^2-2X + 1 + (-3X^2 1X - 1) - (3X^2 + X + 1) -7X^2 4X -5X^2 3X 7. 3X^2 4 15. 3X + 4 + (-3X^2-2) - (-2X - 3) -6X^2 6 5X + 7 8. 3X + 4 16. 5X^2-4 + (2X + 3) - (3X^2 + 2) 5X + 7-8X^2-6
44 Answer sheet for Day 3 (multiplication) Directions: Multiply the polynomials below by drawing a product map and creating a model for the problem. - Compare student s answer to the correct answer and look to see if the product mat is correct. 1. X (X + 1) = X^2 + X 9. (X + 2)(X + 4) = X^2 + 6X + 8 2. (X + 1)(X + 3) = X^2 + 4X + 3 10. X (2X + 5) = 2X^2 + 5X 3. (X + 2)(X + 2) = X^2 + 4X + 4 11. (X + 1)(2X + 2) = 2X^2 + 4X +2 4. (X + 2)(X + 1) =X^2 + 3X + 2 12. (2X + 3)(X + 2) = 2X^2 + 7X +6 5. 3X (X + 1) = 3X^2 + 3X 13. (3X + 1)(X + 1) = 3X^2 + 4X +1 6. X (3X + 2) = 3X^2 + 2X 14. (2X + 2)(X + 1) = 2X^2 + 4X +2 7. X (X + 4) = X^2 + 4X 15. (X + 3)(3X + 4)= 3X^2+13X+12 8. (X + 1)(X + 5) = X^2 + 6X + 5 16. (2X + 5)(X + 2) = 2X^2 +9X+10
45 Answer sheet for Day 4 (Division) Directions: Draw your algebra tiles and product mat and find the missing dimension (length or width) of the rectangle to divide these trinomials - Compare student s answer to the correct answer and look to see if the product mat diagram is correct. 1. (X^2 + 4X) (X + 4) = X 2. (X^2 + 6X + 5) (X + 1) = X+5 3. (X^2 + 6X + 8) (X + 2) = X+4 4. (2X^2 + 5X) (X) = 2X+5 5. (2X^2 + 4X + 2) (2X + 2) = X+1 6. (2X^2 + 7X + 6) (X + 2) = 2X+3 7. (3X^2 + 4X + 1) (X + 1) = 3X+1 8. (2X^2 + 4X + 2) (2X + 2) = X+1 9. (3X^2 + 13X + 12) (X + 3) = 3X+4 10. (2X^2 + 9X + 10) (X + 2) = 2X+5
46 Answer sheet for Day 5 (factoring trinomials) Directions: Complete the following questions drawing tiles to help find the dimensions and find the factors if possible (Compare students answers and look over Algebra Tiles) 1. Use Algebra Tiles to find the dimensions of the rectangle with an area of 4X^2 + 8X +3 Ans. = (2X + 1)(2X + 3) 2. The Area of a rectangle is given by A = X^2 +6X +5. The width of the rectangle is X+5. Use Algebra Tiles to find the length Ans. = (X + 1) 3. The area of a rectangle is given by A=2X^2 + 11X + 15. The length of the rectangle is X + 3. Use algebra tiles to find the width Ans. = (2X + 5) 4. Use Algebra Tiles to factor the trinomial X^2 + 8X +16 Ans. = (X + 4)(X + 4) 5. Use Algebra Tiles to factor the trinomial X^2 + 4X +8 Ans. = Cant be factored 6. Use Algebra Tiles to factor the trinomial 3X^2 + 8X + 4 Ans. = (X + 2)(3X + 2) 7. Use Algebra Tiles to factor the trinomial, if possible: X^2 + 8X + 7 Ans. = (X +1)(X+7) 8. Use Algebra Tiles to factor the trinomial, if possible: X^2 +7X +12 Ans. = (X+3)(X+4) 9. Use Algebra Tiles to factor the trinomial, if possible: 2X^2 + 9X +4 Ans. = (X + 4)(2X + 1)