Basic Operations with Polynomials using Algebra Tiles

A 5 Day Lesson Plan for: Pre-Algebra

8th Grade

Materials needed: Class Set of Algebra Tiles Overhead Algebra Tiles

Overhead Projector

Lesson Unit Objectives NCTM Standards: Use the associative and commutative properties of addition and multiplication and the

distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.

Relate and compare different forms of representation for a relationship. Model and solve contextualized problems using various representations, such as

graphs, tables, and equations. Use visual tools such as networks to represent and solve problems. Use geometric models to represent and explain numerical and algebraic relationships. Organize and consolidate their mathematical thinking through communication. Communicate their mathematical thinking coherently and clearly to peers, teachers,

and others. Use the language of mathematics to express mathematical ideas precisely. Recognize and use connections among mathematical ideas. Understand how mathematical ideas interconnect and build on one another to produce

a coherent whole. Create and use representations to organize, record, and communicate mathematical

ideas. Select, apply, and translate among mathematical representations to solve problems NYS Standards: 8.PS.3 Understand and demonstrate how written symbols represent mathematical

ideas. 8.PS.11 Work in collaboration with others to solve problems. 8.PS.14 Determine information required to solve the problem. 8.CM.3 Organize and accurately label work 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. 8.CM.9 Increase their use of mathematical vocabulary and language when

communicating with others. 8.CM.10 Use appropriate language, representations, and terminology when describing

objects, relationships, mathematical solutions, and rationale. 8.CM.11 Draw conclusions about mathematical ideas through decoding,

comprehension, and interpretation of mathematical visuals, symbols, and technical writing.

8.CN.1 Understand and make connections among multiple representations of the same mathematical idea.

8.CN.5 Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.

8.R.1 Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations.

8.R.2 Explain, describe, and defend mathematical ideas using representations.

8.R.4 Explain how different representations express the same relationship. 8.R.8 Use representation as a tool for exploring and understanding mathematical

ideas. 8.A.5 Use physical models to perform operations with polynomials. 8.A.6 Multiply and divide monomials. 8.A.7 Add and subtract polynomials (integer coefficients). 8.A.8 Multiply a binomial by a monomial or a binomial (integer coefficients). 8.A.9 Divide a polynomial by a monomial (integer coefficients) Note: The degree of

the denominator is less than or equal to the degree of the numerator for all variables. 8.A.11 Factor a trinomial in the form ax2 + bx + c; a=1 and c having no more than

three sets of factors. 8.A.15 Understand that numerical information can be represented in multiple ways:

arithmetically, algebraically, and graphically.

Resources Used 1. Summer 2006 I2T2 Binder. Buffalo State College. Pgs. 52 - 59. Copyright 2006. 2. “NCTM Math Standards and Expectations”. The National Council of Teachers of

Mathematics, Inc. http://standards.nctm.org/. 3. “Mathematics Core Curriculum MST Standard 3”. The University of the State of New

York, The State Education Department. http://math.buffalostate.edu/%7Ewilsondc/MED%20307/MathCore.pdf

Materials and Equipment Needed

1. A class set of “homemade” algebra tiles. 2. One set of algebra tiles for use with the overhead projector. 3. An overhead projector. 4. A class set of rulers. 5. A class set of color pencils (blue, green, yellow, and red). 6. Handouts #1 - #8. 7. Answer keys for handouts #1 - #5, #7, and #8. 8. A transparency copy of handout #6.

Lesson Unit Overview Lesson 1: 1. Introduction of Polynomials a. Monomials, binomials, and trinomials. b. The constant. C. Practice identifying expressions and finding the constant in an expression. 2. Introduction of Algebra Tiles a. Explain what each algebra tile represents. b. Explain each tile has a value based on its area. c. Explain “negative” tiles: negative tiles have the same dimensions, but are red instead. d. Practice modeling with the tiles: look at tile representations of polynomials and identify each as monomial, binomial, or trinomial. 3. Characteristics of Polynomials a. The coefficient. b. Like terms. Lesson 2: 1. Addition of Polynomials with Algebra Tiles a. Explain “zero pairs”. b. Remind students to group like terms. c. Practice problems. d. Find a pattern to do without algebra tiles. Lesson 3: 1. Subtraction of Polynomials with Algebra Tiles a. Explain that subtracting is adding the additive inverse. i. Distributive property. ii. Finding an additive inverse and practice. c. Practice problems. c. Find a pattern to do without algebra tiles. Lesson 4: 1. Multiplying Polynomials with Algebra Tiles A. Using rectangles. B. Multiply a polynomial by a monomial. I. Distributive property. Ii. Commutative property. C. Practice problems. D. Multiplying a polynomial by a polynomial. I. Combine like terms. E. Practice problems. F. Find a pattern to do without algebra tiles. Lesson 5:

1. Dividing Polynomials with Algebra Tiles a. Using rectangles: working backwards (find length and width) b. Patterns. i. Rearranging.

Lesson 1 Objectives: By the end of the day, students should be able to identify monomials, binomials, and trinomials. Students should also understand what each algebra tile represents and be able to use them to represent polynomials. Materials: A class set of algebra tiles, overhead algebra tiles, overhead projector, handout #1 and #2, answer key #1 and #2, and color pencils. Outlined Instructions: 1. Introduce the following terms: monomial, binomial, trinomial, polynomial, and

constant. - Assist students to recall these terms by dissecting the word into prefixes and the

suffix. Once words are dissected, assist students by giving examples with the same prefixes such as bicycle and tricycle. For the suffix, tell students that it means terms. * Prefixes: mono - one, bi - two, tri - three, poly - many * Suffix: nomial - terms 2. Practice identifying expressions as monomial, binomial, or trinomial and identifying

the constant in each expression. (Handout #1) 3. Pass out the algebra tiles and explain what each algebra tile represents and that the

“negative” tiles have the same dimensions, but are red instead. Also at this time, explain that each tile has a value based on its area.

Tile Dimensions Area Color

1 by 1 (1)(1) = 1 Yellow

1 by (1)(x) = x Green

x by x (x)(x) = x^2 Blue Note: “Negative” tiles have the same dimensions, but are red instead. 4. The teacher will now show students how to represent polynomials using the algebra

tiles on the overhead. Do not forget that red tiles are the negative counterparts of the positive tiles with the same dimensions. In addition, it is a good idea to use the term coefficient at this time to explain how many tiles you use. The term like terms should be used at this point, too. For instance, 2 green tiles is 2 x’s and can be written as 2x where 2 is the coefficient or 3 blue tiles is 3 x^2’s and can be written as 3x^2 where 3 is the coefficient.

- Examples the teacher can use: 3x^2, -2x^2, 2x, -4x, 5, -5, x^2 + 5x + 6, and -2x^2 + 3x - 1. - They are represented respectively in the following manner: Expression Algebra Tiles

3x^2

-2x^2

2x

-4x

5

-6

x^2 + 5x + 3

-2x^2 + 3x - 1 5. Handout #2 should now be passed out to the students to practice modeling a few

expressions on their own as the teacher walks around to monitor and assess students’ understanding and progress. Students will use colored pencils to answer the problems on the handout as they cannot glue the class set of algebra tiles to the sheet. Handout #2 will be kept in their notes for reference. Any problems they do not finish on handout #2 will be done for homework.

Handout #1

Name ______________________________________ Date _____________

Answer Key #1 Name ______________________________________ Date _____________

Handout #2

Name ______________________________________ Date _____________

Answer Key #2

Name ______________________________________ Date _____________

Lesson 2 Objectives: At the end of the lesson, students should understand the concept of “zero pairs and be able to add with and without the algebra tiles. Materials: A class set of algebra tiles, overhead algebra tiles, overhead projector, handout #3, answer key #3, and color pencils. Outlined Instruction: 1. Review the concept of adding like terms. 2. Explain the concept of “zero pairs” and that “zero pairs” can be removed from the

group without changing the value of the polynomial the tiles represent.

= x^2 - x^2 = 0

= x - x = 0

= 1 - 1 = 0 3. Now have students simplify the following polynomials by grouping like terms next to

each other, removing the “zero pairs”, and write out the resulting simplified polynomial:

a. -3x^2 + 2x + 1 and 2x^2 + x - 5 - Group like terms next to each other.

- Remove “zero pairs”

- Simplified polynomial: -x^2 + 3x - 4

b. 2x^2 + 4x + 1 and x^2 - x + 1 - Group like terms next to each other.

- Remove “zero pairs”

- Simplified polynomial: 3x^2 + 3x +2 c. -x^2 +-3x and -3x^2 + 4x + 4 - Group like terms next to each other.

- Remove “zero pairs”

- Simplified polynomial: -4x^2 + x + 4 4. At this point, ask the students if they can create a rule, so that they won’t have to use

algebra tiles all the time to add polynomials. (What I want: They will notice that if they add the numbers in front of the same term (the coefficient), they will get the correct solution.)

5. For homework, students will do handout #3.

Handout #3

Name ______________________________________ Date _____________

Answer Key #3

Name ______________________________________ Date _____________

Lesson 3 Objectives: At the end of the lesson, students should be able to use the distributive property, find the additive inverse of a polynomial and subtract polynomials with, and subtract with and without algebra tiles. Materials: A class set of algebra tiles, overhead algebra tiles, overhead projector, handout #4 and #5, answer key #4 and #5, and color pencils. Outlined Instruction: 1. Explain to students subtracting polynomials is merely finding the additive inverse of

the polynomials that will be subtracted and adding the two polynomials together. To find the additive inverse of a polynomial, all you have to do is multiply that polynomial by negative one. Use the tiles to show the students what happens to a polynomial when it is multiplied by negative one (additive inverse).

- Example: 2x^2 - 3 x + 1 additive inverse: -1(2x^2 - 3 x + 1) = -2x^2 + 3x - 1

2. Now have students practice finding the inverse of polynomials using handout #4. - At this point, ask students to find a pattern with the algebra tiles on what happens when you find an additive inverse of a polynomial. Hopefully, they will see that each term in the polynomial changes signs. Thus, each algebra tile switches to its opposite. 3. The students should be able to now subtract polynomials now. Pass out handout #5

and have them practice individually as the teacher walks around to monitor and assess students’ progress and understanding. There are problems for them to practice it with algebra tiles and for them to use the same rule they found for adding polynomials to do it without algebra tiles. Any problems students do not finish on handout #5 will be done for homework.

Handout #4

Name ______________________________________ Date _____________

Answer Key #4

Name ______________________________________ Date _____________

Handout #5

Name ______________________________________ Date _____________

Answer Key #5

Name ______________________________________ Date _____________

Lesson 4 Objectives: By the end of the day, students should be able to multiply polynomials. Materials: A class set of algebra tiles, overhead algebra tiles, overhead projector, handout #6 and #7, answer key #7, rulers, and color pencils. Outlined Instruction: 1. Introduce multiplication of polynomials by giving the students handout #6, which has an “L” bracket on it, which will help in modeling the multiplication. Show students how to use the area of a rectangle to represent the product of two polynomials. One polynomial represents the length and the other polynomial represents the width of the rectangle. We want to match the dimensions of length and width to fill in the rectangle with tiles. Review the students on what happens when multiplying with negative numbers. These rules will apply to variables as well, so algebra tiles will reflect that. Remind students that once they finish multiplying the polynomials, they must combine like terms (removing any “zero pairs”). - Example: (x + 1) (2x + 3)

- Example: (x^2 - 3x + 1) (-2x^2 - 4)

2. After showing those examples to students, remind students that the commutative

property will allow for the two polynomials to switch places in the “L” bracket. It will not change the outcome.

3. Now pass out handout #7 and have students work on a few problems on multiplying polynomials. Have students stop after problem #2.

4. Students will eventually discover a pattern, so that they will not have to use the algebra tiles. Hopefully, they will see that they are using the distributive property in multiplying each term in the polynomials.

5. The rest of handout #7 will be done for homework.

Handout #6

Name ______________________________________ Date _____________

Handout #7

Name ______________________________________ Date _____________

Answer Key #7

Name ______________________________________ Date _____________

Lesson 5 Objectives: By the end of the day, students should be able to divide polynomials. Materials: A class set of algebra tiles, overhead algebra tiles, overhead projector, handout #6 and #8, answer key #8, rulers, and color pencils. Outlined Instruction: 1. Explain to students that when dividing polynomials, it’s simply working backwards

from multiplying polynomials. I will set up in the “L” bracket the polynomial for which I want to divide into, but it must be rearranged in such a way to form one big rectangle. Then, I will set up on the left or the top of the “L” bracket the polynomial that I want to divide by. Next, I will explain that the dimensions of the polynomial that we must find (the other factor) must be such that it will provide the polynomial inside the “L” bracket. In addition, tell the students that “zero pairs” may be needed inside the “L” bracket for it to be possible to divide.

- Example: x^2 - 5x + 6 divided by x - 2

2. Since this is all about practice in rearranging the algebra tiles, I will pass out handout

#6 again and have them work on a few problems from handout #8. Problems they do not finish in class will be done for homework. Because this lesson will take much more thinking on the student’s part, I will have them work in pairs, so that they may assist each other as I walk around to assess their understanding.

Handout #8

Name ______________________________________ Date _____________

Answer Key #8

Name ______________________________________ Date _____________