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Polynomials
Adding, Subtracting, Multiplying and Dividing Through the Use of Algebra Tiles
8th Grade
5 Day Unit Plan
This unit plan utilizes Algebra Tiles
Shari A. Jakubowski
2
Objectives of the unit: Students will use algebra tiles to add polynomials Students will use algebra tiles to subtract polynomials Students will use algebra tiles to multiply a polynomial by a monomial Students will use algebra tiles to multiply a binomial by a binomial Students will use FOIL to multiply a binomial by a binomial Students will use algebra tiles to divide a polynomial by a monomial
New York State Standards
Students will represent and analyze algebraically a wide variety of problem solving situations.
o 8.A.5 Use physical models to perform operations with polynomials
Students will perform algebraic procedures accurately. o 8.A.6 Multiply and divide polynomials o 8.A.7 Add and subtract polynomials (integer coefficients) o 8.A.8 Multiply a binomial by a monomial or a binomial (integer
coefficient) o 8 A.9 Divide a polynomial by a monomial (integer coefficient) Note: The
degree of the denominator is less than or equal to the degree of the numerator for all variables
o NCTM Standards
Students will represent and analyze mathematical situations and structures using algebraic symbols;
Students will develop an initial conceptual understanding of different uses of variables
Resources
Amsco Publication, Integrated Mathematics-Introductory Course, Occhiogrosso, et al., chapter 13, pp565-614, copyright 1995
Materials Needed Class set of algebra tiles Algebra tiles for the overhead projector Overhead projector Guided Practice worksheets (included with plan) Independent Practice worksheets (included with plan) Textbook- Integrated Mathematics-Introductory Course (see Resources section)
Outline of Unit
Day 1 Introduction to algebra tiles, adding polynomials. Polynomials are formally defined
Day 2 Subtracting polynomials using algebra tiles
Day 3 Multiplying a polynomial by a monomial using algebra tiles
Day 4 Multiplying a binomial by a binomial. Using algebra tiles and FOIL
Day 5 Dividing a polynomial by a monomial using algebra tiles
3
Day 1- Introducing Algebra Tiles and Adding Polynomials Objectives:
1. Students will define terms such as, polynomial, monomial, binomial, and trinomial
2. Students will add polynomials with integer coefficients Outline of Activities Anticipatory Set: Teacher will begin lesson by placing a set of algebra tiles with each student. The
directions to the students will be for them to examine the tiles and begin to hypothesize what about the tiles are the same and what about the tiles are different? How might they be used in math?
Modeling/Input: (Modeling Section includes the guided notes and direct instruction portion of the lesson) Upon conclusion of student discussion, the teacher will identify the algebra tiles and
the value for each tile using transparent tiles for the overhead projector.
=2x =x =+1
=!2x =! x =!1
Algebra tiles can be used to represent many different ideas in mathematics. They can
be used to give us another way of representing expressions and equations. Algebra tiles can give us a visual and hands-on way to manipulate our work which basic paper and pencil cannot.
Today we will be using algebra tiles to begin our work with polynomials.
For guided notes using algebra tiles the teacher will be using the overhead projector and overhead algebra tiles throughout the lesson.
Guided Notes Sheet for students is located on pages 4 and 5 Teacher response for guided notes:
Polynomials and Algebra Tiles Guided Notes (Teacher) 1. Polynomials are, algebraic expressions with 1, 2, 3, or more terms.
2. A single terms such as 2
4x is called a monomial.
3. A binomial has 2 terms such as xx 252+ .
4. A trinomial has 3 terms such as 632
2!+ xx .
4
Polynomials and Algebra Tiles-Guided Notes
1. _________________________ are, algebraic expressions with 1, 2, 3, or more
terms.
2. A single terms such as 24x is called a ______________________________.
3. A _________________________ has 2 terms such as xx 25
2+ .
4. A _________________________ has 3 terms such as 632
2!+ xx .
You can use algebra tiles to represent a polynomial. Here are the tiles we will be using over the course of this unit and their values.
=2x =x =+1
=!2x =! x =!1
For example: 322
2+!= xx
• Here we see we have 2 each of the green tiles which gave us 22x .
• We also have 2 each of the red tiles which gave us x2! . • Finally, we have 3 green tiles which gave us +3.
When we add these terms together we created a polynomial, or more specifically a trinomial. It is important to remember our rules of combining like terms.
• Like terms have the same base and the same exponent. We can visualize like terms by the shapes and the colors of the algebra tiles we are using. You simply have to group the tiles according to these two characteristics to get like terms.
Let’s take a look at how to add polynomials.
5
+
542
!+ xx 3222
+! xx Given this arrangement we can use the algebra tiles to find the sum of the two polynomials we have created. Before we go any further we need to identify what a zero pair is. A zero pair is created when you have one positive and one negative tile of the same value. For example: +1 + -1 = 0 and + 2
x + - 2x =0
Now let’s look back at our first example above.
1. Start with your 2x tiles. There are no zero pairs present so we simply combine
like terms or add the tiles together to get: Step 2 Step 1 Step 3
Step 2
2. Next, we look at our x tiles. We see that we have two zero pairs and we will remove these and add what remains to our sum above.
Zero pair
Zero pair
3. Finally, we look at our +1/-1 tiles and remove any zero pairs.
4. If we look at our new arrangement of tiles in #1 we can identify our sum as: 223
2!+x
Or: 542
!+ xx + 322
2+! xx Now try some on your own!
2232
!+ nx Guided practice sheets can be found on page 6 and answers on pages 7. Teacher Note: Guided Practice Sheets are done during class after the guided notes. During guided practice the teacher interacts with the students and offers assistance when needed.
Guided Practice
6
Directions: Using your algebra tiles, find the sum of the polynomials by adding like terms. Draw a model for the set up and sum of your algebra tiles for each polynomial. 1. ( )+++ 53
2xx ( )34
2!+ xx
2. ( ) ( )32232
22 +!+!! xxxx 3. ( ) ( )2242
22!!++!! xxxx
Closure
7
Teacher will ask the students to share with their partners after working independently on their guided practice worksheets. After a brief discussion among the students the teacher will ask for volunteers to present their findings on the over-head.
Wrap-up Teacher:
• Who can explain to the class what a polynomial is? • What are three specific polynomials? • How can we use algebra tiles to find the sum of polynomials? Include zero pairs
and combining like terms in your brief description? Homework for this lesson can be found on page 8 of this document. Answer for homework can be found on pages 9 and 10. Guided Practice Answers from lesson 1: 1. ( )+++ 53
2xx ( )34
2!+ xx = 272
2++ xx
2. ( ) ( )32232
22 +!+!! xxxx = 1532
+! xx 3. ( ) ( )2242
22!!++!! xxxx = 23
2!! xx
Independent Practice Sheet 1 Adding Polynomials with Algebra Tiles
8
Name ______________________________________________ Using algebra tiles, find the sum of the following polynomials. Draw a model of each polynomial and the sum. 1. ( ) ( )1212 !++ xx 2. ( ) ( )2442
22!!+++ xxxx
3. ( ) ( )34214
22!+++!! xxxx
4. ( ) ( )724825
22 ++!+!+ xxxx 5. On the back of this sheet, explain in your own words to a friend how to use algebra tiles to add polynomials. Be sure to include zero pairs and combining like terms in your explanation.
9
Independent Practice Sheet 1 Adding Polynomials with Algebra Tiles Answer Sheet
1. ( ) ( )1212 !++ xx
+ =
2. ( ) ( )2442
22!!+++ xxxx
= +
3. ( ) ( )34214
22!+++!! xxxx
+
2332
+! xx
x4
22!x
10
4. + 14
2!+! xx
5. On the back of this sheet, explain in your own words to a friend how to use algebra tiles to add polynomials. Be sure to include zero pairs and combining like terms in your explanation. Student responses will vary.
( ) ( )72382222 ++!+!+ xxxx
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Day 2- Subtracting Polynomials Objectives:
1. Students will subtract polynomials with integer coefficients Outline of Activities: Anticipatory Set: Teacher will begin lesson by placing a set of algebra tiles with each student. The
directions to the students will be for them to examine the tiles and begin to hypothesize how algebra tiles may be used to subtract polynomials?
Review homework from previous day Modeling/Input:
• In our last lesson we were adding polynomials with respect to using zero pairs and combing like terms. We used our algebra tiles to represent the process required to do this.
• Today, we will be using our algebra tiles again, this time to subtract polynomials.
Subtracting Polynomials-Guided Notes
12
Recall:
=2x =x =+1
=!2x =! x =!1
Let us begin by representing the following two polynomials with our algebra tiles. To set-up a subtraction using algebra tiles, you need to arrange your tiles to represent the first expression in the problem.
( )5322 ++ xx -( )42 ++ xx
When you are subtracting polynomials you need to ask yourself, if I want to subtract 2x do I have enough of that tile to subtract? The same question would go for the other parts of the polynomial as well. Let’s look at another example:
122
++x
13
( ) ( )122233
22 +!!++ nxxx When we being this problem we are able to take away 2
2x during the next step we realize there are no red tiles to allow us to take away .2x! here is were we apply zero pairs to subtracting polynomials. When we add a zero pair it looks like this: Now we have x2! to take away and we continue by subtracting +1 and our final answer looks like: Or: 233
2++ xx
- 1222
+! xx 15
2++ xx Here we need to recall our rules for adding signed numbers. When
we subtract an expression we use Stay-Switch-Switch. The top or first expression stays the same, the minus sign switches to a plus sign, and you switch each sign in the bottom or second expression. Now try some on your own! Guided practice sheet can be found on page 14 of this document. The answers to the guided practice can be found on page 15 of this document.
152
++ xx
14
Guided Practice Directions: Using algebra tiles find the difference and draw your models to represent the problem and the difference. 1. ( ) ( )232354
22 ++!++ xxxx 2. ( ) ( )13422
22!!!!+ xxxx
3. ( ) ( )32323
22 ++!!+! xxxx
15
Closure Teacher will ask the students to share with their partners after working independently on their guided practice worksheets. After a brief discussion among the students the teacher will ask for volunteers to present their findings on the over-head.
Wrap-up Teacher:
• How does subtracting polynomials vary from adding polynomials? • How can we use algebra tiles to find the difference of polynomials? Include zero
pairs and like terms in your explanation. Homework for this lesson can be found on page 16 of this document. Answer for homework can be found on pages 17 and 18. Guided Practice Answers from lesson 2: 1. ( ) ( )232354
22 ++!++ xxxx = 1222
++ xx 2. ( ) ( )13422
22!!!!+ xxxx = 15
2!+ xx
3. ( ) ( )32323
22 ++!!+! xxxx = xx 442!
16
Independent Practice Sheet 2 Subtracting Polynomials with Algebra Tiles Name ______________________________________________ Using algebra tiles, find the difference of the following polynomials. Draw a model of each polynomial and the difference. 1. ( ) ( )1334 +!+ xx 2. ( ) ( )422244
22!+!++ xxxx
3. ( ) ( )524132
22!!!!++! xxxx
4. ( ) ( )23864
22!!!!+! xxxx
5. On the back of this sheet, explain in your own words to a friend how to use algebra tiles to subtract polynomials. Be sure to include zero pairs and like terms in your explanation.
17
Independent Practice Sheet 2 Subtracting Polynomials with Algebra Tiles Answer Sheet
1. ( ) ( )1334 +!+ xx 2. ( ) ( )422244
22!+!++ xxxx
3. ( ) ( )524132
22!!!!++! xxxx
Continued on following page (18)
2+x
6222
++ xx
18
652
2++ xx
4. ( ) ( )23864
22!!!!+! xxxx
5. On the back of this sheet, explain in your own words to a friend how to use algebra tiles to subtract polynomials. Be sure to include zero pairs and like terms in your explanation. Student response will vary.
10352
+! xx
19
Day 3- Multiplying Polynomials by Monomials Objectives:
1. Students will multiply by a monomial 2. Students will apply the laws of exponents for multiplication.
Outline of Activities Anticipatory Set: Teacher will begin lesson by placing a set of algebra tiles with each student. The
directions to the students will be for them to explore how algebra tiles may be used to multiply and divide monomials?
Review homework from previous day Modeling/Input: (Modeling Section includes the guided notes and direct instruction portion of the lesson) We have worked with adding and subtracting polynomials and today we are going to
work with multiplying a polynomial by a monomial. Recall that monomials are polynomials with one term. Before we begin we need to look at our algebra tiles and more specifically their dimensions.
+x 1 + 2
x +11 x x 1 x What is the area of the following rectangle? How can we find this? + 2
x x x Height: is x Base: is x+1 x 1 Area of a rectangle = area of the + 2
x tile and the area of the x tile We know base X height will give us our area so: ( ) xxxx +=!+ 2
1 We have just multiplied a binomial by a monomial. Let’s look at another example.
20
Let us arrange (2) 2x tiles and (1) x tile.
Height is x Base is 2x+1 2x 2
x x x ( )12 +xx = xx +
22
x x 1 When you look at the math we must remember our laws of exponents for multiplication. When you multiply two expressions: Keep the bases and add the exponents If and when there are coefficients, multiply them first, then add the exponents. When we have a polynomial multiplied by a monomial [i.e. ( )12 +xx ] we distribute the monomial through the polynomial and follow the laws of exponents and signed numbers.
Now try a few on your own! Guided practice may be found on page 21. Answers to the guided practice may be found on page 22.
21
Guided Practice Directions: Summarize the result of each model 1. 2. Multiply or divide. 1. ( )123 +xx 2. ( )532
2 +xx 3. ( )xxx 24
2 +
22
Guided Practice- Answers Summarize the result of each model 1. = xx 22
2+
2. = xx +
23
Multiply or divide. 1. ( )123 +xx = xx 36
2+
2. ( )532
2 +xx = xx 1063+
3. ( )xxx 24
2 + = 2384 xx +
23
Closure Teacher will ask the students to share with their partners after working independently on their guided practice worksheets. After a brief discussion among the students the teacher will ask for volunteers to present their findings on the over-head.
Wrap-up Teacher:
• How is the Distributive Property used when multiplying a polynomial by a monomial?
• How can algebra tiles be used to help us better understand multiplying by a monomial?
Homework for this lesson comes from the textbook: page 590 #9. G-I, M-R Answer for homework can be found below. Page 590 #9. G-I, M-R g. ( ) mmmm 22
2!=!
h. ( ) qqqq 35157352!=!
i. ( ) aaaa 82422!=!
m. ( ) yyyyyy 2525
232 +!=+! n. ( ) xxxxxx ++=++ 232 5)15 o. ( ) rrrrrr !!=!!
2321
p. ( ) nnnnnn 15123543232!+=!+
q. ( ) yyyyyy 2461232232 +!=+!
r. ( ) aaaaaa 1520103425232!+=!+
Day 4- Multiplying Two Binomials
24
Objectives:
1. Students will multiply a binomial by a binomial Outline of Activities Anticipatory Set: Teacher will begin lesson by placing a set of algebra tiles with each student. The
directions to the students will be for them to arrange two binomials and explore how one would go about multiplying the two together?
Review homework from previous day Modeling/Input:
• In our last lesson we multiplied a polynomial by a monomial. Recall the laws of exponents we needed to apply.
• Today, we will be using our algebra tiles once again, this time to introduce an idea called FOIL, which will help us to multiple binomials by a binomial.
Multiplying a Binomial by a Binomial- Guided Notes
Recall:
25
=
2x =x =+1
=!2x =! x =!1
Let us look at ( )( )11 ++ xx We are going to use a different set-up then we have before and here is why. Teacher will explain how to use the following lay-out to multiply two binomials. You start by putting each binomial along the side and top of the t-bar. The first binomial is placed along the side and the second binomial is placed along the top. You then fill in the center according to the boundaries created by binomials. The rules of signed numbers determine whether each term is either positive or negative. It is important to go through and point out how each step looks and how the answer is derived.
=
Let’s try another one. Multiply ( )( )31 +!+ xx
Notice how we are still using zero pairs. Another way to multiply two binomials is to use a method called FOIL.
( )3+! x
( )1+x 322
++!= xx
( )1+x
( )1+x12
2++ xx
26
FOIL stands for First-Outer-Inner-Last…here is how it works: Take the last example we used above ( )( )31 +!+ xx In our work with out algebra tiles above the first thing we did was to multiply x by –x. When we did this we got 2
x! . Notice that x and –x are the first terms in each binomial. The next part is to multiply the outer terms of the two binomials. In this case they are x and 3. When you multiply this you get +3x. Notice that x and +3 are the outer most terms. After that, you move to multiply the inner terms, which are right next to each other and in this case are +1 and –x. When you multiply these two terms you get –x. Finally, you multiply the last terms of each binomial, and here they are +1 and +3 and you get +3. Written out you have 33
2+!+! xxx but we are not finished yet because there are like
terms that can be combined. When we combine like terms our final answer is: 32
2++! xx
Let us try to use FOIL again: ( )( )23 +! xx F 2
xxx =! O xx 22 =! I xx 33 !="! L 623 !="! To get your final expression you add all the terms you have to get:
62
!! xx Try a few on your own! Guided Practice Sheet for this lesson can be found on page 27. Answers for the guided practice sheet can be found on pages 28.
Guided Practice
( ) xxx !=!+ 32
27
Directions: Multiply the following binomials using the algebra tiles. Fill in the needed tiles to get the answer. 1. ( )( )12 ++ xx
2. ( )( )123 ++! xx
3. Use FOIL to evaluate: ( )( )122 !+! xx ( )( )110 +! xx ( )( )324 !! xx
Guided Practice-Answers
1+x
( )2+x
12 +x
( )3+! x
28
Directions: Multiply the following binomials using the algebra tiles. Fill in the needed tiles to get the answer. 1. ( )( )12 ++ xx
23
2++= xx
2. ( )( )123 ++! xx
352
2++!= xx
3. Use FOIL to evaluate the following: ( )( ) 242122
2!+!=!+! xxxx
( )( ) 109110
2!!=+! xxxx
( )( ) 6144324
2 +!=!! xxxx
1+x
( )2+x
12 +x
( )3+! x
29
Closure
Teacher will ask the students to share with their partners after working independently on their guided practice worksheets. After a brief discussion among the students the teacher will ask for volunteers to present their findings on the over-head.
Wrap-up Teacher:
• Briefly explain how to use the algebra tiles method we used today to multiply binomials. Compare this method to using FOIL.
• How can algebra tiles be used to help us better understand how to multiply a binomial by a binomial?
Homework for this lesson can be found on page 30 of this document. Answer for the homework can be found below. 1. ( )( ) 15253
2!!=!+ xxxx
2. ( )( ) 8624
2 ++=++ xxxx 3. ( )( ) 321284
2 +!=!! xxxx 4. ( )( ) 232212
2 ++=+! xxxx 5. ( )( ) 1432722
2!!=+!!! xxxx
6. Explain to a friend in your own words how to use the FOIL method to multiply two binomials. Use the back of the sheet for more room. Student responses will vary.
30
Independent Practice Sheet 3 Multiplying a Binomial by a Binomial/FOIL Name ________________________________________ Multiply by using FOIL. Algebra tiles may be used to check your answer. 1. ( )( )53 !+ xx 2. ( )( )23 ++ xx 3. ( )( )84 !! xx 4. ( )( )212 +! xx 5. ( )( )722 +!!! xx 6. Explain to a friend in your own words how to use the FOIL method to multiply two binomials. Use the back of the sheet for more room.
31
Day 5- Dividing a Polynomial by a Monomial Objectives:
1. Students will divide a polynomial by a monomial Outline of Activities Anticipatory Set: Teacher will begin lesson by placing a set of algebra tiles with each student. The
directions to the students will be for them to explore how to arrange a polynomial so that all parts are divided equally.
Review Homework from previous day Modeling/Input:
• When we work with multiplying polynomials we need to observe the laws of exponents and the same goes for dividing.
• Today we will be using our algebra tiles to see how division applies to polynomials.
32
Dividing a Polynomial by a Binomial - Guided Notes Recall:
=2x =x =+1
=!2x =! x =!1
Look at the following arrangement of algebra tiles: This set of algebra tiles represents 963
2++ xx
If we are asked to divide this polynomial by 3 how can you show that with the algebra tiles? This arrangement shows groups containing (1) 2
x , (2) x2 , and (3) +1 tiles. This can also be shown using the expression:
323
9
3
6
3
3
3
963 2
22
++=++=++
xxxxxx
Here we must recall the laws of exponents for division: Keep the bases and subtract the exponents If and when there are coefficients, divide them first, then subtract the exponents.
33
Let us look at another example together: Look at the following arrangement of algebra tiles: This set of algebra tiles represents 642
2+! xx
If we are asked to divide this polynomial by 2 how can you show that with the algebra tiles? This arrangement shows groups containing (1) 2
x , (2) - x2 , and (3) +1 tiles. This can also be shown using the expression:
322
6
2
4
2
2
2
642 2
22
+!=+!
+=+!
xxxxxx
We may also see a variable in the denominator; in this case we follow the same process:
1. Divide each numerator by the denominator 2. Subtract exponents of like bases.
For example:
xxxxxxx
xxx++=!
"
#$%
&+!
"
#$%
&+!
"
#$%
&=
++ ''' 23121314
234
243
3
3
6
3
12
3
3612
Now try a few on your own! Guided practice may be found on page 34. Answers to the guided practice may be found on page 35.
34
Guided Practice
Directions: Draw algebra tiles to represent the expression then arrange into equal groups according to the problem and divide.
1. 3
63 +x 2. 2
2422
++ xx
Divide.
3. 3
362
xx +
4. 3
6182
xx +
5. x
xxx
4
128423+!
6. x
xxx
2
8416234
!+
35
Answers to guided practice sheet from page 34. Divide.
1. 23
63+=
+x
x 2. 122
242 2
2
++=++
xxxx
3. 3
362
xx + = 122+x 4.
3
6182
xx + = xx 262+
5. 324
1284 2
23
+!=+!
xxx
xxx 6. xxxx
xxx428
2
8416 23
234
!+=!+
Closure
Teacher will ask the students to share with their partners after working independently on their guided practice worksheets. After a brief discussion among the students the teacher will ask for volunteers to present their findings on the over-head.
Wrap-up Teacher:
• How do algebra tiles help us to divide a polynomial by a monomial? • Why is it important to understand the law of exponents for division prior to this
lesson? Homework for this lesson comes from the textbook: page 600 #6 A-D #8. A-D Answer for homework can be found below. #6.
a. 325
1510+=
+x
z
b. tttt
244
816 4
4
!=!
c. hhhh
!=! 2
2
45
520
d. 343
912+!=
!
!h
h
#8.
a. yyy
yy23
2
46 2
2
34
+=+
b. zz
z
zz23
3
69 2
3
45
!=!
c. xxx
xx+=
+ 2
23
24
48
d. mm
m
mm23
8
1624 3
2
35
!!=!
+
