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STANDARD I: The student will be able to perform basic operations on algebraic
expressions.
OBJECTIVE 3: Multiply polynomials.
ELIGIBLE CONTENT: Multiplying two quantities in parentheses may be required.
Squaring a quantity in parentheses may be required.
Adding or subtracting may be required.
Raising a quantity to a power may be required.
Fractions may be used.
Adding exponents may be required.
NUMBER OF TEST ITEMS: 4
LOCATION OF OBJECTIVE
SUBJECT WITHIN THE COURSE OF STUDY PAGE
7th Grade Math 43. Simplify and evaluate linear algebraic expressions. 59
8th Grade Math 34. Simplify and evaluate linear algebraic expressions. 67
Introduction
to Algebra 36. Simplify and combine polynomials. 77
4. Use order of operations, including exponentiation, to simplify
numeric and variable expressions. 80
20. Perform basic operations on algebraic expressions. 82
22. Know and use laws of exponents including zero and negative
Algebra I integral exponents. 82
2. Perform operations on rational variable expressions. 94
4. Perform operations involving polynomials including
polynomials with complex coefficients. 94
*Algebra II 37. Simplify expressions involving rational and irrational
with Trig. exponents. 98
*Expansion/Review Material
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PREREQUISITE SKILLS OR RELATED SKILLS FOR THIS
SUBJECT OBJECTIVE WITHIN THE COURSE OF STUDY PAGE
3. Demonstrate proficiency in the use of basic operations on whole
numbers through two-digit multipliers. 42
8. Multiply and divide fractions. 43
9. Add, subtract, and multiply decimals. 43
39. Recognize number sentences that serve as examples of
properties of numbers. 47
5th Grade Math 42. Develop an understanding of the order of operations. 47
6. Demonstrate proficiency in adding, subtracting, and multiplying
decimals. 49
15. Add and subtract integers. 50
40. Apply properties of operations to number sentences. 52
41. Demonstrate an understanding of the addition and subtraction
properties of equality. 52
42. Demonstrate an understanding of exponential notation. 53
6th Grade Math 43. Extend the understanding of the order of operations. 53
2. Add, subtract, multiply, and divide integers. 55
5. Perform basic operations on rational numbers. 55
9. Evaluate powers of whole numbers and roots of perfect squares. 55
37. Demonstrate proficiency in the use of the order of operations. 58
7th Grade Math 48. Exhibit understanding of the properties of rational numbers. 59
1. Demonstrate proficiency in performing basic operations on
rational numbers. 62
6. Apply the laws of exponents to simplify expressions containing
integral exponents. 62
14. Recognize and use absolute value of real numbers. 63
38. Demonstrate proficiency in recognizing the commutative,
associative, and identity properties. 67
th
8 Grade Math 39. Use the properties of rational numbers. 67
2. Demonstrate proficiency with operations on integers and
rational numbers. 74
3. Apply properties of real numbers. 74
4. Recognize, simplify, and use irrational numbers. 74
Introduction 9. Demonstrate proficiency in simplifying rational number
to Algebra expressions using the order of operations. 74
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PREREQUISITE SKILLS OR RELATED SKILLS FOR THIS
SUBJECT OBJECTIVE WITHIN THE COURSE OF STUDY (cont.) PAGE
12. Apply the laws of exponents to simplify expressions containing
natural number exponents. 75
13. Recognize absolute value as distance from zero on the number
Introduction to line. 75
Algebra (cont.) 33. Evaluate algebraic expressions. 77
6. Apply the number properties. 80
7. Recognize absolute value of a number as its distance from zero
Algebra I on a number line. 80
*Algebra II 5. Simplify a number within any subset of the set of complex
with Trig. numbers. 94
*Expansion/Review Material
TEACHER OVERVIEW
The FOIL method is a familiar tool often used when teaching the multiplication of polynomials.
This method ensures that all factors in the first binomial are multiplied by each factor in the
second binomial. For example,
F L
(x - 2)(x + 4)
I
O
The letters in the acronym represent the following:
F = the product of the first terms of each of the binomials
O = the product of the outer terms of each of the binomials
I = the product of the inner terms of each of the binomials
L = the product of the last terms of each of the binomials
This method yields the answer x 2 4 x 2 x 8 x 2 2 x 8.
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The activities that follow will provide a basis whereby the multiplication of polynomials can be
illustrated. All activities should be followed by specific examples of the types of problems
involved in multiplying polynomials. (Examples are included at the end of this section.)
Students should then be given problems to work on their own. These problems should
specifically target the skills required in multiplying polynomials.
ACTIVITY: The Rectangle Game x
+
Purpose: To use algebra tiles to give students concrete 3
experience in multiplying polynomials
x + 5
Materials/Equipment:
Algebra tiles
Math tiles display board
Overhead algebra tiles
Product mat
Procedure:
1. Divide the class into groups of three to four students each.
2. Give each group a random number of algebra tiles.
3. Proceed to give the tiles names that are associated with their area. The correlation will be
geometrically displayed on a poster board.
For example:
x2 x
1
-x2 -x
-1
4. Once the students know the names for the tiles, proceed with an introduction to the
multiplication of polynomials.
Example: ( x 1) ( x 2)
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5. Students must be able to visualize the process of putting the factors to be multiplied on the
outer vertical and horizontal spaces provided with the product mat. They should create the
rectangle in the innermost space.
Same length
and width
x 1 as x and
x + 1 positive
x
Same length as
x -
x, same width
2
-1 as 1, and
-1 Same length as 1, positive
same width as x, Same width as 1,
and negative same length as 1,
because the 1 on and negative
the left is negative because the 1 on the
left is negative
6. Stress that the length and width are determined by the factors on the outer horizontal and
vertical spaces.
7. Once the students have created the innermost rectangle, list the particular tiles that were used.
x -x
Zero Pair
8. Remind the students that zero pairs are formed by pairing one tile with its opposite. In the
example above, there is only one “zero pair” involving the “x” terms.
9. Zero pairs can be added or removed without changing the value of the polynomial.
Therefore, after removing the one “zero pair,” the following algebra tiles are left:
10. Thus, the final answer is x 2 x 2 .
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11. Show the actual problem that students worked on the board, making sure they understand the
procedure for doing this.
x 1 x 2
x x 2x 2
2
x2 x 2
12. Tell each group to model the following problems:
a. ( x 6) ( x 7) (Answer: x 2 13x 42 )
b. ( x 3) ( x 2) (Answer: x2 x 6 )
c. ( x 4) ( x 1) (Answer: x2 3x 4)
Question:
What was noticeable about the length and width of each rectangle?
Reading/Writing Connection:
Reading Comprehension Standard II, Objective 2: Draw conclusions.
Have students write an explanation of how the rectangle manipulative illustrates the
multiplication of polynomials.
Additional Problems:
1. Find the product: x 2x 3 (Answer: x 2 5x 6 )
2. Find the product: 2x 5x 6 (Answer: 2x 2 7x 30 )
3. Find the product: x 5x 8 (Answer: x 2 13x 40 )
( x 3) 2 x ( x 2 ) 3x 2 10x 18
4. Simplify: Answer:
2 4 4
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ADDITIONAL RESOURCES
1. www.forum.swarthmore.edu – Use this home page to search for other activities involving
multiplication of polynomials.
2. www.forum.swarthmore.edu/mathmagic
3. www.cs.uidaho.edu/~casey931/mega-math/index – This is a listing of mathematical topics
with activities included.
4. www.mste.uiuc.edu:591/mathed/completelist.html
5. www.math.upenn.edu/MathSources.html
6. Connected Mathematics, G. Lappan, J. Fey, W. Fitzgerald, S. Friel, and E. Phillips;
available from Dale Seymour Publications, 1-800-552-2259.
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