Polynomial Multiplication Investigation
Mathematical Big Idea and Major Topic: Expand polynomial expressions
More specific statement of the topic: Multiplying/expanding two polynomials
What the students already know:
Context:
Area of a rectangle
Area model of multiplication
Combining like terms
Properties of exponents: multiplying with like bases
Distributive Property
Finding the area of rectangles to help with distribution of polynomials
What part will the students have to figure out on their own? Partial products (areas), process of
combining like terms, multiplying higher
degree polynomials
Context:
Using an area model to represent multiplying polynomial expressions
Statement of Task: Given a diagram of rectangular garden plot, determine the area of the plot.
Where they might get stuck: Multiplying binomial expressions, using variables in the multiplication
process
What questions can I use to prompt them without giving away answers?
How does the number of terms in the factors relate to the number of partial products?
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network.
DuPage Regional Office of Education. May 2007
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Problem 1
a) Find the area of the rectangular the garden:
b) Find the product 7 • 10
10
7
Problem 2
a) Find the area of each rectangular garden:
10
c) Find product 7 • 15
5
7
b) Now, find the area of entire rectangle:
Area = 7 ( ____ + ____ )
Area = 7 ( ____ ) + 7 ( ____ )
Area = ____ + ____
Area = ____
d) What do you notice about the area of the entire rectangular garden and the product 7 • 15?
e) Compare the process you used to find the area to the process you used to multiply 7 • 15.
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network.
DuPage Regional Office of Education. May 2007
2
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network.
DuPage Regional Office of Education. May 2007
3
Problem 3
a) Referring to Problem 2, use the same process to write an expression to find the area of each
rectangular garden:
x
5
7
b) Referring to Problem 2, use the same process to write an expression to find the area of the entire
garden.
c) Referring to Problem 2, use the same process to find the product 7(x + 5)
Problem 4
a) Referring to Problems 2 and 3, use the same process to write an expression to find the area of each
rectangular garden:
x
12
x
b) Referring to Problems 2 and 3, use the same process to write an expression to find the area of the
entire garden.
c) Referring to Problems 2 and 3, find the product x(x + 12)
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network.
DuPage Regional Office of Education. May 2007
4
Problem 5
10
5
a) Referring to Problem 2, use the same process to write an
expression to find the area of the top portion of the garden.
7
b) Referring to Problem 2, use the same process to write an
expression to find the area of the bottom portion of the garden.
6
c) How could you use your answers to parts a) and b) to find the area of the entire garden?
d) How does this relate to finding the product of 13 • 15?
Problem 6
a) Referring to Problem 5, use the same process to write an expression to find the area of each
rectangular garden:
x
3
x
8
b) Referring to Problem 5, use the same process to write an expression to find the area of the entire
garden.
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network.
DuPage Regional Office of Education. May 2007
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c) Referring to Problem 5, use the same process to find the product of (x + 3)(x + 8).
Questions
1. How would you explain this process of multiplying a monomial by a binomial to a student who was
absent yesterday?
2. How would you explain this process of multiplying a binomial by a binomial to a student who was
absent yesterday?
Challenge Yourself With This Problem:
a) Find the area of the large rectangle:
x2
3x
4
x
8
b) Find the product of (x+8)( x2 + 3x + 4)
Improving Student Achievement in Algebra, Grades 7-12. The DuPage1 Mathematics Network.
DuPage Regional Office of Education. May 2007
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