Download Multiply Polynomials * The Area Model

Multiply Polynomials –
The Area Model
Focus 9 Learning Goal – (HS.A-SSE.A.1, HS.A-SSE.A.2, HS.A-SEE.B., HS.A-APR.A.1, HS.A-APR.B.3,
HS.A-REI.B.4) = Students will factor polynomials using multiple methods, perform operations (excluding division) on
polynomials and sketch rough graphs using key features.
4
3
2
1
0
In addition to
level 3,
students make
connections to
other content
areas and/or
contextual
situations
outside of
math.
Students will factor polynomials
using multiple methods, perform
operations (excluding division) on
polynomials and sketch rough
graphs using key features.
- Factor using methods including
common factors, grouping,
difference of two squares, sum
and difference of two cubes, and
combination of methods.
- Add, subtract, and multiply
polynomials,
- Explain how the multiplicity of
the zeros provides clues as to how
the graph will behave.
- Sketch a rough graph using the
zeros and other easily identifiable
points.
Students will factor
polynomials using limited
methods, perform
operations (excluding
division) on polynomials,
and identify key features
on a graph.
- Add and subtract
polynomials.
- Multiply polynomials
using an area model.
- Factor polynomials
using an area model.
- Identify the zeros when
suitable factorizations are
available.
- Identify key features of
a graph.
Students will
have partial
success at a 2 or
3, with help.
Even with help,
the student is not
successful at the
learning goal.
An area model is a useful tool in
performing mental multiplication.
If we want to multiply 23 and 46, assign each number to a side of a rectangle.
Our goal is to calculate the area of the full rectangle by splitting the side lengths into more
manageable numbers, like multiples of 10 and single digits.
23 is the sum of 20 and 3. Subdivide the side of the rectangle.
46 is the sum of 40 and 6. Subdivide the top of the rectangle.
46
23
40
20
3
6
An area model is a useful tool in
performing mental multiplication.
Now we will fill in the area of each internal rectangle.
20(40) = 800
40
6
20(6) = 120
3(40) = 120
20
800
120
120
18
3(6) = 18
Next add up all of the areas:
800 + 120 + 120 + 18 = 1058
3
Use the area model to multiply
binomials.
If we want to multiply (x + 1)(3x – 2), assign each binomial to a side of the rectangle.
Our goal is to calculate the area of the full rectangle by splitting the side lengths into more
manageable parts.
Use “x + 1” and subdivide the side of the rectangle.
Use “3x – 2” and subdivide the top of the rectangle.
3x - 2
x+1
3x
x
1
-2
Use the area model to multiply
binomials.
Now we will fill in the area of each internal rectangle.
x(3x) = 3x2
3x
-2
x
3x2
-2x
1
3x
-2
x(-2) = -2x
1(3x) = 3x
1(-2) = -2
Next add up all of the areas:
3x2 – 2x + 3x - 2 = 3x2 + x - 2
Use the area model to multiply
binomials.
Multiply (3x + 1)(2x – 9)
3x(2x) = 6x2
2x
3x(-9) = -27x
1(2x) = 2x
3x
-9
6x2
-27x
2x
-9
1(-9) = -9
Next add up all of the areas:
6x2 – 27x + 2x - 9 = 6x2 - 25x - 9
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