Introduction to Multiplying Binomials using Partial Products

Background info: Partial Prodcuts

Today's proof uses Algebra, and specifically squaring a binomial.

Have you ever done this:
3 x 48 = 3(40) + 3(8)
= 120 + 24
= 144

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Partial Products: taught in 4th grade

Partial Products

I can split up the number sentence for 2 Digit x 1 Digit Multiplication and complete in PARTS to get the final PRODUCT.

45x3 or 3x45

• Make a rectangle. Break up the 2 digit number into tens and ones.

• Put the one digit number on the left side.

Multiplying a 2 digit number in parts

Step 1: Break up the 2 digit number into tens and ones.

Step 2: Multiply the one digit number by each part of the 2 digit number.

Step 3: Add the products.

Multiple Choice

What goes in the grey box?

• 180

• 1,800

• 18

• 0.18

Multiple Choice

What goes in the box with the question mark?

• 40 x 5

• 30 x 6

• 6 x 40

• 46 x 5

Multiple Choice

The area model represents 43 X 56.

Which answer shows all the correct partial products?

• 20 + 24 + 150 + 18

• 200 + 240 + 150 + 18

• 2000 + 240 + 150 + 18

• 2000 + 240 + 15 + 18

Created by HISD Curriculum using Google Slides

Let’s use base ten blocks to represent 21 x 34.

Step 2:
Multiply:
10 x 10 = 100
10 x 1 = 10
1 x 1 = 1

10

10

1

10

10

10

11 1 1

Base-Ten Blocks by HISD Curriculum using 1,2,3 Math Fonts

Created by HISD Curriculum using Google Slides

Now, let’s use partial product strategy to solve 21 x 34.

Jacob unpacked 21 boxes of baseball cards to display in a sports
store. Each box had 34 cards. How many cards did Jacob unpack?

When we use partial products, we can multiply 21 x 34 by
decomposing each number. Then, add the numbers to find
the product.

Image created by HISD Curriculum using Microsoft Word

Now the same thing, but there will be variables.

Multiply (a - b)(a - b)

Multiple Choice

What is (a + b)(a+b)

• a^2 - 2ab + b^2
• a^2 + ab + b^2
• a^2 - ab + b^2
• a^2 + 2ab + b^2

Multiple Choice

What is (a+b)^2

• a^2 + 2ab + b^2
• a^2 - b^2
• a^2 - 2ab + b^2
• a^2 + 2ab - b^2

that's (a-b) times itself! So...
(a-b)²

(a-b)(a-b)=

a*a -a*b -b*a + -b*-b

a² -2ab + b²

(a-b)(a-b)=

2 Things to Remember for the proof: (put the rest in the back of your mind for Algebra 1)