Writing Products as Sums and Sums as Products

Examples, videos, and solutions to help Grade 7 students learn how to use an area/rectangular array model and distributive property to write products as sums and sums as products.

New York State Common Core Math Grade 7, Module 3, Lesson 3

Download worksheets for Grade 7, Module 3, Lesson 3

Lesson 3 Student Outcomes

• Students use an area/rectangular array model and distributive property to write products as sums and sums as products.
• Students use the fact that the opposite of a number is the same as multiplying by to write the opposite of a sum in standard form.
• Students recognize that rewriting an expression in a different form can shed light on the problem and how the quantities in it are related.

Lesson 3 Closing

• What are some of the methods used to write products as sums? We used repeated use of the distributive property and rectangular arrays
• In terms of a rectangular array and equivalent expressions, what does the product form represent, and what does the sum form represent? The total area represents the expression written in sum form, and the length and width represent the expressions written in product form.

Opening Exercise

Solve the problem using a tape diagram. A sum of money was shared between George and Brian in a ratio of 3:4. If the sum of money was $56.00, how much did George get?

Example 1
Represent 3 + 2 using squares for units.
Represent using the same size square for a unit as above.
Draw a rectangular array for 3(3 + 2).
Draw an array for 3(x + 2).

Exercise 1
Fill in the blanks.

Example 3
Find an equivalent expression by modeling with a rectangular array and applying the distributive property 5(8x + 3).

Exercise 2
For parts (a) and (b), draw a model for each expression and apply the distributive property to expand each expression. Substitute the given numerical values to demonstrate equivalency.
a. 2(x + 1), x = 5
b. 10(2c + 5), c = 1
For parts (c) and (d), apply the distributive property. Substitute the given numerical values to demonstrate equivalency.
c. 3(4f - 1), f = 2
d. 9(-3r - 11), r = 10

Example 4
Rewrite the expression, (6x + 15) ÷ 3, as a sum using the distributive property.

Exercise 3
Rewrite the expressions as a sum.
e. (2b + 12) ÷ 2
f. (20r - 8) ÷ 4
g. (49g - 7) ÷ 7

Example 5
Expand the expression 4(x + y + z)

Exercise 4
Expand the expression from a product to a sum so as to remove grouping symbols using an area model and the repeated use of distributive property: 5(x + 2y + 5z)

Example 6
A square fountain area with side length is bordered by a single row of square tiles as shown. Express the total number of tiles needed in terms of three different ways.

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