1. Lesson 3: Writing Products as Sums and Sums as Products
7th Grade Module 3

2. 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?
∆’s are boxes connected, no box ‘button’ to type down here.
George: ∆∆∆
Benjamin: ∆∆∆∆ } = 56
Draw then explain that each box represents an unknown, an “x” where there are 7 of them and they are equal.
7∆’s = 56, 3∆’s = ?
56÷7=8
1∆ = 8
3∆ = 24
George received $24

3. Vocabulary
Array:
An arrangement of objects, pictures or numbers in rows and columns.

4. Example 1 Represent 3 + 2 using a tape diagram.
Represent x + 2 using a tape diagram.
∆∆∆ ∆∆ One long box divided into five boxes. Mark 3 and 2 on the bottoms with }2 }3
Also make a box, same length but two boxes. One bigger marked 3, one smaller marked 2.
X + 2 is a rectangle labeled x, longer than he two squares that represent the twos, which are the same width

5. Example 1 Draw a rectangular array for 3(3+2) Draw an array for 3(x+2)
We know how to draw 3 + 2; just stack two of them on top.
∆∆∆ ∆∆
∆∆∆ ∆∆ }3
3 2
X∆∆
X∆∆ }3 Discuss the area of 2x3 being six in both drawings. The area of the 3x3 is 9 and the 3•x is 3x.
X∆∆ you can draw the array with x as two blank boxes next to each other, one labeled x and one 2
X 2
This is a visual of the distributive property

6. Vocabulary- Key Term Distributive Property: a(b + c) = ab + ac
The distributive property can be written as the identity
a(b + c) = ab + ac
for all numbers a, b, and c.

7. Exercise 1
Determine the area of each region using the distributive property.
11•16 = 176
11•5 = 55

8. Example 2 – Draw a tape diagram to represent each expression.
(x + y) +(x + y) +(x + y)
(x + x + x) + (y + y + y)
3x + 3y
3(x + y)
∆<> ∆<> ∆<>
∆∆∆ <><><>
∆∆∆<><><> or
∆ <>
∆<>
d. 3{∆<>}
What do you notice about all three of these? They are equivalent.

9. Example 3
Find an equivalent expression by modeling with a rectangular array and applying the distributive property to the expression 5(8x + 3).
Substitute x = 2 into each expression and show they are equal.
Put a 5 to the left, 8x above the left box, 3 above the right. 40x in the left box and 15 in the right.
Both equal 95
Have students do exercise 2 2

10. Exercise 2
For parts a & b draw an array for each expression and apply the distributive property to expand each expression. Substitute the given numerical values to demonstrate equivalency.
2(x + 1), x = 5
2x + 2 12
10(2c + 5), c = 1
20c + 50 70

11. Exercise 2
For parts c and d, apply the distributive property. Substitute the given numerical values to demonstrate equivalency.
3(4f – 1), f = 2
12f – 3 21
9(–3r –11), r = 10
–27r – 99
–369

12. Example 4
Rewrite the expression (6x + 15) ÷ 3 in standard form using the distributive property.
Switch ÷ to x by multiplying by 1/3
2x + 5

13. Exercise 3 Rewrite the expressions in standard form. (2b + 12) ÷ 2
(49g – 7) ÷ 7
b + 6
5r – 2
7g – 1

14. Example 5 Expand the expression 4(x + y + z)
Put a 4 on the left, an x above the first, a y above the second, a z above the third. 4x, 4y, 4z on the inside of each box.
Expanded version is : 4x + 4y + 4z
Students do exercise 4.

15. Exercise 4
Expand the expression from a product to a sum by removing grouping symbols using an area model and the repeated use of distributive property.
Or in other words – distribute the 3 and draw a diagram of it.
3(x + 2y + 5z)
If time is short skip next exercise. Give alternate exit ticket.

16. (3 equivalent expressions)
Example 6
A square fountain area with side length s ft. is bordered by a single row of square tiles as shown. Express the total number of tiles needed in terms of s three different ways.
(3 equivalent expressions)
What if s = 4? How many tiles would you need?
20 b/c I would need 4 for each side (4•4) plus one for each corner.
What if s = 2? How many?
12. 2 for 4 sides plus four, one for each corner.
What kind of general formula can you write that will tell you how many tiles you need.
There are four sides and four corners. 4(s + 1)
4 + s + s + s + s
4s + 4
2s +2(s + 1)

17. Summary
The distributive property is used to write a product as sums. A product of something and parenthesis that has a sum in it.
We can show the distributive property with rectangular arrays. The area of the area represents the the sum form, the length and the width represent the product form.

18. Exit Ticket
Homework is Problem Set from Lesson 3.

19. Alternate Exit Ticket
6(2c + 11)
(80x – 72) ÷ 8

20. Homework Examples
4. Use the distributive property to write the products in standard form. a. 3(2x – 1) g. (40s + 100t) ÷ 10
6x – 3
g. 4s + 10t