1. The Distributive Property
Section 1-7 Part 2

2. Goals
Goal
Rubric
To use the Distributive Property to simplify expressions.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Vocabulary
Term
Constant
Coefficient
Like Terms

4. The Distributive Property
The process of distributing the number on the outside of the parentheses to each term on the inside.
a(b + c) = ab + ac and (b + c) a = ba + ca
a(b - c) = ab - ac and (b - c) a = ba - ca
Example
5(x + 7)
5 ∙ x + 5 ∙ 7
5x + 35

5. Geometric Model for Distributive Property4 5 2
Two ways to find the area of the rectangle.
As a whole
As two parts

6. Geometric Model for Distributive Property4 5 2
Two ways to find the area of the rectangle.
As a whole
As two parts
same

7. Find the area of the rectangle in terms of x, y and z in two different ways.
As a whole
As two parts

8. x y z As a whole As two parts same xy + xz
Your Turn: Find the area of the rectangle in terms of x, y and z in two different ways. x y z
As a whole
As two parts
same
xy + xz

9. Example: Distributive Property with Mental Math
You can use the distributive property and mental math to make calculations easier.
Write the product using the Distributive Property. Then simplify.
5(59)
5(50 + 9)
Rewrite 59 as
5(50) + 5(9)
Use the Distributive Property.
Multiply.
295
Add.

10. Example: Distributive Property with Mental Math
Write the product using the Distributive Property. Then simplify.
9(48)
9(50 - 2)
Rewrite 48 as
9(50) - 9(2)
Use the Distributive Property.
Multiply.
432
Subtract.

11. Your Turn:
Write the product using the Distributive Property. Then simplify.
8(33)
8(30 + 3)
Rewrite 33 as
8(30) + 8(3)
Use the Distributive Property.
Multiply.
264
Add.

12. Your Turn:
Write the product using the Distributive Property. Then simplify.
12(98)
12(100 – 2)
Rewrite 98 as 100 – 2.
12(100) – 12(2)
Use the Distributive Property.
1200 – 24
Multiply.
1176
Subtract.

13. Your Turn:
Write the product using the Distributive Property. Then simplify.
7(34)
7(30 + 4)
Rewrite 34 as
7(30) + 7(4)
Use the Distributive Property.
Multiply.
238
Add.

14. The total cost of 6 CDs at $11.95 each is $71.70.
MENTAL MATH CALCULATIONS
You are shopping for CDs.
You want to buy six CDs
for $11.95 each.
SOLUTION
The mental math is easier if you think of $11.95 as $12.00 – $.05.
6(11.95) = 6(12 – 0.05)
Write as a difference.
= 6(12) – 6(0.05)
Use the distributive property
to calculate the total cost
mentally.
Use the distributive property.
= 72 – 0.30
Find the products mentally.
= 71.70
Find the difference mentally.
The total cost of 6 CDs at $11.95 each is $71.70.

15. Definition Term – any number that is added or subtracted. Example:
In the algebraic expression x + y, x and y are terms.
Example:
The expression x + y – 7 has 3 terms, x, y, and 7. x and y are variable terms; their values vary as x and y vary. 7 is a constant term; 7 is always 7.

16. Definition Coefficient – The numerical factor of a term. Example:
The coefficient of 3x2 is 3.

17. Definition
Like Terms – terms in which the variables and the exponents of the variables are identical.
The coefficients of like terms may be different.
Example:
3x2 and 6x2 are like terms.
ab and 3ab are like terms.
2x and 2x3 are not like terms.

18. Definition
Constant – anything that does not vary or change in value (a number).
In algebra, the numbers from arithmetic are constants.
Constants are like terms.

19. Example:
The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.
Like terms
Constant
4x – 3x + 2

20. 1x2 + 3x Example: Coefficients
A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.
Coefficients
1x2 + 3x

21. Distributive Property
Combining Like Terms
Like terms can be combined. To combine like terms, use the Distributive Property.
= 3x
Distributive Property
ax – bx = (a – b)x
Example
7x – 4x = (7 – 4)x
Notice that you can combine like terms by adding or subtracting the coefficients. Keep the variables and exponents the same.

22. Example: Combining Like Terms
Simplify the expression by combining like terms.
72p – 25p
72p – 25p
72p and 25p are like terms.
47p
Subtract the coefficients.

23. Example: Combining Like Terms
Simplify the expression by combining like terms.
A variable without a coefficient has a coefficient of 1.
and are like terms.
Write 1 as .
Add the coefficients.

24. Example: Combining Like Terms
Simplify the expression by combining like terms.
0.5m + 2.5n
0.5m + 2.5n
0.5m and 2.5n are not like terms.
0.5m + 2.5n
Do not combine the terms.

25. Caution!
Add or subtract only the coefficients. 6.8y² – y² ≠ 6.8

26. Your Turn: Simplify by combining like terms. 3a. 16p + 84p 16p + 84p
16p + 84p are like terms.
100p
Add the coefficients.
3b. –20t – 8.5t2
–20t – 8.5t2
20t and 8.5t2 are not like terms.
–20t – 8.5t2
Do not combine the terms.
3c. 3m2 + m3
3m2 + m3
3m2 and m3 are not like terms.
3m2 + m3
Do not combine the terms.

27. x is the y is the x2 y3 y2 – x2 + 3y3 – 5 + 3 – 3x2 + 4y3 + y + – 3 y2
SIMPLIFYING BY COMBINING LIKE TERMS
Each of these terms is the product of a number and a variable.
Each of these terms is the product of a number and a variable.
terms + – 3 y2 x
variable. + – 3 y2 x
number + – 3 y2 x + – 3 y2 x
x is the
variable.
y is the
–1 is the
coefficient of x.
3 is the
coefficient of y2.
variable
power.
Like terms
Like terms have the same variable raised to the same power.
y2 – x2 + 3y3 – – 3x2 + 4y3 + y
The constant terms –5 and 3 are also like terms. x2 y3

28. 8x + 3x = (8 + 3)x = 11x 4x2 + 2 – x2 = 4x2 – x2 + 2 = 3x2 + 2
SIMPLIFYING BY COMBINING LIKE TERMS
8x + 3x =
(8 + 3)x
Use the distributive property.
= 11x
Add coefficients.
4x2 + 2 – x2 =
4x2 – x2 + 2
Group like terms.
= 3x2 + 2
Combine like terms.
3 – 2(4 + x) =
3 + (–2)(4 + x)
Rewrite as addition expression.
= 3 + [(–2)(4) + (–2)(x)]
Distribute the –2.
= 3 + (–8) + (–2x)
Multiply.
= –5 + (–2x)
Combine like terms
and simplify.
= –5 – 2x

29. Your Turn: Simplify −12x – 5x + 3a + x. Justify each step. Procedure
Justification 1.
–12x – 5x + 3a + x 2.
–12x – 5x + x + 3a
Commutative Property 3.
–16x + 3a
Combine like terms.

30. Your Turn: Statements Justification 1. 14x + 4(2 + x) 2.
Simplify 14x + 4(2 + x). Justify each step.
Statements
Justification 1.
14x + 4(2 + x) 2.
14x + 4(2) + 4(x)
Distributive Property 3.
14x x
Multiply. 4.
14x + 4x + 8
Commutative Property of Addition 5.
(14x + 4x) + 8
Associative Property of Addition 6.
18x + 8
Combine like terms.

31. Joke Time How does the moon get a haircut? Eclipse it.
What’s black & white, black & white, black & white, black & white, black & white, black & white?
A penguin rolling down a hill.
What do you call a fake noodle?
An imPASTA.

32. Assignment
1.7 Exercises Pg. 50: #10 – 64 even