1. Warm Up
Evaluate.
1. 10 · · 10
2. 15 ·
3. ( ) + ( )‏
4. 4(32) – 16(8)
140
196
108

2. Learn to apply properties of numbers and to find counterexamples.

4. Vocabulary
conjecture
counterexample

5. Equivalent expressions have the same value, no matter which numbers are substituted for the variables.
Ex.
a+6 is equivalent to 2+a+4 because
2+a+4
a+(2+4)
a+6
Reading Math

6. Additional Example 1A: Identifying Equivalent Expressions
Use properties to determine whether the expressions are equivalent.
7 · x · 6 and 13x
7 · x · 6 = 7 · 6 · x
Use the Commutative Property.
= (7 · 6) · x
Use the Associative Property.
= 42x
Follow the order of operations.
The expressions 7 · x · 6 and 13x are not equivalent.

7. Additional Example 1B: Identifying Equivalent Expressions
Use properties to determine whether the expressions are equivalent.
5(y – 11) and 5y – 55
Use the Distributive Property.
5(y – 11) = 5(y) – 5(11)
Follow the order of operations.
= 5y – 55
The expressions 5(y – 11) and 5y – 55 are equivalent.

8. Check It Out: Additional Example 1A
Use properties to determine whether the expressions are equivalent.
2(z + 33) and 2z + 66
Use the Distributive Property.
2(z + 33) = 2(z) + 2(33)
Follow the order of operations.
= 2z + 66
The expressions 2(z + 33) and 2z + 66 are equivalent.

9. Check It Out: Additional Example 1B
Use properties to determine whether the expressions are equivalent.
4 · x · 3 and 7x
4 · x · 3 = 4 · 3 · x
Use the Commutative Property.
= (4 · 3) · x
Use the Associative Property.
= 12x
Follow the order of operations.
The expressions 4 · x · 3 and 7x are not equivalent.

10. Additional Example 2A: Consumer Math Applications
During the last three weeks, Jay worked 26 hours, 17 hours, and 24 hours. Use properties and mental math to answer the question.
How many hours did Jay work in all?
Add to find the total.
Use the Commutative and Associative Properties to group numbers that are easy to add mentally.
( ) + 17
= 67
Jay worked 67 hours in all.

11. Additional Example 2B: Consumer Math Applications
Jay earns $7.00 per hour. How much money did he earn for the last three weeks?
7(67)‏
Multiply to find the total.
Rewrite 67 as 70 – 3 so you can use the Distributive Property to multiply mentally.
7(70 – 3)‏
7(70) – 7(3)
Multiply from left to right.
490 – 21 = 469
Subtract.
Jay made $469 for the last three weeks.

12. Check It Out: Additional Example 2A
During the last three weeks, Dosh studied 13 hours, 22 hours, and 17 hours. Use properties and mental math to answer the question.
How many hours did Dosh study in all?
Add to find the total.
Use the Commutative and Associative Properties to group umbers that are easy to add mentally.
( ) + 22
= 52
Dosh studied 52 hours in all.

13. Check It Out: Additional Example 2B
Dosh tutors students and earns $9.00 per hour. How much money does he earn if he tutors students for 21 hours a week?
9(21)‏
Multiply to find the total.
Rewrite 21 as so you can use the Distributive Property to multiply mentally.
9(20 + 1)‏
9(20) + 9(1)
Multiply from left to right.
= 189
Add.
Dosh makes $189 if he tutors for 21 hours a week.

14. A conjecture is a statement that is believed to be true
A conjecture is a statement that is believed to be true. A conjecture is based on reasoning and may be true or false. A counterexample is an example that disproves a conjecture, or shows that it is false. One counterexample is enough to disprove a conjecture.

15. Additional Example 3: Using Counterexamples
Find a counterexample to disprove the conjecture, “The product of two whole numbers is always greater than either number.”
2 · 1
2 · 1 = 2
Multiply.
The product 2 is not greater than either of the whole numbers being multiplied.

16. Check It Out: Additional Example 3
Find a counterexample to disprove the conjecture, “The product of two whole numbers is never equal to either number.”
9 · 1
9 · 1 = 9
Multiply.
The product 9 is equal to one of the whole numbers being multiplied.

17. Lesson Quizzes
Standard Lesson Quiz

18. Lesson Quiz
Use properties to determine whether the expressions are equivalent.
1. 3x – 12 and 3(x – 9) y + 0 and y + 11
3. Alan and Su Ling collected canned goods for 4 days to donate to a food bank. The number of cans collected each day was: 35, 4, 21, and 19. Use properties and mental math to answer each question.
a. How many cans did they collect in all?
b. If each can contains 2 servings, how many servings of food did Alan and Su Ling collect?
4. Find a counterexample to disprove the conjecture, “The quotient of two whole numbers is always less than either number.”