1-3 Properties of Numbers Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

1-3 Properties of Numbers

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes


Warm UpSimplify.

1. 10 · 7 + 7 · 102. 15 · 9 + 61 3. (41 + 13) + (13 + 41)4. 4(32) – 16(8)

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Problem of the Day

Ms. Smith wants to buy each of her 113 students a colored marker. If the markers come in packs of 8, what is the least number of packs she could buy?

15


Learn to apply properties of numbers and to find counterexamples.



Vocabularyconjecturecounterexample


Equivalent expressions have the same value, no matter which numbers are substituted for the variables.

Reading Math


Use properties to determine whether the expressions are equivalent.

Additional Example 1A: Identifying Equivalent Expressions

7 · x · 6 and 13x

Use the Commutative Property.

Use the Associative Property.

7 · x · 6 = 7 · 6 · x

= (7 · 6) · x

= 42x Follow the order of operations.

The expressions 7 · x · 6 and 13x are not equivalent.



Additional Example 1B: Identifying Equivalent Expressions

5(y – 11) and 5y – 55

Use the Distributive Property.

5(y – 11) = 5(y) – 5(11)

= 5y – 55 Follow the order of operations.

The expressions 5(y – 11) and 5y – 55 are equivalent.



Check It Out: Additional Example 1A

2(z + 33) and 2z + 66

Use the Distributive Property.

2(z + 33) = 2(z) + 2(33)

= 2z + 66 Follow the order of operations.

The expressions 2(z + 33) and 2z + 66 are equivalent.



Check It Out: Additional Example 1B

4 · x · 3 and 7x

Use the Commutative Property.

Use the Associative Property.

4 · x · 3 = 4 · 3 · x

= (4 · 3) · x

= 12x Follow the order of operations.

The expressions 4 · x · 3 and 7x are not equivalent.


During the last three weeks, Jay worked 26 hours, 17 hours, and 24 hours. Use properties and mental math to answer the question.

Additional Example 2A: Consumer Math Applications

26 + 17 + 24 Add to find the total.

50 + 17 = 67

Use the Commutative and Associative Properties to group numbers that are easy to add mentally.

How many hours did Jay work in all?

26 + 24 + 17

(26 + 24) + 17

Jay worked 67 hours in all.


Additional Example 2B: Consumer Math Applications

7(67) Multiply to find the total.

490 – 21 = 469

Rewrite 67 as 70 – 3 so you can use the Distributive Property to multiply mentally.

Jay earns $7.00 per hour. How much money did he earn for the last three weeks?

7(70 – 3)

Multiply from left to right.7(70) – 7(3)

Jay made $469 for the last three weeks.

Subtract.


During the last three weeks, Dosh studied 13 hours, 22 hours, and 17 hours. Use properties and mental math to answer the question.

Check It Out: Additional Example 2A

13 + 22 + 17 Add to find the total.

30 + 22 = 52

Use the Commutative and Associative Properties to group umbers that are easy to add mentally.

How many hours did Dosh study in all?

13 + 17 + 22

(13 + 17) + 22

Dosh studied 52 hours in all.


Check It Out: Additional Example 2B

9(21) Multiply to find the total.

180 + 9 = 189

Rewrite 21 as 20 + 1 so you can use the Distributive Property to multiply mentally.

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(20 + 1)

Multiply from left to right.9(20) + 9(1)

Dosh makes $189 if he tutors for 21 hours a week.

Add.


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.


Find a counterexample to disprove the conjecture, “The product of two whole numbers is always greater than either number.”

Additional Example 3: Using Counterexamples

2 · 1

Multiply.2 · 1 = 2

The product 2 is not greater than either of the whole numbers being multiplied.


Find a counterexample to disprove the conjecture, “The product of two whole numbers is never equal to either number.”

Check It Out: Additional Example 3

9 · 1

Multiply.9 · 1 = 9

The product 9 is equal to one of the whole numbers being multiplied.


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


Lesson QuizUse properties to determine whether the expressions are equivalent.

1. 3x – 12 and 3(x – 9) 2. 11 + 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?

not equivalentequivalen

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4. Find a counterexample to disprove the conjecture, “The quotient of two whole numbers is always less than either number.” 2 1 = 2; the quotient 2 is not less than either of the

whole numbers.


1. Which of the following expresssions are equivalent?

A. 2x – 4 = 2(x – 4)

B. 2x – 4 = 2x – 2 + 2

C. 2x – 4 = 2x – 2 – 2

D. 2x – 4 = 2(x + 4)



2. Which of the following expresssions are equivalent?

A. 3x + 4 = 2 + 2 + 3x

B. 3x + 4 = 2 + 2 + 3 + x

C. 3x + 4 = 3(x + 4)

D. 3x + 4 = 3(x + 2)



3. Find a counterexample to disprove the conjecture, “Any number that is divisible by 2 is also divisible by 4.”

A. 20 2 = 10 and 20 4 = 5

B. 18 2 = 9 and 18 4 = 4.5

C. 20 2 = 20 and 20 4 = 80

D. 18 2 = 36 and 18 4 = 72


Documents

1-3 Properties of Numbers Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes