Name Class Date. Associative Property: When you add, you can group any of the numbers. When you multiply, you can group any of the numbers.
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1 Name Class Date Properties of Numbers Extension: Equivalent Expressions Essential question: How do you write equivalent expressions using properties? Properties of Numbers Commutative Property: You can add numbers in any order. You can also multiply numbers in any order. 1-2 video tutor Associative Property: When you add, you can group any of the numbers. When you multiply, you can group any of the numbers. Identity Property: The sum of 0 and any number is that number. The product of 1 and any number is that number. You can use properties of numbers to write equivalent expressions. Equivalent numerical expressions have the same value when they are simplified. 1 CC.7.EE.3 EXPLORE Transforming an Expression Amber practiced playing the trumpet for 2 times per day for 20 minutes each time. She practiced 5 days per week for 6 weeks. One expression for the total number of minutes Amber practiced is An equivalent expression is A Complete the table to transform the first expression into the second expression. For each step, you are allowed to switch any two adjacent factors. The first step has been done for you. Step Expression Starting expression B What property did you use to transform the expression? TRY THIS! Use a property of numbers to write an equivalent expression. Tell which property you used. 1a b. (15 8) 5 Chapter 1 9 Lesson 2
2 REFLECT 1c. Describe two ways you can show that a pair of numerical expressions are equivalent. 2 CC.7.NS.1d EXPLORE Expressions with Multiplication and Addition Three friends went bowling. They each paid $2 to rent shoes and $4 to rent a bowling lane. A One expression for the total amount, in dollars, that the friends spent is 3(2 + 4). Use the order of operations to simplify this expression. 3(2 + 4) = 3( ) = The total amount the friends spent is $. B Jack says that the expression (3 2) + (3 4) can be used to find the total amount, in dollars, that the friends spent. Use the order of operations to simplify this expression. (3 2) + (3 4) = + (3 4) = 6 + = Does Jack s expression give the correct value for the total? C Are the expressions in A and B equivalent? How do you know? D Conjecture How can you write the product of a number and a sum as the sum of two products? TRY THIS! Write each expression as the sum of two products. 2a. 4(8 + 5) 2b. 7(10 + 6) 2c. 6(40 + 2) Chapter 1 10 Lesson 2
3 The Distributive Property states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding. 2(10 + 8) = (2 10) + (2 8) The Distributive Property also applies to multiplication and subtraction. You can use the Distributive Property to help you multiply numbers mentally. 3 CC.7.NS.1d e XAMPLE Applying the Distributive Property A middle school has 8 buses. Each bus can carry 44 students. Use mental math to find the greatest number of students the school s buses can carry. A Method 1: 8(44) = 8(40 + 4) Write 44 as a sum. B Method 2: = (8 ) + ( 4) Use the Distributive Property. = Multiply. = Add. 8(44) = 8(50 6) Write 44 as a difference. = (8 ) ( 6) Use the Distributive Property. = 400 Multiply. = Subtract. TRY THIS! The greatest number of students the school s buses can carry is Use the Distributive Property to write an equivalent expression for each product. Then simplify the expression. 3a. 7(19) 3b. 6(82) 3c. 5(45) REFLECT 3d. In A, the larger factor 44 was written as a sum. Why do you think 44 was written as a sum instead of the smaller factor 8? Chapter 1 11 Lesson 2
4 practice 1. This weekend, Tamara biked for 42 minutes, ran for 36 minutes, and walked for 28 minutes. The expression ( ) + 28 represents the number of minutes she exercised this weekend. Complete the table to transform this expression into 36 + ( ). For each step, list the property you used. Step Expression Property Starting expression ( ) + 28 Starting expression 1 2 Use the Distributive Property to write each expression as the sum or difference of two products. 2. 8( ) 3. 7(30-6) 4. 4(42 + 5) Use the Distributive Property to write an equivalent expression for each product. Then simplify the expression. 5. 6(53) 6. 9(94) 7. 3(49) 8. An office manager buys 9 boxes of pens. Each box holds 6 black pens and 4 blue pens. The expression 9(6 + 4) represents the total number of pens. Use the Distributive Property to write an equivalent expression for the total number of pens. 9. To thaw a frozen turkey in the refrigerator, it takes about 6 hours per pound. The expression 6(18) represents the time, in hours, to thaw an 18-pound turkey. Explain how to use the Distributive Property and mental math to find the time needed to thaw the turkey. 10. What if...? What if the frozen turkey weighed 14 pounds? Find the time needed to thaw a 14-pound turkey in the refrigerator. 11. Is the expression 8(12 + 7) equivalent to ? Explain how you know. Chapter 1 12 Lesson 2
5 Name Class Date 1-2 Additional Practice Tell which property is represented (17 36) 4 17 (36 + 4) (12 5) Simplify each expression. Justify each step (33 45) 7 (33 45) 7 Use the Distributive Property to find each product. 7. 3(26) 8. (18)9 Chapter 1 13 Practice and Problem Solving
6 Problem Solving Problem Solving: Properties of Numbers Write the correct answer. 1. Jo makes and sells jewelry. She sold three bracelets for $45, $17, and $25. Write an expression for the total Jo received. Explain how you can use properties and mental math to simplify the expression. 2. Use parentheses to show two ways of grouping the numbers in Tell which expression you think would be easier to simplify, and why. Then simplify the expression. 3. The distance from Mark s apartment to his job is 27 miles. Mark works 5 days per week. How many miles does Mark drive to and from work each week? 4. Jane said that 6(64) 6(50) 6(14). Is she correct? Use the Distributive Property to explain your answer. Choose the letter for the best answer. 5. Maxine works 8 hours at a rate of $16 per hour. Which expression could not be used to find her total earnings in dollars? A 8 (10 6) B 8 (20 4) C 8 (10 6) D 8 (8 8) 7. Which of the following represents the Identity Property? A (8 4) 3 8 (4 3) B C D 6(26) 6(20) 6(6) 6. Rosemary runs 16 miles on Friday, 8 miles on Saturday, and 14 miles on Sunday. How many miles does Rosemary run in all? F 22 mi G 24 mi H 30 mi J 38 mi 8. Which of the following shows how the Distributive Property could be used to simplify 7(28)? F G 7 (20 8) H 7 (20 8) J (7 20) 8 Chapter 1 14 Practice and Problem Solving
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