Multiplying and dividing integers Student Activity Sheet 2; use with Exploring Patterns of multiplying and dividing integers

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1 1. Study the multiplication problems in the tables. [EX2, page 1] Factors Product Factors Product (4)(3) 12 ( 4)(3) 12 (4)(2) 8 ( 4)(2) 8 (4)(1) 4 ( 4)(1) 4 (4)(0) 0 ( 4)(0) 0 (4)( 1) 4 ( 4)( 1) 4 (4)( 2) 8 ( 4)( 2) 8 (4)( 3) 12 ( 4)( 3) 12 (4)( 4) 16 ( 4)( 4) 16 What patterns do you see in the signs of the factors that help you predict when the product is negative and when it is positive? Answers will vary. They should give the idea that multiplying two positive factors results in a positive product, multiplying a positive factor by a negative factor results in a negative product, and multiplying two negative factors results in a positive product. 2. Complete the statements to generalize the patterns you found for integer multiplication. [EX2, page 2] The product of two positive factors is always positive. The product of one negative factor and one positive factor is always negative. The product of two negative factors is always positive. Page 1 of 7

2 3. Complete the table to review properties of operations. Use the given answer choices. [EX2, page 3] (a b) c = a (b c) 3 + (4 + 5) = (3 + 4) + 5 a b = b a a + ( a) = ( a) + a = ( 5) = ( 5) + 5 = 0 3 (4 5) = (3 4) 5 a + b = b + a 3(4 + 5) = REINFORCE Practice applying the properties of operations by completing the table. Original expression Equivalent expression Properties applied Commutative Property of Addition 4(3 + ( 2)) Distributive Property of Multiplication over Addition (-16 14) (-3) 16 (14 ( 3)) Associative Property of Multiplication 10(4 + 5) or 10( 4 + ( 5)) Distributive Property of Multiplication over Addition Page 2 of 7

3 5. Apply the properties of operations to multiplication of integers. [EX2, page 4] a. Consider the equation a + ( 1) = 0. What must be the value of a? What property of operations can be used to prove your conjecture? For the equation to be true, the value of a must be (+1). Student explanations may vary; for example, if the sum of two integers is 0, the two integers are additive inverses. b. Consider the equation ( 1) ( 1) + ( 1) = 0. How can this equation be true? Can you use the properties of operations to prove your conjecture? Student explanations may vary. Sample response: I can show the equation is true by proving that the term ( 1)( 1) is equal to 1. Each term in the equation has a common factor of 1, so, I can apply the Distributive Property of Multiplication over Addition to prove that ( 1)( 1) = Complete the table to show what you have learned about multiplying integers. Use the answer choices provided. [EX2, page 5] 72 Negative Positive Factor Factor Product Positive Negative Negative Negative Negative Positive Page 3 of 7

4 7. Use the patterns you discovered to make predictions about the sign of a product of more than two factors. Complete the statements. [EX2, page 6] a. When there are two negative factors, the product will be positive. b. When there are three negative factors, the product will be negative. c. When there are five negative factors, the product will be negative. 8. State a rule to predict the sign of the product of any number of negative factors. [EX2, page 6] When there is an even number of negative factors, the product is always positive. When there is an odd number of negative factors, the product is always negative. 9. Complete the table to indicate whether the product of each set of factors will be positive or negative. [EX2, page 7] Expression ( 5)(6)( 2)( 4) Sign of result Negative (3)( 2)(4)( 2)(5)( 1) Negative (2)( 3)( 1)( 4)( 2)(2) Positive 10. REINFORCE Find the value of the variable that makes each equation true. a. 3 7 = r r = 21 b. 9 2 = w w = 18 c. 9 p = 18 w = 2 d. 3 k 4 = 36 k = 3 Page 4 of 7

5 11. Maria builds a table using her computer s spreadsheet program. [EX2, page 8] a. The computer shows an error when she tries to divide by zero. Is it possible to divide a number by zero? If so, how? If not, why not? Student responses will vary. When you divide, you find the number of groups. It is not possible to divide something into zero groups. b. Fill in the blanks to complete the statements. Page 5 of 7

6 c. Now study the rest of the table that Maria created. What sign is the quotient when the signs of the dividend and divisor are different? What sign is the quotient when the signs of the dividend and divisor are the same? When the signs of the dividend and divisor are different, the sign of the quotient is negative. When the signs of the dividend and divisor are the same, the sign of the quotient is positive. 12. Complete the statements to generalize the patterns you saw in the division problems on Maria s computer screen in question 11. [EX2, page 10] a. If the dividend is positive and the divisor is negative, then the quotient is always negative. b. If the dividend is positive and the divisor is positive, then the quotient is always positive. c. If the dividend is negative and the divisor is negative, then the quotient is always positive. d. If the dividend is negative and the divisor is positive, then the quotient is always negative. e. If the dividend and the divisor are different signs, the quotient is always negative. Page 6 of 7

7 13. Complete the table to show your understanding of dividing positive and negative integers. Use the answer choices provided. [EX2, page 11] Positive 10 5 Negative 5 10 Dividend Divisor Quotient Positive Negative Negative Negative Negative Positive Page 7 of 7

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