3 Multiplying and Dividing Integers Unit 1: Number System

Common Core Standards CCSS.Math.Content.7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. CCSS.Math.Content.7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. CCSS.Math.Content.7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts. CCSS.Math.Content.7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers.

Vocabulary/Properties Multiplication Property of Zero The product of a number and 0 is 0. (-4 x 0 = 0) Identity Property of Multiplication The product of a number and the multiplicative identity, 1, is the number. (4 x 1 = 4)

For your notebook:

Why? Why? Why? Why does a negative number times a negative number equal a positive product? Here’s one explanation: Follow the Pattern:

Why? Why? Why? Notice as one factor is multiplied by a decreasing factor, the product decreases: 3X3=9 3X2=6 3X1=3 3X0=0 3X-1=-3 3X-2=-6 3X-3=-9 So following the pattern so far it makes sense that a positive times a negative results in a negative.

Why? Why? Why? Now let’s decrease the first factor but leave the second; notice how the product changes: 3X3=9 3X2=6 3X1=3 3X0=0 3X-1=-3 3X-2=-6 3X-3=-9 3X-3=-9 2X-3=-6 1X-3=-3 0X-3=0 -1X-3=3 -2X-3=6 -3X-3=9 So following the pattern it makes sense that a negative times a negative results in a positive.

Example 1: Multiplying Integers Signs are different. –24 Answer is negative. B. –8(–5)(2) Multiply two integers. –8(–5)(2) Signs are the same. 40(2) Answer is positive. 80

Check It Out! Example 1 Multiply. A. 5(–2) Signs are different. –10 Answer is negative. B. –3(–2)(4) –3(–2)(4) Signs are the same. 6(4) Answer is positive. 6(4) Signs are the same. 24 Answer is positive.

Example 2: Finding Powers of Integers Simplify. A. (–3)4 (-3) (-3) (-3) (-3) Write as a repeated product. Multiply in groups (9)(9) 81 B. -34 Negative sign is not in parenthesis. -(3)4 = -[ (3)(3)(3)(3)] Use order of operations -81 Evaluate power. Multiply by -1

Example 3: Using Multiplication Properties Find the product. A. 6(1) Identity Property of Multiplication 6 B. –15(0) Multiplication Property of Zero

Evaluating an Expression Involving Multiplication MOVIES A stuntman working on a movie set falls from a building’s roof 90 feet above an air cushion. The expression 16t 2 90 gives the stuntman’s height (in feet) above the air cushion after t seconds. What is the height of the stuntman after 2 seconds? – + SOLUTION Evaluate the expression for the height when t 2. = 90 + 16t 2 – 16 ) 2 ( 2 Substitute 2 for t. 90 + = 16 – ) ( 4 Evaluate the power. = - 64 + 90 = 26 26 feet

Zero FOR YOUR NOTEBOOK: Dividing Integers FOR YOUR NOTEBOOK: Zero The quotient of 0 and any nonzero integer is 0. 0 = 0 0 = 0 12 -12 But you can not divide by zero. 12 = undefined

Example 4: Dividing Integers Divide. –18 2 C. Signs are different. –9 Answer is negative. –25 –5 D. Signs are the same. 5 Answer is positive.

Check It Out! Example 4 Divide. –24 3 C. Signs are different. –8 Answer is negative. –12 –2 D. Signs are the same. 6 Answer is positive.

Mean of a Data Set FOR YOUR NOTEBOOK: Mean: Find the sum of all values in a data set and divide by the number of values. Mean = Sum of values # of values

Example 5: Finding Mean (Multiple Choice Practice) One of the coldest places on Earth is a Russian town located near the Arctic Circle. To the nearest degree, what is the mean of the average high temperatures shown in the table for winter in the Russian town? Winter Temperatures Month Dec Jan Feb Mar Average high 41°F – 40°F – 48°F – 18°F – 147°F – 48°F – 43°F – 37°F –

SOLUTION STEP 1 Find the sum of the temperatures. + ( ) 40 – 41 48 18 = 147 STEP 2 Divide the sum by the number of temperatures. 4 147 – = 36.75 To the nearest degree, the mean of the temperatures is 37°F. – ANSWER The correct answer is D.

Evaluate the expression when , and . c ab a = 24, – 4 b 8 Evaluating an Expression, Class Example Evaluate the expression when , and . c ab a = 24, – 4 b 8 c ab 24 – ( ) 8 4 = Substitute values. 192 – 4 = Multiply. 48 = Divide. Same sign, so quotient is positive.

Evaluate the expression when , and . c a2+b a = 6, - 2 b -6 Evaluating an Expression, Example 2 Evaluate the expression when , and . c a2+b a = 6, - 2 b -6 c a2+b 62+(-6) = Substitute values. -2 36+(-6) 2 – = Evaluate exponent. 30 = Divide. Different sign, so quotient is negative. = - 15 - 2