# Unit 7 The Number System: Multiplying and Dividing Integers

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1 Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will understand the rules for multiplication of integers by extending the properties of operations to negative numbers. Students will interpret products and quotients of integers by describing real-world contexts. Students will apply the order of operations to expressions involving brackets, positive and negative numbers, whole-number exponents, and all four operations. Students will substitute negative numbers into equivalent expressions to verify equivalence. A note about models. There are many ways to remember the rules for determining when the product of two integers is positive or negative. Many models can help students remember the rules, but the Common Core State Standards emphasize the importance of students using the properties of operations (e.g., the distributive property) to understand why the rules were made in the first place. If you teach any models such as the patterns we introduce in NS7-26 and NS7-27 or other frequently used models (e.g., number lines or integer tiles), be sure to explain that the purpose of these models is to help students remember the rules, not to help them understand why the rules were made. Fraction notation. We show fractions in two ways in our lesson plans: Stacked: 1 2 Not stacked: 1/2 If you show your students the non-stacked form, remember to introduce it as new notation. In addition to the BLMs provided at the end of this unit, the following Generic BLM, found in section J, is used in Unit 7: BLM 1 cm Grid Paper (p. J-1) Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-1

4 Using patterns to see the rule another way. Write on the board: 5 4 = 5 3 = 5 2 = 5 1 = 5 0 = 5 ( 1) = 5 ( 2) = Have volunteers dictate the answers to the first five multiplications (20, 15, 10, 5, 0), then guide students to look at the pattern in the answers. ASK: What is the next term in the pattern? ( 5) And the next? ( 10) SAY: When the number you multiply 5 by decreases by 1, the product decreases by 5 each time. So because 5 0 is 0, we know that 5 ( 1) = 5, which is 5 less than 0. Fill in the last two blanks on the board. ( 5, 10) Continue the pattern on the board: 5 ( ) = ASK: If we continue this pattern, what is the next number we multiply 5 by? ( 3) How did you get that? (because it is 1 less than 2) And what is the next number in the product column? ( 15) How did you get that? (because it is 5 less than 10) Exercises: Complete the pattern = 10 3 = 10 2 = 10 1 = 10 0 = 10 = 10 = 10 = Answers: 40, 30, 20, 10, 0, 10 ( 1) = 10, 10 ( 2) = 20, 10 ( 3) = 30 Multiplying ( a) b. Write on the board: 2 ( 7) = 7 2 = SAY: When you multiply the same two numbers but in different orders, you always get the same answer. ASK: What property is that called? (the commutative property) Have a volunteer tell you the answer to the first problem. ( 14) ASK: What is the answer to the second problem? ( 14) SAY: When mathematicians were deciding on the rules for multiplying integers, they wanted to make sure that the commutative property would continue to hold. So if a positive number multiplied by a negative number is negative, then so is a negative number multiplied by a positive number. H-4 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

5 Exercises: Use the commutative property to multiply. a) 6 ( 2) =, so 2 6 = b) 4 ( 8) =, so 8 4 = c) 5 ( 10) =, so 10 5 = d) 25 ( 2) =, so 2 25 = Answers: a) 12, 12; b) 32, 32; c) 50, 50; d) 50, 50 Emphasize that 2 ( 7) and 7 2 are both opposite of 2 7 or 7 2. Point out that it is easy to multiply a positive number and a negative number. Just multiply as though they are both positive, then make the answer negative. Exercises: 1. Multiply mentally. a) 8 3 b) 9 ( 1) c) 1 9 d) 2 12 e) 15 ( 2) f) 12 2 g) 13 2 h) 25 ( 3) Answers: a) 24, b) 9, c) 9, d) 24, e) 30, f) 24, g) 26, h) Multiply. a) If = 372, what is 31 ( 12)? b) If = 684, what is 19 36? Answers: a) 372, b) 684 Extension a) How does 7 4 compare to 7 4? b) Predict how 7 ( 4) compares to 7 ( 4). c) If 34 (34) = 1,156, what is 34 ( 34)? Answers: a) 28 and 28. Both multiplications provide a number with the same magnitude but opposite signs (i.e., one is positive and one is negative); b) Changing the sign of one of the numbers in part a) changed the sign of the answer, and that is what we are doing in part b) as well. So since 7 ( 4) = 28, then 7 ( 4) = 28; c) 1,156 Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-5

6 NS7-26 Multiplying Integers Pages Standards: 7.NS.A.2a, 7.NS.A.2c Goals: Students will multiply integers and fractions, determining when the answer is positive or negative. Prior Knowledge Required: Can multiply a positive integer by a negative integer Can multiply a whole number by a fraction Vocabulary: commutative property, integer, negative, opposite, positive Review multiplying a positive and a negative integer. Write on the board: 2 3 = 6 2 ( 3) = 2 3 = Have volunteers fill in the blanks. ( 6, 6) Remind students that if they can multiply two positive numbers, then they can multiply a positive and a negative number too. Using patterns to find the rule for multiplying any number, positive or negative, by 1. Write on the board: 1 4 = 1 3 = 1 2 = 1 1 = 1 0 = 1 ( 1) = 1 ( 2) = 1 ( ) = ASK: Since 1 4 is 4, what is 1 4? ( 4) Have volunteers provide the other answers, up to 1 0. ( 4, 3, 2, 1, 0) Have students look at the pattern. ASK: Are the answers getting bigger or smaller? (bigger) By how much? (1 bigger each time) What is the next answer in the pattern? (1) Write 1 in the blank beside 1 ( 1). Have a volunteer fill in the next answer (2). Now point students to the pattern of the numbers being multiplied by 1. ASK: How are those numbers changing? (they are getting smaller) Have a volunteer finish the last equation. ( 1 ( 3) = 3) H-6 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

7 (MP.8) Ask students to look at the pattern and see if they can find a shortcut for multiplying by 1. (change the sign of the number you are multiplying by or take the opposite number) Remind students that multiplication is commutative, so 1 can be first or second and the same rule applies. Exercises: Multiply. a) 1 2 b) 9 ( 1) c) 8 ( 1) d) 1 ( 13) e) 1 8 f) 15 ( 1) g) 1 ( 10) h) 9 ( 1) Bonus: i) 1 13,000 j) 8,000,000 ( 1) k) 9 3 ( 1) l) 7 8 ( 1) Answers: a) 2; b) 9; c) 8; d) 13; e) 8; f) 15; g) 10; h) 9; Bonus: i) 13,000; j) 8,000,000; k) 27; l) 56 Using patterns to determine the rule for multiplying any number, positive or negative, by any negative number. SAY: The rule for multiplying by 1 is to change the sign. Let s see if we can find the rule for multiplying by 3. Exercises: Copy the chart and complete the pattern Answers: 9, 6, 3, 0, 3 ( 1) = 3, 3 ( 2) = 6, 3 ( 3) = 9 ASK: How does multiplying by 3 compare to multiplying by 3? (the answers are opposite) SAY: So to multiply by 3, first multiply by 3 then change the sign. Tell students that this is what they did when multiplying by 1, too. It s just that the first step of multiplying by 1 wasn t visible because you don t need to change the number at all when you multiply by 1. SAY: You can multiply by any negative number this way: multiply by the positive number, then change the sign. Exercises: Multiply. a) 3 ( 2) =, so 3 ( 2) = b) 4 5 =, so 4 5 = c) 2 ( 8) =, so 2 ( 8) = d) 3 7 =, so 3 7 = Answers: a) 6, 6; b) 20, 20; c) 16, 16; d) 21, 21 Developing and using the rule to multiply two negative numbers. Write on the board: 3 ( 4) = 12, so 3 ( 4) = 12 Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-7

8 SAY: Here s the pattern. A positive number (point to 3) multiplied by a negative number (point to 4) is negative (point to 12), so a negative number (point to 3) multiplied by a negative number (point to 4) is positive (point to 12). Tell students that in the next lesson they will learn more about why two negative numbers multiply to a positive number. Exercises: Multiply mentally. a) 1 ( 1) b) 3 ( 9) c) 4 ( 10) d) 10 ( 10) e) 7 ( 8) f) 7 ( 12) g) 15 ( 3) h) 100 ( 10) Answers: a) 1; b) 27; c) 40; d) 100; e) 56; f) 84; g) 45; h) 1,000 (MP.8) Summarizing all the rules for multiplying integers. Write on the board: 3 ( 2) 4 ( 6) ( 5) ( 6) Have students signal whether the answer is positive (thumbs up) or negative (thumbs down). Write the correct sign under each product after students signal the answer. (+,,, +, +, ) Point to each product in turn and ASK: Do the two numbers being multiplied have the same sign or different sign? Students can signal thumbs up for the same sign and thumbs down for a different sign. Write same or different on the board as students signal their answers. The final picture should look like this: 3 ( 2) 4 ( 6) ( 5) ( 6) same different different same same different ASK: If two numbers have the same sign, is their product positive or negative? (positive) If two numbers have different signs is their product positive or negative? (negative) SAY: When both numbers being multiplied are positive or both are negative, their product is positive. When one number is positive and the other is negative, their product is negative. (MP.7) Exercises: Multiply mentally. a) 2 ( 8) b) 3 0 c) 4 ( 2) d) 12 ( 5) e) 10 9 f) 8 ( 8) g) 19 ( 2) h) 11 ( 6) Answers: a) 16, b) 0, c) 8, d) 60, e) 90, f) 64, g) 38, h) 66 In the Bonus exercises below, fill in the first two squares for part a) as 12 and 24. SAY: 3 4 is 12. So to fill in the next square, I multiply 12 by 2, which is +24 or just 24. You can keep going to find the entire product. Bonus: Keep track as you go along to multiply many numbers. a) 3 4 ( 2) ( 1) 10 = H-8 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

9 b) 2 ( 2) ( 11) 2 ( 1) = Answers: Bonus: a) 12, 24, 24, 240; b) 4, 44, 88, 88 Multiplying a negative fraction and a whole number. Tell students that you can multiply negative fractions by whole numbers the same way you multiply positive fractions by whole numbers. Show the relationship by drawing a number line on the board, as shown below: æ 2ö ç =- çè 5 ø = SAY: If you know how to multiply a whole number by a positive fraction, then multiplying the whole number by the opposite negative fraction gets the opposite answer. Exercises: Multiply. æ 3ö a) 2 - ç çè 5 ø 6 Answers: a) -, b) 5 b) æ 5ö 3 - ç çè 8 ø 15 -, c) c) æ 2ö 4 - ç çè 7 ø SAY: Remember, if you can multiply positive integers, then you can multiply any integers. The same is true for fractions: If you can multiply positive numbers, then you can multiply any numbers. Write on the board: æ 2ö 3 - ç = çè 5 ø 2-3 = 5 æ 2ö - 3 ç - = çè 5 ø Have volunteers provide the answers. æ 6 6 6,, ö ç - - çè ø (MP.3) Exercises: Multiply. Use the same rules you use for multiplying integers to multiply any positive and negative numbers. æ 4ö a) 3 æ 4ö ç - çè 3 b) 1 æ 2 ö ç - ø çè 7 c) 5 1 ç - ø çè 10 d) - 9 ø 2 e) æ 2ö f) - ( 8) g) 12 5 ç çè 3 h) ø 2 4 Answers: a) 4, b) 7, c) 1, d) 9 -, e) , f) 8, g) 8, h) Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-9

10 SAY: Some people write brackets around both numbers when multiplying integers or positive and negative fractions. Then they usually write the + sign in front of a positive number. It looks a bit different but you still multiply the same way. Exercises: Multiply. a) ( 3) (+2) b) (+4) ( 3) c) ( 8) ( 5) æ 1ö d) æ 4 ç + (-5) çè 2 e) ( 3) ö æ 3ö - ç + ø çè 3 f) (- 2) ç - ø çè 5 ø Answers: a) 6, b) 12, c) 40, d) 5 -, e) 4, f) Extensions (MP.3) 1. The product of 1 and 9 is less than Can the product of two negative numbers be 1 less than their sum? Explain. Hint: Start with some examples, such as 2 ( 3) or 6 (- ). 2 Answer: No, because the product will be positive and the sum will be negative. (MP.3) 2. Tell students that once you know the rule for multiplying a positive and a negative number and that 1 ( 1) is 1, you can multiply any two negative numbers. Have students explain each step in the following example: 5 ( 2) = 1 5 ( 1) 2 = 1 ( 1) 5 2 = 1 10 = 10 Answer: 5 ( 2) = 1 5 ( 1) 2 because 5 = 1 5 and 2 = 1 2 = 1 ( 1) 5 2 because 5 ( 1) = ( 1) 5 = 1 10 because 1 ( 1) = 1 and 5 2 = 10 = 10 because 1 any number is itself (MP.7) 3. Tell students that they can think of 3/4 2 as the point that is 3/4 of the distance from 0 to 2, as shown below: = 4 2 Ask students how they would write a multiplication for the point that is 3/4 of the distance from 0 to 2. (3/4 ( 2)) Have students use this method to show the following on a number line: a) - ( 3) b) - ( 5) c) ( 2) H-10 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

12 Answers: a) b) c) Complete the chart by deciding if the product will be positive or negative. a b c a b c Answers: +,,,, +, +, +, (MP.3) 7. Tell students that subtracting is the same thing as multiplying by 1, then adding. For example: 5 ( 3) = 5 + ( 1) ( 3) Have students explain why this is the case. Sample answer: Multiplying by 1 gets the opposite of the number and subtracting a number is the same thing as adding its opposite. H-12 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

13 NS7-27 Using the Distributive Property to Multiply Integers (Advanced) Pages Standards: 7.NS.A.2a, 7.NS.A.2c Goals: Students will develop and apply the formula for multiplying integers. Prior Knowledge Required: Can add and subtract integers Knows and can apply the distributive property Can use repeated addition to multiply whole numbers Can use the order of operations with brackets for these operations: +,,, Vocabulary: commutative property, distributive property, equivalent expression, integer, negative, positive, sign Review the distributive property of multiplication over addition. Write on the board: = 7 2 SAY: If you start with three 2s and add four more 2s, you end up with seven 2s, because 7 is Write on the board: = (3 + 4) 2 SAY: These two expressions are equivalent because multiplication distributes over addition. In fact, the distributive property of multiplication says that you could replace 2, 3, and 4 with any other numbers and the two expressions would still be equivalent. Demonstrate by replacing 3 with 6, 2 with 5, and 4 with 7, as shown below: = (6 + 7) 5 Have students verify the expressions are still equivalent. ( = 65 and 13 x 5 = 65) Exercises: (MP.7) 1. Write an equivalent expression that uses the same numbers. a) b) c) d) e) (2 + 7) 3 f) (3 + 3) 5 g) 2 (4 + 3) Answers: a) (3 + 4) 6, b) 7 (4 + 2), c) (5 + 4) 4, d) 4 (4 + 3), e) , f) , g) Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-13

14 (MP.1) 2. Evaluate the expressions in Exercise 1 to verify that both expressions are equivalent. Answers: a) = 42 and 7 6 = 42, b) = 42 and 7 6 = 42, c) = 36 and 9 4 = 36, d) = 28 and 4 7 = 28, e) 9 3 = 27 and = 27, f) 6 5 = 30 and = 30, g) 2 7 = 14 and = 14 Review the distributive property of multiplication over subtraction. Tell students that multiplication distributes over subtraction too. Write on the board: = 4 2 SAY: If you start with seven 2s and take away three of them, you end up with four 2s, because 7 3 is 4. Write on the board: = (7 3) 2 Exercises: Use the distributive property to write an equivalent expression that uses all the same numbers. a) (5 3) 7 b) c) (3 2) 6 d) (5 5) 6 e) Answers: a) , b) (8 5) 4, c) , d) , e) (7 5) 5 (MP.3) Investigate the distributive law when the subtraction results in a negative integer. Write on the board: 2 (3 7) = ASK: What is 3 7? ( 4) Write on the board: 2 ( 4) = Have volunteers fill in the blanks. (6 14) ASK: What is 6 14? ( 8) Remind students that the answer to 6 14 is the opposite of SAY: We ve just shown that 2 ( 4) is 8 and we only had to multiply positive numbers and subtract them to do so. Tell students that the rules for multiplying integers were chosen as they were so that the distributive property and all the other properties of multiplication continue to hold for negative numbers. (MP.1, MP.7) Exercises: Use the distributive property to multiply 2 ( 4) in different ways. Make sure you always get 8. a) 2 (1 5) b) 2 (0 4) c) 2 (6 10) Answers: a) 2 10 = 8, b) 0 8 = 8, c) = 8 ASK: Using this method of subtracting positive products, what is the easiest way to multiply 2 ( 4)? (subtract 0 8) H-14 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

15 Exercises: Multiply by replacing the negative number with zero minus its opposite, then using the distributive property. a) 7 ( 4) = 7 (0 4) b) 8 ( 3) = 8 (0 3) c) 6 ( 4) d) 3 ( 3) Answers: a) = 0 28 = 28, b) 24, c) 24, d) 9 SAY: Once you know a positive times a negative is negative and you decide to make the commutative property hold, then a negative times a positive must also be negative. Write on the board: 4 7 = 7 ( 4) = 28 SAY: But you can also use the distributive property directly on 4 7 = (0 4) 7. Exercises: Use the distributive property to multiply. a) 4 7 b) 3 5 c) 2 8 Answers: a) 0 28 = 28, b) 0 15 = 15, c) 0 16 = 16 (MP.5) Using the distributive property to multiply two negative integers. Write on the board: ( 3) ( 2) Remind students that we want all the properties of operations to hold for negative numbers. SAY: I want to use properties of operations to show that multiplying two negatives gets a positive. ASK: Can I use the commutative property? (no) PROMPT: Is ( 2) ( 3) any easier than ( 3) ( 2)? (no) SAY: Let s try using the distributive property. Write on the board: ( 3) ( 2) = ( 3) (0 2) = ( 3) 0 ( 3) 2 (MP.1) SAY: I want the distributive property to hold for negative numbers too, so that we can use it to evaluate ( 3) ( 2). Pointing to the last expression on the board, ASK: How can we evaluate this equivalent expression? PROMPT: Do we know how to multiply by 0? (yes) Do we know how to multiply a negative number and a positive number? (yes) Do we know how to subtract a negative number? (yes) SAY: So we know how to do all the components of this problem. We ve broken down one difficult problem that we didn t know how to do into three easier problems that we know how to do. ASK: What is ( 3) 0? (0) SAY: Anything times 0 is 0. ASK: What is ( 3) 2? ( 6) Continue writing on the board: = 0 ( 6) Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-15

16 Remind students that subtracting a negative integer is the same as adding a positive integer. Continue writing on the board: = = 6 SAY: So 3 ( 2) is +6. Exercises: Use the distributive property to multiply. a) 2 ( 5) = 0 2 ( 5) = b) 1 ( 15) = 0 = c) 5 ( 7) = 0 = d) 8 ( 5) = 0 = e) 7 ( 9) = 0 = f) 3 ( 10) = 0 = Selected solutions: a) 0 ( 10) = 10, b) 0 ( 15) = 15 Answers: c) 35, d) 40, e) 63, f) 30 After students finish, point to part a) and ASK: How does 2 ( 5) compare to 2 5? (they have the same answer) Repeat for parts b), c), and d). SAY: You can multiply two negative numbers as though both numbers are positive. You don t even need to change the sign. Write on the board: negative negative = positive Exercises: Multiply without using the distributive property. a) 3 ( 4) b) 2 ( 5) c) 8 ( 4) d) 6 ( 2) e) 7 ( 3) f) 4 ( 4) Answers: a) 12, b) 10, c) 32, d) 12, e) 21, f) 16 (MP.3) Moving negative signs around in a product. Write on the board: 3 4 = 3 ( 4) = Have volunteers dictate the answers. ( 12, 12) SAY: You can move a negative sign from the first term to the second and still get the same answer. Write on the board: 3 4 = 3 4 SAY: These are equivalent expressions because they have the same answer. Exercises: Write an equivalent expression by moving the negative sign to the other number. a) 2 7 b) 8 9 c) 5 4 Bonus: Make several equivalent expressions by moving the negative sign to any other number Answers: a) 2 ( 7); b) 8 ( 9); c) 5 ( 4); Bonus: 2 ( 3) 4 5, 2 3 ( 4) 5, ( 5) H-16 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

18 5. What properties are being used? a) 3 5 = (3 ( 1)) 5 = 3 ( 1 5) = 3 (5 ( 1)) = 3 ( 5) b) 2 ( 4) = 2 (( 1) 4) = (2 ( 1)) 4 = ( 1 2) 4 = 1 (2 4) = 1 8 = 8 Answers: a) associative, commutative; b) associative, commutative, associative 6. Have students complete the table. p q p q = p ( q) = Are the Expressions Equal? When students are done, conclude that products of variables have the same answer when you move a negative sign from one term to another. So these expressions are equivalent, no matter what p and q are: p and q might be both positive, both negative, or one positive and one negative. Answers: p q p q = p ( q) = Are the Expressions Equal? = 10 2 ( 5) = 10 yes = 10 2 ( 5) = 10 yes ( 5) = = 10 yes ( 5) = = 10 yes ( 4) = = 12 yes ( 1) = = 2 yes = 6 3 ( 2) = 6 yes ( 5) = = 10 yes H-18 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

19 NS7-28 Multiplying Integers in the Real World Pages Standards: 7.NS.A.2a, 7.NS.A.3 Goals: Students will represent real-world situations with integer equations, including those involving multiplication and addition. Prior Knowledge Required: Knows the order of operations, without exponents Can add, subtract, and multiply integers Knows when situations can be represented by integers Vocabulary: integer, negative, positive, sign Review real-world situations described by integers. Have students brainstorm things in the real world that they can describe by integers. (sample answers: temperature, electric charge, golf scores, +/ ratings in basketball or hockey, bank balances, gains and losses, elevation above or below sea level) Describing change as positive or negative. Tell students that in most of the situations they suggested above the integer is describing a situation, such as the current temperature or the elevation at a particular spot or the strength of an electric charge. But integers can also represent a change in the situation. Point out that students have already seen this with gains and losses or debits and credits in the way they represent a change in the bank balance. Explain that increases are considered positive and decreases are considered negative. Have volunteers demonstrate the first two exercises below before assigning the rest to all students. Exercises: Describe the change as a positive or negative number. a) A baby gained 14 pounds in his first year. b) A baby lost 0.6 pounds in her first week. c) Ed earned \$60 from working. d) Lynn lost \$40 on the sidewalk. e) Max climbed 1,500 feet. f) The temperature decreased 4.5 F. Bonus: g) The temperature changed from 3 F to +2 F. h) Jane started the day with \$25 and has \$55 at the end of the day. i) Roy started on the 12 th floor of his building and walked down to the 8 th floor. Answers: a) +14; b) 0.6; c) +60; d) 40; e) +1,500; f) 4.5; Bonus: g) +5; h) +30; i) 4 Constant change as repeated addition and multiplication. Tell students that sometimes the same change happens repeatedly. Write on the board: (+10) + (+10) + (+10) = 3 (+10) = +30 or 30 Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-19

20 SAY: If I earned \$10 each hour for 3 hours, then my change in money can be described as +30 dollars. Exercises: Write a multiplication equation to show the amount of change. a) Ted gained \$10 every hour for 5 hours. b) The temperature dropped by 2 F every hour for 6 hours. c) Vicky s bank account balance decreased \$5 every week for 8 weeks. d) A hiker climbed 12 feet every hour for 3 hours. Answers: a) 5 (+10) = +50, b) 6 ( 2) = 12, c) 8 ( 5) = 40, d) 3 (+12) = +36 Representing time using integers by putting 0 as right now. Tell students that they can think of future time as positive and past time as negative. SAY: We ll use zero to represent right now. Write on the board: 3 weeks from now 3 weeks ago Have volunteers tell you what integer can be used to represent 3 weeks from now and 3 weeks ago (+3, 3). SAY: When a problem involves two different units, you need to carefully say what the units are. But when there is no confusion you can ignore the units until you are done. So the same integer can represent many different quantities. Exercises: Describe the time as an integer. a) 2 years ago b) 1 minute ago c) 30 seconds from now d) 30 days from now Answers: a) 2, b) 1, c) +30, d) +30 Writing integer multiplication equations to show real-life situations. Tell students to pretend that somebody is climbing a mountain and their elevation is increasing at a rate of 20 feet per minute. Tell students that their current elevation is at sea level. ASK: What integer do we use to represent sea level? (0) Write on the board: current elevation = 0 ft 5 minutes from now the elevation will be ASK: What will the elevation be 5 minutes from now? (+100 ft) Write that in the blank. ASK: How did the volunteer get the answer? (multiplied 5 20) Tell students that you want to be even more specific and write the multiplication of integers. Write on the board: (+5) (+20) = +100 SAY: The 5 minutes are in the future so we use +5. The 20 feet each minute is an increase in elevation so we use +20. Continue writing on the board: 5 minutes ago the elevation was = H-20 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

24 NS7-29 Dividing Integers Pages Standards: 7.NS.A.2b, 7.NS.A.2c, 7.NS.A.3 Goals: Students will develop and apply the formula for dividing integers. Prior Knowledge Required: Can multiply integers Knows the relationship between multiplication and division Vocabulary: integer, negative, positive, sign (MP.3) Apply the relationship between multiplication and division to negative numbers. SAY: If you can find the missing number in a multiplication, then you can divide. That s true for negative numbers the same as it is for positive numbers. Write on the board: 3 = 12, so 12 ( 3) = SAY: A negative number times something is a positive number. ASK: Is the something positive or negative? (negative) SAY: Once you determine the sign, you just have to look at the numbers. ASK: 3 times what is 12? (4) Write 4 in both blanks. Repeat the questions for ( 4) = ( 20) and 3 = 18. (5, 6) Exercises: Divide by finding the missing number in the product. a) 2 = 14, so 14 2 = b) 3 = 24, so 24 3 = c) 3 = 27, so 27 ( 3) = d) 11 = 88, so = e) 3 = 60, so 60 ( 3) = f) 100 = 200, so 200 ( 100) = Answers: a) 7, 7; b) 8, 8; c) 9, 9; d) 8, 8; e) 20, 20; f) 2, 2 (MP.1, MP.7) Developing a rule for dividing integers. Write on the board: To find 12 3 =, ask: 3 = 12 To find ( ) (+) =, ask: (+) = ( ) SAY: If you can find the missing number in a product, you can answer the division. So if you can find the missing sign in a product, then you can find the correct sign of the division answer. Exercises: 1. Use multiplication to finish writing the signs. a) (+) (+) = b) (+) ( ) = c) ( ) (+) = d) ( ) ( ) = H-24 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

26 SAY: Some people write brackets around both numbers when dividing integers. Then they put positive signs in front of positive numbers. It looks a bit different but you still divide the same way. Exercises: 1. Fill in the blanks. (+7) (+20) = (+7) ( 20) = ( 7) (+20) = ( 7) ( 20) = Answers: 0.35, 0.35, 0.35, Divide. Write your answer as a decimal. a) 25 ( 100) b) (+35) ( 70) c) 3 (20) d) 47 ( 100) e) 15 ( 60) f) ( 19) (+5) Answers: a) 0.25, b) 0.5, c) 0.15, d) 0.47, e) 0.25, f) 3.8 (MP.3) Writing the division answer as a fraction of integers. SAY: You can write the answer to an integer division as a fraction of integers. Write on the board: 7 7 5= 7 5 = = ( 5) = -7-5 Exercises: Write each division answer as a fraction of integers. a) 3 ( 4) b) 8 9 c) 2 ( 15) d) 1 ( 12) 3 Answers: a) - 4, b) - 8-2, c) 9-15, d) 1-12 SAY: If the numerator and denominator have opposite signs, then the value is negative. If they have the same sign, the value is positive. Equivalent fractions of integers. Write on the board: 7 5 = ( 5) = ( 5) = 7 5 For each question, ASK: Is the answer positive or negative? (negative, positive, negative) Write the correct signs in the boxes. Summarize by writing on the board: =- =+ = = H-26 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

27 Exercises: 5 a) Circle the expressions that have the same value as b) Circle the expressions that have the same value as Bonus: Circle the expressions that have the same value as Answers: a) and 3-3 ; b) -5-8 ; Bonus: 2 2 -, 3-3, and -2 3 Average change. Tell students that on December 14, 1924, the temperature at Fairfield, MT, dropped from 63 F to 21 F in 12 hours (from noon to midnight). ASK: How much did the temperature change in total? (84 F) Is the change positive or negative? (negative) How do you know? (because the temperature decreased) Write on the board: 84 F in 12 hours Average change per hour = Remind students that if the total temperature change was 84 F in 12 hours, then the average change in each hour is obtained by division. Ask a volunteer to calculate the average change per hour. ( 7 F) Exercises: Fill in the blanks. Write the average change as an integer. Include the units. a) An elevator rose from 0 ft to 60 ft in 30 seconds. b) Fred s bank account balance changed from \$34 to \$16 in 5 days. c) Milly s bank account balance changed from \$7 to \$17 in 3 days. d) In Spearfish, SD, on January 22, 1943, the temperature went from 4 F to 45 F in 2 minutes. e) The value of a car decreased from its initial value of \$23,000 to a value of \$19,500 after 5 months. f) A skydiver fell from an elevation of 60,000 ft to 45,000 ft in 3 minutes. Answers: a) +2 ft per second; b) \$10 per day; c) +\$8 per day; d) F per minute; e) \$700 per month; f) 5,000 ft per minute Some students might think that the change from 7 to +17 is 20 because, when subtracting positive numbers, the difference between two numbers with the same ones digit is always a multiple of 10. Emphasize that this is not the case when subtracting a negative number from a positive number. Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-27

29 NS7-30 Powers Pages Standards: 7.NS.A.2a, 7.NS.A.2c, 7.NS.A.3 Goals: Students will evaluate powers by using repeated multiplication and will investigate the properties of powers. Prior Knowledge Required: Can multiply negative numbers Understands multiplication as repeated addition Vocabulary: base, exponent, negative, opposite, positive, power, squared Introduce powers as repeated multiplication. Remind students that multiplication is repeated addition. Write on the board: 5 3 = SAY: Just like multiplication is a short form for repeated addition, powers are a short form for repeated multiplication. Write on the board: 3 5 = SAY: 5 times 3 means add five 3s together, and 3 with a raised 5 means multiply five 3s together. Write on the board: base 3 5 exponent Tell students that 3 is called the base, 5 is called the exponent, and 3 5 is called a power of 3. SAY: The base is the number you are multiplying and the exponent is how many of them you are multiplying. Point out that in math a base is the bottom part of something, the same as in English. Write on the board: Have students signal the base, then the exponent for each expression. (4, 3, 2; 2, 4, 5) Write on the board: Teacher s Guide for AP Book 7.1 Unit 7 The Number System H-29

30 Have students signal the base, then the exponent for each expression. (2, 4, 5; 3, 3, 2) Ask a volunteer to write 9 4 as a product. ( ) Ask a volunteer to write as a power. (3 7 ) Exercises: Write the product as a power. a) b) 8 8 c) d) e) f) Bonus: g) 1,500 1,500 1,500 h) 100, ,000 Answers: a) 2 4 ; b) 8 2 ; c) 1 3 ; d) 10 2 ; e) 12 3 ; f) ; Bonus: g) 1,500 3 ; h) 100,000 2 Evaluating powers. Write on the board: 2 3 = = SAY: You can find repeated multiplications by keeping track as you go along. First find the result of 2 2 (write 4 in the first square), then multiply your answer by 2 (write 8 in the next square). So multiplying three 2s gets 8 (write that in the blank). Write on the board: 2 6 = = Ask a volunteer to evaluate the power using the same method. (4, 8, 16, 32, 64) Exercises: 1. Evaluate the power. Keep track of the power as you go along. a) 2 5 = = b) 5 3 = = c) 10 4 = = Answers: a) 32, b) 125, c) 10, Evaluate. a) 2 4 b) 4 2 c) 1 5 d) 3 3 e) 6 2 f) 4 3 g) 0 2 h) 30 1 Bonus: If 2 10 = 1,024, what is 2 11? Answers: a) 16; b) 16; c) 1; d) 27; e) 36; f) 64; g) 0; h) 30; Bonus: 2,048 H-30 Teacher s Guide for AP Book 7.1 Unit 7 The Number System

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