Multiplication and Division of Positive Objective o develop and apply rules for multiplying and dividing positive and www.everydaymathonline.com epresentations eoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive eacher s Lesson Guide eaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Apply previous knowledge of fact families. [Operations and Computation Goal ] Use the order of operations to solve problems. [Patterns, Functions, and Algebra Goal 3] Apply properties of multiplication (Commutative Property and Multiplication Property of 1). [Patterns, Functions, and Algebra Goal 4] Key Activities Students study patterns to derive rules for multiplying and dividing positive and Key Vocabulary Multiplication Property of 1 Materials Math Journal, pp. 14 1 Study Link 3 calculator (optional) slate (optional) 1 4 3 Playing op-it with Positive Student Reference Book, pp. 337 and 338 Math Masters, p. 478 per partnership: complete deck of number cards (the Everything Math Deck, if available), calculator (optional) Students practice adding and subtracting positive and Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 478. [Operations and Computation Goal 1] Math Boxes 4 Math Journal, p. 17 Students practice and maintain skills through Math Box problems. Study Link 4 Math Masters, p. 10 Students practice and maintain skills through Study Link activities. READINESS Using a Multiplication Story Math Masters, p. 11 Students use situations from a story to remember a multiplication property for positive and ENRICMEN Exploring Patterns in Products and Quotients Math Masters, p. 1 Students explore patterns in products and quotients of positive and EXRA PRACICE -Minute Math -Minute Math, p. 34 Students practice finding products and quotients of positive and Advance Preparation eacher s Reference Manual, Grades 4 pp. 71 74, 100 10 48 Unit Number Systems and Algebra Concepts
Getting Started Mathematical Practices SMP4, SMP, SMP7, SMP8 Content Standards.NS. Mental Math and Reflexes Students solve subtraction problems with signed numbers. - - (-) 0 - (-1) 10-1 - -17-1 13 - (-3) 8-1 - (-0) Math Message Complete the problems on journal page 14. Study Link 3 Follow-Up Go over the answers with the class. 1 eaching the Lesson Math Message Follow-Up (Math Journal, p. 14) ELL WOLE-CLASS DISCUSSION ave students work with a partner to look for patterns in each set of number sentences. Ask volunteers to share their findings. Sample answers: One factor remains the same while the other factor decreases by 1. he products in each set also decrease in a pattern. Write key ideas and examples on the board to support English language learners. Be sure that students have circled the correct words in the multiplication rules in Problems 1 4. Discuss the Multiplication Property of -1 in Problem. (he product of a number and -1 is the opposite of the number.) Write additional examples on the board. Point out that the Commutative Property (turn-around rule) is also true for multiplication with For example, (-) = - = -10. Ask students to write a few examples of the turn-around rule for multiplication with Spend a few minutes practicing multiplication. Pose problems that students can solve mentally. hey should write the products on their slates or on scratch paper and then check their answers with a partner. 8 (-) -48 - _ 3 _ - 4_ (-8) 4 1. (-) - -3-1 0. (-) -1 ime Multiplication Patterns In each of Problems 1 4, complete the patterns in Part a. Check your answers with a calculator. hen circle the word in parentheses that correctly completes the statement in Part b. 1. a. 4 3 18 1 0. a. 10 1 0 0 ( 1) ( ) 3. a. 3 1 3 3 0 3 0 1 3 3 1 0 10 b. Positive º Positive Rule: is multiplied by a positive b. Positive º Negative Rule: is multiplied by a negative b. Negative º Positive Rule: is multiplied by a positive 7 4. a. 4 1 4 4 0 0 4 ( 1) 4 4 ( ) 4 (). a. Solve. 1 1 ( 7.7) 1 ( 1 ) 1 m 8 1 7.7 1 m b. Negative º Negative Rule: is multiplied by a negative b. Multiplication Property of 1: For any number a, 1 a a 1 OPP(a), or a. he number a can be a negative number, so OPP(a) or a can be a positive number. For example, if a, then a OPP( ). Math Journal, p. 14 Lesson 4 4
ime Fact Families for Multiplication and Division A fact family is a group of four basic, related multiplication and division facts. Example: he multiplication and division fact family for 3 18 18 / 3, 3, and 18 is made up of the following facts: 3 18 18 / 3 As you already know, when a positive number is divided by a positive number, the quotient is a positive number. Problems 1 and will help you discover the rules for division with Complete the fact families. Check your answers with a calculator. hen complete each rule. 1. a. () b. () c. ( ) 1 1 / () / () 1 / / d. Negative / Negative Rule: e. Negative / Positive Rule: is divided by a negative is divided by a positive. a. 4 () 1 b. 7 ( ) 3 ( 4) 1 1 / () 4 1 / ( 4) ( 7) 3 3/( ) 7 3 / ( 7) c. ( 10) 1 0 10 ( ) 0 0 / ( 10) 0 /( ) 10 3. Solve. Check your answers with a calculator. a. ( 4) 4 (hink: What number multiplied by 4 is equal to 4?) b. 1 c. 48 d. () 7 d. Positive / Negative Rule: is divided by a negative / / e. 1 / f. 48 / () g. 7 / () Math Journal, p. 1 Multiplication fact family for 7, -8, and -: + - -, / 7-8 7 (-8) = - - / 7 = -8-8 7 = - - / -8 = 7 - Deriving Rules for Division of Positive (Math Journal, p. 1) Adjusting the Activity Students use fact families to derive rules for dividing positive and Review the example at the top of journal page 1. hen do Problem 1a as a class. Students can apply multiplication rules to solve the first two problems: (-3) = -1 and -3 = -1. he three numbers used to build this fact family are, -3, and -1. Students can use this information to complete the division facts in the family. Consider using fact triangles with both positive and negative numbers to show patterns. For example, draw a fact triangle for 7, -8, and - on the board. Draw large + (positive) and - (negative) signs at the corners for emphasis. Discuss the pattern of signs for products and quotients. Repeat for other examples, such as -, 8, and -48 and -3, -4, and 1. (See margin.) A U D I O R Y K I N E S E I C A C I L E V I S U A L Spend a few minutes practicing division. 4 / (-) - -4 / 7-7 -0 / (-1) (-8) / (-8) -8 PROBLEM SOLVING Review the rules for division. Discuss the relationship between multiplication and division as shown in Problem 3; that is, any division problem can be solved in terms of multiplication. For example, to solve 3 / (-), think: What number, multiplied by -, is equal to 3? -7 Students complete journal page 1 independently. Partners then compare and check answers. ELL Multiplying and Dividing Positive (Math Journal, p. 1) Review the rules for multiplication and division at the top of journal page 1. ave students complete the journal page independently. Partners then compare and check answers. ave students generate real-life situations that could be modeled by some of the problems on journal page 1. For example, Problem 1 might model this situation: Your parents deduct $7.00 from your monthly allowance for 8 months to help you save for a new bicycle. -$7 8 = -$ 0 Unit Number Systems and Algebra Concepts
Ongoing Learning & Practice Playing op-it with Positive (Student Reference Book, pp. 337 and 338; Math Masters, p. 478) PARNER Distribute a complete deck of number cards (the Everything Math Deck, if available) to each partnership. Students use cards to practice comparing, adding, and subtracting positive and Ongoing Assessment: Recognizing Student Achievement Math Masters Page 478 Use the op-it record sheet to assess students abilities to add and subtract positive and Students are making adequate progress if they can calculate and compare sums and differences of signed numbers. [Operations and Computation Goal 1] ime º, / of Positive A Multiplication Property For all numbers a and b, if the values of a and b are both positive or both negative, then the product a b is a positive number. If one of the values is positive and the other is negative, then the product a b is a negative number. Solve. Use a calculator to check your answers. 1. 7 8. 73 ( 4) 3. ( 10) 70 4. 1 ( 4 ). 0. ( 1). 3.3.3 7. 4 ( 7) 8. () (). 4 / 10. 3 / ( 1) 11. 3 3 ( 4 ) 4 1. 4 / ( ) / () 13. 1 14. 7 / () 1. / 11 1. 1 ( 4 ) 17. ( 4 ) 18. 3 ( 3) 1. ( 4) 0. ( 7 / ) ( 0) 1. ( 3) ( ). (4 ( 7)) (4 ()) Evaluate each expression for y 4. 3. 3 ( y) 4. y / (). y ( 7 3). y (y ) A Division Property For all numbers a and b, if the values of a and b are both positive or both negative, then the quotient a / b is a positive number. If one of the values is positive and the other is negative, then the quotient a / b is a negative number.,8 700 3 8 7. 1 84 3 0 3 4 0 1 1 0 7. ( y ) 8. ( ) ( y) Math Journal, p. 1 3 0 7 Math Boxes 4 (Math Journal, p. 17) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -. he skills in Problems and preview Unit 7 content. Study Link 4 (Math Masters, p. 10) ime ome Connection Students practice multiplication and division of positive and 3 Differentiation Options READINESS Using a Multiplication Story (Math Masters, p. 11) SMALL-GROUP 1 Min Math Boxes 1. Write the reciprocal. a. b. 3 c. 4 7 d..4 1 3 7 1 8 41 3. Lines a and b are parallel. Without using a protractor, find the degree measure of each numbered angle. Write each measure on the drawing. 8 1 1 3 4 1 8 8 1 7 8 1 a b 3. Divide. Simplify if possible. a. 3 1 1 1 4 b. 1 3 1 4 c. 1 8 1 d. 0 10 1 4. If 1 kilogram (kg) is about 1 pounds (lb), then kg is about lb. Circle the best answer. A. 1 1 B. 7 1 C. 10 D. 11 13 371 3 o provide experience with rules for multiplying positive and negative numbers, have students read Math Masters, page 11. hey use the characters and situations in a familiar story to help them remember a multiplication property for multiplication of positive and. Express each decimal as a percent. 104% 8.% 0.3% 400% a. 1.04 b. 0.08 c. 0.003 d. 4.0. Suppose you toss a penny and a nickel together. ow many equally likely outcomes are there? Complete the table. 4 Penny Nickel equally likely outcomes 0 10 13 Math Journal, p. 17 Lesson 4 1
Name ime SUDY LINK Solve. º, / of Signed Numbers A Multiplication Property he product of two numbers with the same sign is positive. he product of two numbers with different signs is negative. 1. 1 º. 3 / 7 3. 4 ( 4) 4. º 4 Study Link Master 0. 0 / 10. º º 8 1,10 7. 48 / ( ) 8. ( º ) 1. 0 º () 10. º (1 1) 4 A Division Property he quotient of two numbers with the same sign is positive. he quotient of two numbers with different signs is negative. 11. (0 10) ( 4) 1. / ( 7) / ( 4) 40 0 13. º ( 7) º ( 4) 14. 40 1 7 ENRICMEN Exploring Patterns in Products and Quotients (Math Masters, p. 1) 1 30 Min o further explore patterns in products and quotients, students multiply and divide by signed numbers that have repeating digits and examine the resulting patterns. hey use these patterns to predict products and quotients. ry his 1. 3 º ) ( 1. (8 º ()) (8 º ()) 1 1 17. 0. º () 18. ( 3 ) 4 ) ( 1 1. Evaluate each expression for b 7. 3 a. ( º b) 7 b. 11 º ( b) 77 c. b / ( 14) 0. d. b (b 1) 1 48 EXRA PRACICE SMALL-GROUP -Minute Math 1 30 Min o offer more practice finding products and quotients of positive and negative numbers, see -Minute Math, page 34. Math Masters, p. 10 eaching Master Name ime eaching Master Name ime A Multiplication Story Patterns with Signed Numbers In many fairy tales and children s stories, there are good characters and bad characters. For example, in the story Little Red Riding ood, the grandmother is a good character; the wolf is a bad character. You can use these character situations to remember a multiplication property for positive and When something good ( ) happens to a good ( ) character, we think it is good ( ). When something bad ( ) happens to a good ( ) character, we think it is bad ( ). 1. Multiply each number in the far left column by each number in the top row. Look for patterns. Use your calculator as few times as possible to complete the table. 11 111 1,111 11 11 1,1 1,1 111 1,1 1,31 13,31 1,111 1,1 13,31 1,34,31 11,111 1,1 1,33,31 1,344,31. Use the patterns from the table above to predict the products below. hen check each prediction with your calculator. 11 111 Prediction 111,111 Answers vary. Actual 111,111 1,,1 1,333,31 When something good ( ) happens to a bad ( ) character, we think it is bad ( ). 3. Divide each number in the far left column by each number in the top row. Look for patterns. Use your calculator as few times as possible to complete the table. Write your own number pattern in last row. Divisor, py g g p When something bad ( ) happens to a bad ( ) character, we think it is good ( ). Example: Solve 4? hink, When something bad ( 4) happens to a good ( ) character, we think it is bad ( ). So, 4 0. Math Masters, p. 11 1 0.1 0.0 1 0.0 0 1 34 0.3 4 0.0 4 0.0 0 4 4 0.4 0.0 4 0.0 0 4 7 0. 7 0.0 7 0.0 0 7 Answers vary. Math Masters, p. 1 Unit Number Systems and Algebra Concepts