Objective 1 To introduce addition involving negative integers. materials Teaching the Lesson Key Activities Students review positive and negative numbers on the number line, thinking of them as reflected across the zero point. They discuss and practice addition of positive and negative numbers as accounting problems, keeping track of “credits” and “debits.” They play the Credits/Debits Game. Key Concepts and Skills • Compare and order integers. [Number and Numeration Goal 6] • Add signed numbers. [Operations and Computation Goal 2] ⵧ Student Reference Book, pp. 60 and 238 ⵧ Study Link 10 5 䉬 ⵧ Teaching Master (Math Masters, p. 320) ⵧ Game Master (Math Masters, p. 468) ⵧ Transparencies (Math Masters, pp. 318 and 321; optional) ⵧ 1 transparent mirror per partnership ⵧ deck of number cards (Everything Math Deck, if available) • Identify a line of reflection. [Geometry Goal 3] See Advance Preparation Key Vocabulary opposite (of a number) • credit • debit Ongoing Assessment: Informing Instruction See page 825. 2 Ongoing Learning & Practice Students solve problems involving fractions, decimals, and percents. Students practice and maintain skills through Math Boxes and Study Link activities. materials ⵧ Math Journal 2, pp. 283 and 284 ⵧ Study Link Master (Math Masters, p. 322) Ongoing Assessment: Recognizing Student Achievement Use journal page 284. [Data and Chance Goal 4] 3 materials Differentiation Options READINESS Students skip count on a calculator to explore patterns in negative numbers. READINESS Students use a number line to add positive and negative integers. ⵧ calculator ⵧ number line ⵧ masking tape See Advance Preparation Additional Information Advance Preparation For Part 1, make and cut apart copies of Math Masters, page 320. Place them near the Math Message. For the second optional Readiness activity in Part 3, use masking tape to create a life-size number line (–10 to 10) on the floor. 822 Unit 10 Reflections and Symmetry Technology Assessment Management System Math Boxes, Problem 1 See the iTLG. Getting Started Mental Math and Reflexes Math Message Pose problems involving comparisons of integers. Suggestions: Take a copy of Math Masters, page 320. Follow the directions and answer the questions. Share 1 transparent mirror with a partner. Are you better off if you have $3 or owe $10? Have $3 Owe $4 or owe $9? Owe $4 Study Link 10 5 Follow-Up 䉬 Owe $20 or owe $100? Owe $20 Which is greater? –10 or 8? 8 5 or –1? 5 10 or –10? 10 Which is colder? –3°C or 10°C? –3°C –9°C or 19°C? –9°C –7°C or –11°C? –11°C Review answers. Have students share some of the patterns they created on their own. An overhead transparency of Study Link 10 5 (Math Masters, page 318) may be helpful. 䉬 1 Teaching the Lesson 䉴 Math Message Follow-Up WHOLE-CLASS DISCUSSION (Student Reference Book, p. 60; Math Masters, p. 320) One way to think about a number line is to imagine the whole numbers reflected across the zero point. Each of these positive numbers picks up a negative sign as it crosses to the other side of zero. The opposite of a positive number is a negative number. Conversely, imagine the negative numbers reflected across the zero point. The sign of each number changes from negative to positive as it crosses to the other side of zero. The opposite of a negative number is a positive number. NOTE In this “flipping” of the number line, the zero point stays motionless, like the fulcrum of a lever. Zero is the only number that equals its opposite. When students place the transparent mirror on the line passing through the zero point on Math Masters, page 320, the negative numbers appear (reversed) across from the corresponding positive numbers. Name LESSON 10 6 䉬 Date Time Positive and Negative Numbers 60 –10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 –1 0 1 2 34 1 2 3 4 1 2 3 4 5 6 7 8 9 10 Place your transparent mirror on the dashed line that passes through 0 on the number line above. Look through the mirror. What do you see? What negative number image do you see . . . 1 above 2? 2 8 above 8? 4321 0 above 1? 109 8 7654321 0 1 2 3 4 5 6 7 8 9 10 12345678901 Math Masters, page 320 Lesson 10 6 䉬 823 Student Page Read and discuss page 60 of the Student Reference Book with the class. The diagram on the page is another way of showing that the opposite of every positive number is a negative number, and the opposite of every negative number is a positive number. Fractions Negative Numbers and Rational Numbers People have used counting numbers (1, 2, 3, and so on) for thousands of years. Long ago people found that the counting numbers did not meet all of their needs. They needed numbers 1 5 for in-between measures such as 2 2 inches and 6 6 hours. Fractions were invented to meet these needs. Fractions can also be renamed as decimals and percents. Most of the numbers you have seen are fractions or can be renamed as fractions. Note Examples Rename as fractions: 0, 12, 15.3, 3.75, and 25%. 0 0 1 12 12 1 153 15.3 10 375 3.75 100 Every whole number (0, 1, 2, and so on) can be renamed as a fraction. For example, 0 can be 0 written as 1. And 8 can 8 be written as 1. 25 25% 100 However, even fractions did not meet every need. For example, 3 1 problems such as 5 7 and 2 4 5 4 have answers that are less than 0 and cannot be named as fractions. (Fractions, by the way they are defined, can never be less than 0.) This led to the invention of negative numbers. Negative numbers are numbers 1 that are less than 0. The numbers 2, 2.75, and 100 are negative numbers. The number 2 is read “negative 2.” Note Numbers like 2.75 and 100 may not look like negative fractions, but they can be renamed as negative fractions. Negative numbers serve several purposes: ♦ To express locations such as temperatures below zero on a thermometer and depths below sea level 䉴 Using Credits and Debits WHOLE-CLASS ACTIVITY to Practice Addition of Positive and Negative Numbers (Math Masters, p. 321) 11 ♦ To show changes such as yards lost in a football game 2.75 4, and ♦ To extend the number line to the left of zero 100 1 100 Links to the Future ♦ To calculate answers to many subtraction problems The opposite of every positive number is a negative number, and the opposite of every negative number is a positive number. The number 0 is neither positive nor negative; 0 is also its own opposite. Students explore subtraction of positive and negative integers in Lesson 11-6. Addition and subtraction of signed numbers is a Grade 5 Goal. The diagram at the right shows this relationship. The rational numbers are all the numbers that can be written or renamed as fractions or as negative fractions. –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 Display a transparency of Math Masters, page 321. Tell students that in this lesson they pretend that they are accountants for a new business. They figure out the “bottom line” as you post transactions. Student Reference Book, p. 60 Discuss credits (money received for sales, interest earned, and other income) as positive additions to the bottom line, and debits (cost of making goods, salaries, and other expenses) as negative additions to the bottom line. Explain that you will label credits with a “” and debits with a “–” to keep track of them as positive and negative numbers. To support English language learners, clarify any misconceptions about the use of the words credits, debits, and bottom line in this lesson as compared with students’ observations of the use of credit and debit cards at stores. Teaching Master Name LESSON 10 6 䉬 Date Time Ledger Transaction Start End/Start of Next Transaction Change Be consistent throughout this lesson in “adding” credits and debits as positive and negative numbers, because Lesson 11-6 uses the same format to show “subtraction” of positive and negative numbers—the effect on the bottom line of “taking away” what were thought to be credits or debits. Following is a suggested series of transactions. Entries in black are reported to the class; entries in color are appropriate student responses. To support English language learners, discuss the meaning of the words transaction and change. –10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 –1 0 1 2 3 4 5 6 7 8 9 10 Math Masters, p. 321 824 Unit 10 Reflections and Symmetry Student Page Transaction Start Change End/Start of Next Transaction New business, start at $0 $0 Credit (payment) of $5 comes in $0 add $5 $5 Credit of $3 $5 add $3 $8 Debit of $6 $8 add $6 $2 Debit of $8 (Be sure to share strategies.) $2 add $8 $6 Debit of $3 $6 add $3 $9 Credit of $5 (At last!) $9 add $5 $4 Credit of $6 $4 add $6 $2 $0 $0 Games Credits/Debits Game Materials 䊐 1 complete deck of number cards 䊐 1 Credits/Debits Game Record Sheet for each player (Math Masters, p. 468) Players 2 Skill Addition of positive and negative numbers Object of the game To have more money after adjusting for credits and debits. Directions You are an accountant for a business. Your job is to keep track of the company’s current balance. The current balance is also called the “bottom line.” As credits and debits are reported, you will record them and then adjust the bottom line. Each player uses one Record Sheet. 1. Shuffle the deck and lay it number-side down between the players. 2. The black-numbered cards are the “credits,” and the blue- or red-numbered cards are the “debits.” 3. Each player begins with a bottom line of $10. Note 4. Players take turns. On your turn, do the following: If both players have negative dollar amounts at the end of the round, the player whose amount is closer to 0 wins. ♦ Draw a card. The card tells you the dollar amount and whether it is a credit or debit to the bottom line. Record the credit or debit in your “Change” column. ♦ Add the credit or debit to adjust your bottom line. ♦ Record the result in your table. 5. At the end of 10 draws each, the player with more money is the winner of the round. Examples Beth has a “Start” balance of $20. She draws a black 4. This is a credit of $4, so she records $4 in the “Change” column. She adds $4 to the bottom line: $20 $4 $24. She records $24 in the “End” column, and $24 in the “Start” column on the next line. Alex has a “Start” balance of $10. He draws a red 12. This is a debit of $12, so he records $12 in the “Change” column. He adds $12 to the bottom line: $10($12) $2. Alex records $2 in the “End” column. He also records $2 in the “Start” column on the next line. Continue until most students respond with ease. Student Reference Book, p. 238 Adjusting the Activity Have students experiment with their calculators to find out how to enter negative numbers and expressions with negative numbers. On the TI-15 students use the (–) key, and on the Casio fx-55, students use the key. A U D I T O R Y 䉬 K I N E S T H E T I C 䉬 T A C T I L E 䉬 V I S U A L 䉴 Playing the Credits/Debits Game PARTNER ACTIVITY (Student Reference Book, p. 238; Math Masters, p. 468) Students play the Credits/Debits Game to practice adding positive and negative numbers. They record their work on Math Masters, page 468. Game Master Name Date Time Credits/Debits Record Sheets 3 5 4 3 22212019181716151413121110 9 8 7 6 5 4 3 2 1 4 6 Game 1 5 7 Change Record Sheet 6 8 Start 7 9 +$10 8 10 As students play, watch for those who are beginning to devise shortcuts for finding answers. For example, most students will probably count up and back on a number line. Some students may notice that when two positive numbers are added, the result is “more positive”; when two negative numbers are added, the result is “more negative”; and when a positive and a negative number are added, the result is the difference of the two (ignoring the signs) and has the sign of the number that is “bigger” in the sense of being farther from 0. End, and next start 0 1 Game 2 2 1 2 Do not try too hard to get explanations; these will evolve over time as students have more experience with positive and negative numbers. 1 9 2 10 Ongoing Assessment: Informing Instruction 1 2 4 3 238 3 4 5 6 Start 7 +$10 8 End, and next start 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Change Record Sheet Math Masters, p. 468 Lesson 10 6 䉬 825 Student Page Date Time LESSON 10 6 䉬 2 Ongoing Learning & Practice Review: Fractions, Decimals, and Percents 1. Fill in the missing numbers in Fraction the table of equivalent fractions, decimals, and percents. Decimal 4 10 Percent 0.4 6 10 75 100 61 62 40% 60% 0.6 0.75 䉴 Solving Fraction, Decimal, 75% and Percent Problems 2. Kendra set a goal of saving $50 in 8 weeks. During the first 2 weeks, she was able to save $10. 10 50 a. What fraction of the $50 did she save in the first 2 weeks? 20% 10 b. What percent of the $50 did she save? c. At this rate, how long will it take her to reach her goal? 3. Shade 80% of the square. (Math Journal 2, p. 283) weeks 80 100 a. What fraction of the square did you shade? Students solve problems involving equivalent fractions, decimals, percents, and discounts. 0.8 20% b. Write this fraction as a decimal. c. What percent of the square is not shaded? INDEPENDENT ACTIVITY 4. Tanara’s new skirt was on sale at 15% off the original price. The original price of the skirt was $60. a. How much money did Tanara save with the discount? b. How much did she pay for the skirt? 䉴 Math Boxes 10 6 $9 $51 INDEPENDENT ACTIVITY 䉬 (Math Journal 2, p. 284) 5. Star Video and Vic’s Video Mart sell videos at about the same regular prices. Both stores are having sales. Star Video is selling its videos at 1 off the regular price. 3 Vic’s Video Mart is selling its videos at 25% off the regular price. Which store has the better sale? Explain your answer. Star Video has the better sale since 13 33 13%, which Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 10-3. The skill in Problem 5 previews Unit 11 content. is more than 25%. So they’re taking more off their regular prices. 283 Math Journal 2, p. 283 Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 1 夹 Use Math Boxes, Problem 1 to assess students’ ability to express the probability of an event as a fraction. Students are making adequate progress 1 3 if they design a spinner that is 4 red and 4 blue. Many students will design a spinner that has 3 consecutive parts red and 9 consecutive parts blue. Some students will explore other possibilities—for example, 2 consecutive red parts, followed by 4 blue parts, 1 red part, and 5 blue parts. [Data and Chance Goal 4] 䉴 Study Link 10 6 䉬 Student Page Date INDEPENDENT ACTIVITY (Math Masters, p. 322) Time LESSON 10 6 䉬 夹 Math Boxes 1. a. Make a spinner. Color it so that if you spin it 36 times, you would expect it to land on blue 27 times and red 9 times. Sample answer: b. Explain how you designed your spinner. red blue Sample answer: If it lands on blue 27 out of the 36 spins, 27 3 then 36, or 4, of the board should be blue. Likewise, 1 9 , or , of the board should 4 36 be red. Home Connection Students compare and order positive and negative numbers and add positive and negative integers. 82–86 2. Complete. Rule: 14 3. Solve each open sentence. in out 8 16 4 16 a. 67.3 p 75.22 p 8 8 6 8 b. 6.86 a 2.94 a c. x 5.69 7.91 x 3 4 2 4 10 12 7 12 15 20 10 20 d. 4.6 n 0.32 n 7.92 3.92 2.22 4.28 READINESS 55 57 acute RUG is an (acute or obtuse) angle. 4. Angle 3 Differentiation Options 34–37 5. Sebastian and Joshua estimated the weight of their mother. What is the most reasonable estimate? Fill in the circle next to the best answer. 䉴 Exploring Skip Counts SMALL-GROUP ACTIVITY 5–15 Min on a Calculator R A. 50 pounds U B. 150 pounds G C. 500 pounds Measure of ⬔RUG 20 ° D. 1,000 pounds 93 142 143 284 Math Journal 2, p. 284 826 Unit 10 Reflections and Symmetry 140 To explore patterns in negative numbers, have students skip count on the calculator. Ask students to start with 30 and count back by 10s on their calculator as they say the numbers aloud. Stop at –10 and ask the following questions: Study Link Master ● What does the calculator display show after zero? –10 ● How do you read this number? negative ten ● Can you predict what number will come next? –20 Name Date STUDY LINK Time Positive and Negative Numbers 10 6 䉬 Write or to make a true number sentence. 1. 3 14 2. 7 7 19 3. 20 8 4. 60 10 List the numbers in order from least to greatest. Have students continue counting back, stopping at –50. Repeat the routine counting back with other numbers such as 2, 5, 25, and 100. Remind students to clear their calculators after each count. SMALL-GROUP ACTIVITY READINESS 䉴 Using a Number Line to 5. 1 4 8 1 1 2 4 3.4 1.7 5 least 6. greatest 14 7 43, 22, , 5, 3, 0 43 3 least 14 0 5 7 Sample answers: 7. Name four positive numbers less than 2. 1 4 1 2 3 4 8. Name four negative numbers greater than 3. 2 1 2 22 greatest 1 1 1 4 Use the number line to help you solve Problems 9–11. 5–15 Min Add Positive and Negative Numbers 1 2 5, 8, , , 1.7, 3.4 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 9. a. 49 13 b. 4 (9) 10. a. 53 8 b. (5) 3 11. a. 15 2 13 b. 11 1 2 3 4 5 2 (2) 13 5 6 c. (4) (9) c. (5) (3) c. 7 8 9 10 11 12 13 14 15 15 13 8 (2) (13) Practice To explore addition of positive and negative integers using a number line model, have students act out addition problems by walking on a life-size number line from –10 to 10. 12. 14. 13.90 13. 7.26 1.94 5.84 8.75 15. 3.38 1.02 12.88 2.91 5.32 0.76 2.62 Math Masters, p. 322 䉯 The first number tells students where to start. 䉯 The operation sign ( or –) tells which way to face: means face the positive end of the number line. – means face the negative end of the number line. 䉯 If the second number is negative, then walk backward. Otherwise, walk forward. 䉯 The second number (ignoring its sign) tells how many steps to walk. 䉯 The number where the student stops is the answer. Example: –4 3 䉯 Start at –4. 䉯 Face the positive end of the number line. 䉯 Walk forward 3 steps. –5 䉯 You are now at –1. So –4 3 –1. –4 –3 –2 –1 0 1 2 4 3 1 Suggestions: ● –6 –3 ? (Start at –6. Face in the positive direction. Walk backward 3 steps. End up at –9.) –10 –9 ● –8 –7 –6 –5 –4 –3 4 –6 ? (Start at 4. Face in the positive direction. Walk backward 6 steps. End up at –2.) –2 –1 0 1 2 3 4 5 Lesson 10 6 䉬 827

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