Explore Adding And Subtracting 10 M Above With Integers .

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LESSON 10 SESSION 1Explore Adding and Subtractingwith Integers10 m abovewater’s surfacePreviously, you learned how to add positive and negativenumbers. In this lesson, you will learn about subtracting positiveand negative fractions and decimals. Use what you know to try to solve the problem below.A pool’s diving platform is 10 m above the water’s surface. Thebottom of the pool is at 25 m, relative to the surface of thewater. What is the distance between the diving platform andthe bottom of the pool?TRYITMath Toolkit grid paper, integer chips, number linesDISCUSS ITAsk: How does yourwork represent thesurface of the water?Share: In my work . . .represents . . .Learning TargetsSMP 1, SMP 2, SMP 3, SMP 4, SMP 5, SMP 6Apply and extend previous understandings of addition and subtraction to add and subtract rationalnumbers; represent addition and subtraction on a horizontal or vertical number line diagram. Apply properties of operations as strategies to add and subtract rational numbers. Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers183

LESSON 10 SESSION 1CONNECT IT1Look Back What is the distance between the diving platform and the bottomof the pool? How do you know?2Look Ahead In the Try It, you found the distance between two elevationsrepresented by integers. You can also find the distance between non-integers.a. Explain how you can use the number line to find the distance between4.5 and 23.75.543210b. You can use both the expression u4.5 2 (23.75)u and the expressionu23.75 2 4.5u to find the distance between 4.5 and 23.75. Why?212223c. You can subtract to find the difference between 4.5 and 23.75. Explain why23.75 2 4.5 is the opposite of 4.5 2 (23.75).324Reflect How is finding the distance between two numbers on the number linelike finding the difference between two numbers? How is it different?184LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 10 SESSION 1Name:Prepare for Subtracting Positive andNegative Numbers1Think about what you know about numbers and absolute value. Fill in each box.Use words, numbers, and pictures. Show as many ideas as you can.What Is It?What I Know About ItabsolutevalueExamples2Non-ExamplesIs 24 the absolute value of 3 2 7? Explain. Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers185

LESSON 10 SESSION 13The top of a molehill is 4 in. above ground level.The bottom of a mole’s burrow is at 29 in. relative to ground level.a. What is the distance between the top of the molehill and the burrow? Showyour work.SOLUTION b. Check your answer to problem 3a. Show your work.186LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 10 SESSION 2Develop Subtracting Positive andNegative Fractions and Decimals Read and try to solve the problem below.During December, the average daily temperature in a town is 21.7 C. The averagedaily temperature in the same town is 3.2 C lower in January. What is the averagedaily temperature in January?TRYITMath Toolkit grid paper, number lines, place-value chartsDISCUSS ITAsk: What did youdo first to find thetemperature inJanuary?Share: I started by . . . Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers187

LESSON 10 SESSION 2 Explore different ways to understand adding or subtractingnegative decimals.During December, the average daily temperature in a town is 21.7 C.The average daily temperature in the same town is 3.2 C lower inJanuary. What is the average daily temperature in January?Model ItYou can write an expression to represent the situation.You can write a subtraction expression:21.7 2 3.2You can also write an addition expression:21.7 1 (23.2)Model ItYou can use a number line to add or subtract negative decimals.The average temperature in January was 3.2 C lower than in December.2121.52222.52323.223.52424.525188LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

CONNECT IT Use the problem from the previous page to help you understandhow to subtract with negative decimals and fractions.1Look at the first Model It. Why can you write both an addition expressionand a subtraction expression for the situation?2Look at the second Model It. How does the number line represent boththe addition expression and the subtraction expression?3What is the average temperature in January?4How is adding two negative decimals like adding two positive decimals?How is it different?5How is subtracting a negative decimal like subtracting a negative integer?6Reflect Think about all the models and strategies you have discussed today.Describe how one of them helped you better understand how to subtractnegative decimals. Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers189

LESSON 10 SESSION 2Apply Itzero tide Use what you learned to solve these problems.7MondayOn Monday, the low tide at Cook Inlet, Alaska, is21    1 ft. On Tuesday, the low tide is 21   1 ft. How does2··3··the low tide on Tuesday compare to the low tide onMonday? Show your work.SOLUTION 8Find 3    1 2 4   1  . Show your work.4··2··SOLUTION 9Find 5.4 2 7.1. Show your work.SOLUTION 190LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 10 SESSION 2Name:Practice Subtracting Positive andNegative Fractions and Decimals Study the Example showing how to subtractnegative fractions. Then solve problems 1–5.ExampleA path from a dry lake bed starts at an elevation of 212   1 ft relative to sea2··1level. The path ends at an elevation of 60    ft above sea level. What number3··represents the change in elevation from the start to the end of the path?You can subtract to find the change in elevation.60   1 2 1 212   1    2 5 60   1 1 12   1  3··2··2··3··5 60    2 1 12   3 6··6··5 72    5 6··The number that represents the change in elevation is 72   5   .6··1You can also find 12   1    2 60   1 5 272   5  . What information does that give you2··3··6··about the path in the Example?2The lowest point of the dry lake bed in the Example has an elevation of 218    3 ft.4··What number represents the change in elevation from the start of the path to thelowest point in the dry lake bed? Show your work.SOLUTION Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers191

LESSON 10 SESSION 23What is 2.6 2 7.3? Show your work.SOLUTION 4When Daria gets to school, the temperature is 5.7 F. The temperature changesby 29.5 F by the time she goes to bed. What is the temperature when Daria goesto bed? Show your work.SOLUTION 5An otter is swimming at 24.2 yd relative to the surface of the water.It dives 8.6 yd deeper. After the dive, what is the otter’s elevationrelative to the surface of the water?A 212.8 ydB 24.4 ydC 4.4 ydD 12.8 yd192LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 10 SESSION 3Develop Adding and Subtracting Positiveand Negative Fractions and Decimalsrises913.9 m Read and try to solve the problem below.Mei releases a lantern for the Lantern Festival. She stands in a field that is 0.5 mbelow sea level. The lantern rises 913.9 m. Then the candle in the lantern goes out.The lantern comes down 925.2 m to land on the surface of a lake. What is theelevation of the lake relative to sea level?TRYITMath Toolkit grid paper, number lines, place-value chartsDISCUSS ITAsk: What did youdo first to find theelevation of the lake?Share: The firstthing I did to find theelevation was . . . Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers193

LESSON 10 SESSION 3 Explore different ways to understand adding and subtractingpositive and negative numbers.Mei releases a lantern for the Lantern Festival. She stands in afield that is 0.5 m below sea level. The lantern rises 913.9 m.Then the candle in the lantern goes out. The lantern comesdown 925.2 m to land on the surface of a lake. What is theelevation of the lake relative to sea level?Picture ItYou can draw a diagram to model the lantern’s movement.Height of lantern913.41913.92925.2Sea Level20.5FieldLakeModel ItYou can add and subtract from left to right to simplify.20.5 1 913.9 2 925.2 5 20.5 1 913.9 2 925.25 913.4 2 925.2Model ItYou can reorder the terms before simplifying.20.5 1 913.9 2 925.2 5 20.5 1 913.9 1 (2925.2)5 20.5 1 (2925.2) 1 913.95 2925.7 1 913.9194LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

CONNECT IT Use the problem from the previous page to help you understandhow to add and subtract positive and negative numbers.1Look at Picture It. Does the lantern end up lower or higher than its startingpoint? How do the arrows show the lantern’s movement?2Look at the first Model It. What is the value of the expression20.5 1 913.9 2 925.2? How does the expression show the movement ofthe lantern?3Look at the second Model It. Why can you change the order of the terms?4How does changing the order of the terms change the way you find the sum?5Consider the expression 1   5 2 2   1 2 1   1  . To find the value, which two terms might6··3··6··you combine first? Why?6Reflect Think about all the models and strategies you have discussed today.Describe how one of them helped you better understand how to solve theTry It problem. Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers195

LESSON 10 SESSION 3Apply It Use what you learned to solve these problems.78A helicopter is 19.25 meters above the top of acanyon wall. It goes down 27.60 meters, passing intothe canyon. Then it goes up 5.25 meters. What is thenew position of the helicopter relative to the top of thecanyon wall?A 252.1 mB 23.1 mC 3.1 mD 52.1 mThe temperature during the day is 5.4 C. After sunset, the temperature falls 7.5 C.Wind chill makes the temperature after sunset feel 6.3 C colder. What temperaturedoes it feel like after sunset? Show your work.SOLUTION 9What is 25    1 2 9   1 1 2   3  ? Show your work.2··4··4··SOLUTION 196LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 10 SESSION 3Name:Practice Adding and Subtracting Positiveand Negative Fractions and Decimals Study the Example showing adding and subtracting positive andnegative fractions. Then solve problems 1–4.ExampleHummingbirds sip 2   1 fl oz of the liquid in a feeder.2··1Then 1    fl oz of food is added to the feeder. Last, hummingbirds2··sip another 3   1 fl oz of food. What is the overall change in the4··amount of food in the feeder?You can write an expression to represent the situation.22   1 1 1   1 2 3   1 2··2··4··You can begin simplifying the expression by reordering the terms.22    1 1 1   1 2 3   1 5 22   1 2 3   1 1 1   1 2··2··4··2··4··2··5 25    3 1 1   1 4··2··5 24    1 4··The overall change in the amount of food in the feeder is 24   1 fl oz.4··1The feeder in the Example had 16 fl oz of food in it to start. How much food does ithave now? Show your work.SOLUTION Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers197

LESSON 10 SESSION 32Is each expression equivalent to 22.7 1 (23.1)?YesNoa. 22.7 2 3.1b. 22.7 1 (23) 1 (20.1)c. 22 1 (20.7) 1 (23) 1 (20.1)d. 22 1 0.7 1 (23.1)3Reth is playing a game. First, he loses 4.8 points. Then he gains 2.5 points.Finally, he loses another 7.8 points. What is the overall change in Reth’s score?Show your work.SOLUTION 4An autonomous underwater vehicle (AUV) is at an elevation of 28.25 ft. It divesdown 6    2 ft to collect a specimen. Then the AUV dives another 15   3 ft. What is the3··4··final elevation of the AUV? Show your work.AUV: autonomous underwater vehicleSOLUTION 198LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 10 SESSION 4Refine Adding and Subtracting Positiveand Negative Numbers Complete the Example below. Then solve problems 1–10.CONSIDER THIS . . .Distance on a numberline is never negative.ExampleWhat is 2.5 2 4.3?Look at how you could use absolute value to find the difference.1.822.5344.35On a number line, 2.5 and 4.3 are 1.8 units apart. The distancebetween two numbers is the same as the absolute value of theirdifference. So, z2.5 2 4.3z 5 1.8.Since 2.5 2 4.3 has a negative result, the difference is anegative number with an absolute value of 1.8.SOLUTION PAIR/SHAREDoes the method inthe Example work forsubtracting negativenumbers?Apply It1Cece’s dog loses 1   1 lb. Then her dog loses another 1   1 lb. Finally, it48····3gains   lb. What is the total change in her dog’s weight? Show your work.8··SOLUTION Curriculum Associates, LLC Copying is not permitted.CONSIDER THIS . . .The result would bethe same if Cece’sdog lost weight, thengained weight, then lostweight.PAIR/SHARESuppose Cece’s dogweighs 38 pounds tostart. Does that changeyour answer? Why?LESSON 10 Add and Subtract Positive and Negative Numbers199

LESSON 10 SESSION 42 What is 28.3 2 (25.4)? Show your work.CONSIDER THIS . . .The inverse ofsubtraction is addition.PAIR/SHAREHow can you checkthat your answer makessense?SOLUTION 3 Which value of x makes the following statement true?CONSIDER THIS . . .Zero is not a positive ora negative number.20.5 2 x 5 a positive numberA x 5 1.5B x 5 0.5C x 5 20.5D x 5 21.5Lulu chose A as the correct answer. How might she have gotten that answer?PAIR/SHAREWhat is anothervalue that makes thisstatement true?200LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

4A submarine 50.2 m below the surface of the oceangoes up 15.6 m. Then it goes down 35.7 m. What is thesubmarine’s new position relative to the surface of theocean? Show your work.SOLUTION 5Which situation could the expression 28 2 10 represent?A An elevator stops at the 8th floor. Then it stops at the 10th floor.B Gabe earned 8. Then he spent 10 on a new book.C At a bus stop, 8 people got off and 10 people got on.D The temperature dropped 8 F. Then it dropped another 10 F.6Without calculating, explain how you can tell if the value of the expression29.45 2 (21.72) 2 2.53 is positive or negative.7The value of 24    5 2 x is positive. What are two possible values of x?8··Show your work.SOLUTION Curriculum Associates, LLC Copying is not permitted.LESSON 10 Add and Subtract Positive and Negative Numbers201

LESSON 10 SESSION 48Is a 2 b a positive number or a negative number? How do you know?b9021a1Esteban stands on a dock 1.25 yd above the water. A fish swims below Esteban at24.61 yd from the surface of the water. How many yards apart are Esteban andthe 666777777888888999999Math Journal Write a subtraction problem involving two negative decimalnumbers. At least one decimal must include hundredths. Explain how to solveyour problem.End of Lesson Checklist INTERACTIVE GLOSSARY Write a new entry for represent. Write at least one synonymfor represent. SELF CHECK Go back to the Unit 2 Opener and see what you can check off.202LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 1Explore Multiplying and Dividingwith Negative NumbersPreviously, you learned about multiplying integers. In thislesson, you will learn about multiplying and dividing withrational numbers. Use what you know to try to solve the problem below.Ryan is washing windows on a skyscraper. He lowers himself six times, going downan equal distance each time. In all, Ryan lowers himself 48 ft. What integerrepresents the change in Ryan’s position each time he lowers himself?What does this integer tell you?TRYITMath Toolkit grid paper, integer chips, number linesDISCUSS ITAsk: How would youexplain what theproblem is asking inyour own words?Share: The problem isasking . . .Learning TargetsSMP 1, SMP 2, SMP 3, SMP 4, SMP 5, SMP 6, SMP 8   Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zeropppdivisor) is a rational number. If p and q are integers, then 2 1  q   2 5  (2q ) 5  (2q)  . Interpret quotients of rational numbers by··········describing real-world contexts.   Apply properties of operations as strategies to multiply and divide rational numbers. Curriculum Associates, LLC Copying is not permitted.LESSON 12 Multiply and Divide with Negative Numbers233

LESSON 12 SESSION 1CONNECT IT1Look Back What integer represents the change in Ryan’s position each timehe lowers himself? How does the integer show that Ryan is going lower?2Look Ahead One way to find the rational number that represents Ryan’schange in position each time he lowers himself is to divide. An integer can bedivided by any integer except 0. Any rational number can be written as thequotient of two integers, where the divisor is not 0. That means every rationalnumber can be written in the form  a   , where a and b are integers and b Þ 0.b··a. Another way to think of a rational number is as the fraction  a   . Explain why itb··makes sense that a can equal 0, but b cannot.b. How can you write 0.07 in the form  a   ? How can you write 1.07 in the form  a   ?b··b··c. When a rational number is negative, you can express it as a fraction in threedifferent forms. For example, you can write 24 as   24   ,   4   , and 2  4   .1 ···21···1·· Express each fraction above as an expression using the division symbol (4).How does this show that the fractions are equivalent?3Reflect Why is ···  23 a rational number?4234LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 1Name:Prepare for Multiplying and Dividingwith Negative Numbers1Think about what you know about fractions and negative numbers. Fill in eachbox. Use words, numbers, and pictures. Show as many ideas as you can.WordIn My Own g says that the opposite of  22 is   5  . Is Doug correct? Explain. Curriculum Associates, LLC Copying is not permitted.5···22···LESSON 12 Multiply and Divide with Negative Numbers235

LESSON 12 SESSION 13Tameka pilots a hot air balloon. She lowers the balloonfive times, going down an equal distance each time. Inall, Tameka lowers the balloon 75 m.a. What integer represents the change in the hot airballoon’s elevation each time Tameka lowers it?What does this integer tell you? Show your work.SOLUTION b. Check your answer to problem 3a. Show your work.236LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 2Develop Multiplying NegativeRational NumbersMount McKinley Trail MapEnd Read and try to solve the problem below.Mt. WhitneyTrailMount Whitney Trail starts at an elevation of 8,350 ft abovesea level and ends at an elevation of 14,500 ft above sea level.The temperature changes by 25.4 F for every thousand feetgained in elevation. What is the change in temperature fromthe starting elevation to the ending elevation of MountWhitney Trail?TRYITStartJohn MuirTrailMath Toolkit grid paper, number linesDISCUSS ITAsk: How do youknow your answer isreasonable?Share: My answer isreasonable because . . . Curriculum Associates, LLC Copying is not permitted.LESSON 12 Multiply and Divide with Negative Numbers237

LESSON 12 SESSION 2 Explore different ways to multiply positive and negative rational numbers.Mount Whitney Trail starts at an elevation of 8,350 ft above sea level and endsat an elevation of 14,500 ft above sea level. The temperature changes by 25.4 Ffor every thousand feet gained in elevation. What is the change in temperaturefrom the starting elevation to the ending elevation of Mount Whitney Trail?Picture ItYou can use a diagram to understand the relationship.11,000 ft 11,000 ft 11,000 ft 11,000 ft 11,000 ft 11,000 ft8,3509,35025.4 F25.4 F10,35011,35025.4 F12,35025.4 F13,35025.4 F1100 ft14,35025.4 F150 ft14,45014,50020.54 F 20.27 FModel ItYou can multiply to find the change in temperature.The change in elevation is 14,500 2 8,350 5 6,150 ft.The temperature decreases 5.4 F every 1,000 ft. That is a rate of  25.4 F per foot.1,000·····change in temperature 5 rate of temperature change 3 change in elevation5   25.4 ( 6,150)1,000·····5   25.4 3 6,150 1,000···········5 25.4 1   6,150   2 1,000·····5 25.4(6.15)238LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

CONNECT IT Use the problem from the previous page to help you understand how tomultiply positive and negative rational numbers.1How does the Picture It model 25.4(6.15)?2What is the temperature change from the beginning to the ending of the trail?What does this number mean in the context of the problem?3Why does 25.4(6.15) have the same product as 2  54  615   ?4You can start solving the problem by finding 25.4 3 6. How is this similar tofinding 5.4 3 6? How is this different from finding 5.4 3 6?5How is multiplying negative rational numbers like multiplying negative integers?6Reflect Think about all the models and strategies you have discussed today.10··100···Describe how one of them helped you better understand how to multiplyrational numbers. Curriculum Associates, LLC Copying is not permitted.LESSON 12 Multiply and Divide with Negative Numbers239

LESSON 12 SESSION 2Apply It Use what you learned to solve these problems.7A peregrine falcon dives for prey. Its elevation changesby an average of 211.5 meters every second. The divelasts for 3.2 seconds. What is the change in the falcon’selevation? What does this mean in the context of theproblem? Show your work.SOLUTION 8Find 2   1 (227). Show your work.9··SOLUTION 9A freezer in a lab is set so that the temperature changes at a rate of 21.6 C everyhour. What is the change in the temperature after 4   1 h? Show your work.4··SOLUTION 240LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 2Name:Practice Multiplying Negative Rational Numbers Study the Example showing how to multiply negative rational numbers.Then solve problems 1–5.ExampleWhat is 26 1 28   1   2 ?2··28   1 5 2  17 22·····The product of two negative rational numbers is positive.26 1 28   1   2 5 2  6 12  17  2 2··1··2···5 511a. Show how to solve the problem in the Example by multiplying decimals.b. Is your answer to problem 1a greater than, less than, or equal to the answer inthe Example? Why?Vocabulary2Find 2   3 121   1    2 . Show your work.5··3··SOLUTION Curriculum Associates, LLC Copying is not permitted.rational numbera number that can beexpressed as afraction  ba where··a and b are integersand b Þ 0. Rationalnumbers includeintegers, fractions,repeating decimals,and terminatingdecimals.LESSON 12 Multiply and Divide with Negative Numbers241

LESSON 12 SESSION 23a. Find 20.2(4.75). Show your work.SOLUTION b. Explain how you can use your answer to problem 3a to find 20.2(24.75).4Amare drops a probe into a lake. The elevation of the probe changesby 21    1 feet every second. The probe takes 50 seconds to reach2··the bottom of the lake. How deep is the lake? Show your work.Elevation changes by121 2 ft every second.SOLUTION 5Salvador drives 6.5 miles to his aunt’s house. For every mile Salvador drives, theamount of gas in the car’s gas tank changes by 20.04 gallon. What is the totalchange in the gallons of gas in the car’s tank? Show your work.SOLUTION 242LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 3Develop Dividing NegativeRational Numbers Read and try to solve the problem below.After a flood, a creek’s water level changes by 22   4 ft in 3   1 h.5··2··What is the average change in the creek’s water level each hour?TRYITMath Toolkit grid paper, number linesDISCUSS ITAsk: What did you dofirst to find the changein the water level?Share: First, I . . . Curriculum Associates, LLC Copying is not permitted.LESSON 12 Multiply and Divide with Negative Numbers243

LESSON 12 SESSION 3 Explore different ways to divide negative rational numbers.After a flood, a creek’s water level changes by 22    4 ft in 3   1 h. What is the average5··2··change in the creek’s water level each hour?Picture ItYou can use a picture to make sense of the problem.Water Level (ft)?2245Time (h)1312Model ItYou can divide fractions.Divide the total change in the level of the water by the number of hours to get theaverage change in the level each hour.22   4 4 3   1 5 2  14 4  7 5··2··5···2··5 2  14  2 5··7··Model ItYou can divide decimals.22   4 5 22   8 5··10···5 22.82443   1 5 3    5 2··10···5 3.522.8 4 3.5LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

CONNECT IT Use the problem from the previous page to help you understand how todivide positive and negative rational numbers.1Look at the second Model It. Without doing any calculations, explain how youcan use 2.8 4 3.5 to find 22.8 4 3.5.2What is the average change in the creek’s height each hour? How does eachModel It help you see if the average change in the creek’s height each hour ispositive or negative?3The product of a positive number and a negative number is negative. Explain whythe quotient of a negative number and a positive number is negative.4How is dividing with negative rational numbers like dividing with positive rationalnumbers? How is it different?5Reflect Think about all the models and strategies you have discussed today.Describe how one of them helped you better understand dividing withnegative numbers. Curriculum Associates, LLC Copying is not permitted.LESSON 12 Multiply and Divide with Negative Numbers245

LESSON 12 SESSION 3Apply It Use what you learned to solve these problems.6What is 16.8 4 (23.5)? Show your work.SOLUTION 7What is 2   3 4 1 2  1   2 ?8··2··A 2  3 4··B 2   3 16··C   3 16··D  3 4··8A helicopter’s altitude changes by 228    1 ft in 3   3 s.8··4··What is the average change in the helicopter’s altitudeeach second? Show your work.SOLUTION 246LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 3Name:Practice Dividing Negative Rational Numbers Study the Example showing how to divide with negative rational numbers.Then solve problems 1–5.ExampleOn a winter day, the temperature changes by 212.6 F in 2    1 h.4··What rational number represents the average change in thetemperature each hour?The total change in the temperature is 212.6 F.The amount of time is 2   1  , or 2.25, h.212.6 4 2.25 5 25.64··The number 25.6 represents the average hourly temperature change.1Explain how to use the Example to find the quotient of 212.6 and 22.25.2What is 26    4 4 1 2  2   2 ? Show your work.5··5··SOLUTION Curriculum Associates, LLC Copying is not permitted.LESSON 12 Multiply and Divide with Negative Numbers247

LESSON 12 SESSION 33Paula is scuba diving. Her elevation changes by 237    4 m in 4.5 min.5··a. What rational number represents the average change in Paula’selevation each minute? Show your work.SOLUTION b. Is Paula going deeper in the water or back toward the surface?How do you know?4What is 57.2 4 (210,000)? Show your work.SOLUTION 5A full bathtub drains at a constant rate. The amount of water in the bathtubchanges by 24    1 gallons in  3 minute. What is the rate, in gallons per minute,2··4··at which the amount of water in the bathtub changes? Show your work.SOLUTION 248LESSON 12 Multiply and Divide with Negative Numbers Curriculum Associates, LLC Copying is not permitted.

LESSON 12 SESSION 4Refine Multiplying and Dividing withNegative Numbers Complete the Example below. Then solve problems 1–10.CONSIDER THIS . . .You can multiply in anyorder.ExampleWhat is the product of 2  3   , 2  2   , and 2  5   ?8··3··2··Look at how you could show your work using multiplication.3823523283655 ···   2  ··   242155 ··   2  ··   42522  ·· 1 2  ··   2 2  ··   5 1 2  ··  2  ··    2 2  ··   12121 21 2PAIR/SHAREHow would the answerchange if 2  3 were  3   ?SOLUTION 8·8·Apply It1Noor, Layla, and Diego play a trivia game. Noor’s score is 22. Layla’s scoreis   3 of Noor’s score. Diego’s score is  2 of Layla’s score. What is Diego’s score?43····Show your work.SOLUTION Curriculum Associates, LLC Copying is not permitted.CONSIDER THIS . . .You can multiply tofind  3 of a quantity.4·PAIR/SHAREHow can you findDiego’s score anotherway?LESSON 12 Multiply and Divide with Negative Numbers249

LESSON 12 SESSION 42 Yolanda measures the depth of the water in a reservoir. In one week,the depth changes from 2,366.74 ft to 2,354.63 ft. What rational numberrepresents the average change in depth each day? Show your work.SOLUTION 3 What is 4.8(22.5) 4 (20.1)?A 1.2CONSIDER THIS . . .Is the change in depthpositive or negative?PAIR/SHAREWhat does the rationalnumber you foundmean in the context ofthe problem?CO

188 LESSON 10 Add and Subtract Positive and Negative Numbers Curriculum Associates, LLC Copying is not permitted. Explore different ways to understand adding or subtracting negative decimals. During December, the average daily temperature in a town is 21.7 C. The ave

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Adding and Subtracting Fractions Objective 1: Adding or subtracting fractions with common denominators When adding or subtracting fractions, all you need to do is add or subtract the numerators if the denominators are the same. Example: Add or subtract. Remember to leave all fractions in lowe

Virginia Department of Education 2018 1 Mathematics Instructional Plan – Algebra I Adding and Subtracting Polynomials Using Algebra Tiles Strand: Expressions and Operations Topic: Adding and subtracting polynomials Primary SOL: A.2 The student will perform operations on polynomials, including b) adding, subtracting, multipl

Adding and Subtracting Rational Expressions Return to Table of Contents Slide 51 / 179 Just as in multiplication and division, when adding or subtracting rationals, use the same rules as basic fractions. Recall: When adding and subtracting fractions, you MUST use common denominators. Adding and

Adding and Subtracting Rational Expressions Add or subtract. Identify any x-values for which the expression is undefined. Example 1A: Adding and Subtracting Rational Expressions with Like Denominators Add the numerators. The expression is undefined at x –4 because this value makes

So adjust answer by adding 1. Answer is 81. 76 40 can be rounded to 80 40 40. 76 is 4 less than 80. So adjust answer by subtracting 4. Answer is 36. Regrouping is another mental math strategy for adding and subtracting numbers. Regroup numbers into 5s or 10s to make calculations

Adding Fractions with Unlike Denominators . 32 Adding Mixed Numbers with Like Denominators .34 Adding Mixed Numbers with Unlike Denominators .36 Subtracting Fractions with Like Denominators .38 Subtracting Fractions with Unlike Denominators .40 Subtracting Fractions from Whole Numbers .

Step 4: Since our fraction part now has equal sized slices, we can subtract their numerators. Subtracting their numerators we have, of a whole. Step 5: Combining our whole number and fraction parts we get, . Exercise 5: Subtract the fractions, Step 1: Subtracting the whole number part, we get Step 2: Subtra

Curriculum Framework. In addition, the Enhanced Scope and Sequence provides teachers with sample lesson plans aligned with the standards and their related essential understandings, knowledge, and skills. School divisions and teachers can use the Enhanced Scope and Sequence as a resource for developing sound curricular and instructional programs. These materials are intended as examples of ways .