· 324 unit 3 • lesson 5 unit 3 • lesson 5 101 name date unit 3 skills maintenance mixed...

324 Unit 3 • Lesson 5

Unit3•Lesson5 101

Name Date

Uni

t 3

SkillsMaintenanceMixedNumbers

Activity1

Addorsubtractthemixednumberswithlikeandunlikedenominators.UsetheLAPSstrategy.

1. 112 + 21

4 33

4

L The problem is aligned and it is addition.

A We will change the first fraction to 4ths.

P 124 + 21

4 = 334

S No simplification needed.

2. 378 − 22

8 15

8

L The problem is aligned and it is subtraction.

A No alteration needed.

P 378 − 22

8 = 158


3. 513 + 61

2 115

6


A We will change the fractions to 6ths.

P 526 + 63

6 = 1156


4. 1256 − 61

2 62

6


A We will change the fractions to 6ths.

P 1256 − 63

6 = 626

S 626 or 61

3 Simplest form has not been taught yet, so it isn’t necessary in the answer.

Lesson5 SkillsMaintenance

Skills MaintenanceMixed Numbers (Interactive Text, page 101)

Activity 1

Students add and subtract mixed numbers with unlike denominators using LAPS.

Skills MaintenanceMixed Numbers

Building Number Concepts: Regrouping With Mixed Numbers

We look at regrouping with mixed numbers. Regrouping with whole numbers is from one place value to another. Regrouping with mixed numbers is from the whole-number component to the fraction component, or vice versa. When you carry in mixed-number addition, you take groupings of fractional parts that equal whole numbers and add them to the whole-number sum.

When you borrow in mixed-number subtraction, you break apart whole numbers into fractional parts. The regrouping step occurs in a different place for addition with mixed numbers (simplify step) than in subtraction with mixed numbers (alter step).

ObjectiveStudents will solve mixed-number addition and subtraction problems that require regrouping.

Monitoring Progress: Quiz 1

Distribute the quiz, and remind students that the questions involve material covered over the previous lessons in the unit.

HomeworkStudents convert improper fractions to mixed numbers, solve problems that involve regrouping, and solve two word problems involving mixed-number operations. In Distributed Practice, students practice whole-number operations.

Lesson Planner

Regrouping With Mixed NumbersMonitoring Progress: Quiz 1

Lesson 5

Unit 3 • Lesson 5 325


When adding mixed numbers, when do we regroup?

We have added and subtracted mixed numbers, but we haven’t had to regroupyet.Sometimeswehavetoregroupthefractionalpartwhenwe add or subtract mixed numbers. This happens in addition when the two fractions add up to more than 1. We do this in the “simplify” step ofLAPS.

L—LOOK at the problem carefully.

• Makesurethenumbersarelinedupcorrectly.

• Decideifadditionorsubtractionissupposedtobeperformed.

Inthisproblem,thefractionsandwholenumbersarelinedup correctly, and we need to add.

A—ALTER the problem if necessary.Inthisproblem,thedenominatorsarethesame,sowedonot have to alter the fractions.

223

+ 123

Regrouping With Mixed Numbers

Regrouping With Mixed NumbersMonitoring Progress: Quiz 1

Lesson 5

with like and unlike denominators without regrouping.

• Explain that we need to regroup when we add mixed numbers when the fractional parts of the two mixed numbers add up to 1 or more

(e.g., 23 + 13 = 33).

• Show the problem 223 + 12

3 . Walk through each step.

L—LOOK• Remind students to make sure the numbers are

lined up correctly and to identify the operation, in this case, addition.

A—ALTER• Remind students that in the alter step, we check

to see if the denominators are the same, as they are in this case.

Building Number Concepts: Regrouping With Mixed Numbers

When adding mixed numbers, when do we regroup?(Student Text, pages 197–199)

Connect to Prior KnowledgeReview regrouping with whole-number operations. Put the following problem on the board:

335 1 27

Ask students to describe the steps required to regroup from one place value to another.

Emphasize that the regrouping happens in the place-value positions with whole numbers. Remind students that there are two parts to a mixed number: the fractional part and the whole-number part.

Link to Today’s ConceptTell students that in today’s lesson, we look at regrouping with mixed numbers.

DemonstrateEngagement Strategy: Teacher ModelingDemonstrate how to regroup with mixed numbers in one of the following ways:

: Use the mBook Teacher Edition for pages 197–199 of the Student Text.

Overhead Projector: Display the

problem 223 + 12

3 on a transparency,

and modify as discussed.

Board: Write the problem 223 + 12

3 on

the board, and modify as discussed.

• Discuss the mixed-number operations we learned to this point. We solved problems

197


Lesson 5


Lesson 5

P—PERFORM the operation.Nowweadd.Westartbyaddingthefractionalpartsofthetwonumbers.

223

+ 123

43

Next, we add the whole numbers.

223

+ 123

343

S—SIMPLIFY the answer.Ouranswerisnotinitssimplestform.Inthisproblem,theanswerincludes an improper fraction. An improper fraction is a fraction that’s equaltoorbiggerthan1.Wehavetoregroupsothatwehaveawholenumber and a fraction that is less than 1.

223

+ 123

343

Thisfractionisbiggerthan 1. We need to regroup.

When adding mixed numbers, when do we regroup? (continued)

Demonstrate• Continue walking through the LAPS steps on

page 198 of the Student Text.

P—PERFORM• Perform the addition. Remind students that

we add the fractional parts, then the whole

numbers. The result is 343 . The fraction

43 is greater than 1 and needs to be

regrouped.

S—SIMPLIFy• Be sure students understand that the

answer includes an improper fraction, or a fraction bigger than 1. This means we must regroup to have a whole number and a fraction less than 1. 198



Lesson 5

We break the fraction 43 into two fractions.

Oneshouldbeequalto1.Inthiscaseit’s33.

The other fraction should be less than 1.

Here it’s 13. After we break up the fraction,

weregroup.

2 23

+ 1 23

33 + 13

We leave the fractional part, or 13, in the

“fractionsplace.”Weregroupbymoving

the 33, or 1, to the ones place.

33 or 1

2 23

+ 1 23

33 + 13

Wefinishtheproblembyaddingthewholenumberstogether.Here,weaddthewholenumber parts 2 + 1 = 3. Then we add the numberweregrouped:3+1=4.Weget4.

1

2 23

+ 1 23

4 13

Our answer is 413 .

Demonstrate• Show students how to regroup, as on page

199 of the Student Text. Make sure students

see that we can break up 43 into 33 + 13 . The

fraction 33 is equal to 1. We add the 1 to

the whole number 3, 1 + 3 = 4. We leave

the remaining 13 in the fractions place. We

rewrite the answer as 413 .

Check for UnderstandingEngagement Strategy: Think Tank

Have students take out a piece of paper to practice this regrouping step. Write the fraction 1 13 Q32

3R on the board, and have students rewrite

the improper fraction as a mixed number. Allow them to use fraction bars, circles, or other pictures at first, if necessary.

Try to move toward a purely numeric representation as students gain familiarity. Ask them to write their solutions and names on their papers. When they finish, collect the papers in a container. Draw out an answer, and read it aloud. If it is correct, congratulate the student. If it is incorrect, invite a student volunteer to share the correct answer. Review the solutions with the

class Q113 S 33 + 33 + 33 + 2

3 = 323R .

Reinforce UnderstandingIf students need more practice regrouping in addition with mixed numbers, have them solve the following problems:

314 + 25

4 Q6 12 R

7 91 0 + 8 1

1 0 (16)

623 + 51

3 (12)

199


Lesson 5


Lesson 5

When subtracting mixed numbers, when do we regroup?

Whenweaddmixednumbers,theregroupingtakesplaceinthe“Simplify”stepofLAPS.Subtractionisdifferent.Whenweregroupinsubtraction, it’s because the fractional part of the top number is smaller thanthefractionalpartofthebottomnumber.Wehavetochangethatinthe“Alter”stepofLAPS.Let’sreviewthestepsforsubtractingwhole numbers.

34 1 8

We cannot work the problem the way it is. The top number in the ones place is smaller than the bottom number in the ones place. We have toregroup.

Let’slookattheregroupinginexpandedform.

We can’t subtract 8 from 4. We have to regroup.

30 4− 10 8

We rewrite the 30 as 20 + 10. Then we borrow the 10 and move it into the ones place.

20 + 10 4− 10 8

We already have 4 in the ones place, so we add the 10 to the 4.

20 10 + 4− 10 8

Now we have 14. We subtract 8 from 14 andget6.Wesubtract20−10inthetensplace.Weget10.Wecombine10+6=16.The answer is 16.

20 14− 10 8

10 6

Answer: 16

When subtracting mixed numbers, when do we regroup?(Student Text, pages 200–202)

Demonstrate• Point out that when adding mixed numbers,

the regrouping takes place in the simplify step of LAPS. Explain that subtraction occurs in a different step.

• Read the paragraph at the top of page 200 of the Student Text with students before reviewing regrouping in subtraction of whole numbers.

• Have students look at the subtraction problem 34 − 18 to review regrouping in subtraction of whole numbers. Elicit from students why we need to regroup.

Listen for:

• The top digit in the ones place is smaller than the bottom digit in the ones place.

• Work through the subtraction in expanded form to demonstrate how to regroup. Remind students that we can rewrite 30 as 20 + 10 to borrow the 10. We add 10 to the 4 in the ones place. Now we can subtract 8 from 14 in the ones place and 10 from 20 in the tens place to get the answer 16.

200



Lesson 5

Theprocessforregroupingmixednumbersinsubtractionissimilartotheprocessforregroupingwholenumbers.Whenwesubtract,weneedtoregroupifthetopfractionalpartissmallerthanthebottomfractional part.

Example 1

Regroup with mixed numbers when subtracting.

L—LOOK at the problem carefully.

• Makesurethenumbersarelinedupcorrectly.

• Decideifadditionorsubtractionissupposedtobeperformed.

Inthisproblem,thefractionsandwholenumbersarelinedupcorrectly,and we need to subtract.

325

− 145

A—ALTER the problem if necessary.The fractions do have a common denominator, so we do not have to changethedenominators.Butwecannotsubtractthefractionsthewaytheyarewritten.Thebottomfractionislargerthanthetopfraction.Weneedtoregroupbeforewecansubtract.

The top fraction, 25, is smaller than the bottom

fraction, 45 ,soweneedtoregroup.

3 25

− 1 45

We break the whole number on the top into a

wholenumberplusafractionequalto1.For

thisproblem,werewrite3as2+1or,using

fifths, 2 + 55.

2 + 55

3 25

− 1 45

Demonstrate• Explain that the process of regrouping when

subtracting mixed numbers is similar to the process for whole numbers. State that we must regroup if the top fractional part is smaller than the bottom fractional part.

• Have students look at Example 1 on page 201 of the Student Text, where we demonstrate regrouping a mixed-number subtraction problem. The problem is

325 − 14

5 . Walk students through the LAPS

steps.

L—LOOK• Remind students to check that the numbers

are lined up correctly and to identify the operation. In this case, the numbers are lined up correctly, and we are subtracting.

A—ALTER• Point out that regrouping for subtraction

takes place in the alter step. We see right

away that we cannot subtract 45 from 25 . We

have to regroup one of the whole numbers. Explain that we break the whole number 3

into 2 and 55. Remind students that 55 = 1.

201


Lesson 5


Lesson 5

We need to move the 55 over and combine

it with the fraction, 25 .Thiswillgiveusa

biggerfractionontop.

2 25 + 55

− 1 45

We add 55 to 25 toget75. Now we

can subtract. 2 7

5

− 1 45

P—PERFORM the operation.Now we subtract.

Westartbysubtractingthefractional parts.

2 75

− 1 45

35

Next, we subtract the whole numbers.2 75

− 1 45

1 35

S—SIMPLIFY the answer.We have the answer we want because it’s a mixed number in itssimplestform.Sowedon’tneedtodoanythinginthisstep.

Answer: 135

Insubtraction,theregroupingtakes place in the “Alter” step. We have to adjust the numbers first before we can perform the operation.

Apply SkillsTurn to Interactive Text, page102.

Reinforce UnderstandingUse the mBook Study Guide to review lesson concepts.

Monitoring ProgressQuiz 1

615 − 33

5 (235 )

434 + 32

4 (8 14)

Ask students to summarize regrouping in addition and subtraction with mixed numbers.

Listen for:

• In addition, the regrouping happens in the simplify step.

• In addition, we take groupings of fractional parts that equal 1, and we add them to the whole numbers.

• In subtraction, the regrouping happens in the alter step.

• In subtraction, we break the whole number into fractional parts to add to the fraction portion.

When subtracting mixed numbers, when do we regroup? (continued)

Demonstrate• Continue working through the regrouping

on page 202 of the Student Text.

• Explain that we add 55 to the 25. Our problem

becomes 275 − 14

5 .

P—PERFORM• Perform the subtraction.

S—SIMPLIFy• The answer is 13

5 . It is already in simplest

form, so we do not need to do anything else.

Check for UnderstandingEngagement Strategy: Look About

Write the problem 413 − 22

3 on the board. Tell

students that they will solve the problem with the help of the whole class. Ask them to write out each step of the process, as well as their solutions, in large writing on a piece of paper or a dry erase board. When students finish their work, they should hold up their answer for everyone to see.

If students are not sure about the answer, prompt them to look about at other students’ solutions to help with their thinking. Review the answers after all students have held up their

solutions. (Rewrite 4 13 as 3 + 33 + 1

3 , or 343. The

new problem is 343 − 22

3 = 123 .)

Reinforce UnderstandingIf you feel students need more practice, have them work the following problems. Again, allow them to use fraction bars, circles, or other pictures if necessary, keeping in mind that they need to move to numeric representations eventually.

202


102 Unit3•Lesson5

Name Date

ApplySkillsRegroupingWithMixedNumbers

Activity1

SolvetheproblemsusingLAPS.

1. 634 + 73

4 142

4


A No alteration needed.

P 634 + 73

4 = 1364

S 1364 = 142

4 or 1412

2. 918 − 23

8 66

8


A Add 1 from 9 to the fraction.

P 898 − 23

8 = 668

S 668 or 62

3

3. 526 − 15

6 33

6


A Add 1 from 5 to the fraction.

P 486 − 15

6 = 336

S 336 or 31

2

Simplest form has not been taught, so it is not yet necessary in the answer.

Lesson5 ApplySkills

Apply Skills(Interactive Text, page 102)

Have students turn to page 102 in the Interactive Text, which provides students an opportunity to practice using LAPS.

Activity 1

Students solve mixed-number problems that require regrouping using LAPS. Monitor students’ work as they complete the activity.

Watch for:

• Do students recognize when a mixed-number problem requires regrouping (e.g., when the top fraction is smaller than the bottom fraction in subtraction and when the sum of an addition problem results in an improper fraction)?

• Do students understand that regrouping in addition occurs in the simplify step?

• Do students understand that regrouping in subtraction occurs in the alter step?

• Can students follow all the procedures for regrouping, perform the operation, and get the correct answer?

Reinforce Understanding Remind students that they can review lesson concepts by accessing the online mBook Study Guide.


Lesson 5

Monitoring Progress: Quiz 1

Assess Quiz 1

• Administer Quiz 1 Form A in the Assessment Book, pages 61–62. (If necessary, retest students with Quiz 1 Form B from the mBook Teacher Edition following differentiation.)

StudentsAssess Differentiate

Day 1 Day 2

All Quiz 1 Form A

Scored80% or above

Extension

Scored Below 80%

Reinforcement

Differentiate• Review Quiz 1 Form A with class.

• Identify students for Extension or Reinforcement.

Extension For those students who score 80 percent or better, provide the On Track! Activities from Unit 3, Lessons 1–5, from the mBook Teacher Edition.

Reinforcement For those students who score below 80 percent, provide additional support in one of the following ways:

■ Have students access the online tutorial provided in the mBook Study Guide.

■ Have students complete the Interactive Reinforcement Exercises for Unit 3, Lessons 1–4, in the mBook Study Guide.

■ Provide teacher-directed reteaching of unit concepts.

Name Date

Uni

t 5

Unit5•Quiz1•FormA 61

Monitoring ProgressAdding and Subtracting Decimal Numbers

Part 1

Solve.

1. 1.24 + 2.321 3.561 2. 10.34 + 5.01 15.35 3. 0.975 − 0.38 0.595

4. 120.5 – 90.6 29.9 5. 4.105 + 3.994 8.099 6. 23.57 – 10.19 13.38

Part 2

Round the decimal numbers.

1. Round to the hundredths place. 12.236 12.24

2. Round to the tenths place. 458.108 458.1

3. Round to a whole number. 34.92 35

Part 3

TheReptileClubisrebuildingitssnakeandlizardhouse. It has to figure out how to arrange the animals. The glass tanksthesnakesandlizardsliveinareallcustom-built, so they have different lengths. Answer the questions based on the table.

1. The Reptile Club wants to put the king cobra next to the reticulated python. How much space do they need for the tanks?

7.27 meters

2. What is the difference in tank length between the sidewinder and the thorny devil?

The sidewinder’s tank is 0.16 meters longer.

3. The Reptile Club wants to make a desert ecosystem exhibit, so they decide to put the gila monster, the sidewinder, and the thorny devil side-by-side. How much space will these reptiles take up?

5.59 meters

ReptileTank Length

in MetersKing Cobra 3.24

Reticulated Python 4.03

Gila Monster 3.55

Black Mamba 3.37

Sidewinder 1.10

Thorny Devil 0.94

Unit 5 Quiz1•FormA

62 Unit5•Quiz1•FormA

Monitoring ProgressAreaofTrianglesandQuadrilaterals

Part 4

Find the area of each triangle, rectangle, and parallelogram.

1.

Area of the triangle 12 square inches

2.

Area of the rectangle 11 square inches

3.

Area of the parallelogram 21 square inches

6 in.

4 in.

11 in.

1 in.

7 in.

3 in.

Unit 5 Quiz1•FormA

Form A


Name Date

©2010 Sopris West Educational Services. All rights reserved.

Final Pages

Unit3•Quiz1•FormB 1

MonitoringProgressMixedNumbers

Part1

Addorsubtractthemixednumbers.

1. 214 + 32

8 54

8 or 512

2. 213 − 11

6 11

6

3. 534 + 21

4 8

4. 412 + 12

5 5 9

10

5. 246 − 11

3 12

6 or 113

Part2

TheScatterPlotsaregoingtowallpaperthebedroomoftheirnewhouse.Theyneedtocutwallpapertherightlengthsothatitgoesfromthefloortotheceiling.Answerthequestionsbasedonthisinformation.

1. The height of the wall in the bedroom is 1012 feet. They have a

strip of wallpaper that is 718 feet. How much wallpaper do they

need?33

8 feet

2. They are putting up their last strip of wallpaper. The strip is

1334 feet long. They only need 101

2 feet. How much should they cut

off from the strip?31

4 feet

Unit 3 Quiz1•FormB

Form B

©2010 Sopris West Educational Services. All rights reserved.

Final Pages

Unit3•Quiz1•FormB 2

Name Date

MonitoringProgressShapes

Part3

Identifythemovementsastranslationsorreflections.

1.

Answer Reflection

2.

Answer Translation

3.

Answer Translation

4.

Answer Reflection

Unit 3 Quiz1•FormB


Lesson 5


Lesson 5

Activity 1

Convert the improper fractions to mixed numbers.

1. 43 11

3 2. 274 63

43. 12

5 225 4. 57

8 718

Activity 2

Use LAPS to add and subtract. Be sure to regroup if necessary.

1. 343

+ 113

2. 512

+ 232

3. 715

− 435

4. 814

− 624

See Additional Answers below..Activity 3

Solve the word problems. Simplify your answers.

Moldingisadecorativestripplacedaroundwindowsorwallstomakearoomlookbetter.TheScatterPlotsarefixingthemoldingonthewindowsinthekitchen.Thatmeansthegroupmemberswillhavetocutpiecesofmoldingsothey fit.

1. Themoldingtheyneedtoputonthebottomofthewindowis413 feet

long.Theyhaveapieceofmoldingthatis414 feetlong.Isthepiecelong

enough?Whatisthedifferencebetweenthepiecetheyhaveandwhatthey need?

Thepieceofmoldingthey have is too short by 1

12 foot.

2. Themoldingalongeachsideofthewindowneedstobe438 feetlong.They

needtwopieces.Whatisthetotallengthofthetwopieces?

Thetotallengthof the two pieces is 83

4 feet.

Activity 4 • Distributed Practice

Solve.

1. 2. 4,002 1,9872,015

3. 4. 90q7,200 803,000+ 8,000

11,000

67 98

6,566

Homework

Homework

Go over the instructions on page 203 of the Student Text for each part of the homework.

Activity 1

Students convert improper fractions to mixed numbers.

Activity 2

Students add and subtract problems using LAPS.

Activity 3

Students solve two word problems involving mixed-number operations.

Activity 4 • Distributed Practice

Students practice whole-number operations. Remind students to use mental math whenever possible (e.g., 13,000 − 5,000 is an extended fact related to the basic fact 13 − 5 = 8). 203

(Additional Answers continue on Appendix, page A5.)

· 324 unit 3 • lesson 5 unit 3 • lesson 5 101 name date unit 3 skills maintenance mixed...

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