em3cug4tlg1 214 u03l11 - everyday math · pdf file214 unit 3 multiplication and division;...

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

214 Unit 3 Multiplication and Division; Number Sentences and Algebra

Additional InformationAn open sentence is a number sentence that contains one or more variables, such as 3 + x = 5. When the variable x is

replaced by a number in 3 + x = 5, the sentence becomes either true or false: 3 + 2 = 5 is true, but 3 + 4 = 5 is false.

Teacher’s Reference Manual, Grades 4–6 pp. 284–297

Open SentencesObjectives To introduce vocabulary and notation for open

sentences; and to provide practice solving open sentences.s

Key Concepts and Skills• Add, subtract, multiply, and divide to solve

open sentences.

[Operations and Computation Goals 1–4]

• Use a “guess-and-check” strategy to make

reasonable estimates for open sentences.

[Operations and Computation Goal 6]

• Identify the solution of an open sentence.

[Patterns, Functions, and Algebra Goal 2]

• Determine whether number sentences are

true or false.


Key ActivitiesStudents learn about open sentences and

their solutions. They participate in the Broken

Calculator activity to reinforce the concept of

open sentences and to practice estimation.

Ongoing Assessment: Informing Instruction See page 215.

Ongoing Assessment: Recognizing Student Achievement Use a Math Log or Exit Slip. [Patterns, Functions, and Algebra Goal 2]

Key Vocabularyvariable � open sentence � solve � solution

MaterialsMath Journal 1, pp. 73 and 74

Study Link 3�10

Math Masters, p. 388 or 389; p. 424

transparency of Math Masters, p. 425

(optional) � slate � calculator � overhead

calculator (optional)

Using a Map ScaleMath Journal 1, p. 75

ruler

Students use a map scale to convert

measurements to actual distances.

Math Boxes 3�11Math Journal 1, p. 76

Students practice and maintain skills

through Math Box problems.

Study Link 3�11Math Masters, p. 99

Students practice and maintain skills

through Study Link activities.

READINESS

Using Fact Triangles to Solve Open SentencesMath Masters, p. 100

º, / Fact Triangles

Students explore the concept of open

number sentences.

ENRICHMENTSolving Open SentencesMath Masters, p. 101

Students find missing values for letters.

EXTRA PRACTICESolving Broken-Calculator ProblemsMath Masters, p. 424

Students practice solving open sentences.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

��

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Lesson 3�11 215

Getting Started

Math MessageIs this sentence true or false? The sum of 10 and some number is 15. Be ready to explain your thinking.

Study Link 3�10 Follow-Up Partners compare answers. Ask students to rewrite Problem 14 so that it is a false number sentence and Problem 13 so that it is a true number sentence.

Mental Math and ReflexesStudents solve addition and subtraction problems mentally and share their strategies. Suggestions:

16 + 5 = 21

8 + 14 = 22

25 - 8 = 17

28 - 9 = 19

70 + 40 = 110

180 + 50 = 230

190 - 60 = 130

210 - 30 = 180

92 + 59 = 151

76 + 25 = 101

92 - 48 = 44

184 - 126 = 58

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

The Math Message is likely to cause some confusion. Students should conclude that they can’t tell because some information is missing, but some students may make good arguments for other conclusions.

Tell students that in this lesson they will explore number sentences with missing information and learn to solve them.

� Exploring the Meaning of WHOLE-CLASSDISCUSSION

Open SentencesAlgebraic Thinking Now write the same sentence with math symbols:

10 + x = 15

In this sentence, the letter x stands for the missing number. A different letter could also be used; for example, 10 + n = 15. Any letter or other symbol that is not a number will do. A letter or symbol that stands for a missing number is called a variable.

Now ask students what number they would write in place of x to change 10 + x = 15 into a true number sentence. 5, because 10 + 5 is equal to 15.

A sentence that has a variable in it, such as 10 + x = 15, is called an open sentence. To solve an open sentence, replace the variable with a number that makes the sentence true. The number that makes the number sentence true is called the solution. The solution of 10 + x = 15 is the number 5.

Ongoing Assessment: Informing Instruction

Watch for students who have difficulty with

variables when they are positioned in

different places. For example, a student may

have little difficulty with a problem such as

15 - x = 9 but struggle with a problem such

as x - 6 = 9. Suggest that students write the

number sentence with their solution to see if

it makes sense.

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Adjusting the Activity


10 + x= 15

Example:

Teacher: 10 + x = 15

Student: 10 + 5 = 15

NOTE The Broken Calculator activity is

a good way to reinforce the idea that the

solution of an open sentence is a number

that makes the sentence true. It is an activity

you can do with students from time to time

to remind them of this basic idea. Broken

Calculator is also an excellent routine for

practicing estimation.

� Finding Solutions of WHOLE-CLASS ACTIVITY

Open SentencesAlgebraic Thinking Use a procedure like the following. (See margin.)

� Write an open sentence on the board.

� Students solve the open sentence.

� On their slates, students write the number sentence with the solution in place of the variable. They circle the solution.

� If students disagree on the solution, they check their solutions on their calculators.

Begin with problems like the following:

� 12 + x = 55 � 36 / p = 9

12 + 43 = 55 36 / 4 = 9

� 2 ∗ x = 18 � 17 = z - 8

2 ∗ 9 = 18 17 = 25 - 8

� 21 - 8 = n � k / 6 = 10

21 - 8 = 1 3 60 / 6 = 10

� 14 = t - 9 � m / 25 = 4

14 = 23 - 9 100 / 25 = 4

Have students restate the open sentences in words. For example,

for 12 + x = 55, ask: What number added to 12 will equal 55? For 2 ∗ x = 18,

ask: 18 is 2 times as many as what number?

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

� Introducing the Broken WHOLE-CLASS ACTIVITY

Calculator Activity(Math Masters, pp. 424 and 425)

Algebraic Thinking Ask students to pretend that the minus key on their calculator is broken. Write the following open sentence on the board, and ask students to solve it using their calculators but without using the minus key:

452 + x = 735


Have the class share solution strategies. Use an overhead calculator, if available. Students who are very skilled in mental computation may have subtracted 452 from 735 in their heads. Others probably replaced the variable x with various numbers until they found a true number sentence. This guess-and-check strategy can be organized in a table like the one shown in the margin.

The solution of 452 + x = 735 is 283, because 452 + 283 = 735 is true.

Pose a few more problems like the following on a transparency of Math Masters, page 425. Have students record their work on Math Masters, page 424.

Lesson 3�11 217

Date Time

Broken Calculator LESSON

3�11

Æ

Solve each open sentence on your calculator without using the “broken” key. Only one key is broken in each problem. Record your steps.

1. 2.

3. 4.

5.6. Make up one for your partner to solve.

Sample answers:

+

÷

– –Broken Key:To Solve: 68 � x � 413

68 � 350 � 418 too much68 � 345 � 413 Got it!

Broken Key:To Solve: z � 643 � 1,210

600 � 643 � 1,243 too much550 � 643 � 1,193 too little560 � 643 � 1,203 closer567 � 643 � 1,210 Got it!

Broken Key:To Solve: d � 574 � 1,437

2,000 � 574 � 1,426 too little2,010 � 574 � 1,436 closer2,011 � 574 � 1,437 Got it!

Broken Key:To Solve: w / 15 � 8

100 � 15 � 6.667 too little120 � 15 � 8 Got it!

Broken Key:To Solve: s � 48 � 2,928

50 � 48 � 2,400 too little60 � 48 � 2,880 closer65 � 48 � 3,120 too much61 � 48 � 2,928 Got it!

Broken Key:To Solve:

Answers vary. Answers vary.

Try This

Math Journal 1, p. 73

Student Page

� Solving Broken Calculator PARTNER ACTIVITY

Problems(Math Journal 1, p. 73)

The journal page contains five Broken Calculator problems and a blank table on which students write problems for their partners to solve.

Ongoing Assessment: Math Log or

Exit SlipRecognizing Student Achievement

Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess

students’ ability to use and explain a strategy for solving open number sentences.

Have students explain the strategy they used to solve Problem 1, 2, 3, or 4 on

journal page 73. Students are making adequate progress if their strategy involves

using estimation to close in on the solution to the open sentence. Some students

may be able to explain how they solved Problem 5, which involves estimating the

product of two 2-digit numbers.


� Solving Open Sentences INDEPENDENTACTIVITY

(Math Journal 1, p. 74)

Have students solve open sentences and rewrite each sentence with the solution in place of the variable.

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BBBBBBBBBBBBBBBBBBBBB EEEMMMMMMMMOOOOOOOOOBBBBBBBBROOOOROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLVINVINVINNNNVINVINNVINVINVINVINVINVINVV GGGGGGGGGGGOOOOLOLOOLOOLOO VINVINVINVLLLLLLLLLVINVINVINVINNVINVINVINVINVINVINVINV NGGGGGGGGGGOLOOLOLOLOLOLOLOOO VVVVLLLLLLLLLLVVVVVVVVVOSOSOOSOSOSOSOSOSOSOOSOSOSOSOOOSOOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVLLLVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIISOLVING

Open Sentence Broken Key Solution

75 + x = 415 340

y + 128 = 563 435

r - 156 = 954 1,110

p / 34 = 27 918

y / 29 = 52 1,508

19 ∗ t = 1,330 70

–Broken Key:

To Solve: 452 + x = 735

452 + 300 = 752 too much

452 + 250 = 702 too little

452 + 280 = 732 very close

452 + 283 = 735 Got it!



2 Ongoing Learning & Practice

� Using a Map Scale INDEPENDENTACTIVITY


Social Studies Link Students measure the distances between locations on a map of Egypt. They use the map scale to convert the measurements to actual distances.

� Math Boxes 3�11 INDEPENDENTACTIVITY


Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-10. The skill in Problem 5 previews Unit 4 content.

Writing/Reasoning Have students write a response to the following: In Problem 1b, you wrote the factor pairs of 16. Is 16 a prime number or a composite number? Explain how you know. Sample answer: Composite. Composite numbers have more than one factor pair and prime numbers have only one factor pair. The number 16 has 3 factor pairs.

Date Time

Estimating DistancesLESSON

3�11

You want to take a trip to Egypt and see the following sights:

� Cairo, the capital, on the Nile River, near the Pyramids at Giza

� Alexandria, a busy modern city and port on the Mediterranean

� The Aswan High Dam across the Nile River, completed in 1970, and Lake Nasser,

which formed behind the dam

� The temples at Abu Simbel, built more than 3,000 years ago and moved to their

present location in the 1960s to escape the rising water of Lake Nasser

You want to know how far it is between locations.

1. The distance between Alexandria and Abu Simbel is about 3 inch(es) on the map.

That represents about 600 miles.

2. The distance between Cairo and Aswan is about 2 inch(es) on the map.


3. The distance between Abu Simbel and Aswan is about inch(es) on the map.


Alexandria

Giza

Suez

Canal

Lake

Nasser

Nile

Luxor

Aswan

Abu Simbel

0 100

1 inch represents 200 miles

200 mi

Cairo

EGYPT

1 _ 2

145

EM3MJ1_G4_U03_53-77.indd 75 1/7/11 1:22 PM


Student Page

Open SentencesLESSON

3�11

Date Time

Solve each open sentence. Copy the entire sentence with the solution

in place of the variable. Circle the solution.

1. 48 + d = 70 2. 51 = n + 29

48 + 22 = 70

51 = 22 + 29

3. 34 - x = 7 4. 32 = 76 - p

34 - 27 = 7

32 = 76 - 44

5. h - 6 = 9 6. b - 7 = 12

15 - 6 = 9

19 - 7 = 12

7. u - 30 = 10 8. 5 ∗ m = 35

40 - 30 = 10

5 ∗ 7 = 35

9. y = 3 ∗ 8 10. 21 / x = 7

24 = 3 ∗ 8

21 / 3 = 7

11. x = 32 / 8 12. 5 = w / 10

4 = 32 / 8

5 = 50 / 10

13. Mr. O’Connor wrote two open sentences on the board.

45 + x = 71

45 + y = 71

Isabel says the two open sentences must have different solutions because

the variables are different.

a. Do you agree with Isabel? no

b. Explain your answer.

Sample answer: In both sentences the variable equals 26.

You can use any variable in a number sentence—different

variables do not necessarily mean different values.

148

Try This

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Student Page

Date Time

5. a. Measure the line segment to the nearest centimeter.

About 11 cm

b. Draw a line segment that is half the length of L__P .

c. How long is the line segment you drew? About 5.5 cm

Math Boxes LESSON

3�11

4. Which of the angles below have a

measure of more than 90 degrees?

Circle them.

1. Complete.

a. Name all the factors of 12.

1 , 2 , 3 , 4 , 6 , 12

b. Name the factor pairs of 16.

1 and 16

2 and 8

4 and 4

2. The areas of which two states differ

by 944 square miles?

Rhode Island and Delaware

3. Use the bar

graph to answer

the questions.

a. How many

students slept

8 hours?

7 b. What is the mode for the number

of hours slept?

9

7

128

State Total Area

Connecticut 5,543 square miles

Rhode Island 1,545 square miles

Delaware 2,489 square miles

New Jersey 8,721 square miles

6 7 8 9 100

2

4

6

8

10

Hours Slept

Number of Hours StudentsSlept Last Night

Nu

mb

er

of

Stu

de

nts

P L

9373

EM3MJ1_G4_U03_53-77.indd 76 1/7/11 1:22 PM


Student Page


� Study Link 3�11 INDEPENDENTACTIVITY

(Math Masters, p. 99)

Home Connection Students tell whether number sentences are true or false, make true number sentences by filling in missing numbers and inserting parentheses, and find solutions for open sentences.

3 Differentiation Options

READINESS PARTNER ACTIVITY

� Using Fact Triangles to 5–15 Min

Solve Open Sentences(Math Masters, p. 100)

To explore the concept of open number sentences, have students use Multiplication/Division Fact Triangles to write and solve open sentences. For example:

Each letter in the animal names on this page has a value.

Some of the values of the letters are known. Some of the values of the letters are unknown.

Use the information below to find the unknown values.

COW is worth 23. KOALA is worth 46. DONKEY is worth 66.

MONKEY is worth 54. LION is worth 35. PANDA is worth 83.

LESSON

3�11

Name Date Time

Solve Open Sentences

C E I L M W Y

8 17 2 12 9 10 4

Each letter in the animal names on this page has a value.

Some of the values of the letters are known. Some of the values of the letters are unknown.

Use the information below to find the unknown values.

COW is worth 23. KOALA is worth 46. DONKEY is worth 66.

MONKEY is worth 54. LION is worth 35. PANDA is worth 83.

LESSON

3�11

Name Date Time

Solve Open Sentences

A D K N O P

13 21 3 16 5 20

C E I L M W Y

8 17 2 12 9 10 4

A D K N O P

Math Masters, p. 101

Teaching Master

Lesson 3�11 219

STUDY LINK

3�11 Open Sentences

148

Name Date Time

Write T if the number sentence is true and F if the number sentence is false.

1. 35 � 7 º 5 2. 43 � 34

3. 25 � 25 � 50 4. 49 � (7 � 7) � 0

Make a true number sentence by filling in the missing number.

5. � 12 / (3 � 3) 6. (60 � 28) / 4 �

7. (3 � 8) � 6 � 8. 30 � (4 � 6) �

Make a true number sentence by inserting parentheses.

9. 4 º 2 � 10 � 18 10. 16 � 16 � 8 º 2

11. 27 / 9 / 3 � 1 12. 27 / 9 / 3 � 9

Find the solution of each open sentence below. Write a number sentence with the

solution in place of the variable. Check to see whether the number sentence is true.

Example: 6 � x � 14 Solution: 8 Number sentence: 6 + 8 = 14

Open sentence Solution Number sentence

13. 12 � x � 32

14. s � 200 � 3

15. 5 º y � 40

16. 7 � x / 4 7 � 28 / 428

5 � 8 � 408

197 � 200 � 3197

12 � 20 � 32 20

204

82

T F

T T

( )

( ) ( )

( )

Practice

17. 366 � 7,565 � 18. 3,238 � 9,784 �

19. 9,325 � 756 � 20. 4,805 � 2,927 � 1,8788,56913,022 7,931

Math Masters, p. 99

Study Link Master

Cole picked up a Fact Triangle and asked, “3 times what number equals 15?”

He wrote 3 ∗ ? = 15; ? = 5

ENRICHMENT INDEPENDENTACTIVITY

� Solving Open Sentences 15–30 Min

(Math Masters, p. 101)

To apply students’ understanding of open sentences, have them determine the unknown values of letters in animal names.

EXTRA PRACTICE INDEPENDENTACTIVITY

� Solving Broken-Calculator 5–15 Min

Problems(Math Masters, p. 424)

To provide practice solving open sentences, have students complete Broken Calculator problems. Use Math Masters, page 424 to create problems to meet the needs of individual students, or have students create and solve their own problems.

�,

15

3

NOTE For practice solving

simple inequalities, see

www.everydaymathonline.com.

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em3cug4tlg1 214 u03l11 - everyday math · pdf file214 unit 3 multiplication and division;...

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