TEKS/STAAR-BASED LESSONS - Having a HOOT in Fourth...

TEKSING TOWARD STAAR © 2014

TEKS/STAAR-BASED

LESSONS

PARENT GUIDESix Weeks 3

®MATHEMATICS

TEKSING TOWARD STAAR

TEKSING TOWARD STAAR2014

TEKSING TOWARD STAARSix Weeks 3 Scope and Sequence

Grade 4 Mathematics

Lesson TEKS/Lesson Content

Lesson 1

4.4E/represent the quotient of up to a four-digit whole number divided bya one-digit whole number using arrays, area models, or equations

4.4F/ use strategies and algorithms, including the standard algorithm, to divide up to a four-digitdividend by a one-digit divisor

4.4G/ round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutionsinvolving whole numbers

Lesson 2

4.4H/solve with fluency one- and two-step problems involving…division, including interpretingremainders

4.5A/represent multi-step problems involving the four operations (division only) with wholenumbers using strip diagrams and equations with a letter standing for the unknown quantity

Lesson 34.5B/represent problems using an input-output table and numerical expressions to generate anumber pattern that follows a given rule representing the relationship of the values in theresulting sequence and their position in the sequence (multiplication and division)

Lesson 4

4.8A/identify relative sizes of measurement units within the customary and metric systems

4.8B/convert measurements within the same measurement system, customary or metric, from asmaller unit into a larger unit or a larger unit into a smaller unit when given other equivalentmeasures represented in a table

Lesson 5

4.7A/illustrate the measure of an angle as the part of a circle whose center is at the vertex of theangle that is "cut out" by the rays of the angle. Angle measures are limited to whole numbers

4.7B//illustrate degrees as the units used to measure an angle, where 1/360 of any circle is onedegrees and an angle that "cuts" n/360 out of any circle whose center is at the angle's vertexhas a measure of n degrees. Angle measures are limited to whole numbers

4.7C/determine the approximate measures of angles in degrees to the nearest whole numberusing a protractor

Lesson 64.9A/represent data on a…dot plot…with whole numbers and fractions

4.9B/solve one- and two-step problems using data in whole number, decimal, and fraction formin a…dot plot

Lesson 74.10B/calculate profit in a given situation

Review

Assessment

NOTES:

STAAR Category 2 GRADE 4 TEKS 4.4E/4.4F/4.4G

TEKSING TOWARD STAAR 2014 Page 1

Lesson 1 - 4.4E/4.4F/4.4G

Lesson Focus

For TEKS 4.4E students are expected to represent the quotient of up to a four-digitwhole number divided by a one-digit whole number using arrays, area models, orequations.

For TEKS 4.4F students are expected to use strategies and algorithms, including thestandard algorithm, to divide up to a four-digit dividend by a one-digit divisor.

For TEKS 4.4G students are expected to round to the nearest 10, 100, or 1,000 or usecompatible numbers to estimate solutions involving whole numbers.

For these TEKS students should be able to apply mathematical process standards todevelop and use strategies and methods for whole number computations and decimalsums and differences in order to solve problems with efficiency and accuracy.

For STAAR Category 2 students should be able to demonstrate an understanding ofhow to perform operations and represent algebraic relationships.

Process Standards Incorporated Into Lesson

4.1.A Apply mathematics to problems arising in everyday life, society, and theworkplace.

4.1.C Select tools, including real objects, manipulatives, paper and pencil, andtechnology as appropriate, and techniques, including mental math, estimation,and number sense as appropriate, to solve problems.

4.1.D Communicate mathematical ideas, reasoning, and their implications usingmultiple representations, including symbols, diagrams, graphs, and languageas appropriate

4.1.E Create and use representations to organize, record, and communicatemathematical ideas

4.1.F Analyze mathematical relationships to connect and communicate mathematicalideas

4.1.G Display, explain, and justify mathematical ideas and arguments using precisemathematical language in written or oral communication

Vocabulary for Lesson

PART I PART II PART IIIdivision partial quotient estimationdividend compatible numbers rounddivisor estimate underestimatequotient multiples rangeequation compatible numbersoriginal amountnumber of sharessize of one share



Math Background Part I - Division

Grade 4 students are expected to divide up to a four-digit number by a one-digitnumber.

Division Words and Symbols

Division is the opposite of multiplication. Multiplication can actually be used everytime you divide. Division, like multiplication, involves a number of groups, thenumber in each group, and a whole.

Use division to separate a group of objects into smaller groups of equal value orwhen you want to separate a whole into groups of equal size.

Division involves equal groups. Three terms in division are dividend, divisor, andquotient. The dividend is the number being divided. The divisor is the number bywhich another number is being divided. The quotient is the result of the division.

Write and Read Division

There are two common ways to write division.

No matter which way you write division, you read it the same way.

Write: or

Say: eighty-one divided by nine equals nine

Reasons to Divide

There are two reasons to divide.When you know the original amount and the number of shares, you divide to find the

size of each share.

EXAMPLE: A large pizza is cut into 12 equal pieces. Four people are sharing thepizza equally. Find the number of pieces of pizza each person will get.

You know:original amount (12)number of shares (4)

When you know the original amount and the size of one share, you divide to find thenumber of shares.

EXAMPLE: One sandwich is made with 2 slices of bread. Find the number of piecesof sandwiches that can be made with 16 slices of bread.

You know:original amount (16)size of 1 share (2)

72 ÷ 9 = 8

dividend

divisor

quotient divisor 9 72 dividend

8

quotient

81 ÷ 9 = 9 9 819

You need to know:size of one share

(12 ÷ 4 = 3)

You need to know:number of shares

(16 ÷ 2 = 8)



Use Division Models and Equations

Strategies Grade 4 students should be able to use represent and solve divisionproblems include pictorial models (equal groups, arrays, and area models) andequations.

TEACHER NOTE: Give each partner pair 30 color tiles and 1 sheet of white paper.

Use Equal Groups

Equal groups can be used to represent and solve a division problem.

EXAMPLE: Harold has decided to divide a bag of 24 baby carrots equally into 6 snackbaggies.

Use color tiles to model the number of carrots he will put into each baggie. Draw 6large circles on your paper to represent the number of baggies. Distribute 24 colortiles equally into the circles to represent the total number of baby carrots he has.

Your model has 6 groups of 4, so Harold will put 4 carrots into each baggie and he hasno carrots left over.

This problem can also be represented using an equation. An equation is a numbersentence that uses the equal sign to show that two amounts are equal.

24 ÷ 6 = 4 or

24 is the original amount (number of baby carrots), 6 is the number of shares(number of baggies), and 4 is the amount of each share (carrots in each baggie).

Remove the counters from your paper, then draw a pictorial model to represent theproblem. Be sure to represent the original number of carrots Harold had, the numberof baggies he had, and the number of carrots in each baggie. Then, below your modelwrite the equation that represents this problem.

Use an Array

An array can be used to model and solve division problems.

EXAMPLE: Maria and 2 friends are playing a game that uses marbles. There are 36marbles for the game. Maria needs to give each player the same number of marblesbefore the game begins.

Use an array to represent the 36 marbles. Place 3 color tiles one under the other torepresent the number of players. These color tiles represent the groups of marbles.

Player 1

Player 2

Player 3

6 244



Distribute the color tiles one-by-one equally into the rows.

Your model has 3 equal groups with 12 color tiles in each group. So, your modelshows that each of the 3 players will begin the game with 12 marbles and there willbe no marbles left over.

This problem can also be represented using an equation.

36 ÷ 3 = 12 or

36 is the dividend (original number of marbles), 3 is the divisor (number of players),and 12 is the quotient (number of marbles each player gets).

Remove the counters from your paper. Then turn your paper over and draw an arrayto represent the problem. Be sure to remember to represent the number of players,and the number of dominoes each person will get. Then, below your model write theequation that represents this problem.

Use an Area Model

EXAMPLE: Diedra made a quilt using 117 fabric squares. She put 9 squares in eachrow of the quilt. What is the number of rows of squares in the quilt?

Divide to solve this problem.117 ÷ 9 = 8

An area model can be used to represent the problem. First, draw 1 row with 9squares in the row to represent the size of one share (number in each row).

Continue to add rows of squares in the area model until you reach 117, the originalamount (number of squares).

The area model shows 13 rows with 9 in each row. The 13 rows represent thenumber of shares. The 9 in each row represent the size of each share.

3 3612

Player 1

Player 2

Player 3

9

18

27

36

45

54

63

72

81

90

99

108

117




117 ÷ 9 = 13 or 9 117

So, the number of rows in the quilt Diedre made is 13.

Two-Digit Whole Number Divided by One-Digit Whole Number

The answer to every possible division problem can not be found in your memory whenyou are dividing by a one-digit divisor. A step-by-step method can be used.

What you know about place value and multiplication facts can be used to divide by aone-digit divisor. The same steps work with any size dividend.

EXAMPLE 1: Jason has 75 baseball cards to add to his collection. He wants to dividethem equally among 5 pages in his collection book. How many cards will he put oneach page?

To solve the problem, divide 75 by 5.

Step 1: Use a model to show 75.

Step 2: Put the 7 tens into 5 equal groups.

There is 1 ten in each group.There are 2 tens and 5 ones left over.

Step 3: Regroup the left over tens and ones.2 tens = 20 ones.

Add the 20 ones to the 5 ones youalready have.

Step 4: Put the 25 ones into 5 equal groups.There are 5 ones in each group.Each group has 1 ten and 5 ones.

Step 5: Count the number of tens and ones ineach group. Each group has 1 ten and 5 ones.

The model shows that Jason will put 15 baseball cards on each of the 5 pages.

13



Divide without using models. Use multiplication and subtraction to help with division.To solve the problem, divide 75 by 5.

Step 1: Divide the tens.75 5 = tens5 1 ten = 50Write 1 in the tens place of the dividend.Write 50 under 75.

Subtract and compare.75 50 = 25There are 25 ones left.25 is greater than 5.Keep dividing.

Step 2: Divide the ones.25 5 = ones5 5 ones = 25Write 5 in the ones place of the dividend.Write 15 under 15.

Subtract and compare.25 25 = 0There are 0 ones left.You are finished dividing.

The division shows that Jason will put 15 baseball cards on each of the 5 pages.

Either way, Jason will put 15 baseball cards on each of the 5 pages in his collectionbook.

EXAMPLE 2: Brad and his brother want to share 28 pencils equally. How manypencils will each boy get?To solve the problem, divide 28 by 2.

Step 1: Divide the tens.20 2 = tens2 1 ten = 20Write 1 in the tens place of the dividend.Write 20 under 28.

Subtract and compare.28 20 = 8There are 8 ones left.8 is greater than 2.Keep dividing.

Step 2: Divide the ones.8 2 = ones2 4 ones = 8Write 4 in the ones place of the dividend.Write 8 under 8.

Subtract and compare.8 8 = 0There are 0 ones left.You are finished dividing.

Each boy will get 14 pencils.

12 28

208

2 1 ten28 20

1

80

142 28

208

2 4 ones

8 8

15 75

5025

5 1 ten75 50

1

250

155 75

5025

5 5 ones

25 25



EXAMPLE 3: Mr. Thompson made 65 bird houses for a craft fair. He displays 5 birdhouses on a shelf in his booth. How many shelves will he need to display all the birdhouses?


Step 1: Divide the tens.60 5 = tens5 1 ten = 50Write 1 in the tens place of the dividend.Write 50 under 65.Subtract and compare.65 50 = 15There are 15 ones left.15 is greater than 5.Keep dividing.

Step 2: Divide the ones.15 5 = ones5 3 ones = 15Write 3 in the ones place of the dividend.Write 15 under 15.Subtract and compare.15 15 = 0There are 0 ones left.You are finished dividing.

Mr. Thompson will need 13 shelves to display the bird houses in his booth.

The problem can also be represented using an equation.

65 ÷ 5 = (50 ÷ 5) + (15 ÷ 5) = 10 + 3 = 13

150

135 65

5015

5 3 ones

15 15

15 65

5015

5 1 ten6550

1



Math Background Part II - Division

The answer to every possible division problem cannot be found in your memory whendividing a three-digit or a four-digit whole number by a one-digit whole number.

Grade 4 students can use strategies and algorithms to divide up to a four-digit dividendby a one-digit divisor.

Three-Digit Whole Number Divided by a One-Digit Whole Number

EXAMPLE: The distance between Kansas City, Kansas to Tulsa, Oklahoma is 248 miles.The Brady family traveled the same number of miles each day for two days to get fromKansas City to Tulsa. How many miles did they travel each day?


Use a model to show division.

Step 1: Use a model to show 248.

Step 2: Put the 2 hundreds into 2equal groups.

Step 3: Put the 4 tens into 2 equalgroups.

Step 4: Put the 8 ones into 2 equalgroups.

Step 5: Count the number ofhundreds, tens, and ones ineach group.

Each group has 1 hundred, 2 tens, and 4 ones.

The model shows that the Brady family will drive 124 miles each day.



Divide without using a model.


Step 1: Divide the hundreds.200 2 = hundreds2 1 hundred = 200Write 1 in the hundreds place of the dividend.Write 200 under 248.Subtract and compare.248 200 = 48There are 48 ones left.48 is greater than 2.Keep dividing.

Step 2: Divide the tens.48 2 = tens2 2 tens = 40Write 2 in the tens place of the dividend.Write 40 under 48.Subtract and compare.48 40 = 8There are 8 ones left.8 is greater than 2.Keep dividing.


The division shows that the Brady family will drive 124 miles each day.

Either way, the Brady family will drive 124 miles each day.

Check Division by Multiplying

Checking an answer helps make sure each step in long division was correctly recorded.Multiplication can be used to check division because multiplication is the opposite, orinverse, of division.

Multiply the quotient by the divisor to find if 124 is the correct quotient for 248 2.

The multiplication shows that 248 2 = 124 because 124 2 = 248.

12 248

20048

2 1 hundred248 200

1

12 248

20048

12

82 2 tens48 40

40

0

12 248

20048

124

82 4 ones

40

8

124

28

40+200

248



Zeros in a Quotient

EXAMPLE: A millipede has 4 legs on each body segment. How many body segmentswould a millipede with 436 legs have?

Divide without using models.


Step 1: Divide the hundreds.400 4 = hundreds4 1 hundred = 400Write 1 in the hundreds place of the dividend.Write 400 under 436.Subtract and compare.436 400 = 36There are 36 ones left.36 is greater than 4.Keep dividing.

Step 2: Divide the tens.36 4 = tens4 0 tens = 0Write 0 in the tens place of the dividend.Write 0 under 36.Subtract and compare.36 0 = 36There are 36 ones left.36 is greater than 4.Keep dividing.


The division shows that a millipede with 436 legs has 109 body segments.

Multiply the quotient by the divisor to find if 109 is the correct quotient for 436 4.

The multiplication shows that 436 4 = 109 because 109 4 = 436.

TEACHER NOTE: Give each pair of students a set of base-ten blocks (1 flat, 20 rodsand 10 unit squares)

14 436

40036

4 1 hundred436 400

1

14 436

40036

10

364 0 tens36 0

0

1094 436

4 9 ones

40036

36

036

0 36 36

109

436

+ 400436



Use a Model

Base-ten blocks can be used to model and solve division of whole numbers with 1-digitdivisors and 3-digit dividends.

EXAMPLE: The Garcia Elementary choir has 189 students. The choir director isorganizing the students in rows during practice for a spring concert. She has decidedto put 9 students in each row. How many rows will the choir director organize?

Divide to find the answer to the problem. 189 9 =

Use place value to place the first digit in the quotient.Look at the hundreds in 189.1 hundred cannot be shared equally among 9 groups. Regroup 1 hundred as 10 tens.

The model now has 18 tens to share equally among 9 groups.The first digit of the quotient will be in the tens place.

Divide the tens equally among 9 groups.

Divide the 9 ones evenly among the 9 groups.

The model shows 9 groups of 21. So, 189 9 = 21

The choir director will organize 9 rows of 21 students.



Use Partial Quotients

In the partial quotient method of dividing, multiples of the divisor are subtractedfrom the dividend and then the partial quotients are added together.

EXAMPLE: Divide 189 students by 9 rows to find the number of students the choirdirector will organize in each row. 189 9

Subtract multiples of the divisor from thedividend until the remaining number isless than the multiple.NOTE: The easiest partial quotients touse are multiples of 10.

Subtract smaller multiples of the divisorfrom the dividend until the remainingnumber is less than the divisor.

Add the partial quotients to find thequotient.

The choir director will organize 9 rows of 21 students.

Use an Area Model

Use an area model to record the partial quotients.

EXAMPLE: Divide 189 students by 9 rows to find the number of students the choirdirector will organize in each row. 189 9

One Way Another Way

(180 ÷ 9) + (9 ÷ 9) = 20 + 1 = 21

(90 ÷ 9) + (90 ÷ 9) + (9 ÷ 9) = 10 + 10 + 1 = 21

Using either area model, the choir director will organize 21 rows of 9 students.

1899

909 99

10

909 90 9

10 10

909 90 9

10 10 1

1899

1809 9

20

1809 9

20 1

STEP 1

STEP 2

STEP 3

189 9 = 21

9 189

–9099

–909

–90

10 × 9

10 × 9

1 × 9

10

10

1++ 21



Four-Digit Whole Number Divided by a One-Digit Whole Number

EXAMPLE: A principal bought 8,512 special pencils to divide equally among 8 teachersto use for student rewards. How many pencils will each teacher get?

Divide without using a model.

To solve the problem, divide 8,512 by 8.

Step 1: Divide the thousands.Share 8 thousands equally among 8 groups.Multiply. 8 1,000 = 8,000Write 1 in the thousands place in the quotient.Subtract. 8,512 8,000 = 512512 is greater than 8.Keep dividing.

Step 2: Divide the hundreds.Share 5 hundreds equally among 8 groups.5 hundreds cannot be shared among 8groups without regrouping.Write 0 in the hundreds place in the quotient.Keep dividing.

Step 3: Divide the tens.Share 51 tens equally among 8 groups.Multiply. 8 60 = 480Write 6 in the tens place in the quotient.Subtract. 512 480 = 3232 is greater than 8.Keep dividing.

Step 4: Divide the ones.Share 32 ones equally among 8 groups.Multiply. 8 4 = 32Write 4 in the ones place in the quotient.Subtract. 32 32 = 0

You are finished dividing.

The division shows the principal will give each of the 8 teachers 1,064 pencils.

Write an equation to represent the problem.

_____ ÷ 8 = (8,000 ÷ ___) + (_____ ÷ 8) + (32 ÷ ___) =

1,000 + _____ + 4 = 1,064Explain why the equation is correct.

18 8,5128,000

5128 1 thousand8,512 8,000

8,000

1 18 8,512

512

0

32480

8,000

1 18 8,512

512

0

8 6 tens512 480

6

08 4 ones3232 32

32480

8,000

1 18 8,512

512

064



Use Partial Quotients

In the partial quotient method of dividing, multiples of the divisor are subtractedfrom the dividend and then the partial quotients are added together.

EXAMPLE: A school ordered 1,686 good citizen awards for students. The school givesthe same number of awards each six weeks. What is the number of good citizenawards that will be given each six weeks? Divide. 1,686 ÷ 6

Subtract multiples of the divisor from thedividend until the remaining number isless than the multiple.NOTE: The easiest partial quotients touse are multiples of 10.

Subtract smaller multiples of the divisorfrom the dividend until the remainingnumber is less than the divisor.

Add the partial quotients to find thequotient.

The school will use 281 good citizen awards each six weeks.

STEP 1

STEP 2

STEP 3

1,686÷ 6 = 281

6 1,686

–6001086–600486–60426–60366–60306–60246–60186–60126–6066

–606

–60

100 × 6 100

100 × 6 100

1 × 6 1++ 281

10 × 6 10

10 × 6 10

10 × 6 10

10 × 6 10

10 × 6 10

10 × 6 10

10 × 6 10

10 × 6 10



Use the Standard Algorithm

A yogurt company uses 8 ounces of yogurt to make each of their yogurt cups. Eachday the company uses 5,800 ounces of yogurt. Find the number of yogurt cups thecompany makes each day.

Divide to solve the problem. 5,800 ÷ 8

Estimate the quotient. 6,000 ÷ 10 = 600

Use the estimate to decide where to place the first digit in the quotient.

Divide the hundreds.

Divide the tens.

Divide the ones.

Since 725 is close the estimate of 600, the quotient is reasonable.

So, the yogurt company can make 725 yogurt cups each day.

NOTE: The standard algorithm for division uses both multiplication and subtraction.

78 5,800–5 6

2

Divide 56 hundreds ÷ 8Multiply 8 7 hundreds = 56 hundredsSubtract 58 hundreds ‒56 hundreds = 8 hundreds2 hundreds cannot be shared among 8 groups without regrouping.

The first digit in the quotient will be in the hundreds place.10 6,000

20–16

4

Divide 20 tens ÷ 8Multiply 8 2 tens = 16 tensSubtract 20 tens ‒16 ones = 4 tens4 tens cannot be shared among 8 groups without regrouping.

728 5,800–5 6

Subtract 40 ones ‒40 ones = 0

Divide 40 ones ÷ 8Multiply 8 5 ones = 40 ones

20–16

40

7258 5,800–5 6

–400



Use Compatible Numbers

Compatible numbers that are found by using basic facts and patterns can be usedto estimate quotients.

45 5 = 9

450 5 = 90

4,500 5 = 900EXAMPLE: The lowest depth of a cavern is 253 feet below ground. An elevator takestourists from ground level to that depth in 3 minutes. About how many feet does theelevator travel per minute?Estimate. 253 3.

Use two sets of compatible numbersto find two different estimates.

253 3

240 3

253 3

300 3

Use patterns and basic facts to helpestimate.

24 3 = 8

240 3 = 80

30 3 = 10

300 3 = 100

The elevator travels about 80 to 100 feet per minute.The more reasonable estimate is 80 feet because 240 is closer to 253 than 300.

So, the elevator travels about 80 feet per minute.

Use Estimation to Divide

With this strategy for dividing, different estimates can be made, but the answer willalways be the same.

EXAMPLE: Divide 720 by 3.

Estimate the quotient.

Multiply your estimate by the divisor.If the product is less than the dividend, subtract.If the product is greater than the dividend, try a lower estimate.

Keep estimating, multiplying, and subtracting until thedifference is less than the divisor.

Add your estimates to get the whole number part of thequotient.

4

The quotient is 24. So, 720 3 = 24.

basic fact

STEP 1

STEP 2

203 720

3 720–600120

3 720–600120–6060

–600

20

2

20

2+24



Multiply to Check Division

Checking an answer is important. Checking helps make sure each step in the divisionwas correctly recorded. One way to check division is to use multiplication.

Multiplication can be used to check division because multiplication is the opposite, orinverse, of division. Multiply the quotient by the divisor to check your answer to adivision problem.

EXAMPLE: Marissa found a quotient of 72 when she divided 648 by 9. Is 72 acorrect quotient for 648 9?

648 9 = 72 and 72 9 = 648, so Marissa's division is correct.

729 648

630

1818

0

divisor

quotient

dividend

729

18630648



Math Background Part III - Rounding to Estimate Quotients

Sometimes an exact answer to a problem is not needed. Estimation can be used tofind an answer that is close to the exact answer. Estimation can be used to solvethese problems that ask about how many or approximately how much.

One way to estimate an answer to a problem is to round the numbers before workingthe problem. You can round numbers to the nearest ten, nearest hundred, or nearestthousand. A number line or a set of rounding rules can be used to round numbers.

Rounding Rules

When rounding a number to the nearest thousand, look at the hundreds place.If the digit in the hundreds place is 0 to 4, the digit in the thousands place stays the

same. Change the digits in the ones, tens, and hundreds place to zeros.If the digit in the hundreds place is 5 to 9, the digit in the thousands place rounds

to the next-higher thousand. Change the digits in the ones, tens and hundredsplaces to zeros.

When rounding a number to the nearest hundred, look at the tens place.If the digit in the tens place is 0 to 4, the digit in the hundreds place stays the same.

Change the digits in the ones and tens place to zeros.If the digit in the tens place is 5 to 9, the digit in the hundreds place rounds to the

next-higher hundred. Change the digits in the ones and tens places to zeros.

When rounding a number to the nearest ten, look at the ones place.If the digit in the ones place is 0 to 4, the digit in the tens place stays the same.

Change the digit in the ones place to a zero.If the digit in the ones place is 5 to 9, the digit in the tens place rounds to the next-

higher ten. Change the digit in the ones place to a zero.

Use Compatible Numbers to Estimate a Quotient

Compatible numbers can be helpful when estimating the answer to a division problem.Changing the numbers to numbers that are easy to divide can help you solve theproblem in your head.

EXAMPLE 1: Estimate the quotient of 177 3.Find a number close to 177 that you can divide by 3 in your head.

Use the basic fact 18 3 6 to help solve the problem in your head.

Think: 180 3 60

The quotient of 177 3 is approximately 60.

EXAMPLE 2: A roller coaster holds 6 people per car. When the ride was almost full,38 people were riding the roller coaster. About how many cars are on the roller coaster?Use the operation of division to solve the problem.Estimate the quotient of 38 6.Find a number close to a multiple of 6 that you can divide by 6 in your head.

38 is close to 36, which is a multiple of 6.Think: 36 6 6The quotient of 38 6 is approximately 6.So, there are about 6 cars on the roller coaster.



NOTE: This is an underestimate because one factor was rounded down. Six carswould only hold 36 people, so 7 cars are on the roller coaster to hold 38 people.

EXAMPLE 3: There are 437 people signed up for a city tour in a double-decker buses.The tour company will use 6 buses. About how many people will ride on each bus?

Use the operation of division to solve the problem.Estimate the quotient of 437 6.Look for a basic fact.

437 6

420 6 70

About 70 people will ride on each bus.

NOTE: The estimate of 70 is less than the actual quotient. This is an underestimate.This means that more than 70 people will ride on each bus.

Understanding Range Estimates

An estimate can be a range of numbers.

EXAMPLE 1: Lance took pictures at the holiday parade. He used 6 rolls of film. Thenumber of pictures that were taken with each roll of film was from 12 pictures to 24pictures. What could be the total number of pictures Lance took?

Find the least number of pictures Lance could have taken with the 6 rolls.He could have taken up to 5 rolls with 12 pictures and 1 roll with 24 pictures.Find the product of 5 12 and then add 24 to find the least number of picturesLance could have taken.

5 12 = 60

60 24 = 84

The least number of pictures Lance could have taken is 84.

Find the greatest number of pictures Lance could have taken with the 6 rolls.He could have taken up to 5 rolls with 24 pictures and 1 roll with 12 pictures.Find the product of 5 24 and then add 12 to find the greatest number of picturesLance could have taken.

5 24 = 120

120 12 = 132

The greatest number of pictures Lance could have taken is 132.

Lance took at least 84 pictures, but not more than 132 pictures.

So, the total number of pictures Lance took is between 84 and 132.

NOTE: A reasonable estimate of the number of pictures Lance took is between 80and 130.

Think: 43 is close to 4242 6 7

STAAR Category 2 GRADE 4 TEKS 4.4H/4.5A


Lesson 2 - 4.4H/4.5A

Lesson Focus

For TEKS 4.4H students are expected to solve with fluency one- and two-stepproblems involving multiplication and division, including interpreting remainders.

For this TEKS students should be able to apply mathematical process standards todevelop and use strategies and methods for whole number computations and decimalsums and differences in order to solve problems with efficiency and accuracy.

For TEKS 4.5A students are expected to represent multi-step problems involving thefour operations with whole numbers using strip diagrams and equations with a letterstanding for the unknown quantity. Focus for this lesson is division.

For this TEKS students should be able to apply mathematical process standards todevelop concepts of expression and equations.




4.1.B Use a problem-solving model that incorporates analyzing given information,formulating a plan or strategy, determining a solution, justifying the solution,and evaluating the problem-solving process and the reasonableness of asolution.




PART I PART IIremainder strip diagramequation equationdividend unknown quantitydivisorquotientoriginal amountnumber of sharesamount of each share



Mathematics Background Part I - Division with Remainders

Grade 4 students learn to divide whole numbers that do not divide evenly. When anumber cannot be divided evenly, the amount left over is called the remainder.

Use Models

Objects can be used to make a model to solve division problems.

EXAMPLE 1: Maria and 2 friends are playing a game of dominoes. There are 19dominoes in a set. Maria needs to give each player the same number of dominoesbefore the game begins.

Use an array to represent the 19 dominoes. Place 3 color tiles one under the other torepresent the number of groups in the problem. These color tiles represent each ofthe three players.

Distribute the color tiles one-by-one equally into the rows.

The model has 3 equal groups of 6 color tiles and 1 of the 19 color tiles left over.So, the model shows that each of the 3 players will begin the game with 6 dominoesand there will be 1 domino left over.

This problem can also be represented using an equation. An equation is a numbersentence that uses the equal sign to show that two amounts are equal.

19 ÷ 3 = 6 remainder 1 or

19 is the dividend (number of dominoes), 3 is the divisor (number of players), 6 isthe quotient (number of dominoes each player gets), and 1 is the remainder(number of dominoes left over that could not be divided equally).

Remove the counters from your paper, then sketch an array to represent theproblem. Be sure to remember to represent the number of players, the number ofdominoes each person will get, and the left over domino. Then, below the sketch,write an equation that represents the problem.

EXAMPLE 2: Harold has decided to divide a bag of 29 baby carrots equally into 5snack baggies to take in his lunch this week.

Use color tiles to model the number of 29 carrots he will put into each baggie. Turnyour paper onto the other side. Draw 5 large circles on your paper to represent thenumber of baggies. Distribute 29 color tiles equally into the circles to represent thetotal number of baby carrots he has.

Player 1

Player 2

Player 3

Player 1

Player 2

Player 3

Left over

3 196 R1



The model has 5 groups of 5 with 4 left over, so Harold will put 5 carrots into eachbaggie and he has 4 carrots left over.


29 ÷ 5 = 5 remainder 4 or

29 is the original amount (number of carrots), 5 is the number of shares(number of baggies), 5 is the amount of each share (carrots in each baggie), and4 is the number left over (number of carrots that could not be divided equally intothe baggies).

Remove the counters from your paper, then sketch a model to represent the problem.Be sure to represent the original number of carrots Harold had, the number ofbaggies he had, the number of carrots in each baggie, and the number of left overcarrots. Then, below the sketch, write an equation that represents the problem.

Interpreting Remainders

Sometimes remainders in division problems must be left as a whole number in thequotient, sometimes they should be written as a fraction or a decimal in the quotient.

EXAMPLE 1: In the domino problem, a domino cannot be divided into equal parts, sothe leftover domino must be interpreted as a whole number.

19 ÷ 3 = 6 R1EXAMPLE 2: In the baby carrot problem, a baby carrot can be divided into equalparts, so the remainder can be interpreted as a whole number or a fraction.

If Harold does not want to divide the left over carrots, then he will have 4 wholecarrots left over.

29 ÷ 5 = 5 R4If Harold decides he wants to divide the left over 4 carrots, then he will divide the 4whole left over carrots.

29 ÷ 5 = 5 R4Remember, each baggie of the 5 baggies has 5 carrots. The 4 left over carrots can bedivided equally among the 5 baggies. First, divide each carrot into 5 equal pieces.

Each piece is 1 out of 5 equal pieces of a carrot.

So, each piece is15

of a carrot.

5 295 R4



Group the pieces of the 4 carrots equally into 5 groups to represent how many pieceswill go into each of the 5 baggies.

Now, represent the number of whole carrots and pieces of carrot that will go into eachof the 5 baggies.

Four15

pieces of carrot will go into each of the 5 baggies.

Each baggie will have 5 +15

+15

+15

+15

= 545

carrots.

EXAMPLE 3: Keisha has been wrapping holiday gifts. She has a length of 73 inchesof a roll of wrapping paper left. She wants to cut the remaining wrapping paper into 8equal pieces to use to wrap 8 small gifts. Find the length of each piece if she uses allof the remaining wrapping paper.

REMEMBER: When you solve a division problem with a remainder, the way theremainder is interpreted depends on the situation and the question.Use division to find the answer to this problem. Divide 73 (the original length) by 8(the number of pieces with equal lengths).

73 ÷ 8 = 5Draw a sketch to represent the problem.

The sketch shows 73 ÷ 8 = 9 R1

Write the remainder as a fraction.To write the remainder as a fraction, use the remainder as the numerator, and thedivisor as the denominator.

remainderdivisor

=18

Write the quotient as a mixed number by including the fraction as a part of thequotient.

So, the quotient is 918

. Each of the 8 equal pieces will be 918

inches long.



EXAMPLE 4: Ramon and 3 of his friends spent a total of $58 at the school carnival.They each spent the same amount of money. What is the amount each of the friendsspent at the carnival?

REMEMBER: When you solve a division problem with a remainder, the way theremainder is interpreted depends on the situation and the question.Use division to find the answer to this problem. Divide $58 (the original amount) by 4(the number of friends).

$58 ÷ 4 = 5

Draw a sketch to represent the dividend.

Divide the tens into 4 equal groups.

Regroup the remaining ten.

Divide the ones equally into the 4 groups.

The model shows 14 in each group with a remainder of 2 to divide equally into the4 groups.

The remainder of 2 represents two dollars, so divide each dollar equally into 4 parts.

Each dollar can be equally divided into four 14

dollars.

Divide the 14

dollars equally into the 4 groups.

The sketch shows 56 ÷ 4 = 14 24

. So, each friend spent 14 24

dollars.

Write the quotient as a decimal.

You know 14

of a dollar is $0.25, so 24

of a dollar is $0.50.

So, each friend spent $14.50 at the school carnival.

10 10 10 10 10 1 1 1 1 1 1 1 1

10 10 10 10 10 1 1 1 1 1 1 1 1

10 10 10 101 11 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

10

1 1 1 11 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1

10

1 1 1 1



Math Background Part II - Division Problems

Two-step division problems can be represented using strip diagrams and equationswith a letter standing for the unknown quantity.

EXAMPLE 1: Evan weighs 8 times as much as his baby brother. Together, theyweigh 144 pounds. Find Evan's weight and his brother's weight.

Record the information given in the problem.

Evan and his brother weigh a total of 144 pounds.Evan weighs 8 times as much as his brother.

Draw a strip diagram to represent the problem.

Divide the total weight of Evan and his brother by the total number of boxes in thediagram.

Write the quotient in each of the boxes in the strip diagram.

Multiply the quotient by 8 to find Evan's weight.

So, Evan weighs 128 pounds and his baby brother weighs 16 pounds.

144 poundsEvan's baby brother

Evan

169 144

–954

–540

144 poundsEvan's baby brother

Evan

16

16 16 16 16 16 16 16 1611

168

48+80128



EXAMPLE 2: The Longoria family is preparing corn on the cob for a community picnic.They have 3 crates with 32 ears of corn in each crate. After the corn is cooked, theywill divide the corn equally among 8 containers that will keep the corn warm untilserving time. Find the number of earns of corn that will be put into each container.

Record the information given in the problem.

There are 3 crates with 32 ears of corn in each crate.

The total number of ears of corn will be divided equally into 8 containers.

Draw a strip diagram for each step and use equations to solve the problem.Step 1: Draw a strip diagram to find the total number of ears of corn. Use a letter,

e, to stand for the unknown number, the total number of ears of corn.

32 × 3 = e

96 = e

So, the total number of ears of corn is 96.

Step 2: Draw a strip diagram to find the number of ears of corn that should go intoeach container. Use a letter, c, to stand for the unknown number, thenumber of ears of corn to put in each container.

96 ÷ 8 = c

12 = c

So, the total number of ears of corn that will go into each container is 12.

32 32 32

e

96

STAAR Category 2 GRADE 4 TEKS 4.5B


Lesson 3 - 4.5B

Lesson Focus

For TEKS 4.5B students are expected to represent problems using an input-outputtable and numerical expressions to generate a number pattern that follows a givenrule representing the relationship of the values in the resulting sequence and theirposition in the sequence. The focus for this lesson is multiplication and divisionnumber pattern rules.

For these TEKS students should be able to apply mathematical process standards todevelop concepts of expressions and equations.





4.1.D Communicate mathematical ideas, reasoning, and their implications usingmultiple representations, including symbols, diagrams, graphs, and languageas appropriate.

4.1.E Create and use representations to organize, record, and communicatemathematical ideas.

4.1.F Analyze mathematical relationships to connect and communicate mathematicalideas.


PART I PART IIsymbol number patternvariable termexpression positionequation value

rulesequenceinput-output tablefunctionrelationship



Math Background Part I - Multiplication and Division Relationships

Relationships can be described mathematically by replacing words and sentences withnumbers, symbols, expressions, and equations. Describing relationships with numbers,symbols, expressions, and equations can help to solve problems.

Symbols, Variables, and Expressions

A symbol is something that represents something else in mathematics.The symbol × means multiply. The symbol ÷ means divide.

If something varies, that means it changes. Most things, like your height andweight, do not stay the same. In mathematics, to describe things that change, orvary, letters are used instead of numbers. When a letter is used this way, it is calleda variable. Any letter in the alphabet can be used as a variable.

EXAMPLES :n (number of inches tall you are)t (amount of time you spend on homework)c (number of cents in your pocket)

In language, an expression can be a short way to describe an idea or feeling.

In mathematics, an expression is a short way to describe an amount.

An expression is a variable or combination of variables, numbers, and symbolsthat represents a mathematical relationship.

Sometimes an expression is just a number, like 6.Sometimes an expression is just a variable, like w.Sometimes an expression is a combination of numbers, variables, and operations,like 6 × 3 or n ÷ 3.

Multiplication and Division Expressions

To write an expression that describes what is going on in a word problem, think aboutthe problem in words. Use numbers when you know what they are. Use variableswhen you do not know the numbers.

Problem Expression in Words ExpressionYou have 3 full boxes of pencils. Write anexpression to represent the total numberof pencils.

full box 3 b 3

You have a box of pencils shared equallyamong 3 people. Write an expression torepresent the number of pencils eachperson will get.

full box 3 people b 3

Multiplication and Division Equations

An equation is a mathematical sentence that tells you that two expressions are equal.

7 × n = 21variable

expressionexpression

equation



To write an equation, think about which two amounts are equal in a problem. Thenwrite an expression for each amount.

Problem Equal Expressions in Words EquationYou have 6 reams of paper. Writean expression to represent thetotal number of pieces of paper.

ream × 6 = pieces r × 6 = p

You have a ream of paper. Writean expression to represent howmany pieces of paper each personwill get if the paper is sharedequally among 6 people.

ream ÷ 6 = pieces r ÷ 6 = p



Math Background Part II - Number Patterns

A number pattern is set of numbers that is related to each other by a specific rule.Each number in the pattern is called a term. Each term in the pattern has a positionand each term has a value. The pattern can be described by a specific rule.

A number pattern can be represented by a sequence of numbers.

EXAMPLE 1: 5, 10, 20, 40, 80, 160, … is a number pattern. Find and describe therule for this number pattern.

Decide if each number in the pattern is greater or less than the number before it.

5, 10, 20, 40, 80, 160, …

Each number in this pattern is greater than the number before it.

Decide the rule for the pattern.

5, 10, 20, 40, 80, 160, …

The rule for this pattern is multiply by 2.

The first term in this pattern is 5. The sixth term in this pattern is 160. To find thenext term, or the seventh term in this pattern, multiply by 2. The next term in thispattern is 160 × 2, so the next term is 320.

EXAMPLE 2: 243, 81, 27, 9, 3, … is a number pattern. Find and describe the rule forthis number pattern.

Decide if each number in the pattern is greater or less than the number before it.

243, 81, 27, 9, 3, …

Each number in this pattern is less than the number before it.

Decide the rule for the pattern.

243, 81, 27, 9, 3, …

The rule for this pattern is divide by 3.

The first term in this pattern is 243. The fifth term in this pattern is 3. To find thenext term, or the sixth term in this pattern, divide by 3. The next term in thispattern is 3 ÷ 3, so the next term is 1.

EXAMPLE 3: 6, 9, 12, 15, 18, … is a number pattern.

Divide 8 by 4, then multiply the quotient by 3 to get the first term in the pattern.

(8 ÷ 4) × 3 = 2 × 3 = 6

Divide 12 by 4, then multiply the quotient by 3 to get the second term in the pattern.

(12 ÷ 4) × 3 = 3 × 3 = 9

Divide 16 by 4, then multiply the quotient by 3 to get the third term in the pattern.

(16 ÷ 4) × 3 = 4 × 3 = 12

So, the rule for the pattern is divide by 4, multiply by 3.

×2 ×2 ×2 ×2 ×2

÷3 ÷3 ÷3 ÷3



Generating a Number Pattern

An input-output table can be used to generate a pattern. A pattern is called afunction when one quantity depends on the other. An input-output table shows therelationship between the inputs and outputs of a function. A rule can be written todescribe this relationship. The rule can be an expression or an equation.

EXAMPLE 1: One gallon of water is equivalent to 4 quarts of water. Thisrelationship between gallons and quarts will not change.This relationship is called a function because the number of quarts is a function of thenumber of gallons.

There are several ways a function can be shown.

A function can be represented with a diagram. A function is like a machine. Whenthe input changes, the output will also change.

For this machine, the rule for the relationship between gallons and quarts is × 4.When the input changes, the output will also change. But for any input there willonly be one possible output. When the number of gallons is 9, the number of quartsis 9 × 4, so the number of quarts is 36.

A function can be shown using an equation. This is the rule for the function.

g × 4 = q

A function can be shown using an input-output table. The input is the position inthe sequence of numbers in the pattern. The input is also called the term. Theoutput is the value of each position in the sequence. The output is also called thevalue of the term.

The Input is number of gallons. The Output is number of gallons.

6Gallons

Input24Quarts

× 4

RULE

Output

Gallons Quarts

QuartsGallons Input,Position

FunctionRule

Output,Value

g g × 4 q0 0 × 4 01 1 × 4 42 2 × 4 83 3 × 4 124 4 × 4 165 5 × 4 20

Value of termFirst term

Value of termSecond term



For any input position, there is only one possible output value.

The first term in the pattern is 0. The value of the first term is 0.For any term, there is only one possible value.

EXAMPLE 2: Paul divides any money he earns into 2 equal parts. He puts 1 equalpart in savings and keeps 1 equal part to spend. Paul earns $2 each time he walks hisneighbor's dog. How much money did Paul put in his savings if he earned $26 walkinghis neighbor's dog this month?

This problem can be represented with a function machine diagram.

For this machine, the rule for the relationship between the amount earned and theamount saved is ÷ 2. When the amount earned is $26, the amount saved is 26 ÷ 2,so the amount saved is $13.

This function can be represented by a rule.

e ÷ 2 = s

This function can be shown in an input-output table.

The Input is the amount earned. The Output is the amount saved.The first term in the pattern is 2. The value of the first term is 1.

The sixth term in the pattern represents $12 earned, so Paul will put $6 in savingswhen he earns $12.

$26Amount Earned

Input$13

Amount Saved

÷2

RULE

Output

Amount Earned Amount Saved

Output

Amount SavedAmount Earned Input,Position

FunctionRule

Output,Value

e e ÷ 2 s2 2 ÷ 2 14 4 ÷ 2 26 6 ÷ 2 38 8 ÷ 2 410 10 ÷ 2 512 12 ÷ 2 6

First term

OutputSecond term

STAAR Category 3 GRADE 4 TEKS 4.8A/4.8B


Lesson 3 - 4.8A/4.8B

Lesson Focus

For TEKS 4.8A students are expected to identify relative sizes of measurement unitswithin the customary and metric systems.

For TEKS 4.8B students are expected to convert measurements within the samemeasurement system, customary or metric, from a smaller unit into a larger unit or alarger unit into a smaller unit when given other equivalent measures represented in atable.

For these TEKS students should be able to apply mathematical process standards toselect appropriate customary and metric units, strategies, and tools to solve problemsinvolving measurement.

For STAAR Category 3 students should be able to demonstrate an understanding ofhow to represent and apply geometry and measurement concepts.




4.1.D Communicate mathematical ideas, reasoning, and their implications usingmultiple representations, including symbols, diagrams, graphs, and languageas appropriate




PART I PART IIlength convertcustomary dividemetric multiplycapacityliquid volumemassbalanceweightscale



Math Background Part I - Measurement

Grade 4 students are expected to apply mathematical process standards to selectappropriate customary and metric units to solve problems involving measurement,including units of length, capacity or liquid volume, weight, and mass.

Measuring Length

Length is the measure of the distance between two points. When you measure thedistance between the beginning and the end, you are finding the length. To find thelength of an object, use a ruler. The STAAR Grade 4 Reference Materials has acustomary ruler and a metric ruler.

Benchmarks for Length

The chart shows some common units of length used in the customary and metricsystems. The benchmarks will help you understand the size of units of length in thecustomary system and in the metric system.

Customary Units of Length Metric Units of Length

inch(in.)

millimeter(mm)

foot(ft)

centimeter(cm)

yard(yd)

meter(m)

mile(mi)

kilometer(km)

To understand which is the best unit to describe length, you need to understand thebasic units of measure.

For example, you could measure the length of a football field in inches, but it wouldtake you a very long time. A more reasonable measure of the length of a football fieldis yards or meters.

A small paper clip has alength of about 1 inch

A car license plate hasa length of about 1 foot

A dime has a thicknessof about 1 millimeter

A fingertip has a widthof about 1 centimeter

A baseball bat has a length ofabout 1 yard A door has a width

of about 1 meter

The distance you can walk inabout 20 minutes is a mile

The distance you can walk inabout 12 minutes is a kilometer



EXAMPLE 1: What unit of measure should Norton use to describe the length of apygmy chameleon?Norton knows that the units mile, yard and foot are too large.He knows that the unit inch is a good choice because the length of a chameleon

could be measured with a small paper clip.Inch is the best units to use. The length of a pygmy chameleon is about 4 inches.

EXAMPLE 2: What unit of measure should Norton use to describe the width of acomputer mouse?Norton knows that the unit kilometer is too large because 1 kilometer is about six-

tenths of a mile.Norton knows that the unit millimeter is too small because the thickness of a dime is

about 1 millimeter.He decides that the unit centimeter is a good choice because the length of a

centimeter is about the width of a finger.Centimeter is the best unit to use. The width of a computer mouse is about 6

centimeters.

Measuring Capacity or Liquid Volume

Capacity or liquid volume is a measure of how much a container can hold. Capacityor liquid volume can be measured in customary or metric units.Customary units of capacity or liquid volume include gallons, quarts, pints, cups, and

ounces.Metric units of capacity or liquid volume include liters and milliliters.The chart shows some common units of capacity or liquid volume used in the customaryand metric systems.

Metric Customary

Milliliter Cup

Pint

Liter

Quart

Gallon

1 c

1 gal

1 pt

MIL 1 qt

COLA1 L

10 drops – about 1 mL



These benchmarks model the size of units used to measure the capacity of liquids inthe customary system:

A milk carton from the school cafeteria holds 1 cup of liquid.A tomato sauce can from the grocery store holds 1 pint of liquid.A carton of orange juice from the grocery store holds 1 quart of liquid.A large jug of milk from the grocery store holds 1 gallon of liquid.

These benchmarks model the size of units used to measure the capacity of liquids inthe metric system:

An eyedropper from the pharmacy holds about 1 milliliter of liquid – about 10 drops.A bottle of water from the grocery story holds 1 liter of water – about 4 cups.

Measuring Mass or Weight

Mass is the measure of the amount of matter in an object. A balance is the toolused to measure mass. An object is placed on one of the balance pans. Masses areplaced on the other balance pan so that the two balance pans hold the same amountof matter.

Weight is the measure of how heavy an object is. A scale is the tool used to measureweight. A scale contains a spring that is pulled down by the force of gravity andmeasures how heavy an object is.

Comparing Mass and Weight

Mass and weight are similar, but they are not the same. Gravity influences weight,but does not affect mass. Your weight could change if you were on another planetwith more or less gravity than Earth. Your mass, or the measure of the amount ofmatter in your body, would not change.

Step on a scaleto measure weight.

Place objects on a scaleto measure weight.

Use a balance to measure mass.



Suppose you took a trip to the moon. If you got on a scale on the moon you wouldweigh much less than on Earth. The moon has less gravity than Earth. Since theamount of gravity affects weight, you would weigh less on the moon than on Earth.The moon pulls down on your body less than Earth does, so your weight would be lesson the moon.

There is still the same amount of you no matter where you are. Your mass on themoon would be the same as on Earth.

Benchmarks for Mass and Weight

The chart shows some common units of weight and mass used in the customary andmetric systems. The benchmarks will help you understand the size of units of weightin the customary system and units of mass in the metric system.

Customary Units of Weight Metric Units of Mass

ounce gram

pound kilogram

ton

To understand which is the best unit to describe the weight or mass of an object, youneed to understand the basic units of measure.

For example, you could weigh a medium size car in ounces, but the number of ounceswould be very large. You could weigh a medium size car in tons, but you may not beable to be very accurate since a subcompact car is about 1 ton and a medium size caris probably more than 1 ton, but probably not as much as 2 tons. A more reasonablemeasure of the weight of a car would be to measure in pounds because you can bemore accurate.

A shoestringhas a mass of about 1 gram

A slice of breadweighs about 1 ounce

A loaf of breadweighs about 1 pound

A textbook has a mass ofabout 1 kilogram

A subcompact carweighs about 1 ton



EXAMPLE 1: What unit of measure should June use to describe the weight of a nickel?June knows that the unit pound is too large.She knows that the unit ounce is a good choice because the weight of a nickel is

about the same as the weight of 1 slice of bread.Ounces are the best units to use. The weight of a nickel is about 1 ounce.

EXAMPLE 2: What unit of measure should June use to describe the mass of a nickel?June knows that the unit kilogram is too large because 1 kilogram is about the mass

of a math textbook.June knows that the unit milligram is too small because 1000 milligrams is the same

as 1 gram and 1 gram is about the mass of a shoestring.She decides that the unit gram is a good choice because when she used a balance

she found the mass of a nickel is about the same as the mass of 5 shoestrings.Grams are the best units to use. The mass of a nickel is about 5 grams.



Math Background Part II - Measurement Conversions

This part of the lesson begins with Hands-On Activity 3, an activity in whichstudents use actual measurement tools to record relationships between twounits of measure within the same system, such as centimeters and meters.

Converting Measures Within the Same Measurement System

Some problems involve more than one measurement unit. To solve these problems,you may need to convert from one unit of measurement to another, such as feet toinches or yards to feet. When changing from one unit of measure to another, therelationships between the two units of measure are very important. For example, tochange inches to feet you need to know that 1 foot = 12 inches.

Rules for Measurement Conversions

To convert a smaller unit to a larger unit, divide by the number of smaller unitsthere are in one of the larger units. Because you are converting from a smaller unitto a larger unit, you can expect to have a smaller number.

To convert a larger unit to a smaller unit, multiply by the number of smaller unitsthere are in one of the larger units. Because you are converting from a larger unit toa smaller unit, you can expect to have a larger number.

EXAMPLE 1: Jonah needs 29 feet of decorative lights for a display. Lights come in4-yard, 7-yard, and 10-yard strands. Which length should he purchase?

To decide which length he should purchase, convert the number of yards in eachstrand to feet. The Grade 4 Reference Materials shows 1 yard = 3 feet.To convert from a larger unit to a smaller unit use the operation of multiplication.

4 yards 3 feet per yard = 4 3 = 12 feet7 yards 3 feet per yard = 7 3 = 21 feet

10 yards 3 feet per yard = 10 3 = 30 feet

Jonah wants the length of the strand of lights to be as close as possible to the exactlength he needs. There are 30 feet in 10 yards. For 29 feet of lights, the 10-yardstrand seems to be the right choice.

EXAMPLE 2: Jerissa measured her room and decides that she needs 48 feet of awallpaper border. The border is sold by the yard. How many yards will she need?

To decide how many yards she will need, convert the number of yards in each strandto feet.

To convert from a smaller unit to a larger unit use the operation of division.

There are 3 feet in one yard. Divide the number of feet by 3 to find the number ofyards.

Large Units Small Units



48 3 = 16

There are 16 yards in 48 feet.

Jerissa needs to buy 16 yards of wallpaper border for her room.

Reference Materials for Conversions

The Grade 4 Reference Materials lists units of measure for the metric and customarysystems. The Reference Materials also lists abbreviations for the measurement labels.

GRADE 4 REFERENCE MATERIALSLENGTH

Customary Metric

1 mile (m) = 1,760 yards (yd)1 yard (yd) = 3 feet (ft)1 foot (ft) = 12 inches (in.)

1 kilometer (km) = 1,000 meters (m)1 meter (m) = 100 centimeters (cm)1 centimeter (cm) = 10 millimeters (mm)

VOLUME AND CAPACITYCustomary Metric

1 gallon (gal) = 4 quarts (qt)1 quart (qt) = 2 pints (pt)1 pint (pt) = 2 cups (c)1 cup (c) = 8 fluid ounces (fl oz)

1 liter (L) = 1,000 milliliters (mL)

WEIGHT AND MASSCustomary Metric

1 ton (T) = 2,000 pounds (lb)1 pound (lb) = 16 ounces (oz)

1 kilogram (kg) = 1,000 grams (g)

1 gram (g) = 1,000 milligrams (mg)

A similar Reference Materials will be provided on Grade 4 STAAR. Students are NOTexpected to memorize the information in the chart. Students ARE expected to beable to utilize the chart.

STAAR Category 3 GRADE 4 TEKS 4.7A/4.7B/4.7C


Lesson 5 - 4.7A/4.7B/4.7C

Lesson Focus

For TEKS 4.7A students are expected to illustrate the measure of an angle as the partof a circle whose center is at the vertex of the angle that is "cut out" by the rays ofthe angle. Angle measures are limited to whole numbers.

For TEKS 4.7B students are expected to illustrate degrees as the units used tomeasure an angle, where 1/360 of any circle is one degree and an angle that "cuts"n/360 out of any circle whose center is at the angle's vertex has a measure of ndegrees. Angle measures are limited to whole numbers.

For TEKS 4.7C students are expected to determine the approximate measures ofangles in degrees to the nearest whole number using a protractor.

For these TEKS students should be able to apply mathematical process standards tosolve problems involving angles less than or equal to 180 degrees.

For STAAR Category 3 students should be able to demonstrate an understanding ofhow to represent and apply geometry and measurement concepts.



4.1.B Use a problem-solving model that incorporates analyzing given information,formulating a plan or strategy, determining a solution, justifying the solution,and evaluating the problem-solving process and the reasonableness of asolution.





4.1.G Display, explain, and justify mathematical ideas and arguments using precisemathematical language in written or oral communication.


PART I PART I PART IImeasure clockwise vertexangle counterclockwise rayfractional part elapsed anglecenter whole protractorvertex full outer scale"cut out" degrees inner scaleray(s) n degrees



Math Background Part I - Angles

The measure of an angle is a fractional part of a circle whose center is at thevertex of the angle that is "cut out" by the rays of the angle.

TEACHER NOTE: This lesson begins with Hands-On Activity 1. Students discover

how many112

fraction pieces come together in the center of a circle.

Angles as Fractions of a Clock Face

Fractions and angles can be related to the hands of a clock. The clock is a circle. Thehands of the clock represent the rays of an angle that "cut out" a fraction of the clockface.

EXAMPLE 1: Each 5-minute mark represents a112

turn clockwise.

An angle formed in a circle using a 112

fraction piece is like a 112

turn on a clock.

EXAMPLE 2: The minute hand makes a 14

turn clockwise. 15 minutes has elapsed.

An angle formed in a circle using a14

fraction piece is like a14

turn on a clock.

EXAMPLE 3: The minute hand makes a34

turn clockwise. 45 minutes has elapsed.

An angle formed in a circle using a 34

fraction piece is like a 34

turn on a clock.

EXAMPLE 4: The minute hand makes a full turn clockwise. 60 minutes has elapsed.



An angle formed in a circle using a whole circle piece is like a full turn clockwise.

Angles as Fractions of a Circle

Two rays that create an angle with a vertex at the center of a circle "cut out" afraction of the circle.

EXAMPLE 1: The rays of this angle have a vertex at the center of the circle. Theshaded part of the circle shows the part of the circle "cut out" by the angle.

This angle "cuts out" 12

of the circle.


This angle "cuts out"14

of the circle.


This angle "cuts out"34

of the circle.

EXAMPLE 4: The rays of this angle have a vertex at the center of the circle. The

rays show a12

clockwise turn.

The measure of the distance between the rays of the angle is 12

of the circle.



EXAMPLE 5: The rays of this angle have a vertex at the center of the circle. Therays show a full clockwise turn.

The measure of the distance between the rays of the angle is the whole circle.

EXAMPLE 6: The rays of this angle have a vertex at the center of the circle. The

rays show a 34

clockwise turn.

The measure of the distance between the rays of the angle is 34

of the circle.

Degrees Related to Fractional Parts of a Circle

What you know about angles and fractional parts of a circle can be used to understandangle measurement. Angles are measured in units called degrees.

Think of a circle divided into 360 equal parts. An angle that cuts1

360out of a circle

measures 1 degree.

The measure of this angle is written as 1°.

So, an angle that cuts360

nout of a circle measures n degrees.

EXAMPLE 1: The angle between two spokes on Jackson's bicycle wheel "cuts out" or

"turns through" 10360

of a circle. Find the measure of the angle between the spokes.

Each spoke on the bicycle wheel represents a ray of an angle.

Each1

360turn measures 1 degree or 1°.

Ten 1360

turns measure 10 degrees or 10°.

So, the measure of the angle between the spokes is 10°.



There are 36 angles that measure 10° between the spokes on Jackson's bicyclewheel.The measure of each angle on the wheel is 10°. 36 10° = 360°

So, the measure of the angle that turns through the whole wheel is 360°.

EXAMPLE 2: What you know about angles and fractional parts of a circle can be usedto find the measure of a right angle.

Think: What fraction of a circle does a right angle "cut out"?

Write14

as an equivalent fraction with 360 in the denominator.

To write an equivalent fraction, multiply the numerator and denominator by thesame factor.

4 × 9 = 36, so 4 × 90 = 360

14

=90360

Write 90360

in degrees.

An angle that cuts1

360out of a circle measures 1°.

An angle that cuts90360

out of a circle measures 90°.

So, a right angle measures 90°.

right angle symbol



Math Background Part II - Measuring Angles

A ray is a part of a line that has one endpoint and extends indefinitely in onedirection. When two rays meet at a common endpoint, they form an angle.

Point E is called the vertex of DEF , or E .A protractor is a tool for measuring the number of degrees in an angle.Two protractors are shown.

A protractor has two scales.The outer scale starts at 0° on the left and increases clockwise to 180° on the right.The inner scale starts at 0° on the right and increases counterclockwise to 180° on

the left.

Measuring an Angle with a Protractor

Step 1: Place the center of the protractor on the vertex of the angle.

Step 2: Line up one ray of the angle with a 0-degree mark on either theouter scale or the inner scale.

Step 3: Read the angle measure where the second ray crosses the scale.Be sure to look at the same scale that you used in Step 2.

Step 4: The measure of the angle can be written in two ways.One Way: the measure of angle B is 55° Another Way: mB = 55°.Either way is read as the measure of angle B is 55 degrees.

Step 5: Check to make sure the angle measurement is reasonable.If the angle is larger than a right angle, it should measure greater than 90°.If the angle is smaller than a right angle, it should measure less than 90°.

Outer Scale Inner ScaleCenter

Outer Scale Inner ScaleCenter

D

EF



Measuring an Angle Drawn On a Protractor

Step 1: Read the angle measure where the first ray crosses the scale.Use either the outer scale or the inner scale.

Step 2: Read the angle measure where the second ray crosses the scale.Be sure to look at the same scale that you used in Step 1.

Step 3: Find the difference between the two readings.This is the measure of the angle.

Step 4: Check to make sure the angle measurement is reasonable.If the angle is larger than a right angle, it should measure greater than 90°.If the angle is smaller than a right angle, it should measure less than 90°.

EXAMPLE 1: Find the measure of BAD .

Step 1: Read the angle measure where the first ray crosses the scale.The angle measure where ray BA crosses the outer scale is 55°.

Step 2: Read the angle measure where the second ray crosses the scale.The angle measure where ray AD crosses the outer scale is 150°.

Step 3: Find the difference between the two readings.150° 55° = 95°So, the measure of BAD is 95°.

Step 4: Check to make sure the angle measurement is reasonable.The angle is a little larger than a right angle, so 95° is reasonable.

EXAMPLE 2: Find the measure of CAD .

Step 1: Read the angle measure where the first ray crosses the scale.The angle measure where ray AC crosses the inner scale is 65°.

Step 2: Read the angle measure where the second ray crosses the scale.The angle measure where ray AD crosses the inner scale is 30°.

Step 3: Find the difference between the two readings.65° 30° = 35°So, the measure of CAD is 35°.

Step 4: Check to make sure the angle measurement is reasonable.The angle is much smaller than a right angle, so 35° is reasonable.

B

C

D

A



EXAMPLE 3:

138 122 16AEB

122 74 48BEC

74 45 29CED

138 74 64AEC

138 45 93AED

122 45 77BED

A

B

C

D

E



Lesson 6 - 4.9A/4.9B

Lesson Focus

For TEKS 4.9A students are expected to represent data on a frequency table, dot plot,or stem-and-leaf plot marked with whole numbers and fractions. Focus for this lessonis dot plot.

For TEKS 4.9B students are expected to solve one- and two-step problems using datain whole number, decimal, and fraction form in a frequency table, dot plot, or stem-and-leaf plot. Focus for this lesson is dot plot.

For these TEKS students should be able to apply mathematical process standards tosolve problems by collecting, organizing, displaying, and interpreting data.

For STAAR Category 4 students should be able to demonstrate an understanding ofhow to represent and analyze data and how to describe and apply personal financialconcepts.







PART Idot plotdataclusterednumber line



Math Background Part I - Dot Plots

A dot plot is a useful tool to display data. A dot plot uses dots to record each piece ofdata above a number line. A dot plot makes it easy to see how data is groupedtogether or clustered.

EXAMPLE: Jackson is collecting and identifying insects for a science project. This dotplot represents the lengths of the insects in his collection.

Each dot on the plot represents 1 insect.The lengths for 20 insects is shown on the dot plot.

The numbers under the number line represent lengths of insects.

The least length is 14

inch and the greatest length is 11

2inches.

The data shows Jackson has 2 insects with a length of 14

inch, 5 insects with a

length of 12

inch, 6 insects with a length of 34

inch, 4 insects with a length of 1 inch,

1 insect with a length of 11

4inches, and 2 insects with a length of 1

12

inches.

Create a Dot Plot

Make a dot plot when you need to show data that is grouped together or clustered.

Follow these steps to make a dot plot:Draw a number line.

The number line must include all the numbers in the first column of the data table.

Draw dots above the number line to represent the data.

After all the data is recorded, title the dot plot.

EXAMPLE: A school bus driver school collected data about the distance students whoride her bus live from school. The table shows the data she collected.

14

12

34

11

14

11

2

Insect Length(in inches)



Make a dot plot to represent the data.Draw a number line.

The number line must include all the numbers in the first column of the data table.The least number in the data table is 2.1 and the greatest number in the data tableis 2.8. So, begin the number line with 2.1 and end the number line with 2.8.

Draw dots above the number line to represent the data.Six students live 2.1 miles from school. Put 6 dots above 2.1 on the number line.Four students live 2.2 miles from school. Put 4 dots above 2.2 on the number line.Five students live 2.3 miles from school. Put 5 dots above 2.3 on the number line.Three students live 2.4 miles from school. Put 3 dots above 2.4 on the number line.Four students live 2.5 miles from school. Put 4 dots above 2.5 on the number line.Three students live 2.6 miles from school. Put 3 dots above 2.5 on the number line.Zero students live 2.7 miles from school. Put 0 dots above 2.7 on the number line.Four students live 2.8 miles from school. Put 4 dots above 2.8 on the number line.

After all the data is recorded, title the dot plot.

Title this dot plot "Distance Students Live from School". Be sure to include "in miles"below the title to show the measure that was used for the data.

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Distance Students Live from SchoolDistance(in miles)

Numberof Students

2.1 62.2 42.3 52.4 32.5 42.6 32.7 02.8 4

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Distance Students Live from School(in miles)



Math Background Part II - Solving Problems With Dot Plots

Data from a dot plot can be used to compare data and solve problems.

EXAMPLE: A class of twenty fourth grade students took a survey of the number ofletters in their first names. This dot plot represents the data they gathered.

Do more students have first names with 3-5 letters or 6-8 letters?

The total of first names with 3-5 letters is 2 + 5 + 6 = 13.

The total of first names with 6-8 letters is 4 + 1 + 2 = 7.

So, more students have first names with 3-5 letters than 6-8 letters.

What is the difference between the least number of letters in the first names and thegreatest number of letters?

The least number of letters in the first names is 3 and the greatest number of lettersin the first names is 8.

The difference between 8 and 3 is 8 ‒ 3 = 5.

So, the difference between the least number of letters in the first names and thegreatest number is 5.

What is the difference between the number of students with 3 or 8 letters in their firstname and the number of students with 5 letters?

The number of students with 3 letters in their first name is 2 and number of studentswith 8 letters in their first name is 2. 2 + 2 = 4.

The number of students with 5 letters in their first name is 6.

The difference between 6 and 4 is 6 ‒ 4 = 2.

So, the difference between the number of students with 3 or 8 letters in their firstname and the number of students with 5 letters is 2.

Letters in First Name

3 4 5 6 7 8



Lesson 7 - 4.10B

Lesson Focus

For TEKS 4.10B students are expected to calculate profit in a given situation.

For this TEKS students should be able to apply mathematical process standards tomanage one's financial resources effectively for lifetime financial security.

For STAAR Category 4 students should be able to demonstrate an understanding ofhow to describe and apply personal financial concepts.





PART Iprofit



Math Background Part I - Profit

Profit is the amount of money left after all the expenses are subtracted from theamount of money received from selling an item or a service.

EXAMPLE 1: Simeon runs his own lawn mowing business. His expenses for oneweek are shown below.

Lawn Mowing Expenses June 4 - 11Expense Cost

Gas for truck and tools $63.20Other truck maintenance $59.38New tools $189.74

Simeon charges $45 to mow a lawn. During the week of June 4 - 11, he mowed 15lawns. Find the amount of profit he made for the week.

First, find the total expenses for the week.

$63.20 + $59.00 + $189.74 = $312.32

Next, find the amount he received for mowing lawns.

15 × $45 = $675.00

The find the difference between his expenses and the amount he received.

$675.00 $312.32 = $362.68

So, Simeon's profit for the week of June 4 - 11 was $362.68.

EXAMPLE 2: During the week of August 15 - 23, Simeon mowed and edged 12lawns. His expenses for the week are shown.

Lawn Mowing Expenses August 15 - 23Expense Cost

Gas for truck and tools $45.60Other truck maintenance $62.17New lawnmower $378.79

Simeon is now charging $55 to mow and edge a lawn. Find the amount of profit hemade for the week.

First, find the total expenses for the week.

$45.60 + $62.17 + $378.79 = $486.56

Next, find the amount he received for mowing lawns.

12 × $55 = $660.00

Then find the difference between his expenses and the amount he received.

$660.00 $486.56 = $173.44

So, Simeon's profit for the week of August 15 - 231 was $173.44.

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