Name2

Divide by 2-Digit Divisors

SKILL

S61

OBJECTIVE Divide by 2-digit divisors.

You can use estimation to help you place the first digit

in the quotient. Then, you can divide. To check your answer,

multiply the quotient by the divisor and add the remainder.

Then, compare the result to the dividend.

Divide: 63)2,637

STEP 1

Use compatible numbers to

estimate the quotient.

STEP 2

Divide the tens.

63)2,637

÷

=

The first digit will be in the

place.

tens ÷ 63

Divide.

Multiply. 63 ×

tens =

tens

tens –

Subtract.

11 tens

tens =

Check. 11 tens cannot be shared among

63 groups without regrouping.

© Houghton Mifflin Harcourt Publishing Company

STEP 3

Regroup any tens left as ones.

Write the remainder to the

right of the whole number

part of the quotient.

63)2,637

-2 52

117

ones ÷ 63

Divide.

Multiply. 63 ×

Subtract.

54 ones

ones =

ones –

ones

ones =

Check. 54 ones cannot be shared among

63 groups without regrouping.

Try This!

Divide.

1. 89)1,597

Skill S61

2. 46)5,624

S61

Name

SKILL

S62

2

Estimate Decimal Sums and Differences

OBJECTIVE Use rounding or benchmarks to estimate decimal sums and differences.

You can use rounding or benchmarks to estimate decimal sums

and differences.

A

2.4 rounds to

Use rounding to estimate the sum

or difference.

1.67 rounds to

.

Estimate 2.4 + 1.67 + 3.18.

Round addends to the nearest

whole number.

Write the rounded addends. Then add.

3.18 rounds to

.

To use rounding to estimate the

difference, follow the same steps.

.

2.4

1.67

+ 3.18

B

Use benchmarks to estimate the sum

or difference.

Identify the closest benchmark for

each decimal. Round each decimal to

the closest benchmark.

Subtract the rounded decimals.

0

0.25

0.50

0.53 is closer to

.

0.22 is closer to

.

–

1

0.75

=

0.53 – 0.22 is about

.

© Houghton Mifflin Harcourt Publishing Company

Estimate 0.53 – 0.22.

Locate and label a point on the

number line for each decimal.

Try This!

Use rounding or benchmarks to estimate.

1.

3.48

-2.15

2.

–

0.52

+0.16

+

0

S62

0.25

0.50

=

0.75

1

Skill S62

Name

SKILL

S63

2

Model Decimal Addition

OBJECTIVE Use a model to add decimals to the hundredths.

You can use a decimal model to add decimals. A hundredths model

shows hundredths.

Add: 1.44 + 0.87.

STEP 1

Shade squares to show 1.44.

Each model shows one whole.

STEP 2

Shade squares to show 0.87.

Since there are only

56 squares left in the second

model, use the third model to

shade the remaining squares

from the second decimal.

STEP 3

Find the sum.

There are

Count the total number of

squares shaded.

whole squares shaded and

one-hundredths squares shaded.

© Houghton Mifflin Harcourt Publishing Company

So, 1.44 + 0.87 =

.

Try This!

Use the decimal models to find the sum.

2.

1.

0.33 + 0.41 =

Skill S63

0.98 + 0.75 =

S63

Name

SKILL

S64

2

Add Decimals

OBJECTIVE Use place value to add decimals to hundredths.

You can use a place-value chart to help you add decimals.

Adding decimals is similar to adding whole numbers. Decimals

can be added place-by-place starting with the least place.

What is the sum of 2.35 and 1.82?

STEP 1

Estimate 2.35 + 1.82.

Round each decimal to the nearest

whole number. Add the whole

numbers to estimate the sum.

2.35 + 1.82

+

STEP 2

Line up the place values for each

number in a place-value chart.

Add the hundredths first.

Then, add the tenths.

Finally, add the ones.

2 +

Estimate:

2

=

Ones

Tenths

Hundredths

t

+

t

t

Regroup as needed.

STEP 3

Draw a quick picture to check

your work.

Use your estimate to see if your

answer is reasonable.

is close to the estimate, 4.

The answer is reasonable.

Try This!

Estimate. Then find the sum.

1. Estimate:

2. Estimate:

1.30

+

S64

+ 0.12

1.51

+

+ 1.22

Skill S64

© Houghton Mifflin Harcourt Publishing Company

2.35 + 1.82 =

Name

SKILL

S65

2

Model Decimal Subtraction

OBJECTIVE Use a model to subtract decimals to the hundredths.

You can use a decimal model to subtract decimals.

Subtract: 1.73 – 0.48.

STEP 1

Shade squares to show 1.73.

Each model shows one whole.

STEP 2

Subtract the second number.

Circle and cross out squares to

show subtracting 0.48.

STEP 3

Find the difference.

Count the total number of shaded

squares that are not crossed out.

There is

whole square and

hundredths shaded squares that

are not crossed out.

So, 1.73 – 0.48 =

.

Try This!

© Houghton Mifflin Harcourt Publishing Company

Use the decimal models to find the difference.

2.

1.

0.88 – 0.27 =

Skill S65

1.09 – 0.33 =

S65

Name

SKILL

S66

2

Subtract Decimals

OBJECTIVE Use place value to subtract decimals to hundredths.

You can use a place-value chart to help you subtract

decimals. Subtracting decimals is similar to subtracting whole

numbers. Decimals can be subtracted place-by-place starting

with the least place.

Find 12.65 – 4.32.

STEP 1

Estimate. Round each

decimal to the nearest whole

number and subtract.

STEP 2

Line up the place values for

each number in a

place-value chart.

Subtract the hundredths first.

Subtract the tenths next.

Then subtract the ones.

Regroup as needed.

Use your estimate to see if

your answer is reasonable.

12.65 – 4.32

– 4= 9

Estimate: 13

–

Tens

Ones

1

2

4

Tenths

Hundredths

•

6

5

•

3

2

•

is close to the estimate,

.

The answer is reasonable.

8.33 + 4.32 =

© Houghton Mifflin Harcourt Publishing Company

Use addition to check

your answer.

Try This!

Estimate. Then find the difference.

1. Estimate:

2. Estimate:

4.82

–

S66

– 2.14

15.82

–

– 1.22

Skill S66

Name

SKILL

S67

2

Algebra • Multiplication Patterns

with Decimals

OBJECTIVE Use patterns to place decimal points when multiplying by 10, 100, and 1,000.

You can use patterns to help you place the decimal

point in a product when you multiply a decimal by

10, 100, and 1,000.

Use a pattern to find 1,000 × 0.85.

STEP 1

Multiply 1 × 0.85.

Write the product.

STEP 2

The decimal point moves 1 place to the

right as you multiply by 10, 100,

and 1,000.

Multiply 10 × 0.85.

STEP 3

Complete the pattern. Write

the products.

1 × 0.85 =

10 × 0.85 =

100 × 0.85 =

1,000 × 0.85 =

So, 1,000 × 0.85 is

.

Try This!

© Houghton Mifflin Harcourt Publishing Company

Complete the pattern.

1. 1 × 2.81 =

2. 1 × 34.25 =

10 × 2.81 =

10 × 34.25 =

100 × 2.81 =

100 × 34.25 =

1,000 × 2.81 =

1,000 × 34.25 =

Skill S67

S67

Name

SKILL

S68

2

Multiplication with Decimals and

Whole Numbers

OBJECTIVE Use place value to multiply decimals through hundredths.

You can use strategies based on place value to find the product

of a 1-digit whole number and a decimal.

Find 5 × 2.25.

STEP 1

Multiply 2.25 by 5 as you would multiply

225 by 5.

225

×

5

_

Multiply the ones.

Regroup when necessary.

STEP 2

Multiply the tens.

2

Remember: Add the regrouped ones.

Regroup when necessary.

225

×

5

_

5

STEP 3

Multiply the hundreds. Add the

regrouped ten.

12

225

×

5

_

25

Think: There are 2 decimal places in the

decimal factor, 2.25.

© Houghton Mifflin Harcourt Publishing Company

Place the decimal point. Count the

number of decimal places in each

original factor. The number of decimal

places in the product equals the

total number of decimal places

in the factors.

5 × 2.25 =

Write the product.

Try This!

Find the product.

1. 2.33

×

3

_

2.

6.2

×

8

_

3. 4.21

×

7

_

4. 5.64

×

4

_

5.

1.87

×

6

_

6.

S68

21.31

×

4

__

Skill S68

Name

2

Multiply Decimals

SKILL

S69

OBJECTIVE Use place value to multiply two decimals.

You can use strategies such as patterns and place value to place

the decimal point in the product when you multiply two decimals.

Find 1.3 × 2.8.

1 place value

28

×1

3

_

A Use place value to place the

decimal point.

Multiply 1.3 × 2.8 as you would

multiply whole numbers.

×

1 place value

2 place values

+

Write the product.

Rewrite the multiplication

as a decimal.

Think: Tenths are being multiplied by tenths.

Use the pattern 0.1 × 0.1 = 0.01.

The decimal point should be placed

so the value of the decimal

is

.

Place the decimal point.

2.8

B Use an estimate to place the

decimal point. Estimate by rounding

each factor to the nearest

whole number.

×

2.8

1.3

×1.3

_

=

28

×13

_

Multiply as with whole numbers.

© Houghton Mifflin Harcourt Publishing Company

×

Use the estimate to place the decimal

point in the product.

+

Think: The product should be close to

your estimate.

Try This!

Find the product.

1.

0.7

×

0.6

_

Skill S69

2.

6.8

×2.3

_

S69

Name

SKILL

S70

2

Algebra • Division Patterns with Decimals

OBJECTIVE Find patterns in quotients when dividing by 10, 100, and 1,000.

You can use patterns to find quotients when dividing

by 10, 100, and 1,000. A place-value pattern can help you

determine where to place the decimal point in the quotient.

Use a pattern to find 152 ÷ 1,000.

STEP 1

As you divide by 10, 100, and 1,000 the

decimal point moves to the left in

the quotient.

STEP 2

Complete the pattern. Write

the quotients.

Think: Write zeros to the left of the digits as

Write the number of places to the left

the decimal point moves.

Divide By

1

placeholders when moving the decimal

point to show the correct place value.

Move Decimal Point

0

152 ÷ 1 =

places to left

10

place to left

152 ÷ 10 =

100

places to left

152 ÷ 100 =

1000

places to left

152 ÷ 1,000 =

Try This!

Complete the pattern.

146 ÷ 10 =

124 ÷ 10 =

146 ÷ 100 =

124 ÷ 100 =

146 ÷ 1,000 =

124 ÷ 1,000 =

3. 18 ÷ 1 =

S70

2. 124 ÷ 1 =

© Houghton Mifflin Harcourt Publishing Company

1. 146 ÷ 1 =

4. 121 ÷ 1 =

18 ÷ 10 =

121 ÷ 10 =

18 ÷ 100 =

121 ÷ 100 =

18 ÷ 1,000 =

121 ÷ 1,000 =

Skill S70

Name

SKILL

S71

2

Estimate Decimal Quotients

OBJECTIVE Use compatible numbers to estimate decimal quotients.

You can estimate decimal quotients by using compatible numbers.

When choosing compatible numbers, you can look at the

whole-number part of a decimal dividend or you can rename the

decimal dividend as tenths or hundredths.

A

Estimate 52.6 ÷ 6.

Use a whole number compatible with

6 that is close to 52.6.

Divide using the compatible number.

is close to 52.6 and is

compatible with 6.

÷6=

4.4 =

B

Estimate 4.4 ÷ 7

Rewrite 4.4 as tenths.

Use a number compatible with 7 that

is close to 44.

Divide the tenths using the

compatible number.

tenths

is close to 44 and is compatible

with 7.

tenths ÷ 7 =

tenths

or

Try This!

Use compatible numbers to estimate the quotient.

© Houghton Mifflin Harcourt Publishing Company

1. 34.2 ÷ 7 is about

2. 76.9 ÷ 4 is about

.

is close to 34.2 and is

compatible with 7.

is close to 76.9 and is

compatible with 4.

÷7=

÷4=

3. 3.1 ÷ 8 is about

3.1 =

4. 4.6 ÷ 8 is about

.

4.6 =

tenths

is close to 31 and is

compatible with 8.

tenths ÷ 8 =

or

Skill S71

.

.

tenths

is close to 46 and is

compatible with 8.

tenths ÷ 8 =

tenths

tenths

or

S71

Name

SKILL

S72

2

Fractions of a Whole

OBJECTIVE Model, read, and write fractions that represent more than one part of a whole.

A fraction can tell how many equal parts a whole has been divided

into. A fraction can name more than 1 equal part of a whole.

The denominator, or bottom number in a fraction, tells how many

equal parts are in the whole. The numerator, or top number in a

fraction, tells how many equal parts are being counted.

Write the fraction that names the shaded part.

1.

Think: Each part is __

4

STEP 1

Count the number of equal parts in

the whole. This is the denominator.

STEP 2

Count the number of shaded parts

in the whole. This is the numerator.

There are

shaded parts.

There are

equal parts.

Say: three fourths

Try This!

1.

2.

Each part is

.

eighths

S72

© Houghton Mifflin Harcourt Publishing Company

Write the fraction that names each part. Then write a fraction

in words and in numbers that names the shaded part.

Each part is

.

sixths

Skill S72

Name

2

Fractions on a Number Line

SKILL

S73

OBJECTIVE Represent and locate fractions on a number line.

You can use number lines to show fractions. The distance from one

whole number to the next whole number represents one whole.

__.

Complete the number line. Draw a point to show 3

4

STEP 1

The denominator is 4, so use

fraction strips for fourths. Place

four 1_4 -fraction strips end to end

above the number line.

0

1

1

4

1

4

STEP 2

At the end of each fraction strip, draw

a mark on the number line. Label the

marks on the number line as fourths.

1

4

1

4

0

4

4

STEP 3

Draw a point on the number line

to represent the distance from 0 to 3_4 .

Try This!

Complete the number line. Draw a point to show the fraction.

__

2. 5

8

© Houghton Mifflin Harcourt Publishing Company

1

1. __

6

0

1

1

6

0

Skill S73

1

6

1

6

1

6

1

6

1

6

0

1

1

8

6

6

0

1

8

1

8

1

8

1

8

1

8

1

8

1

8

8

8

S73

Name

SKILL

S74

2

Generate Equivalent Fractions

OBJECTIVE Find fractions that are equivalent to given fractions.

Equivalent fractions are fractions with different denominators

that name the same value. You can find equivalent fractions

by multiplying the numerator and denominator by the

same number.

3?

How many tenths are in __

5

STEP 1

Compare fifths and tenths.

1.

Shade both models to show __

5

STEP 2

Find how many tenths you need

to make 3 fifths.

3.

Shade both models to show __

5

You need

tenth-size parts

to make 1 fifth-size part.

1

__ = ____

5

10

Complete the multiplication and write

the equivalent fraction.

3×

© Houghton Mifflin Harcourt Publishing Company

STEP 3

Multiply the numerator and the

denominator of 3_5 by the same factor

to get tenths.

=

5×

Try This!

Write the equivalent fraction.

3?

1. How many twelfths are in __

4

3×

4×

S74

1?

2. How many eighths are in __

2

1×

=

12

2×

=

8

Skill S74

Name

2

Equivalent Fractions and Simplest Form

SKILL

S75

OBJECTIVE Use division to find equivalent fractions and simplest form.

A fraction is in simplest form when the only common factor for

the numerator and denominator is one. You can use fraction

strips or division to find an equivalent fraction in simplest form.

__. Write the fraction in simplest form.

Write equivalent fractions for 2

4

A Use fraction strips.

Line up fraction strips to show 2_4 .

Line up other fraction strips beneath to show

the same amount as 2_4 .

Draw to show your work.

2

__ =

4

B Divide.

2÷

1

2

__ = ________ = __

4

2

4÷

Divide the numerator and the denominator

by the same number.

Find the simplest form by dividing until 1 is

the only number that can be divided into the

numerator and the denominator.

© Houghton Mifflin Harcourt Publishing Company

___, = ___, = ___

__ is

Simplest form of 2

4

___ .

Try This!

Write an equivalent fraction. Then write the fraction in simplest form.

__ =

1. 6

8

___

6÷

6 = ________ =

__

8

__ =

2. 4

6

8

Skill S75

4÷

___

4 = ________ =

__

6

8÷

6 in simplest form is

__

___

___.

___

6÷

4 in simplest form is

__

6

___.

S75

Name

2

Add and Subtract Fractions with

Equal Denominators

SKILL

S76

OBJECTIVE Add and subtract fractions with equal denominators.

You can add and subtract fractions with equal denominators.

3 + __

2.

A Add: __

8

8

Use fraction strips to model

the addends.

1

1

8

1

8

Count the number of shaded eighths.

Write the addition equation.

5 – __

2.

B Subtract: __

6

6

5.

Use fraction strips to model __

6

Take away 2 sixths. Circle and cross

out the sixths you take away.

1

8

1

8

1

8

1

8

1

8

1

8

eighths in all

____

____

____

1

1

6

1

6

1

6

1

6

1

6

sixths.

Take away

Count the number of shaded sixths left.

1

6

sixths left

Write the subtraction equation.

____

____

____

© Houghton Mifflin Harcourt Publishing Company

Try This!

Use fraction strips to model. Shade or cross out to show your work.

Find the sum or difference. Write the equation.

3 + __

1

1. __

5

5

1

5

1

5

1

2 – __

2. __

3

3

1

5

1

5

1

5

1 1 1

3 3 3

Take away

fifths in all

____ + ____ = ____

S76

third.

third left

____ – ____ = ____

Skill S76

Name

SKILL

S77

2

Rename Fractions and Mixed Numbers

OBJECTIVE Rename mixed numbers as fractions greater than 1 and fractions

greater than 1 as mixed numbers.

You can write a mixed number as a fraction greater than 1 and

rename fractions greater than 1 as a mixed number.

A Write a mixed number as a fraction.

Use fraction strips to model 2 2_3 . Find how many 1_3 -size pieces

are in each whole. Find the total number of 1_3 -size pieces in 2 2_3 .

22 =

3

1

1

3

22 =

3

1

3

1

1

3

1

3

1

3

1=

1=

There are

© Houghton Mifflin Harcourt Publishing Company

1

3

1

3

1

3

3

1-size pieces in 22

2 = ____

__

__. 2__

3

3

3

B Write a fraction greater than 1 as a

mixed number.

Use fraction strips to model 9_5 . Find how

many wholes are in 9_5 and how many

fifths are left over.

Write 9_5 as a mixed number.

9

__ =

5

1

3

1

3

____

0

5

1

5

1

5

2

5

1

5

3

5

1

5

4

5

1

5

____ = 1

5

5

1

5

6

5

1

5

7

5

1

5

8

5

1

5

9

5

1

5

____

Try This!

Write the fraction as a mixed number and the

mixed number as a fraction.

1 = ____

1. 1__

7

Skill S77

__ = ____

2. 15

6

S77

Name

2

Add and Subtract Mixed Numbers with

Equal Denominators

SKILL

S78

OBJECTIVE Add and subtract mixed numbers with equal denominators.

You can add and subtract mixed numbers with equal denominators

by focusing on the whole parts and the fraction parts separately.

A

B

1 + 1__

1

Find the sum. 2__

4

4

Draw pictures to show the

mixed numbers.

5 – 1__

4

Find the difference. 3__

6

6

5.

Draw a picture to show 3__

6

Add the fractions.

1 + __

1=

__

4 4

Cross out the part

you subtract.

Subtract the fractions.

Add the whole numbers.

–

2+1=

Subtract the whole numbers.

Add the sums.

+

=

=

=

Add the differences.

+

=

© Houghton Mifflin Harcourt Publishing Company

Try This!

Find the sum or difference. Show your work.

5

7 – 2__

2. 5__

8

8

2 + 3__

1

1. 4__

5

5

S78

+

=

–

=

+

=

–

=

+

=

+

=

Skill S78

Name

SKILL

S79

2

Subtract Mixed Numbers with

Equal Denominators

OBJECTIVE Rename mixed numbers with equal denominators to subtract.

You can use fraction strips to help you model mixed numbers

when subtracting.

3.

1 – 2__

Find the difference 3__

4

4

STEP 1

Model the number you are

subtracting from, 3 1_4 .

STEP 2

There aren’t enough fourths

to subtract 3_4 from 1_4 without

renaming. Change one of the

1-whole strips to four 1_4 strips.

Rename the fraction as 2 5_4 .

© Houghton Mifflin Harcourt Publishing Company

STEP 3

Subtract by crossing out three

1

_ strips and two 1-whole strips.

4

Write the difference.

1

1

1

1

1

4

1

1

4

1

4

1

4

1

4

1

4

3=

1 – 2__

3__

4

4

Try This!

Find the difference.

2 – 2__

4=

1. 3__

5

5

1 – 2__

2=

2. 4__

3

3

1 – 1__

5=

3. 4__

6

6

3 – 2__

5=

4. 5__

8

8

1 – 1__

3=

5. 3__

8

8

1 – 2__

2=

6. 4__

5

5

Skill S79

S79

Name

2

SKILL

S80

Estimate Fraction Sums and Differences

OBJECTIVE Make reasonable estimates of fraction sums and differences.

You can estimate fraction sums and differences by rounding

fractions to benchmark points such as 0, 1_2 , or 1.

3 + __

7

Estimate the sum. __

5 8

0

5

STEP 1

Find 3_5 on the number line. Determine

if it is closest to 0, 1_2 , or 1.

1

5

2

5

3

5

0

4

5

1

1

2

3 is closest to

__

.

5

STEP 2

Find 7_8 on the number line. Determine

if it is closest to 0, 1_2 , or 1.

5

5

0

8

1

8

2

8

3

8

4

8

0

5

8

6

8

7

8

8

8

1

1

2

7 is closest to

__

.

8

STEP 3

Add the two rounded numbers.

=

3 + __

7 is about

So, __

5

8

.

© Houghton Mifflin Harcourt Publishing Company

+

Try This!

Use the number lines. Estimate the sum or difference.

0

6

1

6

2

6

0

3

6

1

2

5 + __

1

1. __

6 8

5 is closest to

__

6

1 is closest to

__

8

+

S80

4

6

.

.

=

5

6

6

6

0

8

1

0

1

8

2

8

3

8

4

8

5

8

6

8

8

8

1

1

2

7 – __

2

2. __

8 6

7 is closest to

__

8

2 is closest to

__

6

–

7

8

.

.

=

Skill S80

Name

2

Add and Subtract Fractions with

Unequal Denominators

SKILL

S81

OBJECTIVE Use equivalent fractions to add and subtract fractions.

To add or subtract fractions with unequal denominators, you

can rename them as fractions with equal denominators. You

can make a list of equivalent fractions, and rename the

given fractions.

A

5 + __

1.

Add: ___

12

8

5

Write equivalent fractions for __

12 .

Write equivalent fractions for

_1 .

8

Think: Continue until you have two fractions with

the same denominator.

Add using the equivalent fractions.

5

___

12 ,

,

1

__

8,

,

,

5 + __

1=

___

12

B

3 – __

1.

Subtract: __

5

2

Write equivalent fractions for 3_5 .

3

__

Write equivalent fractions for 1_2 .

1

__

+

8

5,

2,

=

,

,

,

,

,

,

Think: Continue until you have two fractions with

the same denominator.

© Houghton Mifflin Harcourt Publishing Company

Subtract using the equivalent fractions.

3 – __

1=

__

5

–

2

=

Try This!

Find a common denominator. Then find the sum or difference.

Show your work.

3 + __

1

1. __

4

8

7 – __

2

2. __

9

3

3,

__

7,

__

8

1,

__

4

9

2

__,

3

+

Skill S81

=

–

=

S81

Name

SKILL

S82

2

Model Multiplication with Fractions and

Whole Numbers

OBJECTIVE Model the product of a fraction and a whole number.

You can make a model to multiply a fraction by a

whole number or a whole number by a fraction.

9 × 3. Think: Find ___

9 of 3.

Multiply: ___

12

12

STEP 1 Draw 3 rectangles to represent the factor 3.

9

STEP 2 The denominator of the fraction __

12 is 12.

Divide the 3 rectangles into 12 equal parts.

9

STEP 3 The numerator of the fraction __

12 is 9. Shade 9 parts.

STEP 4 There are

shaded parts.

This is the numerator of the product.

Each rectangle is divided into

© Houghton Mifflin Harcourt Publishing Company

equal parts.

This is the denominator of the product.

9 × 3 = _____

___

12

Write the fraction as a mixed number: _____ =

_____ .

Try This!

Find the product.

3=

1. 2 × __

4

S82

5× 3=

2. __

6

Skill S82

Name

2

Model Division with Fractions

and Whole Numbers

SKILL

S83

OBJECTIVE Divide a whole number by a unit fraction and a unit fraction by a whole number.

You can use number lines and models to show divison of a whole

number by a unit fraction or a unit fraction by a whole number.

A

__ .

Divide: 4 ÷ 1

2

Think: How many halves are in 4 wholes?

Draw a number line from 0 to 4.

0

4

Divide the number line into halves.

Label each section.

__ =

4÷ 1

2

Skip count by halves from 0 to 4. Count

the number of skips to find the quotient.

Record the quotient.

B

1

__ ÷ 4.

Divide: 1

2

1

2

Place a __1 -fraction strip under a

2

whole strip.

1

8

1

8

1

8

1

8

Think: I am dividing a part into more parts. Each of

© Houghton Mifflin Harcourt Publishing Company

these parts is how much of the whole?

Place 4 of the same fraction strips

so that they fit beneath the __12 -fraction

strip with no gaps or overlaps.

of the whole.

Each part is

1

__ ÷ 4 =

2

Each part is __18 of the whole.

Record the quotient.

Try This!

Divide. Draw a number line or use fraction strips.

1 =

1. 2 ÷ __

3

1 =

2. 3 ÷ __

4

1 ÷ 4=

3. __

3

1 ÷ 3=

4. __

2

Skill S83

S83

Name

SKILL

S84

2

Solve Two-Step Problems

OBJECTIVE Solve two-step problems using the four operations.

Sometimes, you will have to use two steps to find the answer

to a problem. This may mean using more than one operation.

Jasmine had 18 apple slices. She ate 3 slices and gave 5 slices

each to some friends. With how many friends did Jasmine share?

Draw to show your work.

STEP 1 Use 18 counters to show all

of the apple slices.

STEP 2 Take away 3 counters from the

18 counters to show the number

of slices Jasmine ate.

18 –

=

Subtract to show how many

slices are left.

STEP 3 Make groups of 5 counters to

show the number of slices Jasmine

shared with each friend.

15

=

friends

So, Jasmine shared her apple

slices with

friends.

Try This!

Solve. Write the number sentence for each step.

1. Marcus had $33. He put $3 in

his coin bank. Then, with the rest

of his money, he bought 2 DVDs that

cost the same amount. How much

did each DVD cost?

S84

2. Six friends are playing a game with

52 cards. Each player gets the same

number of cards, and 4 cards are

left over. How many cards does

each player get?

=

=

=

=

Skill S84

© Houghton Mifflin Harcourt Publishing Company

Complete the division sentence

and solve.

Name

2

Algebra • Describe Patterns in a Table

SKILL

S85

OBJECTIVE Identify and describe a number pattern in a function table.

You can identify a number pattern in a table. You can look for

a pattern across the rows of the table. You also can look for a

pattern in the columns of the table.

Describe a pattern in the table. Find the unknown number.

Number of Albums

1

2

3

4

5

6

Cost (in $)

5

10

15

20

25

■

STEP 1

Look for a pattern across the rows.

The number of albums increases by

. The cost increases by $

© Houghton Mifflin Harcourt Publishing Company

Write the pattern.

5, 10,

,

.

,

STEP 2

Look for a pattern in the columns.

1×

=5

2×

= 10

Complete the multiplication

sentences to show the pattern.

3×

= 15

4×

= 20

5×

= 25

STEP 3

Use the patterns to find the

unknown number.

25 +

=

6 ×

=

Describe the pattern in the columns.

The pattern in the table is

.

Multiply the number of albums by

.

Try This!

Complete the table. Describe a pattern.

1.

Skill S85

Beds

2

3

4

5

Pillows

6

9

12

15

6

2.

Teachers

1

2

3

Students

15

30

45

4

5

S85

Name

2

Algebra • Number Patterns

SKILL

S86

OBJECTIVE Determine a rule and use it to extend a pattern.

You can use a rule within a pattern to extend it.

Write a rule for the pattern. Then predict the

next two numbers in the pattern.

Think: How do the numbers change?

A

Look at the numbers. 8, 16, 24, 32, 40

8

16

24

32

40

each time.

The numbers increase by

A rule is to

.

Use the rule to find the next two numbers. 40,

,

B

Look at the numbers. 60, 54, 48, 42, 36

54

48

42

36

The numbers decrease by

each time.

A rule is to

.

Use the rule to find the next two numbers. 36,

,

© Houghton Mifflin Harcourt Publishing Company

60

Try This!

Write a rule for the pattern. Then write the next two numbers.

1. 13, 19, 25, 31, 37

2. 87, 78, 69, 60, 51

Rule:

Next two numbers:

Rule:

,

3. 81, 76, 71, 66, 61

S86

,

4. 45, 52, 59, 66, 73

Rule:

Next two numbers:

Next two numbers:

Rule:

,

Next two numbers:

,

Skill S86

Name

SKILL

S87

2

Algebra • Order of Operations

with Parentheses

OBJECTIVE Evaluate an expression by using the order of operations.

The order of operations gives the order in which calculations

are done in an expression.

Evaluate 3 × 7 – (4 ÷ 2).

STEP 1

First, do the operation inside

the parentheses.

STEP 2

Next, multiply and divide from left

to right.

STEP 3

Then, add and subtract from left

to right.

3 × 7 – (4 ÷ 2)

3× 7- 2

3 × 7 – (4 ÷ 2) =

Try This!

© Houghton Mifflin Harcourt Publishing Company

Use the order of operations to find the value of the expression.

Show each step.

1. 8 + (4 – 2) × 5

2. (9 – 5) × 6 ÷ 3

3. (6 + 2) – 3 × 1

4. 5 × 4 + (12 ÷ 2)

Skill S87

S87

Name

SKILL

S88

2

Line Segments and Angles

OBJECTIVE Identify and draw line segments and angles.

A line segment is a portion of a line with two endpoints. An angle

is a figure made of two rays that share the same endpoint. A ray

is part of a line that has one endpoint and continues without end

in one direction.

Draw a line segment and draw an angle.

A

Line Segment

Start by drawing two points.

Use a straightedge to connect the

two points.

Angle

B

Start by drawing a ray.

Draw a second ray from the endpoint

of the first ray.

Try This!

1.

2.

3.

4.

S88

© Houghton Mifflin Harcourt Publishing Company

Name the figure.

Skill S88

Name

SKILL

S89

2

Classify and Measure Angles

OBJECTIVE Classify angles by the size of the opening between the rays.

An angle is formed by two rays that have the same endpoint.

Four types of angles are shown below.

A right angle

forms a

square corner

and measures

exactly 90°.

An acute angle

has a measure

greater than 0° and

less than 90°.

An obtuse angle

has a measure

greater than 90°

and less than 180°.

A straight

angle measures

exactly 180°.

How can the angle at the right be classified?

Think: How does the angle

© Houghton Mifflin Harcourt Publishing Company

compare to a square corner?

STEP 1 Determine if the angle is a

right angle.

The angle

STEP 2 Determine if the angle

measure is greater than or less than 90°.

The angle measure is

than 90°.

STEP 3 Classify the angle.

The angle is

a right angle.

.

Try This!

Classify the angle. Write acute, right, or obtuse.

1.

Skill S89

2.

S89

Name

SKILL

S90

2

Describe Sides of Polygons

OBJECTIVE Describe line segments that are sides of polygons.

Lines that meet are intersecting lines. Intersecting lines can

meet to form right angles. These are called perpendicular lines.

Lines that never meet and are always the same distance apart are

called parallel lines.

Write intersecting, perpendicular, or parallel to describe

the numbered sides.

STEP 1

Look at the sides. Do sides 1 and 2

meet?

STEP 2

Do sides 1 and 2 appear to form a

1

right angle?

STEP 3

Do sides 1 and 2 never meet and are

always the same distance apart?

2

© Houghton Mifflin Harcourt Publishing Company

STEP 4

Write the words that describe

sides 1 and 2.

Try This!

Write intersecting, perpendicular, or parallel to describe the numbered sides.

1.

2.

1

1

2

2

S90

Skill S90

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