Capella University Math Algebra Calculus Worksheet

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.
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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
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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
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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
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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 $
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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,
,
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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!
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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
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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
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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
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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