Using the Partial-Products Strategy to Multiply
(Two-Digit Numbers)
7.1
Emma is painting the concrete floor of a playground.
She needs to know the area of the playground
to figure out how much paint to buy. The dimensions
are shown to the right.
2.Color the tens part red and the ones part blue. Then write each product.
Add the two partial products and write the total.
Width is 6 yards
40
b.
Mary drew this grid to help. She split 24 into tens and ones
then multipled 6 × 20 and 6 × 4.
E
SA
M
4×
d.
50
=
3×
=
3×
=
6×
=
6×
=
8
Total
Step Ahead
sq units
Write the dimensions of a rectangle that will give a product
close to but not exactly 360. Then write the multiplication
sentences to match.
×
=
×
=
×
=
=
=
sq units
ORIGO Stepping Stones 4 • 7.1
© ORIGO Education.
Area
=
=
© ORIGO Education.
6×
7×
4
6 × 78
70
6 × 24
6×
=
Total
3 × 64
60
7×
8
6
=
Area
152
3×
Total
1.Write the dimensions around the grid. Color the tens part blue
and the ones part yellow. Write the product for each part then add
the products to figure out the area of the grid.
4 × 17
7 × 58
3
4×
b.
c.
PL
4
How could you use this strategy to figure out 3 × 28?
a.
=
7
7
You can split a rectangle
into parts to find the
partial products.
6
Step Up
3×
Total
How could you figure out the exact area?
20
3 × 47
3
sq yards
Area is
Estimate the area of the playground.
Would it be more or less than 100 sq yards?
a.
Length is 24 yards
ORIGO Stepping Stones 4 • 7.1
153
7.2
Using the Partial-Products Strategy to Multiply
(Three-Digit Numbers)
Compare these dimensions of two paper strips.
2.Write the dimensions around the rectangle. Figure out each partial product.
Then add to figure out the total.
a.
STRIP A
Width – 4 in
Length – 176 in
Which strip has the greater area?
How do you know?
4 × 289
STRIP B
Width – 7 in
Length – 124 in
How could you figure out the exact area of each strip?
Look at this diagram.
×
=
×
=
×
=
Total
280
24
100
70
6
How has the rectangle been split?
You can split a rectangle
into parts to find the
partial products.
c.
SA
M
What does each of the red numbers represent?
How could you use the diagram to figure out
the total area of Strip A?
b.
I would add the areas of the smaller rectangles.
That's 400 + 280 + 24. The total area is 704 sq inches.
How could you figure out the exact area of Strip B?
6 × 354
6
300
50
6×
=
6×
=
6×
=
4
Total
154
ORIGO Stepping Stones 4 • 7.2
Step Ahead
© ORIGO Education.
a.
×
=
×
=
×
=
Total
6 × 391
×
=
×
=
×
=
Total
1.Figure out each partial product. Then add to figure out the total.
© ORIGO Education.
Step Up
7 × 534
E
400
PL
4
Split the rectangle to show 7 × 307.
Then write number sentences to figure out the total area.
Area
ORIGO Stepping Stones 4 • 7.2
155
7.3
Reinforcing the Partial-Products Strategy for Multiplication
(Three-Digit Numbers)
These marbles are numbered from 0 to 9.
6
7
Imagine you pick four marbles without looking.
2
8
5
Write four possible numbers in these boxes.
3
9
0
2.Choose more numbers from the marbles on page 156.
Complete these to figure out the products.
1
a.
4
×
=
×
Look at the expression you wrote. How could you figure out the product?
Total
Draw lines to split the blue rectangle to show the hundreds, tens, and ones.
Label the dimensions of each part.
Then write each partial product to figure out the total.
b.
=
SA
M
PL
E
×
What are some other multiplication
sentences that you could figure
out using this strategy?
3. Complete each number sentence. Then write the total of the partial products.
a.
Working Space
×
3 × 146
3 × 100 =
1.Choose four different numbers from the marbles above.
Use these numbers to write an expression. Then split the rectangle
into hundreds, tens, and ones to help you figure out the product.
Step Up
Total
=
300
156
c.
5 × 703
5 × 700 =
3 × 40 =
8 × 70 =
5×3=
3×6=
8×4=
Total
Total
© ORIGO Education.
© ORIGO Education.
ORIGO Stepping Stones 4 • 7.3
8 × 374
8 × 300 =
Step Ahead
Total
b.
Write these digits in the spaces below to create an expression
that will give the greatest possible product. Use each digit once
only. Then write the product.
×
ORIGO Stepping Stones 4 • 7.3
Total
=
5
3
2
7
157
7.4
Using the Partial-Products Strategy to Multiply
(Four-Digit Numbers)
Jamal is planning a summer vacation.
He buys three package deals to the Grand Canyon.
2.Label the dimensions for each part and write number sentences to figure
out each partial product. Then add the partial products to figure out the total.
a.
GRAND CANYON
Estimate the total amount that he will pay.
3 × 2,178
VACATION
PACKAGES
$1, 849
Do you think it will be more or less than $5,000?
×
=
×
=
×
=
×
=
How could you figure out the exact cost?
Lora drew this diagram to help her thinking.
Total
3
800
40
9
3 × 1,000 =
How did she split the rectangle?
3 × 800 =
Complete the number sentences to show
each partial product. Then add the partial
products to find the total.
Compare the parts of the rectangle in the diagram
to each partial product. Color the part of the rectangle
that shows 3 × 800.
Step Up
SA
M
3 × 40 =
3,000
=
×
=
×
=
×
=
Total
c.
Total
2 × 3,208
×
=
×
=
×
=
Total
4 × 2,000 =
4 × 100 =
Step Ahead
4 × 20 =
20
5
Color the part of the rectangle yellow that shows 8 × 20.
Color the part of the rectangle blue that shows 8 × 500.
Color the part of the rectangle red that shows 8 × 3,000.
4×5=
Total
ORIGO Stepping Stones 4 • 7.4
© ORIGO Education.
100
© ORIGO Education.
4
2,000
×
3×9=
1.Figure out each partial product.
Then write the total of the four products.
4 × 2,125
158
4 × 1,795
PL
1,000
E
b.
8
3,000
ORIGO Stepping Stones 4 • 7.4
500
20
6
159
Reinforcing the Partial-Products Strategy for Multiplication
(Four-Digit Numbers)
7.5
2.Write the expressions to match.
Then use the working space to figure
out each product.
What multiplication expression could you write to match this diagram?
a.
×
3
2,000
600
70
=
7
1,000
Write the expression in the boxes below.
300
50
5
80
9
b.
×
=
E
×
3
How could you figure out the product?
PL
What number sentences would you write?
6,000
Working Space
3.Figure out each product. Show your thinking.
a.
Record your thinking
in the working space.
=
4
1,000
500
60
6 × 2,503 =
Working Space
a.
×
SA
M
2 × 4,825 =
1.Write the multiplication expressions
to match. Then use the working
space to figure out each product.
Step Up
b.
Step Ahead
3
Aran figured out 3 × 3,065. He added these partial products.
Describe his mistake in words.
b.
2,000
700
20
8
Working Space
ORIGO Stepping Stones 4 • 7.5
© ORIGO Education.
5
160
9,000
1 ,800
+
15
10,8 1 5
=
© ORIGO Education.
×
ORIGO Stepping Stones 4 • 7.5
161
7.6
Using the Partial-Products Strategy to Multiply
(Two Two-Digit Numbers)
2.Write a multiplication sentence to show each part.
Then write the total of the four partial products.
a.
New turf is being laid in a playground.
This diagram shows the dimensions of the playground.
19 × 32
Estimate the amount of turf needed.
43 yd
10
I know 40 x 3 is 120.
40 x 30 is ten times more,
so about 1,200 sq yards of
turf will be needed.
9
27 yd
30
×
=
×
=
×
=
2
Total
b.
How did he split the rectangle?
800
40
280
40
How could you figure out the total area
of the playground?
SA
M
What does each red number represent?
What is the unknown value? How do you know?
46 × 35
PL
Jose drew this diagram.
What does his diagram show?
3
?
21
20
7
6
30
×
=
×
=
×
=
×
=
5
Total
1.Figure out each partial product.
Then write the total of the four products.
Step Up
Step Ahead
36 × 24
30 ×
600
6×
6×
6
20
4
4
20
4
=
600
Write the dimensions around the rectangle.
Write a multiplication sentence to show
each part. Then add the partial products
to figure out the total.
=
=
=
Total
ORIGO Stepping Stones 4 • 7.6
© ORIGO Education.
30
20
© ORIGO Education.
30 ×
162
=
E
How could you figure out the exact amount of turf to order?
×
28 × 42
×
=
×
=
×
=
×
=
Total
ORIGO Stepping Stones 4 • 7.6
163
7.7
Reinforcing the Partial-Products Strategy for Multiplication
(Two Two-Digit Numbers)
2.Write the dimensions around the rectangle. Figure out the product for each
part. Then add the four partial products to find the total.
Mia split this rectangle into partial products
to help figure out the total.
What multiplication sentence is she trying
to figure out?
34 × 37
30
Color the part blue that shows 30 × 8.
Color the part yellow that shows 2 × 40.
2
40
8
E
Color the part red that shows 30 × 40.
What does the
last part show?
Total
PL
3.Figure out each product. Show your thinking.
a.
17 × 25 =
Write each partial product in the working space.
Then add the partial products to find the total.
Step Up
SA
M
How could you figure out the total?
b.
38 × 27 =
c.
41 × 29 =
Working Space
1.Write the dimensions around the rectangle. Write a multiplication
sentence for each part. Then add the four partial products to find
the total.
26 × 45
×
=
×
=
×
=
20 × 10 = 200
Total
164
Write a number sentence for the missing partial product.
Then write the total.
a.
24 × 16 =
ORIGO Stepping Stones 4 • 7.7
© ORIGO Education.
=
© ORIGO Education.
×
Step Ahead
4 × 10 = 40
4 × 6 = 24
ORIGO Stepping Stones 4 • 7.7
b.
51 × 32 =
c.
18 × 36 =
50 × 30 = 1,500
10 × 30 = 300
50 × 2 = 100
10 × 6 = 60
1 × 30 = 30
8 × 6 = 48
165
7.8
Solving Multi-Step Word Problems Involving Multiplication
2.Solve these word problems. Show your thinking.
a. Sweaters cost $39 each. This is
$15 more than the price of a cap.
Over the season, 725 caps were
sold. What was the total sales
from caps?
The Bay City Tigers need to buy 25 pairs of shorts.
$72
How could you figure out the total cost of the shorts?
Alex wrote the partial products to figure out the total.
b. A stadium parking lot has
38 rows. There are 42 spaces
in each row. 200 spaces are
reserved for staff. How many
spaces are there for supporters?
$45
25 × 38
20 × 30 = 600
20 × 8 = 160
5 × 30 = 150
5 × 8 = 40
$
$
1.Use the uniform prices above. The Cincinnati Chargers
need to buy 16 complete uniforms. What is the total cost?
$
ORIGO Stepping Stones 4 • 7.8
Step Ahead
Write a word problem to match this
equation. Then figure out the product.
42 × 13 =
© ORIGO Education.
Working Space
166
d. A team of 18 players bought
boots for $36 and socks for
$9. What was the total cost
for the team?
$85
© ORIGO Education.
Step Up
spaces
c. Sports bags cost $29 each.
If this price was reduced by
$3, how much would be saved
when buying 24 bags?
SA
M
I will call the total cost of
the shirts and shorts T.
T = (45 + 38) x 12
$12
$
PL
The Mountain Warriors need to buy 12 team shirts
and 12 pairs of shorts. What will be the total cost?
E
$3 8
What is the total cost of the shorts? How do you know?
ORIGO Stepping Stones 4 • 7.8
167
Subtracting Common Fractions (Number Line Model)
7.9
Logan went to the movies and bought a small box of popcorn.
At the start of the movie, the box was 87 full.
At the end of the movie, there was 28 of the box left over.
2. Use this number line to help you write the differences.
0
6
POPCORN
a.
How much popcorn did Logan eat during the movie?
What equation could you write?
−
1
PL
0
a.
SA
M
a.
0
6
6
6
9
10
4
10
Difference
a.
12
6
0
8
8
8
16
8
c.
168
12
4
16
4
ORIGO Stepping Stones 4 • 7.9
© ORIGO Education.
8
4
© ORIGO Education.
c.
4
4
b.
15
4
b.
11
18
−
=
12
12
= 18 −
6
1
12
4
c.
5
8
d.
23
8
Difference
3
8
1
Difference
16
−
4
20
−
8
34
−
8
26
= 8
c.
2
=
3
14
− 3
Complete each equation so that the difference is between 2
and 3. Use the number lines on pages 168 and 169 to help you.
Step Ahead
a.
0
4
17
2
=
−
6
6
f.
21
16
− 6 =
6
Difference
b.
16 12
4
− 4 = 4
4
c.
20
8
= 6 − 6
24
6
4.Write the missing fraction in each equation.
1. Draw and label jumps to match each equation.
13
4
9
− 8 = 8
8
9
= 6 − 6
Layla and her friend bought a box of popcorn to share.
They each ate 83 of the popcorn in the box.
How much popcorn was left over? How could you figure out the amount?
11
7
4
− 6 = 6
6
23
e.
18
6
3.Use what you know about subtracting fractions to calculate the difference
between each pair of numbers.
How could you show the difference
on this number line?
Step Up
b.
15
4
=
−
6
6
d.
=
12
6
E
When you subtract fractions
what happens to the numerator?
What happens to the denominator?
6
6
b.
=
25
−
6
=
d.
=
22
−
4
=
Working Space
ORIGO Stepping Stones 4 • 7.9
169
Calculating the Difference Between Mixed Numbers
7.10
2.Calculate the difference. Draw jumps on the number line to show your thinking.
a.
3
One bunch of bananas weighs 5 4 pounds.
Another bunch weighs 3 24 pounds.
How could you figure out the difference
in mass between the two bunches?
2
1
3 4 − 1 4 =
0
1
2
3
4
b.
6
3
2 8 − 1 8 =
0
1
2
3
3
2
4
How could subtraction be used to find the difference?
+
+1
a.
3
4
7
4
5
5
3
4
b.
3
c.
4
5 10
4
6 8 − 2 8 =
SA
M
0
2
4
2
3.Calculate the difference. Show your thinking.
PL
How was addition used to calculate the difference on this number line?
+
E
When I add mixed numbers, I add the
whole numbers and fractions separately
then combine their totals. I think this
will work for subtraction too.
3
I would start with 5 4 then take away
2
3 4 in smaller jumps. One jump would
2
be 3 and the next jump would be 4 .
1
−
2
4 10
10
7
7
7
6 12 − 1 12 =
=
6
d.
e.
5
3
10 6 − 2 6 =
f.
8
2
9 12 − 8 12 =
6 10 − 1 10 =
1.Calculate the difference. Draw jumps on the number line to show
your thinking.
Step Up
a.
4
2
3 6 − 1 6 =
0
1
2
3
4
Step Ahead
3
0
170
1
2
3
4
5
ORIGO Stepping Stones 4 • 7.10
© ORIGO Education.
4
4 5 − 3 5 =
© ORIGO Education.
b.
4
1
12
9
+3 12
ORIGO Stepping Stones 4 • 7.10
Write the missing numbers on this trail.
7
−2 12
8
+5 12
4
−8 12
171
Calculating the Difference Between Mixed Numbers
(Decomposing Whole Numbers)
4
8
7
8
Amos has two pet lizards. One is 3 inches long and the other is 1
How could you figure out the difference in their lengths?
How could you use addition to help you calculate
the difference?
1
1
7
8
=
The difference is 1
1
8
+
+1
1
5
8
a.
1
7 6
inches.
4
8
4
8
−
4
2 6
1
2
3
b.
5
6
4
5
c.
=
=
d.
2
6 5
−
e.
=
4
5 5
3
3
9 10 − 4 10 =
2
14 8
−
3
1 8
4
16 12
f.
−
9
11 12
6
=
= 5 − 3 8
4
8
4
3
4
8
Step Ahead
1.Calculate the difference. Draw jumps on the number line to show
your thinking.
Step Up
4
3. Calculate the difference. Show your thinking.
5
8
3
7
8
3
3
0
4
+
2
4 5 − 3 5 =
7
8
3
2
1
−
3
8
− 1 1
b.
E
12
8
+
7
8
+
12
8
2
0
SA
M
1
1
5 3 − 2 3 =
2 − 1 = 1
Look at the number lines below.
What is the same about the two methods shown?
What is different?
2
inches long.
is the same as
2 1
a.
3 84 − 1 87
Maka figured it out like this.
What did she do to make the subtraction easier?
+1
2. Calculate the difference. Draw jumps on the number line to show your thinking.
PL
7.11
Look at these
related equations.
4+2=6
2+4=6
6−2=4
6−4=2
Each sentence describes the same two
parts (4 and 2) that make a total (6).
Write the related equations for this.
0
172
1
2
3
4
ORIGO Stepping Stones 4 • 7.11
© ORIGO Education.
5
© ORIGO Education.
2
3 6 − 2 6 =
3 87 + 2 84 = 6 83
ORIGO Stepping Stones 4 • 7.11
173
7.12
Solving Word Problems Involving Mixed Numbers
and Common Fractions
2. Solve these problems. Show your thinking.
a. Alisa cut 5 oranges into sixths for a
picnic. Afterward, there was only 64
of an orange left. How many oranges
and part oranges were eaten?
Chang used a watering can and poured 3 43 quarts
of water onto his seedlings which were in a garden
bed that was 5 43 feet long. Afterward, the watering
can had 4 43 quarts of water left in it.
b. A baker used 3 127 sticks of licorice
5
4
and had 1 12
sticks left so she ate 12
of a stick. How many sticks did the
baker have at the start?
How much water was in the watering can at the start?
How could you figure it out?
Which information is important?
oranges
Two identical cakes were baked for a big party. Each cake
3
was cut into twelfths. Halfway through the party, 12
of one
cake had been eaten. The other cake had 4 pieces missing
and 5 people were standing near it.
1. Figure out the answer to each problem. Show your thinking.
a. Bixy and Boxy are cats. Bixy weighs
4 101 kilograms. The total weight of
7
the two cats is 9 10
kilograms.
How much does Boxy weigh?
b. A bucket held 3 21 gallons of water.
1 21 gallons was used on the lettuce
then 21 gallon on the carrots. How
much water was left in the bucket?
kg
gal
E
Step Ahead
Write a subtraction word story that involves mixed numbers
and common fractions.
hr
ORIGO Stepping Stones 4 • 7.12
© ORIGO Education.
in
yd
d. Luis visited his dad. It took 2 41 hr
to get there when it usually takes
1 34 hr. How much later than usual
did he arrive?
© ORIGO Education.
c. The builder cut 2 87 inches off a
length of lumber. The piece left was
5 83 inches long. How long was the
piece of lumber at the start?
174
SA
M
Step Up
c. Mom had 7 62 yards of fabric rolled up. She cut off 63 yards for a quilt she
was making. Some squares on that quilt were 61 yard long on each side.
She also cut 2 64 yards off the roll to make a second quilt. How much fabric
was left on the roll?
PL
How much cake was left over in total?
Which operations will you use to figure it out?
sticks
ORIGO Stepping Stones 4 • 7.12
175
Reading and Writing Six-Digit Numbers
(without Teens and Zeros)
8.1
2.Look at the abacus. Write the matching number on the expander.
a.
Imagine you start at 1,000 and skip count by 1,000. What numbers would you say?
What number would you say after 99,000?
What numbers would you say after that number?
Thousands
H
T
b.
Ones
O
H
T
H T O
Thousands
O
Look at this place-value chart.
What do you notice about each group
of three places?
H
T O
Ones
c.
H T O
Thousands
H
T O
Ones
H T O
Thousands
H
T O
Ones
H T O
Thousands
H
T O
Ones
d.
E
Look at the number on this abacus.
How do you know where to write the digits
on this expander?
How do you read the first three digits of the number?
How do you read the whole number?
Step Up
H
T O
Ones
T O
Ones
e.
SA
M
H T O
Thousands
H
PL
H T O
Thousands
f.
H T O
Thousands
H
T O
Ones
1.Draw extra beads on the abacus to match the number
on the expander.
a.
b.
Step Ahead
This abacus shows the number of people who watched the first
episode of a new reality TV show.
1.How many people
watched Episode 1?
176
T O
Ones
5 2 3
H T O
Thousands
4 3
7
H
T O
Ones
2 4 5
ORIGO Stepping Stones 4 • 8.1
a.How many people
watched Episode 2?
© ORIGO Education.
2 6 4
H
© ORIGO Education.
H T O
Thousands
2.Cross out one bead from each place to show
the number of people who watched Episode 2.
b.How many more people saw
Episode 1 than Episode 2?
ORIGO Stepping Stones 4 • 8.1
H T O
Thousands
H
T O
Ones
177
8.2
Reading and Writing Six-Digit Numbers
on Expanders and in Words
2.Calculate the values and write the matching number on the expander.
Then write the number in words.
Imagine you used all three of these cards
to show a single number.
a.
Where would you write the digits for
the number on the expander below?
How do you know?
2 × 100,000
5 × 100
b.
How would you read the number on the open expander?
E
6 × 100
The first three digits are all thousands, so you can put
these places together and read the number of thousands.
Step Up
PL
7×1
1.Write the matching number on the expander.
Then write the number in words.
a.
5 hundred thousands
2 hundreds
4 × 10
c.
SA
M
Write the same number on this expander.
How would you read the number?
3 × 100,000
8 × 100,000
Step Ahead
a.
b.
Figure out the value of each set of cards, then record the value
on the expanders below.
b.
178
© ORIGO Education.
7 ten thousands
ORIGO Stepping Stones 4 • 8.2
© ORIGO Education.
6 hundred thousands
ORIGO Stepping Stones 4 • 8.2
179
8.3
Reading and Writing Six-Digit Numbers
(with Teens and Zeros)
2.Calculate the values and write the matching number on the expander.
Then write the number in words.
Write digits on the expander to match the number
shown on the abacus.
a.
2 × 100,000
9 × 1,000
H T O
Thousands
H
T O
Ones
3 × 100
How could the expander help you figure out how to say the number name?
b.
Write the number name.
7 × 1,000
E
1 × 100
PL
5 × 100,000
c.
What would you write on the expander below to match this abacus?
2 × 10
How would you say the number name?
How could the expander help you?
Step Up
SA
M
3 × 100,000
H T O
Thousands
H
4 × 1,000
T O
Ones
3.Read the number name. Write the matching number on the expander.
a. t wo hundred five thousand
nine hundred forty
1.Look at the abacus. Write the matching number on the expander.
Then write the number in words.
Step Ahead
Write the value shown in the ten thousands place in each number.
180
T O
Ones
© ORIGO Education.
H
ORIGO Stepping Stones 4 • 8.3
© ORIGO Education.
a.463,759
H T O
Thousands
c.604,050
ORIGO Stepping Stones 4 • 8.3
b. e
ight hundred fourteen thousand
six hundred two
b.815,240
d. 390,11 1
181
Locating Six-Digit Numbers on a Number Line
Start at 7. Say each number on the card.
How do the numbers change?
Say the same pattern again starting at 3.
What numbers belong at each mark on this number line?
How do you know?
Look at each number line carefully. Write the number that is shown by each arrow.
2.
a.
7
70 3
700 30
7,000300
30 0 0
70,000
70 000
7007,000
00 000
100,000
SA
M
0
100,000
Draw marks on this part to show multiples of 1,000.
d.
160,000
170,000
e.
f.
g.
h.
3.
a.
b.
c.
d.
0
Look closely at this part of the same number line.
c.
150,000
PL
0
b.
E
8.4
100,000
200,000
e.
f.
g.
h.
4.
a.
b.
c.
d.
What numbers belong at these marks? Write two numbers.
389,500
e.
1.Draw a line to connect each number to its position
on the number line.
Step Up
403,000
408,000
411,000
f.
182
412,000
415,000
ORIGO Stepping Stones 4 • 8.4
a.
© ORIGO Education.
409,000
420,000
© ORIGO Education.
405,000
410,000
g.
389,700
h.
417,000
Step Ahead
400,000
389,600
Odometers measure distance. These are odometer readings from
vehicles that have just been serviced. If they are serviced every
50,000 miles, write the next service reading.
3 0 0 0 0 0
1 5 0 0 0 0
ORIGO Stepping Stones 4 • 8.4
b.
NEXT
1 7 0 0 0 0
1 5 0 0 0 0
NEXT
183
Working with Place Value
8.5
2. Write the numbers that would be shown on these odometers.
a.
Write the number that is shown on this abacus.
3 6 9 2 8 4
0 3 2 5 9 8
8 8 0 3 4 9
3 6 9 2 8 4
0 3 2 5 9 8
8 8 0 3 4 9
3 6 9 2 8 4
0 3 2 5 9 8
8 8 0 3 4 9
3 6 9 2 8 4
0 3 2 5 9 8
8 8 0 3 4 9
100
more miles
Draw one more bead in the hundred-thousands place.
b.
Write the new number.
H T O
Thousands
Draw one more bead in the ten-thousands place.
H
T O
Ones
1,000
more miles
Write the new number.
E
c.
PL
Cross out one bead in the hundreds place.
10,000
more miles
Write the new number.
d.
Step Up
SA
M
What bead will you cross out now so
that the new number will be 836,335?
100,000
more miles
1.Mechanical odometers start with all of the places at zero and
change as the car travels. These odometers show the the distance
in miles that different cars have traveled. Write what the odometers
would show if the cars travel more miles.
a.
3 6 9 2 8 4
0 3 2 5 9 8
Step Ahead
8 8 0 3 4 9
Look at this odometer.
1 2 0 1 8 9
1.If the vehicle travels another 12 miles,
what will the odometer show?
1 more mile
3 6 9 2 8 4
0 3 2 5 9 8
8 8 0 3 4 9
184
© ORIGO Education.
10 more miles
ORIGO Stepping Stones 4 • 8.5
© ORIGO Education.
b.
2.a.What is the greatest number that
the odometer can show?
b.Think about what the odometer
will do if the vehicle travels one
mile more than the greatest
number. Write what you will see.
ORIGO Stepping Stones 4 • 8.5
185
8.6
Comparing and Rounding Six-Digit Numbers
This table shows approximate populations of another six cities.
Use this table for Questions 4 to 8.
These tables show the approximate populations of ten cities.
City
Billings, MT
Cary, NC
Everett, WA
Fargo, ND
Green Bay, WI
Population
104,1 70
135,234
103,01 9
105,549
104,057
City
Lansing, MI
McKinney, TX
Palm Bay, FL
Springfield, MA
Sunnyvale, CA
City
Cape Coral, FL
Charleston, SC
Flint, MI
High Point, NC
Kansas City, KS
Lafayette, LA
Population
11 4,297
131 ,1 1 7
103,1 90
153,060
140,081
How can you figure out which city has the greatest population?
Which city has the least population?
a. Charleston
E
PL
Which cities have populations that are about 110,000?
How can you figure it out? What helps you?
c. High Point
SA
M
Draw an arrow on this number line to show the approximate location
of Springfield’s population.
b. Which city has the least population?
Flint
b. Kansas City
Cape Coral
Flint
d. Lafayette
Charleston
b. Cape Coral
7. Round each city’s population to the nearest thousand.
a. Kansas City 1. Round each city’s population to the nearest hundred thousand.
a. High Point
Use the tables and number line above to help you complete these.
a. Springfield 6. Round each city’s population to the nearest hundred thousand.
200,000
If you had to round Sunnyvale’s population to the nearest ten thousand,
what number would you write? Why?
Step Up
4. a. Which city has the greatest population?
5. Write the population of each city. Then write < or > to complete each sentence.
Which city has the greater population: Everett or Palm Bay? How do you know?
100,000
Population
154,305
120,083
102,434
104,371
145,786
120,623
b.Charleston
8. Round each city’s population to the nearest hundred.
b.Lansing
a.Flint
b.Lafayette
2. Round each city’s population to the nearest ten thousand.
b.Cary
Step Ahead
a. McKinney 186
© ORIGO Education.
3. Round each city’s population to the nearest thousand.
b.Fargo
ORIGO Stepping Stones 4 • 8.6
© ORIGO Education.
a. Green Bay
Use each digit once.
Write a number that is closest to 250,000.
5
ORIGO Stepping Stones 4 • 8.6
3
9
0
1
6
187
Exploring the Relationship Between Meters
and Centimeters
8.7
2.Write the missing lengths in meters and centimeters. Then draw lines to show
where the other lengths are located on the measuring tape.
Jack has to cut paper streamers for a party.
Each streamer has to be about 70 centimeters long.
The whole roll is 4 meters long.
0 m 87 cm
Will there be enough on the roll for 10 streamers?
How do you know?
I know that 100 centimeters
is the same as 1 meter. So
how many centimeters is equal
to 4 meters? What is the total
length of 10 streamers at
70 centimeters each?
In the word centimeter,
centi means one-hundredth.
A related word is cent,
because one cent is
one-hundredth of a dollar.
20
30
40
50
60
70
How could you say the length of Gavin’s arm span?
How could you write the length of Gavin’s arm span?
80
90
100
110
E
120
meter
b.
cm
50 m is
4 m 0 cm
310 cm
m
cm
m
230 cm
220 cm
320 cm
4 m 39 cm
cm
cm
240 cm
3 m 35 cm
330 cm
m
340 cm
cm
410 cm
400 cm
cm
Step Ahead
420 cm
430 cm
440 cm
450 cm
Write these heights in centimeters.
cm
13 m is
cm
d.
130 m is
cm
e. 280 m is
cm
f. 4,300 m is
cm
c.
188
5 m is
300 cm
2 m 27 cm
m
140 cm
130 cm
120 cm
1. Write each distance using centimeters.
ORIGO Stepping Stones 4 • 8.7
a. Spinosaurus
4 m 30 cm
© ORIGO Education.
a.
290 cm
3 m 18 cm
1 m 34 cm
cm
centimeters
You could also abbreviate the units.
Step Up
cm
© ORIGO Education.
10
cm
cm
210 cm
200 cm
190 cm
m
SA
M
0
m
PL
This picture shows the length of Gavin’s arm span in centimeters.
1 m 90 cm
cm
110 cm
100 cm
90 cm
0 cm
m
cm
ORIGO Stepping Stones 4 • 8.7
b. Ceratosaurus
3 m 70 cm
cm
189
8.8
Introducing Millimeters
3. Complete these.
Some types of ants are
shorter than one centimeter.
a.1 centimeter is the same length as
millimeters.
b.100 centimeters is the same length as
One millimeter is one-tenth of the length of a centimeter.
How many millimeters are the same length as one centimeter?
How many millimeters are are the same length as five centimeters?
4. Use the information in Question 3 to help you complete these.
a.40 centimeters is the same length as
millimeters.
b.85 centimeters is the same length as
millimeters.
E
1.List things in your classroom that are a little less than one
millimeter thick and a little more than one millimeter thick.
A little less than one millimeter thick
millimeters.
A little more than one millimeter thick
PL
How long is each ant from head to tail? How do you know?
Step Up
c.1 meter is the same length as
A short way to
write millimeter
is mm.
millimeters.
c.125 centimeters is the same length as
millimeters.
SA
M
5. Write these lengths in millimeters.
a.5 cm 4 mm is
mm
b.13 cm 8 mm is
mm
2.Measure and label the dimensions of these stickers in millimeters.
a.
b.
c.
Step Ahead
mm
mm
mm
mm
mm
3
×10
mm
mm
mm
ORIGO Stepping Stones 4 • 8.8
© ORIGO Education.
mm
40
45
© ORIGO Education.
mm
f.
Millimeters (mm)
1
e.
d.
190
Complete the missing
numbers in this machine.
mm
mm
Centimeters (cm)
1
92
ORIGO Stepping Stones 4 • 8.8
191
Exploring the Relationship Between Meters
and Millimeters
8.9
2. Look at this floor plan. Write each dimension in millimeters.
This block measures 10 cm.
mm
mm
1
4m
4 2 m
mm
How many millimeters are in 10 cm?
How do you know?
a.
b.
6 meters
is the same length as
mm
mm
e.
11 meters
is the same length as
g.
is the same length as
mm
is the same length as
mm
1
1
E
mm
3m
Bathroom/
Laundry
Toilet
1
3 2 m
3m
mm
mm
is the same length as
Step Ahead
mm
i.
1
Bedroom 1
7 21 meters
mm
23 2 meters
Bedroom 3
4 meters
mm
is the same length as
h.
1
8 2 meters
Bedroom 2
is the same length as
f.
15 meters
mm
192
c.
9 meters
is the same length as
d.
12 meters to millimeters
6m
1. Complete each of these.
1
m
2
mm
3,500 mm to meters
SA
M
Step Up
PL
How would you change these?
mm
1
46 2 meters
230 mm
Write numbers to complete this addition trail.
+3 m
mm
is the same length as
mm
ORIGO Stepping Stones 4 • 8.9
© ORIGO Education.
How many millimeters are in one meter?
How did you figure it out?
Dining/Kitchen
Living Room
5 2 m
In the word millimeter,
milli means one-thousandth.
A related word is millipede,
a creature with so many
legs it was guessed that
they have about 1,000.
© ORIGO Education.
You could check
by placing 10 tens
blocks along one
side of a meter stick.
2 2 m
mm
How many centimeters are in one meter? How can you check?
mm
ORIGO Stepping Stones 4 • 8.9
+60 m
+5 m
mm
mm
+11 m
193
8.10
Exploring the Relationship Between Meters,
Centimeters, and Millimeters
Step Up
Where might you use a meter stick, tape measure,
or other measuring device to measure length?
1. Write the metric unit of length that you would use for each of these.
a. length of your pencil
What metric units of measurement could you use?
b. length of a paper clip
What are some things that you would
measure in centimeters?
c. length of a car
What unit of length would you use
to measure a paper clip?
d. thickness of your ruler
2. Complete the missing numbers in this table to show equivalent lengths.
A decimeter is a metric unit of length that is not
used often but it helps show an important pattern.
Look at this diagram. What do you notice?
× 10
1 mm
× 10
1 cm
÷ 10
A decimeter is equal to
10 centimeters. A short way
to write decimeter is dm.
Meters (m)
100
1
2
1
2
400
4,500
b. The sides of a triangle measure
76 mm, 8 cm, and 10 cm. What is
the difference between the lengths
of the two shortest sides?
a. The sides of a triangle measure
64 mm, 3 cm, and 7 cm. What
is the total length of the two
longest sides?
1m
mm
mm
÷ 10
Step Ahead
How would you figure out how many decimeters make 2 meters?
How would you figure out how many centimeters make 2 meters?
© ORIGO Education.
ORIGO Stepping Stones 4 • 8.10
Write these measurements in order from least to greatest.
700 cm
9,000 mm
40 cm
500 mm
© ORIGO Education.
8m
How would you figure out how many millimeters make 2 meters?
194
Centimeters (cm)
3. Solve each problem. Show your thinking.
× 10
1 dm
÷ 10
SA
M
PL
E
Millimeters (mm)
ORIGO Stepping Stones 4 • 8.10
195
8.11
Introducing Kilometers
Step Up
Where have you heard of kilometers before?
a.
My mom and dad
do a 5-kilometer
fun run every year.
1. Complete these.
b.
1 kilometer
is the same length as
m
Rapid City
11 miles
100 kilometers
is the same length as
m
m
2. a.These hiking trails are in Yellowstone National Park. Loop the trails that are
between 1,000 and 6,000 meters long. Use the information above to help you.
18 km
Look at a meter stick.
What do you remember about
the decimeter?
E
In the word kilometer, kilo means
one thousand. A related word
is kilogram, which is equal to
1,000 grams. A short way to
write kilometer is km.
SA
M
How many meter sticks would you need
to make one kilometer?
Howard Eaton
3
11 10 km
PL
Kilometers are used to measure long distances.
How is “kilo” different from “milli”?
Look at this diagram. What do you notice?
3. Write these lengths in meters.
÷ 10
1 hm
÷ 10
1 km
Step Ahead
÷ 10
How is the relationship between kilometers and meters the same as the relationship
between meters and millimeters?
ORIGO Stepping Stones 4 • 8.11
© ORIGO Education.
1 dam
b.5 km 40 m is
m
m
× 10
© ORIGO Education.
1m
× 10
Two Ribbons
2 km
Lava Creek
6
5 10 km
Rescue Creek
8
12 10 km
a.16 km 8 m is
× 10
Pelican Valley
8
10 10 km
Ice Lake
1
2 km
Beaver Ponds
8 km
b.Juan’s family hiked about 15,000 m. Which trails might they have walked?
Write two different combinations.
A dekameter is equal to 10 meter
s.
A short way to write dekameter is
dam.
A hectometer is equal to 100 meter
s.
A short way to write hectometer is
hm.
Some other metric units of length are not
used often but help show the relationship
between metric units of length.
Garnett Hill
8
11 10 km
Mystic Falls
4 km
Duck Lake
6
1 10 km
I have seen kilometers
used on some road signs.
196
c.
10 kilometers
is the same length as
mm
Complete the table below to show equivalent distances.
cm
dm
m
600
ORIGO Stepping Stones 4 • 8.11
dam
hm
km
6
10
197
8.12
Solving Word Problems Involving Metric Length
2. Solve each problem. Show your thinking.
a. There were three alligators at a
zoo. The smallest was 1 21 meters
long. The second was 68 cm
longer than the smallest one. The
largest was 2 meters 10 cm longer
than the second one. How long
was the largest alligator?
Two friends live at opposite ends of the same straight street.
They arranged to meet at a store on their street.
Teena lives 34 meters from the store and Megan
lives half a kilometer from the store.
How many meters is it from Teena’s home to Megan’s home?
b. A carpenter is cutting a piece
of lumber for some shelves. The
piece of lumber is 2 meters long.
Each shelf needs to be 645 mm
long and there are two shelves.
How much lumber will be left over
after the shelves are cut?
What do you need to find out?
What information will help you?
How could you figure out the distance?
1.Figure out the answer to each problem. Show your thinking
and be sure to use the correct units in your answer.
a. James has two pet lizards, Apollo
and Dino. Apollo is half a meter long
from head to tail and Dino is 38 cm
long. Which lizard is longer?
b. Ribbon A is 500 mm long. Ribbon B
is taped to the end of Ribbon A
so that the total length is 63 cm.
How long is Ribbon B?
PL
c. Liam jumped forward 3 times and
measured the length of each jump.
The first jump was 1 m 34 cm, the
second was 1 m 46 cm, and the
third was 1 m 15 cm. How far did
he jump in total?
SA
M
Step Up
d. Sofia’s grandparents live 60 km
away. If she visits them twice in
one month how far will she
travel in total?
a. 10 cm +
198
ORIGO Stepping Stones 4 • 8.12
© ORIGO Education.
© ORIGO Education.
Who ran farther,
Rita or Grace?
How far did she run?
m
cm = 170 mm
km + 540 m = 10,540 m
c. 3,000 cm +
km
d. Rita ran three times around a
400-m track. Grace ran 1 21 km.
Complete each equation.
Check your answers.
Step Ahead
b.
m
mm
cm
mm
c. Jacob rode 450 meters to Ramon’s
house. Together they rode 3 km
to the mall. How many meters did
Jacob ride in total to the mall?
cm
E
I need to think about how many meters make 1 kilometer.
Then I can figure out how many meters make half a kilometer.
d.
ORIGO Stepping Stones 4 • 8.12
m = 3,200 cm
mm + 3 m = 4,863 mm
Working Space
199
Developing a Rule to Calculate the Area of Rectangles
9.1
2. Calculate the area of each rectangle. Show your thinking.
a.
Each small square in this large rectangle
measures 1 yard by 1 yard.
b.
4 yd
6 yd
6 yd
What are the dimensions of the large rectangle?
12 yd
8 yd
How could you use the dimensions to figure out
the area of the rectangle?
A short way to write square
units is to use a small numeral 2.
So, 370 square yards can be
written as 370 yd².
yd2
Area
a.
Does the order in which you multiply matter? Explain.
PL
What rule could you write to calculate the area of any rectangle?
b.
yd
yd
Area is 36 yd2
yd
yd
SA
M
Use your rule to calculate the area of a rectangle that is 7 yards wide and 9 yards long.
Step Up
1.Imagine that each small square inside these large rectangles
measures 1 yd by 1 yd. Write the dimensions of the whole rectangle.
Then write how you will use the dimensions to calculate the area.
a.
b.
yd2
Area
3. Write possible dimensions for each rectangle.
E
The width is 6 yards and
the length is 8 yards.
15 yd
Area is 120 yd2
4. Write how you figured out each dimension in Question 3.
Step Ahead
Figure out the area of this rectangle.
Area
yd2
8 yd
Length
yd
Width
yd
Length
yd
Width
yd
200
yd2
Area
yd2
ORIGO Stepping Stones 4 • 9.1
© ORIGO Education.
Area
© ORIGO Education.
9 yd
15 yd
ORIGO Stepping Stones 4 • 9.1
Working Space
201
Working with the Area of Rectangles
7m
Area
14 m
7 × 10 = 70
Daniel figured it out like this.
7 × 4 = 28
so
7 × 14 = 98 m2
8 × 7 = 56 m2
and
6 × 5 = 30 m2
7−5=2m
2 × 6 = 12 m2
What steps did Isabelle use?
What steps did Daniel use?
Is there another way you could figure out the area?
Which way do you like best? Why?
Area
yd2
Working Space
Step Ahead
40 yd
202
16 yd
ORIGO Stepping Stones 4 • 9.2
© ORIGO Education.
20 yd
© ORIGO Education.
13 yd
Barn
Calculate the area of the shaded part.
30 yd
10 yd
Barnyard
yd2
8 yd
6 yd
Area
15 yd
Area
yd2
1.Imagine you wanted to lay turf in this barnyard.
Write how you would calculate the area.
Step Up
yd2
25 yd
20 yd
E
SA
M
56 + 12 = 68 m2
PL
98 − 30 = 68 m
Area
d.
40 yd
25 yd 15 yd
14 − 6 = 8 m
4 yd
8 yd
yd2
c.
Isabelle figured it out like this.
15 yd
4 yd
Carpet
How can you figure out the area of
floor that will be covered with carpet?
2
11 yd
20 yd
Tiles
b.
30 yd
8 yd
5m
a.
5 yd
6m
10 yd
This diagram shows the floor area
of a room that will be covered with
tiles and carpet. The shaded area
will be tiled.
2. Calculate the area of each shaded part. Use the working space below.
7 yd
9.2
Area
ORIGO Stepping Stones 4 • 9.2
9 yd
yd2
Working Space
203
9.3
Developing a Rule to Calculate the Perimeter of Rectangles
2. Calculate the perimeter of these. Show your thinking.
a.
What are the dimensions of this mirror frame?
b.
What do you call the distance around a rectangle?
12 in
How could you figure out the perimeter of this mirror frame?
15 in
18 in
12 in
12 + 12 + 6 + 6 = 36 inches
You could multiply the length and
width by 2. Then add them together.
That«s 2 � 12 + 2 � 6.
6 in
Frame A
in
Frame B
Length is 15 in
Width is 8 in
b.
Perimeter
7 in
18 in
204
Perimeter
in
Length is 25 in
Width is 16 in
1. Calculate the perimeter of each frame.
a.
2×7 =
2×9 =
2 × 15 =
in
Perimeter
Perimeter
in
in
Figure out the perimeter of each polygon. For each shape,
all sides are the same length.
a.
b.
c.
6 in
9 in
5 in
in
ORIGO Stepping Stones 4 • 9.3
© ORIGO Education.
2 × 18 =
Perimeter
Step Ahead
15 in
© ORIGO Education.
9 in
Perimeter
3. Calculate the perimeter of each frame. Show your thinking.
SA
M
What rule could you write to figure out the perimeter of a rectangle?
PL
E
What is another way you could figure out the perimeter?
Step Up
21 in
Perimeter
ORIGO Stepping Stones 4 • 9.3
in
Perimeter
in
Perimeter
in
205
9.4
Working with Rules to Calculate the Perimeter
of Rectangles
How could you figure out the perimeter of this field?
2. Complete these to figure out the perimeter of each rectangle.
a.
25 yd
b.
43 yd
Damon figured out the perimeter like this.
73 yd
19 yd
28 yd
P = Perimeter
P = (2 × L) + (2 × W)
P = (2 × 45) + (2 × 25)
45 yd
P = 90 + 50
P = 140 yd
b.
13 yd
P=
yd
Perimeter
yd
Perimeter
yd
59 yd
Step Ahead
Color the
beside each rule that you could use to calculate
the perimeter of a rectangle.
32 yd
Add all the distances around the sides.
18 yd
ORIGO Stepping Stones 4 • 9.4
Add the length and width. Then multiply the total by 2.
© ORIGO Education.
+
yd
49 yd
25 yd
b. P = (2 × 32) + (2 × 18)
P=
P=
28 yd
P = 140 yd
© ORIGO Education.
yd
yd
25 yd
P = 2 × 70
1.
Complete the calculation to figure out the perimeter
of each rectangle.
a. P = (2 × 25) + (2 × 13)
206
a.
SA
M
P = 2 × (45 + 25)
PL
P = 2 × (L + W)
You could add the length and width
first. Then multiply the total by 2.
P=
P=2×
3. Calculate the perimeter of each rectangle. Show your thinking.
Is there a more efficient way
to figure out the perimeter?
+
P=2×
E
How many steps did it take him to calculate
the perimeter?
P=
P = 2 × (73 + 19)
P=
What steps did he follow?
Step Up
P = 2 × (43 + 28)
Multiply the length by the width.
ORIGO Stepping Stones 4 • 9.4
207
9.5
Exploring the Multiplicative Nature of Common Fractions
(Area Model)
2.Each large shape is one whole.
Shade each shape to match the equation then write the product.
Three friends share one pizza that is cut into eighths.
If they each eat one slice of pizza, how much pizza will they eat in total?
a.
b.
c.
d.
How could you figure it out?
Because there are three
people and they have 81
of the pizza each, that's
the same as 81 + 81 + 81 .
There are three
people with 81 of the
pizza each. I would
multiply 81 by 3.
3 2
2× 8 =
What multiplication sentence could you write
if each person had 2 slices?
5
1× 6 =
2 × 12 =
E
3. Write the product for each of these.
a.
b.
c.
PL
2
2× 6 =
Draw a picture to show your thinking.
=6×
2
10
d.
4
3 × 10 =
=4×
5
8
4. Solve each problem. Show your thinking.
a. There were 2 glasses. Each was
5
6 full of juice. How much juice
was there in total?
SA
M
Imagine there were two pizzas and they each
ate 3 slices. What multiplication sentence could
you write to figure out the total?
4
3 × 12 =
b. Each straw was 23 foot long.
Paige laid 5 of them end to end.
What was the total length?
When you multiply a fraction by a whole number, what do you notice?
Step Up
1. Each large shape is one whole. Complete each equation.
a.
b.
2
3 × 10 =
c.
d.
glasses of juice
e.
Step Ahead
Complete each equation.
208
4
2 × 10 =
2
4× 8 =
ORIGO Stepping Stones 4 • 9.5
© ORIGO Education.
1
© ORIGO Education.
a.
4× 6 =
Which whole number
is that closest to?
Which whole number
is that closest to?
4
2 × 12 =
b.
1
3× 8 ×2=
ORIGO Stepping Stones 4 • 9.5
feet
c.
2
4× 3 ×7=
3
6× 5 ×3=
209
Exploring the Multiplicative Nature of Common Fractions
(Number Line Model)
9.6
2.Multiply each fraction by the number in the hexagon.
Write the products in the circles.
Eva needs 7 pieces of string that are each 43 of a foot long.
What is the total length of string she needs?
a.
b.
How does this number line show the problem?
Write the missing numerators in the fractions below the line.
+
0
3
4
+
1
3
4
4
3
4
+
2
3
4
+
3
4
+
3
3
4
+
4
3
4
5
3
12
2
10
5
12
6
10
6
6
3. Write the first ten multiples of each fraction.
4
4
4
4
4
4
a.
What multiplication sentence could you write
to show the total length of string?
SA
M
b.
What do the jumps on the number line help you identify?
3
Look at the multiples of 4 shown below the number line.
What do you notice about the numerators?
Step Up
7
5
E
3
4
4
11
8
4
8
3
5
PL
+
1
6
The jumps help me see
the fractions that are
multiples of 34.
5
12
4. Loop the fractions in Question 3 that are between 1 and 2.
5. Write a word story to match this equation. Then write the product.
7
5× 8 =
1. T
he distance between each whole number is one whole.
Draw jumps to show the equation. Then write the product.
a.
5
4× 8 =
0
1
2
Step Ahead
3
b.
1
2
3
4
5
6
7
ORIGO Stepping Stones 4 • 9.6
b.
2
© ORIGO Education.
0
© ORIGO Education.
a.
4
5× 3 =
210
Write the missing numbers in each equation.
8
× 3 = 3
ORIGO Stepping Stones 4 • 9.6
c.
5×
5
= 4
18
5 =
3
× 5
211
9.7
Multiplying Mixed Numbers
3
A groundskeeper is laying new turf in
a rectangular section of the playing field.
1
The section measures 5 yards by 3 6 yards.
How many square yards of turf will be needed?
1
6
Step Up
yd
Complete each calculation.
a.
1
3 rows of 5 and 3 rows of 4
1
4
5 yd
5
(3×
)+(3×
)=
3
b.
1
6
3
3
(4×
Then he wrote this number sentence
that was easier to figure out.
c.
5
SA
M
Evan wrote this number sentence
to represent the problem.
)+(4×
PL
What do the numbers in her picture mean?
One dimension is
three whole feet and
1
6 of a foot. The other
dimension is 5 foot.
E
Zoe drew a picture like one she used
to multiply whole numbers.
1
4 rows of 3 and 4 rows of 6
1
6
4
)=
3
16
6
3
5 rows of 6 and 5 rows of 16
(5×
)+(5×
)=
5
d.
7
1
5 × 3 6 =
1
8
1
3 rows of 7 and 3 rows of 8
(3×
1
(5 × 3) + (5 × 6 ) =
)+(3×
3
)=
How are the sentences different? How are they the same?
What is the solution?
Look at Zoe’s and Evan’s methods.
Step Ahead
Write the missing numbers on this trail.
© ORIGO Education.
Which method do you prefer? Why?
Is there another way you could figure out the answer?
212
ORIGO Stepping Stones 4 • 9.7
© ORIGO Education.
How are they the same? How are they different?
2 121
ORIGO Stepping Stones 4 • 9.7
×8
6
−9 12
×3
213
Reinforcing the Multiplication of Mixed Numbers
9.8
1
Akari is painting a wall that is 7 feet high and 5 4 feet long.
What is the area of the wall? How could you figure it out?
5
2. Complete the missing numbers to calculate each product.
a.
1
4
(5×
Mato drew this picture to help him figure it out.
What numbers should you write below to match his picture?
( ×
) + ( ×
+
c.
1
4
so the final answer makes sense?
5
2
3
c.
d.
5
2
4
5
3
8
Step Ahead
3
12
=
3
4 × 7 8
)
(4×
=
)+(4×
+
)
=
4
© ORIGO Education.
in
b.
3
5 10
=
6
10 10
c.
ORIGO Stepping Stones 4 • 9.8
b. Each fence picket is 2 34 inches wide.
What is the total width of 8 pickets?
Complete these.
a.
© ORIGO Education.
214
)
lb
×
3
d.
a. Each bag of apples weighs 4 21 lb.
How much do 6 bags weigh?
SA
M
1
b.
2
6
+
)+(9×
+
)+(3×
3. Solve each of these. Show your thinking.
1. Write the partial products in each picture.
a.
(3×
2
PL
What is the area of the wall?
)
9 × 3 10
E
What do you need to do to the product of 7 and
5
3 × 10 12
=
(9×
What is the value of each partial product?
Step Up
)+(5×
7
)
b.
1
5 × 3 4
× 1 4
2×
= 1 8
d.
5
15 8
=
ORIGO Stepping Stones 4 • 9.8
×
1
3 8
1
5=
2
Working Space
215
9.9
Reviewing Customary Units of Length
2. Write the name of a classroom object to match each length. Then use an inch
ruler or yardstick to check the length of each object that you wrote.
What unit of measurement would you
use to describe the length of a piece of string?
Length
Classroom Object or Distance
Measured Length
8 in
Pencil case
bit more than 9 in
15 in
If the string was short, I would
describe the length in inches.
1 ft
3 ft
What unit do you use to describe the distances that you travel in a car?
Step Up
the distance around a sporting field
580 inches
the distance of a plane flight
465 inches
c.
the length of a baseball bat
32 inches
d.
465 yards
465 miles
32 yards
32 miles
50 feet
50 yards
c. the distance to the nearest hospital
d. the distance to the nearest airport
e. the distance to the nearest train station
Step Ahead
36 feet
Investigate the length
of these distances.
Estimate (mi)
Actual Distance
(mi)
a.from your school to the nearest shopping
mall by road
50 miles
b. from your home to the nearest beach by road
the length of a school bus
36 inches
216
32 feet
580 miles
Actual Distance (mi)
b. the distance to the nearest fire station
the width of a basketball court
50 inches
e.
465 feet
580 yards
Estimate (mi)
a. the distance to the nearest town
36 yards
36 miles
ORIGO Stepping Stones 4 • 9.9
© ORIGO Education.
b.
580 feet
© ORIGO Education.
a.
1. Loop the distance that makes sense.
3. Estimate each distance. Then ask your teacher to confirm your estimates.
SA
M
There are 5,280 feet in one mile
and 1,760 yards in one mile.
5 yd
PL
A short way to write
mile is mi.
E
2 yd
c. from the East Coast to the West Coast by air
ORIGO Stepping Stones 4 • 9.9
217
Converting Feet to Inches
9.10
1. Draw
on the line plot to show each length at the bottom of page 218.
Snake Length
A zoo keeper compares the length of two snakes.
The first snake is 2 feet long. The second snake is 21 inches.
Which snake is longer? How do you know?
There are 12 inches in 1 foot.
Complete this table.
Inches
12
2
3
5
10
15
1
22 22 2
20
23
E
1
1
23 2
24
1
24 2
1
25 25 2
26
Number of inches
1
26 2
27
1
27 2
PL
Feet
2. Use the line plot above to answer these questions.
a. What is the most common length of snake?
How did you figure out the number of inches in 10, 15, and 20 feet?
How many inches is that? How do you know?
I know there are 12 inches
in one foot, so there must
1
be 6 inches in 2 foot.
23 2 inches
26 inches
27 inches
22 inches
1
25 2 inches
1
27 2 inches
23 inches
26 inches
23 2 inches
26 inches
25 2 inches
1
1
1
25 2 inches
23 inches
d. What is the difference in length between
the shortest and longest snakes?
e. If all the snakes grew by 21 inch,
how many snakes would be 2 feet long?
26 inches
Step Ahead
25 inches
Write the length of these larger snakes in inches.
2 feet
Rattlesnake
1
26 inches
23 2 inches
2 feet
25 inches
ORIGO Stepping Stones 4 • 9.10
King Cobra
1
7 feet
© ORIGO Education.
1
218
c.
How many snakes are longer than 2 feet?
The lengths of 20 snakes are shown below. Use this data to complete
the line plot on page 219.
© ORIGO Education.
Step Up
b. How many snakes are less than 26 inches long?
SA
M
1
The Australian taipan, an extremely poisonous snake, is 2 2 feet long.
25 feet
16 2 feet
inches
ORIGO Stepping Stones 4 • 9.10
Python
inches
inches
219
Converting Yards to Feet and to Inches
9.11
2. Convert yards to feet and then inches. Write your thinking below.
a.
Two friends compare their running jumps.
Lilly jumped 2 yards. Dakota jumped 5 feet.
b.
4 yards
7 yards
is the same length as
There are 3 feet
in 1 yard.
What is the difference in length between
their jumps? How do you know?
is the same length as
feet
feet
is the same length as
is the same length as
Complete this table.
1
Feet
3
2
3
5
15
20
inches
35
× 12
1 inch
How many inches in 2 yards?
How do you know?
Step Up
a.
×3
1 foot
1 yard
3. Solve these word problems. Show your thinking.
SA
M
How many inches are in 1 yard?
PL
How did you figure out the number of feet in 15, 20, and 35 yards?
What does this diagram show?
inches
E
Yards
a. Trina’s golf ball is 3 yards from the
hole. Janice’s ball lands 10 feet
from the hole. Whose ball is closer
to the hole?
× 36
b. Tyler kicked a ball 42 feet. His dad
kicked the ball 3 yards farther. How
many feet did his dad kick the ball?
1. Convert yards to feet. Show your thinking below.
b.
5 yards
is the same length as
ft
c.
9 yards
is the same length as
ft
6 yards
is the same length as
ft
ft
Step Ahead
Figure out the length of each jump.
220
ORIGO Stepping Stones 4 • 9.11
© ORIGO Education.
© ORIGO Education.
Lara jumped 2 yards. Andre jumped 1 foot farther than Lara. Carlos jumped 1 yard
less than Andre. How far did each person jump?
Lara
ORIGO Stepping Stones 4 • 9.11
ft
Andre
ft
Carlos
ft
221
9.12
Converting Miles to Yards and to Feet
2. Convert miles to yards. Remember there are 1,760 yards in one mile.
a.
Carmen rides one mile to school each day. Cody walks 1,200 yards.
Who lives closer to the school? How do you know?
There are 1,760 yards in one mile.
How could you figure out the number of yards in 5 miles?
What number sentences could you write?
What steps did he follow?
What is the total?
×3
What does this diagram show?
=
×
=
×
=
×
=
×
=
×
=
×
=
yards
yards
a.
× 1,760
1 yard
1 mile
b.
3 miles
SA
M
1 foot
How could you figure out
the number of feet in one mile?
×
3. U
se your answers from Question 2 to figure out the number of feet
in each distance. Remember there are 3 feet in one yard.
Total
How could he figure out the number
of yards in 7 miles?
=
E
7 miles
×
PL
Brady wrote these number sentences.
5 × 1,000 = 5,000
5 × 700 = 3,500
5 × 60 = 300
5×0=0
b.
3 miles
7 miles
×
1. C
omplete these number sentences to figure out the number
of yards in each distance.
b.
4 miles
4 × 1,000 =
6 × 1,000 =
4 × 700 =
6 × 700 =
8 miles
feet
feet
8 × 1,000 =
8 × 700 =
4 × 60 =
6 × 60 =
8 × 60 =
4×0=
6×0=
8×0=
yd
222
c.
6 miles
yd
Step Ahead
Mary walked over 7,000 yards as she played 18 holes of golf.
About how many
miles did she walk?
yd
ORIGO Stepping Stones 4 • 9.12
© ORIGO Education.
a.
© ORIGO Education.
Step Up
miles
ORIGO Stepping Stones 4 • 9.12
Working Space
223
10.1
Relating Multiplication and Division
2. Complete each of these.
a.
What do you know about this rectangle?
How can you figure out the length
of the rectangle?
?
Write two number sentences that you could use
to help you.
×
=
÷
=
Area is 36 m²
What number sentences
could you write?
yd
4×
28 ÷ 4 =
224
= 28
=
÷
=
÷
=
Area is 48 ft²
e. 54 ÷
ft
ORIGO Stepping Stones 4 • 10.1
=
÷
=
b. 4 =
= 9 f.
Step Ahead
= 48
48 ÷ 8 =
×
a. 36 ÷ 9 =
8 ft
8×
d.
Area is 27 yd²
3 yd
yd
×
=
÷
=
3. Figure out the missing number in each fact.
© ORIGO Education.
Area is 28 yd²
© ORIGO Education.
b.
×
cm
SA
M
4 yd
6m
1.Complete the two number sentences that you could use to help
figure out the unknown dimension. Then label the diagram.
a.
=
E
?
8 in
Area is 42 cm²
6 cm
in
×
c.
PL
What thinking would you use to figure out the length
of the unknown side?
Area is 8 in²
m
What do you know about this square rectangle?
Step Up
Area is 63 m²
7m
Area is 45 ft²
5 ft
b.
÷ 9 = 9 g. 30 ÷
÷ 9
d. 35 ÷
= 5 h. 9 =
=7
÷2
Write three pairs of possible dimensions for a rectangle that
has an area of 600 ft2.
×
= 600 ft²
×
= 600 ft²
×
= 600 ft²
ORIGO Stepping Stones 4 • 10.1
÷ 8 c. 1 =
Working Space
225
10.2
Using the Partial-Quotients Strategy to Divide
(Two-Digit Dividends)
1.These rectangles have been split into parts to make it easier to
divide. Write the missing numbers. Then complete the equation.
Step Up
a.
Three friends share the cost of this gift.
How can you figure out the amount that each person will pay?
80
40
Julia used a sharing strategy.
What do the blocks at the top of the chart represent?
c.
What steps will she follow?
2
Nina used a different strategy. She followed these steps.
She split the rectangle into
two parts so that it was
easier to divide by 3.
63
3
60
?
Why did she split the rectangle into two parts?
Why did she choose the numbers 60 and 3?
3
She thought:
3 × 20 = 60
3×1=3
then 20 + 1 = 21
3
60
a.
9
+
d.
2
48 ÷ 4 =
4
40
+
8
+
b.
93 ÷ 3 =
3
3
77 ÷ 7 =
7
20+ 1
+
Step Ahead
Why did she add 20 and 1?
© ORIGO Education.
How could you use these strategies to figure out 96 ÷ 3?
ORIGO Stepping Stones 4 • 10.2
915
+
Break each number into parts that you can easily divide by 3.
b.
612
c.
396
© ORIGO Education.
a.
I'll call the amount that each person pays A.
To find the amount, Julia thinks 63 Ö 3 = A
and Nina thinks 3 x A = 63.
226
60
2.Inside each rectangle, write numbers that are easier to divide.
Divide the two parts then complete the equation.
SA
M
She drew a rectangle to
show the problem. The
length of one side becomes
the unknown value.
3
Step 3
3
+
60
PL
What division sentence could you write?
6
62 ÷ 2 =
E
What amount will each person pay? How do you know?
Step 2
69 ÷ 3 =
$63
2
Step 1
b.
86 ÷ 2 =
ORIGO Stepping Stones 4 • 10.2
227
Reinforcing the Partial-Quotients Strategy for Division
(Two-Digit Dividends)
10.3
2.Inside each rectangle, write numbers that are easier to divide.
Divide the two parts then complete the equation.
How can you figure out the length of this rectangle?
a.
90 ÷ 6 =
Area is 75 ft²
5 ft
I know that 5 x 10 = 50.
That leaves 25 left over.
?
b.
6
51 ÷ 3 =
3
+
Grace split the rectangle into two parts like this.
50
5
Why did she choose the numbers 50 and 25?
What is the length of the unknown side?
How could this help you figure out 45 ÷ 3?
b.
56 ÷ 4 =
40
5
16
50
72 ÷ 6 =
6
60
12
+
228
+
+
b.
85 ÷ 5 =
÷ 5 plus
c.
15
÷ 3 plus
÷5=
d.
96 ÷ 8 =
÷ 8 plus
48 ÷ 3 =
is the same as
is the same as
+
d.
7
is the same as
65 ÷ 5 =
+
c.
a.
÷3=
84 ÷ 6 =
is the same as
÷ 6 plus
÷8=
÷6=
84 ÷ 7 =
7
70
Step Ahead
14
+
ORIGO Stepping Stones 4 • 10.3
© ORIGO Education.
4
91 ÷ 7 =
3.Break each starting number into parts that you can easily divide.
Then complete the equations.
1.These rectangles have been split into two parts to make it easier
to divide. Write the missing numbers. Then complete the equation.
© ORIGO Education.
a.
4
SA
M
3
How did you break 45 into two parts?
60 ÷ 4 =
PL
Write numbers inside the rectangle to show the parts.
d.
E
10+ 5
Split this rectangle into two parts so that it is easier to figure out 45 ÷ 3.
Step Up
c.
25
+
a.
42 ÷ 3 =
ORIGO Stepping Stones 4 • 10.3
Use the same thinking to complete these equations.
b.
95 ÷ 5 =
c.
84 ÷ 4 =
d.
102 ÷ 6 =
229
10.4
Using the Partial-Quotients Strategy to Divide
(Three-Digit Dividends)
1.These rectangles have been split into parts to make it easier
to divide. Divide each part then complete the equation.
Step Up
Jamal paid for this laptop in 3 monthly payments.
He paid the same amount each month.
a.
606 ÷ 6 =
9
$63
What amount did he pay each month? How do you know?
b.
600
6
6
3
963 ÷ 3 =
900
60
+
I would break 639 into parts
that are easier to divide.
d.
484 ÷ 4 =
E
c.
+
4
Describe how this rectangle has been split.
What amount does Jamal pay each month?
600
200
30
+
9
PL
3
What is special about the numbers 600, 30, and 9?
10 + 3
+
530 ÷ 5 =
5
+
+
+
2.Estimate each answer in your head.
Then write number sentences to figure out the exact amount.
a.
SA
M
Alisa’s laptop was $546. She paid the same amount each month for 6 months.
3
How can you figure out the amount that she paid each month?
b.
742 ÷ 7 =
693 ÷ 3 =
c.
630 ÷ 6 =
546
It's easier to divide if you think
of 546 as 54 tens and 6 ones.
Complete the equations to figure out the
amount that she paid each month.
54 tens
6 ones
540 ÷ 6 =
Step Ahead
6÷6=
a.
Use this strategy to figure out 279 Ö 3.
ORIGO Stepping Stones 4 • 10.4
© ORIGO Education.
© ORIGO Education.
546 ÷ 6 =
230
Write the missing numbers.
b.
ORIGO Stepping Stones 4 • 10.4
÷ 4 = 132
÷ 6 = 104
Working Space
231
10.5
Reinforcing the Partial-Quotients Strategy for Division
(Three-Digit Dividends)
2.Write number sentences to figure out each answer.
a.
Megan paid $453 to buy three concert tickets.
Each ticket costs the same amount.
How could you estimate the price of each ticket?
I thought of a number that will
give me 450 when multiplied by 3.
d.
+
g.
Jack paid $296 to buy four theme park tickets. Each ticket costs the same amount.
How could you figure out the price of each ticket?
You could break 296
into parts that are
easier to divide by 4.
28 tens
What is the price of each ticket? How do you know?
15 ÷ 5 =
÷4=
÷3=
16 ones
ones
tens
h.
426 ÷ 3 =
j.
i.
786 ÷ 6 =
600 ÷ 6 =
6÷3=
496 ÷ 8 =
k.
489 ÷ 3 =
400 ÷ 4 =
180 ÷ 6 =
÷6=
847 ÷ 7 =
568 ÷ 4 =
÷4=
÷4=
l.
524 ÷ 4 =
296 ÷ 4 =
c.
136
+
Step Ahead
ones
184
tens
+
Figure out the cost of buying one two-day pass for each theme
park. Then loop the theme park that is least expensive.
2-DAY PASS
ones
ORIGO Stepping Stones 4 • 10.5
© ORIGO Education.
tens
+
f.
126 ÷ 3 =
16 ÷ 4 =
© ORIGO Education.
b.
176
176
e.
342 ÷ 6 =
1. Break each number into parts that you can easily divide by 4.
a.
232
90 ÷ 3 =
120 ÷ 3 =
280 ÷ 4 =
Step Up
108 ÷ 3 =
320 ÷ 4 =
300 ÷ 3 =
SA
M
296
Use this strategy to figure out 258 ÷ 3.
364 ÷ 4 =
3
+
How do the two parts help you divide by 4?
c.
250 ÷ 5 =
E
How do the parts in this diagram help
you divide by 3?
265 ÷ 5 =
PL
What numbers could you write in this diagram
to help figure out the exact price of each ticket?
b.
T O R TOWN
GA
5 passes cost $480
ORIGO Stepping Stones 4 • 10.5
2-DAY PASS
2-DAY PASS
4 passes cost $336
3 passes cost $324
Dream Land
233
10.6
Using the Partial-Quotients Strategy to Divide
(Four-Digit Dividends)
2.Write number sentences to figure out each of these.
The Hornets have 6,936 members.
They have three times as many members as the Wild Cats.
a.
3,603 ÷ 3 =
b.
8,032 ÷ 4 =
c.
3,930 ÷ 3 =
3,000 ÷ 3 =
8,000 ÷ 4 =
3,000 ÷ 3 =
32 ÷ 4 =
900 ÷ 3 =
How many members do the Wild Cats have?
600 ÷ 3 =
There must be more than 2,000
members because 6,000 Ö 3 = 2,000.
3÷3=
James wrote these number sentences
to figure out the answer.
Complete each of the sentences.
900 ÷ 3 =
30 ÷ 3 =
6÷3=
I would group the tens and
ones together. 36 Ö 3 is easy
to figure out.
Step Up
a.
g.
6,036 ÷ 6 =
SA
M
Can you think of another way to break 6,936 into parts?
PL
6,000 ÷ 3 =
How did he break 6,936 into parts that are easier
to divide by 3?
f.
5,050 ÷ 5 =
h.
5,525 ÷ 5 =
i.
1,815 ÷ 3 =
6,936 ÷ 3 =
1. Break each number into parts that are easy to divide by 4.
c.
4,240
Step Ahead
3,236
a.
ORIGO Stepping Stones 4 • 10.6
© ORIGO Education.
8,016
© ORIGO Education.
b.
234
e.
9,036 ÷ 3 =
E
d.
4,824 ÷ 4 =
30 ÷ 3 =
b.
ORIGO Stepping Stones 4 • 10.6
Write the missing numbers.
÷ 4 = 2,106
÷ 3 = 2,307
Working Space
235
10.7
Reinforcing the Partial-Quotients Strategy for Division
(Four-Digit Dividends)
2.Write number sentences to figure out each of these.
a.
1,720 ÷ 4 =
A beachside apartment costs $5,236 to rent for four weeks.
What is the price of one week?
Would it cost more or less than $1,000 a week? How do you know?
Ashley wrote these number sentences
to figure out the price.
Complete each of the sentences.
b.
1,600 ÷ 4 =
4,000 ÷ 4 =
120 ÷ 4 =
1,926 ÷ 6 =
1,500 ÷ 3 =
1,800 ÷ 6 =
÷3=
÷6=
÷6=
3.Estimate each answer in your head.
Then write number sentences to figure out the exact amount.
36 ÷ 4 =
E
What is another way to break 5,236 into parts?
a.
5,612 ÷ 4 =
Another apartment costs $1,620 for four weeks rent.
What is the price of one week?
1,620
c.
7,830 ÷ 6 =
e.
4,206 ÷ 3 =
f.
9,640 ÷ 8 =
20 ones
SA
M
16 hundreds
b.
8,407 ÷ 7 =
PL
5,236 ÷ 4 =
You could break 1,620 into
parts that are easier to
divide by 4. This diagram
shows you how.
1,659 ÷ 3 =
÷3=
1,200 ÷ 4 =
How did she break 5,236 into parts that are easier
to divide by 4?
c.
1,600 ÷ 4 =
d.
4,650 ÷ 5 =
20 ÷ 4 =
Complete the sentences.
1,620 ÷ 4 =
Step Up
1. Break each number into parts that are easy to divide by 5.
ones
b.
236
Step Ahead
2,505
hundreds
1,525
hundreds
Color the numbers that you can divide equally by 4.
ones
ones
ORIGO Stepping Stones 4 • 10.7
© ORIGO Education.
hundreds
c.
3,550
© ORIGO Education.
a.
3,216
ORIGO Stepping Stones 4 • 10.7
4,810
1,720
5,204
5,642
237
10.8
Solving Word Problems Involving Division
Look at the prices on page 238. Solve these word problems.
Show your thinking.
Step Up
a. Zola buys the cell phone. She pays
$50 first then pays 4 equal monthly
payments. How much does she
pay each month?
TAKE HOME TODAY!
Buy Now – Pay Later
350
$1,
E
$
86
5
$8
PL
$4
$
c. Victor buys 6 cameras for his class.
He makes equal payments over 5
months. What amount does he pay
each month?
SA
M
6
$78
b. Dixon buys the laptop and camera.
He makes equal monthly payments
over 7 months. What amount does
he pay each month?
d. Sheree buys the cell phone and
laptop. She makes equal monthly
payments over 6 months. What
amount does she pay each month?
Imagine you buy one of these items and pay for it over several months.
How would the store figure out the amount you need to pay each month?
Imagine you buy the television and pay equal
monthly amounts over six months.
How much would you pay each month?
$
A
ORIGO Stepping Stones 4 • 10.8
$8
© ORIGO Education.
$
each month
ORIGO Stepping Stones 4 • 10.8
47
5
$63
paid over
paid over
paid over
7 months
8 months
© ORIGO Education.
What amount will you pay each month?
C
B
0
$52
Imagine you buy the cell phone and pay equal monthly amounts over six months.
238
Calculate the monthly payments for each phone.
Then draw a beside the plan that you would choose.
Step Ahead
How could you break 786 into parts
that are easy to divide by 6?
$
$
each month
5 months
$
each month
239
Exploring Points, Lines, Line Segments, and Rays
10.9
Step Up
1.Name five unique line segments you can see on the line below.
A straight line continues in both directions forever.
When you draw a straight line, it is just a part of a longer
continuous line. This part is called a line segment.
R
A line segment has
a start point and an
end point.
Look at the line below. The arrows show that it continues
in both directions forever. Points A, B, and C are all
on the same line.
S
T
U
V
RV
2.Look at the line above. Name a pair of rays that start at each of these end points.
B
C
All the points beginning at Point A and ending at Point B form one line segment AB.
Point B splits the line into two parts. Each part
is call a half-line or a ray. A ray is named with its
start point written first, followed by another
point that the ray goes through.
and
and
A
SA
M
C
H
I
J
L
K
M
G
F
E
D
B
A ray is part of a line that
begins at a point and
continues on forever.
Polygons can be described by naming the line segments that make their sides
or the points that are the vertices. This can help identify shapes.
N
R
P
O
Q
S
Write the points that make up the vertices of each shaded polygon.
Blue
Green
Red
Orange
A
D
C
B
Step Ahead
Look at the picture in Question 3. Find other examples of each
polygon below. Write the points that are the vertices of each shape.
H
I
ORIGO Stepping Stones 4 • 10.9
© ORIGO Education.
F
© ORIGO Education.
E
G
240
and
3.Look at the picture below.
Look at the line above.
If Point B is the start point, the two rays BC and BA go in opposite directions.
What other polygons can you
see and describe?
Point U
PL
What other line segments are part of this line?
Use a color pencil to trace over
the polygon made by joining the
points A, C, G, and F.
What shape is it?
Point T
E
A
Point S
a.triangle
b. quadrilateral
c.pentagon
d. hexagon
ORIGO Stepping Stones 4 • 10.9
241
Identifying Parallel and Perpendicular Lines
What do you know about parallel lines?
Where might you see parallel lines?
B
E
Parallel sides
F
G
C
Cut out the shapes from the support page and paste them in the correct
spaces below. Some shapes do not belong in any of the spaces.
When two lines are the same
distance apart for their entire
lengths, they are parallel.
Which two line segments below are parallel?
How do you know?
A
Step Up
H
D
E
10.10
PL
Parallel line segments do not have to be directly opposite each other or the same length.
If the lines that they are part of are parallel, then the line segments will be parallel too.
The line segment JK below is parallel to line segment ST and also to line segment TU.
K
Which other line segments are parallel?
Perpendicular lines make a right angle
with each other. The blue line is
perpendicular to the purple line.
T
U
© ORIGO Education.
Perpendicular line segments do not need to intersect one another.
However, the lines that they are part of must intersect.
242
Perpendicular sides
Perpendicular lines do not have to be
vertical or horizontal. These lines are
also perpendicular to each other.
ORIGO Stepping Stones 4 • 10.10
Step Ahead
© ORIGO Education.
S
L
SA
M
J
Draw a square and a
non-square rectangle.
One side of each has
been drawn for you.
Use a protractor to
check your drawings.
ORIGO Stepping Stones 4 • 10.10
243
10.11
Reflecting Shapes
2. Draw the reflection on the other side of the dashed line.
Imagine you were wearing this shirt and looked in the mirror.
What would the shirt look like?
a.
b.
c.
d.
What words can you use to describe what mirrors do?
When I look in the mirror,
I see my reflection.
Step Up
SA
M
What other letters have two sides that are a reflection of each other?
PL
E
Some shapes have parts that are a reflection of each other.
Draw the other half of the letter M on the other side of the
dashed line. How will you know it is a reflection?
1. Draw the reflection of each shape on the other side of the dashed line.
a.
b.
c.
d.
e.
244
ORIGO Stepping Stones 4 • 10.11
Draw what the numbers 7, 35, and 86 would look like when
reflected in a mirror to the left.
7
© ORIGO Education.
© ORIGO Education.
Step Ahead
f.
ORIGO Stepping Stones 4 • 10.11
35
86
245
Identifying Lines of Symmetry
Draw a line of symmetry on each shape so that one
side of the shape is a mirror image of the other.
2. Find and draw the line of symmetry on each shape.
A line of symmetry splits a
whole shape into two parts
that are the same shape
and the same size.
a.
E
10.12
c.
b.
e.
c.
246
Step Ahead
a.
Draw one shape that has a line of symmetry. Show the line
of symmetry on the shape. Then draw one shape that has
no lines of symmetry.
b.
f.
© ORIGO Education.
d.
1. Find and draw the line of symmetry on each of these.
ORIGO Stepping Stones 4 • 10.12
© ORIGO Education.
a.
SA
M
How did you know where to draw the line on each shape?
Step Up
d.
PL
Try cutting and folding shapes
like these to check your work.
b.
ORIGO Stepping Stones 4 • 10.12
247
11.1
Exploring Equivalent Fractions with Tenths
and Hundredths
2. E
ach square is one whole. Draw lines and shade parts to show the first fraction.
Then draw extra lines to help you identify the equivalent fraction.
Look at these pies.
a.
b.
c.
How has each pie been divided?
Apple pie
Write fractions to complete these
equivalence statements.
Pecan pie
1
is equivalent to
2
Peach pie
is equivalent to 3
5
4
5
PL
e.
100
×
248
=
100
100
g.
=
6
100
4
100
×
h.
×
80
=
4
×
×
5
3
=
5
×
=
100
×
=
10
×
90
100
×
c.
=
10
4
5
=
10
ORIGO Stepping Stones 4 • 11.1
© ORIGO Education.
5
5
© ORIGO Education.
10
2
100
d.
×
×
Step Ahead
=
=
10
f.
=
10
1. E
ach square is one whole. Draw lines and shade parts to show
the first fraction. Then draw extra lines to help you identify the
equivalent fraction in tenths.
1
5
7
10
100
×
×
12
How could you change the picture to show hundredths?
b.
100
×
SA
M
How many tenths do you need to shade?
a.
=
c.
×
4
=
2
shaded?
b.
×
1
How could you change the picture so that
Step Up
100
E
a.
What other fraction describes the area
that is shaded?
3
5
1
4
3. Complete the diagrams to show the equivalent fractions.
This square is one whole.
What fraction of the square is shaded?
it shows
=
Complete each sequence to show equivalent fractions.
a.
6
10 = 20 = 100
b.
c.
1
5
d.
= 10 = 25 = 100
ORIGO Stepping Stones 4 • 11.1
6
10 = 60 = 100
4
=
2
8
= 20 = 100
249
11.2
Introducing Decimal Fractions
1. E
ach square is one whole. Read the fraction name and shade the
squares to match. Write the decimal fraction on the open expander.
Step Up
Look at this picture.
a.
two and five-tenths
b.
one and seven-tenths
c.
one and three-tenths
d.
two and six-tenths
Each square is one whole.
What amount is shaded?
What are the different ways you can write
this number without using words?
4
10
can be written like this.
Ones
tenths
2
4
The red dot is called a decimal point. The decimal point
is a mark that identifies the ones place.
SA
M
Where have you seen numbers written with a decimal point?
PL
A number such as 2
E
When fractions have a denominator that is a power of 10
they can easily be written in a place-value chart. Powers
of 10 include numbers such as 10, 100, 1,000 and so on.
2. Read the fraction name. Write the amount as a common fraction or mixed
number. Then write the matching decimal fraction on the expander.
Sometimes packets of
food use a decimal point
for weights like 3.5 lb.
I«ve seen a decimal
point used for
prices like $3.99.
Look at the expanders below.
a. four and two-tenths
b. sixty-three tenths
c. five and eight-tenths
How would you say the number that each expander shows?
2
4
How do these numbers relate to mixed numbers
and common fractions?
Why do you need to show the decimal point
when the expander is completely closed?
250
2
Step Ahead
4
ORIGO Stepping Stones 4 • 11.2
Read the clues. Write the numeral on the expander to match.
a. I am greater than three and less
than four. The digit in my tenths
place is less than the digit in my
ones place.
b. I am less than five and greater than
one. The digit in my ones place is
twice the value of the digit in my
tenths place.
© ORIGO Education.
4
© ORIGO Education.
2
A decimal fraction is a
fraction that is written with
no denominator visible. The
position of a digit after the
decimal point tells what the
invisible denominator is.
ORIGO Stepping Stones 4 • 11.2
251
Locating and Comparing Tenths
11.3
Use the masses of these fruit and vegetables to answer the questions on this page.
Look at the number line below. The distance between each whole number is one whole.
ORANGES
6.4 lb
0
1
2
POTATOES
6.3 lb
What fraction is the orange arrow pointing to? How do you know?
Write it as a common fraction
and as a mixed number.
Complete these expanders to show
the same fraction.
AVOCADOS
3.5 lb
CARROTS
4.3 lb
ONIONS
5.4 lb
APPLES
6.2 lb
GRAPES
3.4 lb
10
What fraction is the green arrow pointing to?
Can you write it as a common fraction and as
a mixed number? Why?
What would it look like on an expander?
Think about how you compare 267 and 305
to figure out which number is greater.
Which place do you look at first?
PL
a. apples
c. apples
A zero is used in the ones
place when the amount is
less than 1. This makes it easy
to quickly see whether it is a
whole number or a fraction.
Think about the fractions indicated by the arrows on the number line above.
What do they look like as decimal fractions?
Which is greater?
How can you tell by looking at their places?
Step Up
252
ORIGO Stepping Stones 4 • 11.3
or
potatoes
d. carrots
or
grapes
potatoes
b.
onions
grapes
c.
oranges
apples
d.
avocados
onions
Step Ahead
© ORIGO Education.
© ORIGO Education.
2
or avocados
b. oranges
avocados
1.On this number line, the distance between each whole number is
one whole. Write the decimal fraction that is shown by each arrow.
1
onions
a.
lb
0
or
3. Write the masses. Then write < or > to make the sentence true.
SA
M
10
E
2. In each pair below, loop the box of fruit or vegetables that is heavier.
3
ORIGO Stepping Stones 4 • 11.3
Write the masses of the fruit and vegetables in order from
least to greatest. Then draw a line to connect each mass
to its approximate position on the number line.
lb
lb
4
lb
5
lb
lb
6
lb
7
253
11.4
Exploring Hundredths
2.
Shade each picture to match the description. Then write how much more
needs to be shaded to make one whole.
Each large square represents one whole.
a.
How many columns are in the shaded square?
What fraction of one whole does each
column show?
2 tenths plus
4 hundredths
b.
4 tenths plus
9 hundredths
c.
9 tenths plus
5 hundredths
Shade the first four columns of the other red square. What is the total shaded now?
Start from the bottom and shade five small squares in the next column.
How much is shaded now? What number is now shown by the shaded parts?
How many hundredths are in one whole? How do you know?
b.
c.
3
tenths plus
tenths plus
tenths plus
2
hundredths
hundredths
hundredths
32
hundredths
hundredths
d.
E
hundredths
d.
SA
M
a.
1.Each large square represents one whole. Write the missing
numbers to describe the shaded part of each large square.
hundredths
254
tenths plus
hundredths
hundredths
6 tenths plus
0 hundredths
f.
4 tenths plus
15 hundredths
tenths plus
tenths plus
tenths plus
hundredths
hundredths
hundredths
Step Ahead
tenths
Draw lines to match the numbers.
Some numbers do not have a match. hundredths
ORIGO Stepping Stones 4 • 11.4
© ORIGO Education.
hundredths
© ORIGO Education.
63 hundredths
hundredths
e.
tenths plus
hundredths
e.
tenths
0 tenths plus
3 hundredths
PL
Step Up
tenths plus
70 hundredths
ORIGO Stepping Stones 4 • 11.4
7 tenths
2 tenths
8 hundredths
10 tenths
28 hundredths
6 tenths
3 hundredths
70 tenths
7 hundredths
1
255
11.5
Writing Hundredths as Decimal Fractions
(without Teens or Zeros)
2. Complete the missing parts. Each large square is one whole.
a.
Each large square represents one whole.
How can you color them to show one and
seventy-six hundredths without counting
each hundredth?
I would color all the first
square to show one whole.
Then I«d color 7 columns
to show 7 tenths and then
color 6 small squares to
show 6 hundredths.
100
b.
9
3
100
E
Write the number above on these two expanders.
2
PL
c.
Which expander helps you to read the number? Why?
Step Up
SA
M
Step Ahead
85
100
b.I am less than nine and greater than
four. The digit in the tenths place
is a multiple of 3. The digit in the
hundredths place is greater than
the digit in the ones place.
© ORIGO Education.
© ORIGO Education.
100
Read the clues. Write a matching numeral on the expander.
a. I am greater than five and less than
seven. I have more in the tenths
place than in the hundredths place.
I have more in the ones place than
in the tenths place.
ORIGO Stepping Stones 4 • 11.5
7
1
1. E
ach large square is one whole. Color the squares to show the number.
Then write the number on the expanders and as a mixed number.
100
4
d.
two and twenty-eight hundredths
256
1
hundredths
tenths
Ones
Tens
Write one and seventy-six hundredths
on the chart.
Hundreds
What do you notice about the places
on either side of the ones place?
Thousands
Look at this place-value chart.
Ten Thousands
Which digit is in the tenths place? ... hundredths place?
ORIGO Stepping Stones 4 • 11.5
257
11.6
Writing Hundredths as Decimal Fractions
(with Teens and Zeros)
2. Read the number name. Then write the number on the expander.
a.
Each large square represents one whole.
two and fourteen hundredths
How much has been shaded?
b.
Write the amount on each expander below.
c.
Which of these labels matches each expander above?
How do you know?
137
100
37
1 + 100
d.
13
7
+
10 100
ninety-four hundredths
four and twenty hundredths
E
What numbers are shaded below?
How will you write each number on the expander?
six and two hundredths
3. Write each number in words.
PL
a.3.19
b.9.40
1. Complete the missing parts.
a.
b.
SA
M
Step Up
c.7.06
d.12.15
4. Write the matching decimal fraction and mixed number.
a.
six and
seventeen hundredths
two and five hundredths
258
ORIGO Stepping Stones 4 • 11.6
© ORIGO Education.
© ORIGO Education.
Step Ahead
0.705
ORIGO Stepping Stones 4 • 11.6
b.
six and
seventy hundredths
c.
six and
seven hundredths
Loop the numerals that are the same as 705 hundredths.
5
7 10
7.05
5
7 100
0.75
259
Comparing and Ordering Hundredths
11.7
1. Write the greatest distance that each student threw.
Blake
Six students had a throwing competition using a ball made of scrunched paper.
They measured the distance of their throws in meters and fractions of a meter.
This table shows the results.
Student
Distance (m)
Lela
m
Anna
Cole
Peta
Franco
Sumi
Amos
2.21
1.84
3.49
1.22
4.10
3.13
Kayla
m
2. W
rite the distance of these
students’ throws for Round 2
and Round 3.
Then write < or > to make
each statement true.
E
4
5
When I said each
number name aloud
it was really easy
to figure it out.
SA
M
Was Sumi’s throw longer or shorter than Peta’s? How did you figure it out?
I looked at the value
of the digit in the
ones place first.
Which student threw the greatest distance? How do you know?
Mark and label all the throws on the number line.
260
Five students had a throwing competition. They played three rounds.
Use this data to help you answer the questions on page 261.
Blake
Lela
Kayla
Carter
Luis
Round 1 (m)
3.45
4.06
3.38
3.21
4.30
Round 2 (m)
3.87
4.15
3.50
3.86
4.51
Round 3 (m)
3.18
4.27
3.42
2.97
4.04
ORIGO Stepping Stones 4 • 11.7
m
Round 3
Blake
m
m
Lela
m
m
Kayla
m
m
Carter
m
m
Luis
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
Step Ahead
© ORIGO Education.
Student
© ORIGO Education.
Step Up
m
3. W
rite the names of the students from shortest throw to longest throw for each
round. Write the distance below each name.
ROUND 3 ROUND 2 ROUND 1
3
PL
2
m
Luis
Round 2
Mark the length of Peta’s throw on this number line. How did you figure out the position?
1
Carter
a. U
se 0, 1, and 2 to write four different decimal fractions.
Use each digit once in each numeral.
b.Loop the least number in blue.
ORIGO Stepping Stones 4 • 11.7
c. Loop the greatest number in red.
261
Exploring the Relationship Between Kilograms and Grams
11.8
2. Read the scales carefully. Then write each mass in grams.
a.
Look at this balance picture. Each small box has the same mass.
How could you figure out the mass
of each one?
kg
11 kg
b.
c.
I know that 1,000 grams is
the same mass as 1 kilogram.
Look at this scale.
How could you write the mass shown?
g
g
g
PL
E
3.Write the missing numbers to show the same mass in each row.
The first row has been done for you.
1 would write 121 kg or 1.5 kg.
How could you write the same mass in grams?
1
1
2
kg is the same as
g
1
10
SA
M
Complete these statements.
kg is the same as
Grams
Kilograms
(common fraction)
Kilograms
(decimal fraction)
100
1
10
0.1
4
10
0.4
2,600
g
2.6
3
What are some other kilogram masses that you can say in grams?
Step Up
1.8
1. Read the scales carefully. Write each mass in grams.
a.
b.
7
3 10
c.
COPY
PAPER
BLOCKS
Step Ahead
Look at each balance picture. Draw a
3.6 kg
262
g
ORIGO Stepping Stones 4 • 11.8
on the picture that is true.
2,900 g
3.1 kg
© ORIGO Education.
g
© ORIGO Education.
1,900 g
g
4.3
4 10
ORIGO Stepping Stones 4 • 11.8
263
11.9
Solving Word Problems Involving Mass
2. L ook at the table on page 264. Figure out the weight gain for these dogs.
Show your thinking.
This table shows the birth weight and adult weight of some dogs.
a.
Sadie
Coco
b.
Oakleigh Puppy Hospital
Adult Weight (kg)
Rex
400
5.2
Star
600
4.5
Buster
550
6
Duke
500
5.9
Sadie
480
6.5
Coco
390
6.4
Which dogs weighed more than
1
2
g
E
Birth Weight (g)
a. At the start of May, Zoe’s dog
weighed 5.5 kg. At the end of June,
the dog weighed 7.2 kg. How many
grams did the dog gain?
kilogram at birth?
Which adult dogs weighed more than 5,000 grams?
SA
M
How could you figure out the amount of weight that Rex gained?
g
3. Write number sentences to solve each problem. Then write the answer.
PL
Dog Name
b. Jacob bought 4 kg of dog biscuits.
He feeds his dog 200 g of biscuits
each day. How many days will the
dog biscuits last?
I would change the adult weight into grams then
figure out the difference. That«s 5,200 Ð 400.
Step Up
a.
1. L ook at the table above. Figure out the amount of weight gained
by these dogs. Show your thinking.
Star
b.
grams
Buster
Step Ahead
264
g
ORIGO Stepping Stones 4 • 11.9
© ORIGO Education.
g
© ORIGO Education.
1.4 kg
Write the missing weights on this trail.
+ 250 g
+ 380 g
+ 1,200 g
ORIGO Stepping Stones 4 • 11.9
days
+ 1.7 kg
265
11.10
Reviewing Liters and Introducing Milliliters
2.This juice box holds 250 mL. Think about the real size of each
container below. Estimate the amount each container holds then
draw a line from each container to a label that shows the amount.
Do you think the lid of this milk bottle
would hold more or less than one liter?
250 mL
What are some other containers that hold
less than a liter?
5 mL
590 mL
1,000 mL
150 mL
20 mL
Which metric unit is used to describe
an amount that is less than a liter?
tablespoon
There are 1,000 milliliters
in 1 liter. A short way to
write milliliter is mL .
E
Milliliters are used
to describe amounts
less than 1 liter.
eye dropper
How is the relationship between kilograms and grams similar to relationship
between liters and milliliters?
SA
M
Did you know that a 1-cm cube fills up as
much space as exactly one milliliter of water?
PL
soda bottle
flower vase
single-serve
yogurt tub
mug
How many milliliters would 1,000 centimeter cubes hold? How do you know?
Step Up
1.Think about the real size of each container.
Then draw a line to a matching label.
travel-sized shampoo
medicine cup
soda can
Step Ahead
Write the names of three containers to match each liquid volume.
266
50 mL
15 mL
ORIGO Stepping Stones 4 • 11.10
Holds about 500 mL
© ORIGO Education.
335 mL
© ORIGO Education.
Holds about 250 mL
ORIGO Stepping Stones 4 • 11.10
267
Exploring the Relationship Between Liters and Milliliters
11.11
2. Look carefully at the scale. Then write the amount in each container.
a.
What amount of juice is in this pitcher?
b.
Imagine the juice was poured equally into two containers.
How many milliliters would be in each container?
c.
2 L
2 L
2 L
1 L
1 L
1 L
I know there is 1,000 mL in 1 liter.
500 mL is half of 1,000 mL.
mL
Imagine the juice was poured equally
into ten containers.
How many milliliters would be in each container?
How do you know?
1
L is the same as
1
10
mL
E
SA
M
Complete these statements.
PL
Milliliters
1 L
L is the same as
Liters (common fraction)
b.
10 L
7
900
0.9
8
2,800
2.8
2 10
3.2
6
10
10 L
Step Ahead
For each of these, look at the amount of water in both the containers.
Then write an amount that could be in the second container.
mL
ORIGO Stepping Stones 4 • 11.11
© ORIGO Education.
mL
© ORIGO Education.
a.
268
1.7
1 10
c.
10 L
mL
2.4
2 10
1. L ook carefully at the scale on each container.
Then write the amount of water in each.
a.
Liters (decimal fraction)
4
2,400
mL
What are some other liter amounts that you can say in milliliters?
Step Up
mL
3.Write the missing numbers to show the same capacity in each row.
The first row has been done for you.
2 L
This container holds more than one liter.
How much juice is in the container?
1
2
mL
b.
1.2 L
ORIGO Stepping Stones 4 • 11.11
L
c.
400 mL
L
1.7 L
mL
269
11.12
Solving Word Problems Involving Liquid Volume
Kimie has two bowls that she can use to hold
this punch. The first bowl holds 4 liters.
The second bowl holds 5 liters.
Which bowl should she use? Why?
2. Use this punch recipe to solve each word problem. Show your thinking.
FRUIT PUNCH
2.5 L of orange juic
2 L of lemonade
1.7 L of apple juice
1.5 L of cranberry juice
2 L of lemonade
600 mL of pineapple juice
200 mL of lime juice
e
There are 1,000 mL in 1 liter. I can change
liters to milliliters to figure out the total
amount. That«s 2,000 + 1,500 + 600.
y juice
ade
700 mL of pin
100 mL of lem
eapple juice
on juice
Step Ahead
mL
270
Recipe B
2.5 L of lime juice
300 mL of lemon juice
1.5 L of lemonade
mL
ORIGO Stepping Stones 4 • 11.12
Recipe C
600 mL of orange juice
500 mL of apple juice
2.5 L of lemonade
500 mL of lime juice
For each of these, look at the amount of water in both the containers.
Then write an amount that could be in the second container.
a.
© ORIGO Education.
mL
c. If you pour all the punch equally
into 10 glasses, how much punch
will be in each glass?
© ORIGO Education.
b. What is the difference between
the amount of cranberry juice and
pineapple juice in the recipe?
on each recipe that will make more than 4 L of punch.
Recipe A
1.5 L of lemonade
800 mL of
pineapple juice
1.2 L of cranberry juice
a. If you follow the recipe exactly,
how much punch will you make?
3 L of cranberr
1.2 L of lemon
3. Draw a
SA
M
1. T
his is a different punch recipe. Solve each word problem.
Show your thinking.
PL
How many liters of punch did she make?
c. If you poured all the punch equally
into 4 containers, how much punch
will be in each container?
E
b. What is the difference between
the amount of apple juice and lime
juice in the recipe?
Kimie used the 5-liter bowl and added the ingredients. She then tasted the punch
and decided to double the amount of pineapple juice.
Step Up
a. Koda used a 7-L bowl and followed
the recipe exactly. How much more
punch could the bowl hold?
b.
1,900 mL
ORIGO Stepping Stones 4 • 11.12
L
c.
900 mL
L
2L
mL
271
Locating Decimal Fractions on a Number Line
12.1
3.The distance between each whole number is one whole. Draw a line to join each
numeral to its approximate position on the number line. Be as accurate as possible.
The distance between each whole number on these number lines is one whole.
a.
What number is the orange arrow pointing to? What helped you figure it out?
1
3.42
What other decimal fraction describes that position? How do you know?
b.
b.
c.
2
e.
2.
a.
b.
g.
c.
e.
272
0.23
0.46
0.75
1.08
1.8
1
0.10
g.
Athlete
h.
d.
h.
ORIGO Stepping Stones 4 • 12.1
1.24
1.4
1.63
1996 Olympic Games Men’s 200 Meters
Time (sec)
Athlete
Time (sec)
Michael Johnson
19.32
Jeff Williams
20.17
Frank Fredericks
19.68
Ivan Garcia
20.21
Ato Boldon
19.80
Patrick Stevens
20.27
Obadele Thompson
20.14
Michael Marsh
20.48
MJ
1
f.
2
Some athletes are so fast that the last person in a race may only
be one second slower the first person. Their times are recorded
in whole seconds and hundredths of a second.
Draw arrows and write the initials to show the approximate position of each
athlete’s time on the number line. The first one has been done for you.
d.
0
4.9
Step Ahead
3
f.
4.50
© ORIGO Education.
a.
The distance between each whole number is one whole. Write the decimal
fraction that is shown by each arrow. Think carefully before you write.
© ORIGO Education.
1.
4.1
0.01
SA
M
Step Up
5
3.17
0
What number do you think the green arrow is pointing to? How could you figure it out?
4.60
3.05
PL
5
4.28
4
E
Look where the red arrow is pointing. Which two decimal fractions describe that position?
4
3.82
3
2
3
3.7
19
ORIGO Stepping Stones 4 • 12.1
20
273
Comparing Tenths and Hundredths
12.2
Look at these six decimal fractions.
Which number is greater, C or F?
How could you figure it out?
A
B
C
D
E
F
3.41
3.38
2.6
3.8
3.04
2.43
These eight decimal fractions are between 1 and 4. Use the data in the table
to answer Questions 2 and 3. Use the number line or what you know about
equivalence to help you.
P
Q
R
S
T
U
V
W
1.96
2.91
3.4
3.12
2.19
2.03
3.2
2.3
Logan thought it would be easier to compare the numbers if they had the same
denominator. How should he change the numbers? Do you need to change only one
number or both numbers?
0
I would think about where the numbers
would be on a number line.
1
S
E
a.
PL
e.
274
H
T
O
T
O
t
8
100
1 75
100
1 7
10
b.
f.
7
10
4 1
10
70
100
4 1
100
c.
g.
125
100
2 3
10
14
10
2 30
100
d.
275
100
h.
2 10
100
U
b.
T
W
c.
Q
T
d.
R
S
e.
U
W
f.
T
S
g.
U
Q
h.
W
V
i.
P
U
t
3.Write the decimal fractions in order from least to greatest.
h
1.Write <, >, or = to make each sentence true. Use what you know
about equivalence to help you.
5
10
4
h
Step Ahead
275
10
5 9
10
ORIGO Stepping Stones 4 • 12.2
0.2
© ORIGO Education.
a.
H
© ORIGO Education.
Step Up
SA
M
Write each number in these place-value charts.
How do the charts help you figure out the
greater number?
3
2.Write the fraction from the table. Then write < or > to complete each sentence.
Between which two whole numbers are the numbers C and F?
How could you show the locations of the numbers on this number line?
I would think about the place
value of each number.
2
Write these numbers in order from greatest to least.
0.58
greatest
ORIGO Stepping Stones 4 • 12.2
0.6
1.4
1.07
2.00
0.09
least
275
12.3
Relating Common Fractions and Decimal Fractions
2. Complete the diagrams below.
a.
Each large square is one whole.
b.
×
1
2
×
1
=
100
What fraction of this whole is
shaded? How do you know?
c.
=
1
2
=
10
Shade
of each.
1
5
b.
4
5
0.
of each.
Shade
d.
4
5
=
Shade
3
4
of each.
276
0.
3
4
=
100
1
5
3
5
1
2
4
5
3
4
a.
0.
ORIGO Stepping Stones 4 • 12.3
0.2
0.7
b.
0.
of each.
=
0.35
0.5
0.95
0.75
and
Step Ahead
8
10
c.
0.
and
d.
0.
and
0.
and
Write each set of numbers in order from least to greatest.
0.75
2
5
b.
0.6
1
2
0.95
1
4
0.55
© ORIGO Education.
100
=
×
4. Write four pairs of equivalent fractions from Question 3.
0.
© ORIGO Education.
=
×
100
1
0.25
a.
1
5
=
20
0
=
100
100
E
=
1.Shade each large square to show the fraction.
Then complete the equivalence statement.
Shade
1
2
=
7
10
PL
a.
1
4
SA
M
Step Up
×
8
=
5
×
3. Draw a line to show where each fraction is located on the number line.
What are two ways you can write hundredths?
Complete this sentence to show
how the fractions are equivalent.
100
d.
×
3
=
4
×
What fraction of this whole is
shaded? How do you know?
c.
ORIGO Stepping Stones 4 • 12.3
277
12.4
Adding Tenths
Step Up
Jacob and Claire are going on a 5-km fun run.
1.Calculate the total distance for each of these.
a.
START
b.
3.4 km + 2.3 km =
2 km
km
c.
1 km
km
e.
E
PL
SA
M
+
+
4.5 km + 1.4 km =
km
5.4 km + 3.5 km =
km
j.
km
a. Write how far each checkpoint is from the start.
Checkpoint 1 Checkpoint 2 Checkpoint 3 Checkpoint 4 Checkpoint 5
FINISH
b.
The finish is located 1.5 km after the last checkpoint.
=
ORIGO Stepping Stones 4 • 12.4
km
Emma and Mary ran a relay. Emma ran the first 3.1 kilometers then
Mary ran the last 3.3 kilometers.
a. Did they run more than or less than 6.05 kilometers in total?
© ORIGO Education.
b. Write how you know.
© ORIGO Education.
Can you think of another method?
How long is the fun run?
Step Ahead
It«s like adding mixed
numbers. I would add
the whole numbers and
fractions separately then
add the totals together.
I could start with 5.3, then add 2, then add 0.4.
278
km
START
=
On another fun run, the total distance is 10 km.
If you were at the mark for 5.3 km, where will you be after you run 2.4 km
farther along the track? How can you figure it out?
1 would add the ones
together, then add the
tenths together, then
add the totals. I use the
same strategy for adding
two-digit whole numbers.
3.2 km + 3.5 km =
2. There are checkpoints located every 3.1 km along a fun run.
Shade 1.3 km of the track from the start.
What is two-tenths of a kilometer more? How can you figure it out?
What equation with decimal fractions could you
write to show what is happening?
km
i.
5.3 km + 2.3 km =
How has each kilometer been divided?
What fraction of one kilometer does the orange part show?
km
h.
1.6 km + 4.2 km =
What equation with mixed numbers
and common fractions could you write?
km
g.
3 km
6.3 km + 1.4 km =
f.
6.1 km + 2.3 km =
5 km
km
d.
2.7 km + 4.2 km =
4 km
2.1 km + 3.5 km =
ORIGO Stepping Stones 4 • 12.4
279
12.5
Adding Hundredths
2. Write the total cost. Show your thinking.
1.36 m
a.
A new downspout is being made to attach to the side of a building.
This sketch shows the pipes that are needed.
1 would add the ones together, then
the tenths, then the hundredths.
c.
a.
b.
m
e.
3.72 m + 3.15 m =
m
280
$1.50
$
f.
$5.00
$1.24
$0.65
$
Step Ahead
a.
Write each decimal fraction as a mixed number or common fraction
then write the total. The first numbers have been done for you.
4.35 + 1.62
b.
2.17 + 3.41
c.
1.62 + 1.05
e.
1.40 + 0.08
f.
0.04 + 0.60
62
4 100 + 1 100 =
2.84 m + 5.03 m =
m
d.
ORIGO Stepping Stones 4 • 12.5
0.02 + 0.07
© ORIGO Education.
d.
$4.20
$
m
© ORIGO Education.
1.65 m + 0.23 m =
d.
$
$3.71
35
c.
$
$2.31
1. Add the lengths and write the total. Show your thinking.
4.32 m + 3.65 m =
$3.45
E
$2.06
PL
It«s easy to think about
this. The whole numbers are
dollars and the fractions
are cents.
SA
M
Step Up
$5.24
$
These two items are needed for the downspout.
What is their total cost? How could you figure it out?
$4.05
b.
$1.42
2.53 m
How could you figure out the total length of pipe?
$1.64
$3.56
ORIGO Stepping Stones 4 • 12.5
281
12.6
Adding Tenths and Hundredths
2. Use what you know about equivalence to calculate each total.
a.
Mia drew these pictures to help figure
out the total of 0.4 and 0.23.
d.
How could you use the pictures to help you?
g.
Lilly wrote the numbers as common fractions to help her think about the problem.
She realized the denominators were different and knew that
adding fractions was easier when they had the same denominator.
4
10
How could she change the fractions?
What helps him identify the places correctly?
E
a.
The decimal point
tells me where
the ones place is.
Then it«s easy.
b.
a.
SA
M
b.
0.3 + 0.25 =
c.
c.
e.
1.0 + 0.43 =
1.34 =
0.1 + 0.11 =
e.
0.6 + 0.20 =
1
33
4
3 100 + 10 =
f.
d.
0.4 + 0.03 =
e.
© ORIGO Education.
b.
2.47 =
d.
2.96 =
f.
h.
+
f.
1
+
1.2
+
+
+
1.50 =
+
+
0.1 =
+
+
0.27
Figure out which pairs of numbers add to a total that is
a whole number. Use the same color to show matching pairs.
Some numbers have no match.
0.95
© ORIGO Education.
ORIGO Stepping Stones 4 • 12.6
0.04
+
+
1.0 =
+
+
+
0.67 =
g.
0.3
+
1.45 =
0.1
282
1
7
10 + 2 100 =
i.
2
9
100 + 10 =
c.
+
Step Ahead
d.
f.
1
30
2 10 + 100 =
2
45
10 + 100 =
4.Show each decimal fraction as the sum of three numbers.
1.Complete each equation. You can use the pictures to help you.
Each large square is one whole.
a.
0.5 + 0.34 =
h.
14
6
1 100 + 10 =
c.
5
5
10 + 100 =
3. Choose six totals from Question 2. Write each as a decimal fraction.
How would you use each of these methods to figure out the total of 2.05 and 0.8?
Step Up
e.
8
12
10 + 100 =
PL
Daniel thought about the value of each
place and knew if he added like places
he would find the total.
23
100
b.
6
15
10 + 100 =
0.6
0.09
0.8
ORIGO Stepping Stones 4 • 12.6
1.2
3.5
2.0
0.90
0.50
1.05
2.40
0.01
283
12.7
Solving Word Problems Involving Decimal Fractions
2. Solve each problem. Show your thinking.
a. On Monday, Amber ran 3 41 km.
On Tuesday, she ran 2.3 km.
On Wednesday, she ran 4.1 km.
How far did she run in total on
Monday and Wednesday?
Paige is wrapping two packages to send. She knows that
the store closes in half an hour. One package weighs
5.2 lb and the other weighs a quarter of a pound.
b. In 1952 the winning time for the
women’s 100-m race was 11.5
seconds. In 1968 it was 11.0 seconds
and in 1980 it was 11.06 seconds.
Which was the fastest time?
lb
What is the total weight of the packages?
What information in the story is necessary to help
you answer the question?
What steps will you follow to figure it out?
a. There are three bags of dog
biscuits. Each bag weighs 1.25 kg.
What is the total weight?
b. Kylie drove 4.6 miles before lunch.
She also drove 8.3 miles after lunch.
How far did she drive in total?
kg
284
a.
Which ice cream is the better buy?
ORIGO Stepping Stones 4 • 12.7
b. Write how you know.
1.75 qt
1.5 qt
$4.99
$4.99
© ORIGO Education.
$
What is the total cost? $
Step Ahead
d. There is $2.48 in a money box. If
you put in three more dimes, how
much money will there be inside?
L
d. Damon has $2 to buy some candy.
The red bag costs $0.57, the blue bag
costs $1.62, and the brown bag costs
$1.20. Which two bags can he buy?
gal
mi
© ORIGO Education.
c. Kettle A holds 1.7 liters. Kettle B
holds 2.2 liters. How much water
do they hold in total?
c. If you put 2.5 gallons of water into an
empty 3.5-gallon bucket then add a
quarter gallon of liquid fertilizer, how
much liquid will be in the bucket?
PL
1. Figure out the answer to each problem. Show your thinking.
SA
M
Step Up
s
E
I will need to make sure I am adding
the same type of fractions first.
km
ORIGO Stepping Stones 4 • 12.7
285
12.8
Reviewing Pounds and Introducing Ounces
2. Loop the bag of items that weighs more than 1 lb in total.
Do you think an apple weighs more or less than one pound?
What are some other items that weigh less than one pound?
7 oz
6 oz
8 oz
4 oz
A granola bar would weigh
less than one pound.
6 oz
4 oz
4 oz 2 oz
3. Loop the bag of items that weighs more than 2 lb in total.
Step Up
PL
There are 16 ounces in 1 pound.
A short way to write pound is lb.
A short way to write ounce is oz.
“Ounce” comes from the old
Italian word onza.
1.Three baseballs weigh about 1 pound in total. Think about the real
mass of each item below. Then write the name of each item in the
matching column of the table.
stapler
cell phone
laptop
school bag
banana
eraser
pencil
bowling ball
Weighs less than one pound
1 lb
5 oz
7 oz
8 oz
1 lb
a. Bag A weighs 2 lb. Bag B weighs
3 oz less than Bag A. How much
does Bag B weigh?
© ORIGO Education.
© ORIGO Education.
1 lb
b. A bag of groceries weighs 2 lb.
It holds 4 identical items.
How much does each item weigh?
1 lb
ORIGO Stepping Stones 4 • 12.8
oz
Look at each balance picture. Loop the picture that is true.
9 oz
7 oz
ORIGO Stepping Stones 4 • 12.8
7 oz
9 oz
oz
Weighs more than one pound
Step Ahead
286
8 oz
4. Solve these word problems. Show your thinking.
SA
M
I have seen oz written on
jars and packets of food.
E
What unit of measure is used to describe something that weighs less than one pound?
3 oz
1 lb
10 oz
3 oz
287
12.9
Exploring the Relationship Between Pounds and Ounces
2. Figure out the difference for each of these. Show your thinking.
a.
How could you figure out the difference in mass
between these two bags?
2
7 oz
2 lb
1
2
b.
1
lb
5 2 lb
10 oz
5 oz
I would change the pounds into
ounces to find the difference.
That«s 32 Ð 5.
oz
oz
Complete these statements.
ounces
How could you figure out the difference in mass
between these two boxes?
What number sentences would you write?
Step Up
2
1
2
a. Some cookies weigh 4 oz less than
some cakes. The cakes weigh 1 21 lb.
How much do the cookies weigh?
ounces
12 oz
lb
E
ounces
1
4 pound is
PL
1
2 pound is
SA
M
1 pound is
3. Solve these word problems. Show your thinking.
oz
1. F
or each pair of bags, figure out the difference in mass.
Write number sentences to show your thinking.
a.
b.
3 lb
9 oz
12 oz
Step Ahead
a.
5 lb
b.
15 oz
2 lb
288
ORIGO Stepping Stones 4 • 12.9
© ORIGO Education.
© ORIGO Education.
oz
oz
Write the missing mass in each balance picture.
3 lb
oz
c.
oz
b. 2 21 lb of flour is poured equally
into 4 containers. How much flour
is in each container?
oz
20 oz
d.
oz
11 oz
ORIGO Stepping Stones 4 • 12.9
1
1 2 lb
19 oz
oz
1
2 2 lb
289
12.10
Reviewing Gallons, Quarts, and Pints
and Introducing Fluid Ounces
2.a.Draw a
Draw a
on items that hold more than 1 qt.
on items that hold less than 1 pt.
This table shows the relationship between quarts, pints, and gallons.
What do you notice?
Size of Container
Number of Containers
Soda
20 fl oz
Gallon
Pickles
80 fl oz
Canola
Oil
64 fl oz
Cream
13 fl oz
Detergent
150 fl oz
Juice
12 fl oz
Quart
E
b. W
rite the name of the item above
that contains more than one gallon.
Pint
PL
3. Solve each word problem. Show your thinking.
a. There are 150 fl oz in a bottle of
detergent. Each load of laundry
uses 2 fl oz. How much detergent is
left in the bottle after 3 loads?
Complete this statement.
quarts or
pints
SA
M
1 gallon is
How many pints
are in one quart?
b. A bottle of canola oil holds 64 fl oz.
How many bottles of oil would you
need to make one gallon?
What are some other units of measure that hold less than one pint?
Cups and fluid ounces
hold less than a pint.
290
16
Step Ahead
1
2
fl oz
1 quart =
fl oz
1 gallon =
fl oz
Working Space
ORIGO Stepping Stones 4 • 12.10
© ORIGO Education.
1 pint =
fl oz
1. Figure out the number of fluid ounces in each of these units.
© ORIGO Education.
Step Up
There are 16 fluid ounces
in 1 pint. A short way to
write fluid ounce is fl oz.
bottles
Draw lines to connect equivalent amounts.
gal
8 pints
6 pints
1
2
qt
3 pt
1 2 qt
1.5 gal
16 fl oz
2 qt
1 gal
1
ORIGO Stepping Stones 4 • 12.10
291
12.11
Exploring the Relationship Between Gallons,
Quarts, and Fluid Ounces
2. Solve these word problems. Show your thinking.
a. There is half a gallon of water in
a sink. Another quart of water is
poured into the same sink. How
much water is in the sink now?
Which of these containers would hold the most water?
How do you know?
7 qt
b. Brady bought two 1-qt bottles of
juice. Julia bought 10 bottles of juice
that each held 8 fl oz. Who bought
the greater amount of juice?
2 gal
I know there are
4 quarts in 1 gallon.
qt
What thinking did you use to figure out which container has the greater capacity?
E
c. Teena opens a one-gallon bottle of milk. She fills 4 glasses with milk.
Each glass holds 16 fl oz. How much milk is left in the bottle?
How many fluid ounces
are in one quart?
2 qt
What does this diagram show?
× 32
1 fluid
ounce
×4
1 quart
÷ 32
SA
M
70 fl oz
PL
How could you compare the capacity of these two containers?
Step Ahead
Write numbers to make these balance pictures true.
a.
b.
3 qt
2 gal
1 gallon
÷4
qt
c.
4 fl oz
fl oz
qt
c.
1 qt =
fl oz
1 gal =
fl oz
ORIGO Stepping Stones Grade 4 • 12.11
© ORIGO Education.
b.
© ORIGO Education.
a.
292
12 qt
2 qt
45 fl oz
fl oz
1
2 qt
40 fl oz
fl oz
qt
1. Write the missing amounts.
e.
1 gal =
5 gal
d.
1 qt
Step Up
fl oz
f.
1 gal
ORIGO Stepping Stones Grade 4 • 12.11
fl oz 80 fl oz
293
12.12
Solving Word Problems Involving Liquid Volume (Capacity)
Step Up
What can you see in this picture?
Solve each word problem. Show your thinking.
a. Each bottle of soda holds 1 qt.
Each can holds 12 fl oz. How much
soda is in 1 21 bottles and 3 cans?
b. Bottled water is sold in packs of 6.
Each small bottle holds about
17 fl oz. About how much water
will there be in 2 packs?
fl oz
GRAPE
JUICE
Imagine each small glass holds 8 fl oz.
E
PL
SA
M
59 fl oz
c. Kayla buys three 2-gallon bottles
of water. She pours all the water
equally into 12 pitchers. How much
water is in each pitcher?
Step Ahead
fl oz
Write numbers to complete each word story.
Make sure the stories make sense.
a.
Jamal buys
How much grape juice
would be left over?
bottles of water. Each bottle holds
He pours the water equally into
There is now
ORIGO Stepping Stones 4 • 12.12
© ORIGO Education.
© ORIGO Education.
How many large glasses could you fill from the bottle of water? How do you know?
gallons.
containers so there is no water left over.
fl oz in each container.
b.
Rita buys one carton of juice. The carton holds
Imagine each large glass holds 16 fl oz.
294
d. Anna opens a carton of juice that
holds 59 fl oz. She fills 4 glasses of
juice that each holds 12 fl oz. How
much juice is left in the carton?
fl oz
How many small glasses could you fill from one bottle of soda? How do you know?
How many glasses could you fill with grape juice?
fl oz
fluid ounces. She fills
glasses with juice from the carton. Each glass holds
ounces. There are
ORIGO Stepping Stones 4 • 12.12
fluid
fl oz left in the carton.
295
COMMON CORE STATE STANDARDS for MATHEMATICS
– with modifications by the California State Board of Education shown underlined.
Grade 4 Overview
In Grade 4, instructional time should focus on three critical areas: (1)
developing understanding and fluency with multi-digit multiplication,
and developing understanding of dividing to find quotients involving
multi-digit dividends; (2) developing an understanding of fraction
equivalence, addition and subtraction of fractions with like denominators,
and multiplication of fractions by whole numbers; (3) understanding that
geometric figures can be analyzed and classified based on their properties,
such as having parallel sides, perpendicular sides, particular angle
measures, and symmetry.
(1) Students generalize their understanding of place value to 1,000,000,
understanding the relative sizes of numbers in each place. They apply their
understanding of models for multiplication (equal-sized groups, arrays,
area models), place value, and properties of operations, in particular the
distributive property, as they develop, discuss, and use efficient, accurate, and
generalizable methods to compute products of multi-digit whole numbers.
Depending on the numbers and the context, they select and accurately apply
appropriate methods to estimate or mentally calculate products. They develop
fluency with efficient procedures for multiplying whole numbers; understand
and explain why the procedures work based on place value and properties
of operations; and use them to solve problems. Students apply their
understanding of models for division, place value, properties of operations,
and the relationship of division to multiplication as they develop, discuss,
and use efficient, accurate, and generalizable procedures to find quotients
involving multi-digit dividends. They select and accurately apply appropriate
methods to estimate and mentally calculate quotients, and interpret
remainders based upon the context.
• Use the four operations with whole numbers to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
Number and Operations in Base Ten
• Generalize place value understanding for multidigit whole numbers.
se place value understanding and properties of operations to perform
• U
multi-digit arithmetic.
Number and Operations—Fractions
• Extend understanding of fraction equivalence and ordering.
• B
uild fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
• Understand decimal notation for fractions, and compare decimal fractions.
Measurement and Data
olve problems involving measurement and conversion of measurements
• S
from a larger unit to a smaller unit.
• Represent and interpret data.
• G
eometric measurement: understand concepts of angle and measure
angles.
Geometry
• Draw and identify lines and angles, and classify shapes by properties of
their lines and angles.
M
PL
(2) Students develop understanding of fraction equivalence and operations
with fractions. They recognize that two different fractions can be equal
(e.g., 15/9 = 5/3), and they develop methods for generating and recognizing
equivalent fractions. Students extend previous understandings about how
fractions are built from unit fractions, composing fractions from unit fractions,
decomposing fractions into unit fractions, and using the meaning of fractions
and the meaning of multiplication to multiply a fraction by a whole number.
Operations and Algebraic Thinking
E
Mathematics | Grade 4
(3) Students describe, analyze, compare, and classify two-dimensional shapes.
Through building, drawing, and analyzing two-dimensional shapes, students
deepen their understanding of properties of two-dimensional objects and the
use of them to solve problems involving symmetry.
Mathematical Practices
1. Make sense of problems and persevere in solving them.
2. R
eason abstractly and quantitatively.
3. C
onstruct viable arguments and critique the reasoning of others.
4. M
odel with mathematics.
5. U
se appropriate tools strategically.
6. A
ttend to precision.
7. L
ook for and make use of structure.
SA
8. L
ook for and express regularity in repeated reasoning.
Operations and Algebraic Thinking
4.OA
Use the four operations with whole numbers to solve problems.
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7
as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.
Represent verbal statements of multiplicative comparisons as multiplication
equations.
2. M
ultiply or divide to solve word problems involving multiplicative
comparison, e.g., by using drawings and equations with a symbol for the
unknown number to represent the problem, distinguishing multiplicative
comparison from additive comparison.1
3. S
olve multistep word problems posed with whole numbers and having
whole-number answers using the four operations, including problems in
which remainders must be interpreted. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the
reasonableness of answers using mental computation and estimation
strategies including rounding.
Gain familiarity with factors and multiples.
SOURCE
Common Core State Standards Initiative (2012) Common Core State
Standards for Mathematics Available at: http://www.corestandards.org/thestandards/download-the-standards (Accessed: 8th April 2013).
California State Board of Education (no date) California’s CCSS for
Mathematics Available at: http://www.cde.ca.gov/re/cc/ccssmathtemp.asp
(Accessed: 8th April 2013).
4. F
ind all factor pairs for a whole number in the range 1–100. Recognize that
a whole number is a multiple of each of its factors. Determine whether a
given whole number in the range 1–100 is a multiple of a given one-digit
number. Determine whether a given whole number in the range 1–100 is
prime or composite.
Generate and analyze patterns.
5. G
enerate a number or shape pattern that follows a given rule. Identify
apparent features of the pattern that were not explicit in the rule itself. For
example, given the rule “Add 3” and the starting number 1, generate terms
in the resulting sequence and observe that the terms appear to alternate
between odd and even numbers. Explain informally why the numbers will
continue to alternate in this way.