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ORIGO Stepping Stones 3 • 6.1ORIGO Stepping Stones 3 • 6.1

Look at this array. What do you see?

What strategy can you use to figure out the product?

Look at the next array. What do you see?

What strategy can you use to figure out the product?

2 × 6 = = 6 × 2

Step Up 1. Look at these pictures. Write the products.

a.

2 × 9 =

double, double, double 9

8 × 9 =

c.b.

double, double 9

4 × 9 =

Introducing the Eights Multiplication Facts6.1

2 × 6 = = 6 × 2

4 × 6 = = 6 × 4

2 × 6 = = 6 × 2

4 × 6 = = 6 × 4

8 × 6 = = 6 × 8

I used the double, double strategy.

8 sixes is the same as double, double, double 6.

Look at the next array. What do you see?

How can you use the fours fact to help you figure out the product for the eights fact?

Is this an easy strategy to use?

What other eights facts could you solve using this strategy?

2. Write the products for these.

c.

2 × 5 =

4 × 5 =

8 × 5 =

b.

2 × 7 =

4 × 7 =

8 × 7 =

a.

2 × 3 =

4 × 3 =

8 × 3 =

d.

2 × 8 =

4 × 8 =

8 × 8 =

3. Use a doubling strategy to complete this table.

Number Double (×2)

Double Double (×4)

Double Double Double (×8)

6

7

20

36

double 9

Step Ahead Write numbers in the squares so that the numbers in each row and column multiply to give the product in the circle.

8 24 5

16

20

1032

56

12

2428

40

36

2072

a. c.b.SAM

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ORIGO Stepping Stones 3 • 6.2ORIGO Stepping Stones 3 • 6.2

2. Complete each picture.Reinforcing the Eights Multiplication Facts6.2

Amber bought 8 of these packs of pencils.

How many pencils did she buy in total?

How could you figure it out?

Jamal bought 8 different packs of pencils. Each pack had 2 pencils. How many pencils did he buy?

How could you figure it out?

What other problems could you solve using these strategies?

Double 3 is 6, double 6 is 12, and double 12 is 24. So 8 packs of 3 is 24.

Step Up 1. Show how you could use the double, double, double strategy to solve each word problem.

a. Zoe bought 6 packs of stickers. There are 8 stickers in each pack. How many stickers did she buy in total?

double is

double is

double is stickers

b. Ramon bought 8 packs of trading cards. There are 9 cards in each pack. How many trading cards did he buy in total?

double is

double is

double is cards

3. Complete each fact. Then write the turnaround fact for each.

8 × 3 =

× =

a.8 × 6 =

× =

b.8 × 2 =

× =

c.

4. Think about the fours and eights facts. Write the missing number in each equation.

× 2 × 2× 2

What should you write in each of the boxes below to show how many pencils she bought?

× 2

×

× 2× 2a.

× 2

×

× 2× 2b.

× 2

×

× 2× 2c.

× 2

×

× 2× 2d.

It is easier to think double 8 than double, double, double 2.

5 7

4 8

Step Ahead

Use repeated doubling to complete the missing numbers.

× 2× 2 × 2 × 2

4

× 8

3

a. 4 × 7 =

d. 24 = × 6

g. 7 ×

= 56

b. 16 = 2 ×

e. × 8 = 64

h. 32 =

× 8

c. × 4 = 36

f. 40 =

× 10

i. 3 × 8 =

× 16

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132 ORIGO Stepping Stones 3 • 6.3ORIGO Stepping Stones 3 • 6.3

Look at this hundred chart.

Exploring Patterns with the Eights Multiplication Facts6.3

Color all the numbers above that you know are products of the eights facts.

There are two numbers in the bottom two rows that are the products of 8 × 11 and 8 × 12.

Is there a pattern you can see that could help you figure out the numbers?

Imagine the chart continued to 200. What are some other numbers you would color to continue the pattern?

How do you know?

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

What strategy could you use to check your answer?

Step Up Use the hundred chart on page 132 to answer these questions.

1. Look at the color pattern of numbers. Are the numbers odd or even?

2. Look at the numbers in a single column. What do you notice?

3. Look at the numbers that lie along a sloping line. What change happens in the ones digit from one number to the next?

Step Ahead Complete these facts. Then write about the pattern you notice.

8 ×

= 32

4 ×

= 32

2 ×

= 32

4. a. Loop all the numbers in the chart that are products of the fours facts.

b. What pattern do you notice?

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ORIGO Stepping Stones 3 • 6.4ORIGO Stepping Stones 3 • 6.4

Step Up 1. Complete the picture to match the problem. Then complete the related number fact.

Introducing the Ones Multiplication Facts6.4

What do you see in this picture?

Write a number sentence to describe your picture.

What number sentence could you write to describe the row of vehicles?

What else might you see in one row? Draw a picture to match.

7 birds sitting on a fence How many birds in total?

× 1 =

a. 6 cookies in a jar How many cookies in total?

× 1 =

b.

2. Complete the number fact to match each picture.

3. In each story write a number greater than 1 but less than 10. Then draw a matching picture and write the related number fact.

a.

bananas in a bunchHow many bananas in total?

× =

b.

stamps in a rowHow many stamps in total?

× =

c.

muffins on a trayHow many muffins in total?

× =

d.

flowers in a vaseHow many flowers in total?

× =

1 × =

1 line of ducks How many ducks in total?

b.

× 1 =

1 ball for each student How many balls in total?

a.

Step Ahead Write a rule you can use when you multiply by 1.SAM

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ORIGO Stepping Stones 3 • 6.5ORIGO Stepping Stones 3 • 6.5

Introducing the Zeros Multiplication Facts6.5

Step Up 1. Draw the picture. Then write the multiplication fact.

Describe what you see in each row.

What number sentences could you write to describe each row?

Look at Row D. What happens when you multiply by 0?

d. 0 cookies in each jar c. 1 cookie in each jar

2. Draw rows of 5 stars on the flag to match each multiplication fact.

3. Draw jumps on the number line to show each fact. Then write the products.

4 × 2 =

4 × 1 = 4 × 0 =

10 2 3 4 5 6 7 8 9 10

a.

5 × 2 =

5 × 1 = 5 × 0 =

10 2 3 4 5 6 7 8 9 10

b.

Step Ahead Write a rule you can use when you multiply by 0.

3 × =

3 × =

a. 3 cookies in each jar b. 2 cookies in each jar

3 × =

3 × =

Row A

Row B

Row C

Row D

2 × 5 = 10

a. b.3 × 5 = 15

0 × 5 = 0

c.1 × 5 = 5

d.

8

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ORIGO Stepping Stones 3 • 6.6ORIGO Stepping Stones 3 • 6.6

2. Multiply the two numbers across each row and write the product in the matching circle. Then multiply the two numbers down each column and write the product in the matching circle.

Reinforcing the Ones and Zeros Multiplication Facts6.6

Tyler planted seedlings in one row of 6.

How many seedlings did he plant? How do you know?

Gavin had 6 mushrooms on his plate. He ate none of them.

How many mushrooms did Gavin eat? How do you know?

Rita had 6 pencils on her desk. Then her friend gave her another pencil.

How many pencils did Rita have in total? How do you know?

Claire had 6 packets of stickers. She gave stickers to each of her friends until the packets were empty.

How many stickers did Claire have left? How do you know?

Step Up 1. Read the equation carefully. Then write the answer.

3. Figure out what numbers must be in each row and column to make the product in the matching circle.

Step Ahead Write a story problem to match this fact.

4 4

15

512

2 2

10

102

2 2

18

94

1 1

4

14

5 10

0

00

3 0

1 4

2 6

8 0

2 1

3 5

215

56

What can you say about the math involved in each story?

a. c.b.

a. c.

d. f.e.

b.

0 0

7

50

1 × 9 = 9

a. 5 × 1 =

d. 1 + 7 =

g. 37 × 1 =

b. 9 + 1 =

e. 0 × 4 =

h. 97 + 0 =

c. 8 × 0 =

f. 5 − 0 =

i. 0 × 58 =

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140 ORIGO Stepping Stones 3 • 6.7ORIGO Stepping Stones 3 • 6.7

2. Figure out the solution to each problem. Show your thinking.

Step Up 1. Write an equation to match each story. You can use a letter for the unknown amount. Then figure out the solution for each problem.

There are 6 plates on a picnic table.

Each plate has 4 strawberries.

There was also one banana on 2 of the plates.

Solving Word Problems Involving Multiplication6.7

How many strawberries are there in total? How do you know?

What numbers in the story helped you?

Which numbers would you use to figure out how many pieces of fruit there are in total?

b. Every day Mika and 2 of her friends jog 3 laps around the athletics track. How many laps will Mika jog in 8 days?

×

=

laps

a. The teacher has 5 rulers. Each ruler is 1 foot long and 3 of them are plastic. What is the total length of the rulers if they are laid end to end?

×

=

feet

d. Chairs in an auditorium were set out in 6 rows. There were 8 chairs in each row. Only 5 people sat in each row. How many people are there in total?

×

=

people

c. Paige arranged a bookcase so that each shelf had 9 books. There are 4 shelves of books. 7 of the books are about cats. How many books are there in total?

×

=

books

a. Isabelle had 4 pieces of ribbon. Each ribbon was 3 feet long. She needed twice the total length that she had. How many feet of ribbon did Isabelle need in total?

b. A fruit store sells 3-kilogram bags of apples and oranges. Dad bought 2 bags of oranges and 6 bags of apples. What was the total weight of all the fruit Dad bought?

c. Evan baked a tray of cookies. The tray held 4 rows of cookies with 7 cookies in each row. He had some cookie dough left over so he baked another tray that had only one row of 7 cookies on it. How many cookies did he bake?

Step Ahead Write a story that matches this number sentence.

feet

kg

cookies

4 × 6, then add 3

I need to find the total number of strawberries. I can use the letter T to represent the total number of strawberries.

So T = 6 x 4.

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ORIGO Stepping Stones 3 • 6.8ORIGO Stepping Stones 3 • 6.8

Step Up

Janice had three strips of paper. The picture shows how she folded two of them into parts of equal size.

The first strip is one whole. What would you label each part of the other strips?

When you write 1 3 what does the 3 tell you?

Look at the folds that made sixths. How are they related to the folds that made thirds?

Exploring Related Partitions (Fraction Strips)6.8 3. Write the fraction that matches the fraction words. Then write how many parts are needed to fill the whole.

The 3 tells me that three of the parts are needed to fill the whole.

The paper strips belong to the 1 3 fraction family because you can make sixths by first making thirds.

1. Draw a fold line on the second strip to show halves. Draw fold lines on the last strip to show fourths. Then label each part of the strips.

2. Draw fold lines on the second strip to show thirds. Draw fold lines on the last strip to show ninths. Then label each part of the strips.

1 1

1 Fraction Word Fraction Parts in One Whole

one-third 1 3 3

one-fourth

one-eighth

Step AheadUse paper strips to find a different fraction that belongs in each of the 1

2 and 1 3 fraction families. Draw lines to show the folds

and label each part of the strips.

1 1

4. This strip is one whole.

a. Draw lines to split the strip into sixths. Then shade 4 6 of the strip.

b. What does the 6 tell you?

c. What does the 4 tell you?

The first strip in each of these is one whole.

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ORIGO Stepping Stones 3 • 6.9ORIGO Stepping Stones 3 • 6.9

2. For each number line, the distance from 0 to 1 is one whole. Write the answer to each problem. Draw jumps on the number line to show your thinking.

Exploring the Additive Nature of Common Fractions6.9

Look at this number line.

The distance from 0 to 1 is one whole.

Show how you would mark 1 6 on the number line.

a.Mary has a bottle that holds 1

6 of a gallon of water. How many bottles will she need to make 4 6 of a gallon of water?

b.Ella’s family bought a pizza cut into eighths. Ella ate 3

8 of the whole pizza. How many pieces of pizza did she eat?

3. Write an addition equation to match the jumps on the number line.

0 1

0 1

=

c.

0 1

=

b.

0 1

a.

Step Ahead Complete each equation.

a. 1 3 +

1 3 +

1 3 =

3c.

1 2 +

1 2 =

2b.

1 8 +

1 8 +

1 8 +

1 8 +

1 8 =

8

Wesley is making a cake.

He needs 3 4 cup of sugar

but only has a 1 4 measuring cup.

What can he do to measure the correct amount of sugar?

What does this number line show?

0 13 4

2 4

1 4

Step Up1. Look at how each number line has been split up. The distance

from 0 to 1 is one whole. Write the fraction that each arrow is pointing to.

a. b.

I would split the line from 0 to 1 into 6 equal parts. The distance from 0 to the first mark will be 1

6 of the total distance from 0 to 1.

0 1

0 1 0 1

+ 1 4+ 1

4+ 1 4

0 1

+ 1 6 + 1

6

= 6+

1 6

1 6

cup1 4

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ORIGO Stepping Stones 3 • 6.10ORIGO Stepping Stones 3 • 6.10

0 24 3

5 3

2 3

1 3 1

On each number line, the distance from 0 to 1 is one whole. Write the fraction that should be in each box. Draw jumps to help you.

One batch of 12 muffins needs 2 3 cup of mashed banana.

Logan wants to make 2 batches but he only has a 1 3 measuring cup.

What can he do to measure the correct amount of banana for 2 batches of muffins?

Exploring Improper Fractions (Number Line Model)6.10

How could you show your thinking on a number line?

What fraction could you write to show the total amount of banana?

What do you notice about the fraction 4 3 ?

Logan can use the 1 3 measuring cup two times for one

batch, so he can use it four times for two batches.

The numerator is greater than the denominator. I can see on the number line that 4

3 is greater than 1.

Step Ahead Complete each equation.

1 6 +

1 6 +

1 6 +

1 6 +

1 6 +

1 6 +

1 6 = 6

c. d.1 2 +

1 2 +

1 2 +

1 2 +

1 2 = 2

a.1 4 +

1 4 +

1 4 +

1 4 +

1 4 = 4

b.1 3 +

1 3 +

1 3 +

1 3 = 3

Step Up 1. On this number line, the distance from 0 to 1 is one whole. Write the fraction that should be in each box. Draw jumps to help you.

0 21

5. Use the fractions you wrote on the number lines above.

a. List the fractions that are less than 1. b. List the fractions that are greater than 1 but less than 2.

+ 1 3 + 1

3 + 1 3 + 1

3

2.

0 21c.b.a. d.

3.

c.b.a. d.0 31 2

4.

c.b.a. d.0 21

a. 4

b. 4

c. 4

d. 4

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148 ORIGO Stepping Stones 3 • 6.11ORIGO Stepping Stones 3 • 6.11

2. Each large shape is one whole. Color the shapes to show each fraction.

On this number line, the distance from 0 to 1 is one whole.

Exploring Improper Fractions (Area Model)6.11

Fractions greater than 1 can also be shown with shapes.

Each large square on the right is one whole.

Each whole is split into four parts of equal size.

How many fourths are shaded in total?

What fraction is shown?

0 21

Step Up 1. Each large shape is one whole. Write the fraction that is shaded.

c.

b.a.

a.

c.

g.

b.

d.

f.

h.

Step AheadEach square is one whole. Draw lines to split the squares into parts of equal size. Then color parts to show 9

4 .

e.

6 4

5 2

10 4

8 3

9 8

9 6

6 3

13 8

What fraction is the arrow pointing to?

How do you know?

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ORIGO Stepping Stones 3 • 6.12ORIGO Stepping Stones 3 • 6.12

Identifying Fractions6.12 For these number lines, the distance from 0 to 1 is one whole. Draw a line to connect each fraction to its position on the number line.

5. Use the fractions above to answer these questions.

a. List the fractions that equal 1.

c. What are three other fractions that would equal 2?

b. List the fractions that equal 2.

Step Ahead Draw shapes to show 7 2 . Be sure to explain what one whole

looks like.

8 8 is the same as one whole. ýOne wholeý is the same as 1.

3.

0 31 2

4.

0 31 2

e. 9 4

c. 8 4

a. 5 4

b. 4 4

d. 12 4

e. 15 6

c. 6 6

a. 3 6

b. 8 6

d. 12 6

Eight friends shared a pizza. Each friend had one-eighth of the whole pizza.

What fraction can you write to show how many eighths were eaten?

What is another way of describing how much pizza was eaten?

Step Up 1. Each red shape is one whole. Color parts to show each fraction.

d.

e.

a.

b.

c.

2. Loop the fractions above that are equal to 1 or 2.

f.

g.

1 3

2 3

3 3

4 3

5 3

6 3

7 3

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