Recognizing Division Situations 1

OPEN-ENDED

The pictures below show the division 14.5 4 2 5 7.25 in 2 ways.

• Explain why the same division can represent both situations.

How long is half of a 14.5 cm string?

How many 2 cm pieces can I cut from a 14.5 cm long string?

• Create 4 problems that can be solved by dividing 18.56 by 2. Make them as different as you can. Do not solve them.

• Explain what makes each problem a division situation. Record your work below and on the next page.

Problem 1:

How do you know this is a division problem?

Recognizing Division Situations

• A lot of measurement situations involve hundredths.

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Pathway 1

Copyright © 2012 by Nelson Education Ltd.80

Leaps and BoundsRecognizing Division Situations, Pathway 1

e.g., The drive from home to Lisa's grandmother's house was a total of 18.56 km both ways. How far away is Lisa's grandmother's house?

e.g., The total is split into 2 equal distances, or groups and I want to know the distance of each one.

Problem 2:


Problem 3:


Problem 4:


• How could you solve one of your problems using multiplication?


Leaps and Bounds Recognizing Division Situations, Pathway 1

e.g., I bought two identical games that cost $18.56 altogether. How much did each cost?

e.g., The total is split into 2 equal amounts, or groups and I want to know the value of each.

e.g., There was 18.56 L of juice to be poured into 2 L containers. How many full containers and what fraction of another container could be filled?

e.g., It's about finding out how many equal groups of 2 are in 18.56.

e.g., There was 18.56 m of wood and I had to cut off 2 m chunks. How many chunks could I cut off? What would be left?

e.g., It's about finding out how many equal groups of 2 are in 18.56.

e.g., For Problem 1, I'd figure out what to multiply by 2 to get 18.56.

GUIDED

Division always involves making equal groups from a known total amount. There are 2 types of division situations.

You might know the number of groups and want to know how many are in each group.

You might know the size of the groups and want to know how many groups there are.

• Division can mean “How many are in each group?” For example:

4 equal-sized bags of sand have a total mass of 22.56 kg. What is the mass of each bag?

22.56 kg 4 4 bags 5 5.64 kgEach bag has a mass of 5.64 kg.

This makes sense since 4 3 5.64 5 22.56.

22.56 kg5.64 kg

5.64 kg

5.64 kg

5.64 kg

A special example of a division problem that answers the question “How many are in each group?” is when you calculate a unit rate. That is when you know the total amount for several items and you divide by the number of items to calculate the amount for one item.

• Division can also mean “How many groups are there?” For example:

There is 22.56 kg of sand. You are filling bags so each has 4 kg. How many bags can you fill?

22.56 kg 4 4 kg 5 5.64 bagsThat’s 5 bags of 4 kg each, and a 6th bag that is 0.64 of 4 kg full. 0.64 of 4 kg is 2.56 kg.

4 kg 2.56 kg4 kg4 kg4 kg

• Which meaning of division would you use to solve each egg problem shown in the illustration above? Why?

Recognizing Division Situations

How many eggs does each of us get, if 3 of us share?

How many 3-egg omelettes can I make?

• Division is about making equal groups from a known total amount, or number. Multiplication is about figuring out the total of a number of equal groups.

• Since division is the opposite of multiplication, you can use multiplication to check division. e.g., 20 4 4 5 5, so 5 3 4 5 20.

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Pathway 1


Recognizing Division Situations, Pathway 1 Leaps and Bounds

(oral) For the first problem, I would use the How many are in each group? meaning. For the second problem, I would use the How many groups are there? meaning.

Try These 1. Circle the problems that can be solved using division.

Write a division expression for each but do not solve.

A. There are 6 lengths of ribbon, and each is 2.14 m long. If they were lined up end to end, what would be the total length?

D. There was 7.5 L of water in a container, and then 4.3 L was poured out. How much water is still in the container?

B. A container holds 8.23 L of water. The water is to be shared equally into 8 cups. How much will each cup get?

E. Six apples have a total mass of 0.942 kg. What is the mass of each apple?

C. A container holds 8.23 L of water that is to be poured into 0.75 L glasses. How many glasses are needed?

F. Hannah drove 234.6 km in 3 h. What is her average speed in kilometres/hour?

2. Choose one of the problems you circled in Question 1.Explain why it can be solved using division.

3. Josey says that Problem E in Question 1 is a division problem. Kara says it’s not. How might both girls be right?

4. Rhea bought 1.2 kg of trail mix to put into 0.15 kg bags.

a) Why can you use division to figure out how many bags Rhea could make?

b) What numbers would you divide? _________________________


Recognizing Division Situations, Pathway 1Leaps and Bounds

8.23 ÷ 8

8.23 ÷ 0.75

0.942 ÷ 6

234.6 ÷ 3

e.g., B, it`s a known total amount, 8.23 L, being shared equally among 8 groups to figure out how much is in each group.

e.g., Josey either thinks the apples have equal masses or that it's about the average mass of the apples. Kara thinks the apples have different masses.

e.g., It's a “How many groups are there? problem, since the total 1.2 kg is divided into equal groups of 0.15 kg.

1.2 ÷ 0.15

5. The area of a playground is 356.28 m2. It is to be separated into 4 equal sections for different activities. What is the area of each?

a) What number sentence or equation would solve the problem?

b) How do you know it is a division problem?

6. a) Create a problem about pouring 3.45 L of water into several containers that is not a division problem.

b) Why is your problem not a division problem?

7. Finish each sentence below to make 2 division problems:

– Make one problem about figuring out how many equal groups there are.

– Make one problem about figuring out how much is in each group.

Will had 4.25 kg of potatoes

Nik had 4.25 kg of potatoes

8. What clues can you use to decide if a problem can be solved using division?

3.45 L3.45 L

Recognizing the different types of division problems when decimals and whole numbers are involved will help you recognize division problems when other types of numbers, such as fractions, are involved.

FYI


Recognizing Division Situations, Pathway 1 Leaps and Bounds

356.28 ÷ 4 = [ ]

e.g., It's a known total amount, 356.28 m2, being shared equally among 4 equal groups to figure out how much is in each group.

e.g., I poured 3.45 L of water from a pitcher into a 2 L container

and a 1 L container. How much water is left in the pitcher?

e.g., The groups are not equal.

that he divided into bags that

each can hold 1.1 kg. How many bags could he make?

that he split into two equal size

bags. What is the mass of each bag?

e.g., If the total amount is given and there are equal groups involved, it's probably division.

e.g.,

e.g.,

OPEN-ENDEDPathway 2

The pictures below show 5 3 4.23 5 21.15 in 2 ways.

• Explain why the same multiplication can represent both situations.

Each of us has $4.23. Jonah has $4.23. Kali has 5 times as much. How much do we have altogether? How much does Kali have?

• Create 4 problems (not about money) that can be solved by multiplying 6 3 3.24. Make them as different as you can. Do not solve them.

• Explain what makes each problem a multiplication situation. Record your work below and on the next page.

Problem 1:

How do you know this is a multiplication problem?

Recognizing Multiplication Situations


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Leaps and Bounds Recognizing Multiplication Situations, Pathway 2

e.g., There were 6 pieces of wood, each 3.24 m long. How long would they be if they were put end to end?

e.g., There are 6 equal lengths, or groups of 3.24 m and I want the total length.

Problem 2:


Problem 3:


Problem 4:


• How could you solve one of your problems using addition?


Leaps and BoundsRecognizing Multiplication Situations, Pathway 2

e.g., A rectangular room is 6 m long and 3.24 m wide. What is its area?

e.g., You multiply length and width to figure out the area of a rectangle.

e.g., You walk 3.24 km to school and another 3.24 km home every day. How far would you walk in 3 days?

e.g., There are 6 equal distances or groups of 3.24 km and I want the total distance.

e.g., A jug holds 3.24 L. Another container holds 6 times as much. How much does the container hold?

e.g., You multiply what the jug holds, 3.24, by 6 to figure out what the container holds.

e.g., For Problem 4, I'dd add 3.24 + 3.24 + 3.24 + 3.24 + 3.24 + 3.24.That's what you do when you multiply, you add the same number over and over.

GUIDEDPathway 2

Multiplication always involves calculating the total of a number of equal groups. There are different types of problem situations that can be solved using multiplication.

• Multiplication can be used to solve problems involving the total of equal amounts or measures. For example:

Liv runs a 2.12 km long course 5 days a week. How far does she run in a week?

This diagram shows 5 equal “groups” of 2.12 km.

10.60 km

2.12 km 2.12 km 2.12 km 2.12 km 2.12 km

5 3 2.12 km 5 10.60 km

Liv runs 10.60 km a week.

• Multiplication can be used to solve problems involving the area of a rectangle. For example:

A rectangle is 4 m wide and 9.1 m long. What is its area?

The diagram shows that each “group” is 9.1 m2 and there are 4 of them.

9.1 m

4 m9.1 m2

4 3 9.1 m2 5 36.4 m2

The area is 36.4 m2.

• Multiplication can be used to solve problems involving rates. For example:

Axl’s lunch cost $4.29. His sister Stella’s lunch cost 3 times as much. How much was Stella’s lunch?

The diagram shows that each “group” is $4.29 and Stella’s lunch has 3 groups.

$4.29

3 3 $4.29 5 $12.87

Stella’s lunch was $12.87.

Recognizing Multiplication Situations

• Multiplying always involves equal groups.

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Try These 1. Circle the problems that can be solved using multiplication,

without any other operations. Write a multiplication expression for each but do not solve.

A. Arshad bought 4 binders. Each one cost $2.39. How much did he spend?

D. A triangle has sides that are 5.8 cm, 6 cm, and 2 cm. What is its perimeter?

B. Six identical pieces of ribbon are 13.24 m long, in total. How long is each piece?

E. A rectangular garden is 4.2 m wide and 5.1 m long. What is its area?

C. Two 1.6 L containers and two 0.8 L containers are filled with juice. How much juice is there altogether?

F. Bretta has 1.23 m2 of fabric, but she needs 3 times that much to make a curtain. How much does she need?

2. Choose one of the problems you circled in Question 1.Explain why it can be solved using multiplication.

3. Bryn says that Problem C in Question 1 is a multiplication problem. Hassan says it’s not. How can both be right?

4. Four cans each hold 1.89 L of juice. They cost $2.29 each.

a) What multiplication expression describes the total amount of juice?

____________________

b) What multiplication expression describes the total cost?

____________________

c) How do you know parts a) and b) are multiplication situations?


Leaps and BoundsRecognizing Multiplication Situations, Pathway 2

4 x 2.39

e.g., 6 x ___ = 13.24

e.g., 3 x 1.6 or 6 x 0.8

4.2 x 5.1

3 x 1.23

e.g., E is about calculating the area of a rectangle by multiplying length

and width.

e.g., Hassan says that since there are 2 different sizes, you need to

multiply twice. Bryn realizes you can combine the two 0.8 L sizes to

have 3 equal groups of 1.6.

4 x 1.89

4 x 2.29

e.g., They are about finding the total of 4 equal groups.

5. How are these 2 problems the same? How are they different?

• What is the area of the rectangle at the right?• Six members of the photo club donated $12.30 to help with

the costs for an exhibit. How much money was collected?

same:

different:

6. a) What would you do to change the rectangle problem in Question 5 a little so that it can no longer be solved by multiplication?

b) Why is it no longer a multiplication problem?

7. a) Continue the story below to make it a multiplication problem.

A box of paper clips costs $2.59

b) Continue the story another way to make it a different operation.

A box of paper clips costs $2.59

What operation did you use? Explain your thinking.

8. What clues can you use to decide if a problem can be solved using multiplication?

12.3 cm

6.0 cm

Recognizing the different types of multiplication problems when decimals and whole numbers are involved will help you recognize multiplication problems when other types of numbers, such as fractions, are involved.

FYI



e.g., Both involve finding the total of 6 equal groups of 12.3 and

can be solved using 6 x 12.3.

e.g., One is about area and the other is about money.

e.g., I'd have each of the 6 club members donate a different amount.

e.g., The group sizes are not equal anymore.

e.g., ... and I want to buy 4

boxes. How much will it cost altogether?

e.g., ... and I paid with a $5 bill. How much change will I get?

e.g., Subtraction; I need to know what to add to $2.58 to make $5.00 to figure out the change.

e.g., If there are a number of equal groups and you want to know the

total, it's probably multiplication.

OPEN-ENDED

The pictures below show 3.00 2 1.98 5 1.02 in 2 ways.

• Explain why the same subtraction can represent both situations.

How much ribbon is left on the spool after I cut a 1.98 m piece?

How much longer is the 3 m ribbon than the 1.98 m ribbon?

• Create 4 problems that can be solving by subtracting 2.34 from 4.0. Make them as different as you can. Do not solve them.

• Explain what makes each problem a subtraction situation. Record your work below and on the next page.

Problem 1:

How do you know this is a subtraction problem?

Recognizing Subtraction Situations


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Pathway 3


Leaps and BoundsRecognizing Subtraction Situations, Pathway 3

e.g., If you figure out what to add to 2.34 to get 4.00, the difference is the change and adding is the opposite of subtracting.

e.g., I paid for something that cost $2.34 with 2 toonies. How much change do I get?

Problem 2:


Problem 3:


Problem 4:


• How could you solve one of your problems using addition? Explain.


Leaps and Bounds Recognizing Subtraction Situations, Pathway 3

e.g., I had 4.0 m of fabric and cut off a piece that was 2.34 m long. What length is left?

e.g., You are taking away and figuring out what is left.

e.g., I ran 4.0 km and my brother ran 2.34 km. How much farther did I run?

e.g., I am figuring out how much more one thing is than another.

e.g., I had 4.0 L of apple juice in a pitcher. I poured 1.0 L into one container and 1.34 L into another. How much juice is left in the pitcher?

e.g., I am figuring out what's left even though I did it in two steps.

e.g., For Problem 2, I could have figured out what to add to 2.34 to get to 4. I'd add in parts: first hundredths, then tenths and then ones.

GUIDED

In a subtraction situation, you always know the whole and a part of the whole, and you want to know the other part.

For 6 – 2 = 4, 6 is the whole and

2 and 4 are the parts.

There are different types of problem situations that involve subtraction.

• You can use subtraction to solve take-away situations.

Sometimes you have a whole amount, and you take away part of it and want to know what part is left. For example:

You have 2.4 L of juice and pour 1.25 L of it into a punch bowl. How much juice is left?

2.4 L 2 1.25 L 5 1.15 L

There is 1.15 L of juice left in the pitcher.

• You can use subtraction to figure out how much more is needed.

Sometimes you have a part of a whole and need to know how much more is needed to make up the whole. For example:

You have already gone 1.7 km in a 5 km race. How much farther do you have to go?

5 km 2 1.7 km 5 ❚ can be solved using 1.7 km 1 ❚ 5 5 km.

0 1 1.7 2 3

3

3.3 km

0.3

4 5

Since 1.7 km 1 3.3 km 5 5.0 km, then 5.0 km 2 1.7 km 5 3.3 km.

You have 3.3 km left to go.

2.4 L

Recognizing Subtraction Situations

• Subtraction is the opposite of addition. That’s why you can solve a subtraction problem using addition. e.g., 7 2 3 5 ,so 3 1 5 7.

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Pathway 3



• You can use subtraction in situations that involve comparing.

Sometimes you want to know how much more one thing is (you can think of it as the whole) than another (you can think of it as one of the parts). For example:

Jayson has 2 pet Bernese Mountain Dogs from the same litter. Bernie is 47.2 kg and Boris is 44.9 kg. How much heavier is Bernie than Boris?

47.2 kg 2 44.9 kg 5 2.3 kg

47.2 kg44.9 kg

2.3 kg

Bernie Boris

Bernie is 2.3 kg heavier than Boris.

• In each type of subtraction situation shown above and on the previous page, what is the whole and what is the known part?

Try These 1. Circle the problems that can be solved using subtraction.

Write a subtraction expression for each but do not solve.

A. Pete bought 4 binders. Each binder cost $2.39. How much did Pete spend?

D. Liam ran 3.2 km and Mark ran 2.9 km. How much farther did Liam run?

B. Louisa bought a binder for $2.39. How much change did she get from $5?

E. Jane had 10.7 m of fabric to make curtains. She used 1.9 m for 1 panel. How much fabric is left?

C. One package of meat is 0.492 kg and another is 0.398 kg. How much meat is there altogether?

F. Kyle is building a 29.3 m2 deck. He has finished 14.8 m2 of it. How much does he have left to do?



(oral) Juice: The whole is 2.4 L and the known part is 1.25 L. Race: The whole is 5 km and the known part is 1.7 km. Masses: The whole is 47.2 kg and the known part is 44.9 kg.

5.00 - 2.39

3.2 - 2.9

10.7 - 1.9

29.3 - 14.8

2. a) Choose 2 of the problems you circled in Question 1. Explain why each can be solved using subtraction.

b) Show or explain how to use addition to solve each problem.

3. Choose a problem you did not circle in Question 1. Explain why it cannot be solved using subtraction.

4. How do you know this is a subtraction problem?

There is 1.2 kg of trail mix. The raisins are 0.35 kg and the nuts are 0.29 kg. What is the mass of the other ingredients?

5. How are these problems the same? How are they different?

• Lisa lives 2.1 km from school. She has already walked 0.9 km. How much farther does she have to go?

• I cut 0.9 m of ribbon off a 2.1 m piece. How much ribbon is left?• A recipe calls for 2.1 kg of meat. Another recipe uses 0.9 kg

of meat. How much more meat does the first recipe use?

same:

different:



e.g., In B, figuring out change can mean figuring out what to add to one part to make a whole. In E, you are finding out what is left

after you take away a part from the whole.

e.g., B: 5.00 - 2.39 = � 2.39 + � = 5; I'd start at 2.39 and add 0.01 to get to 2.4, then I'd add 0.1 to get to 2.5, then I'd add 2.5 to get to 5. That's adding 2.61 altogether.

e.g., E: 10.7 - 1.9 = � 1.9 + � = 10.7. I'd start at 1.9 and add 0.1 to get to 2 and then 8 to get to 10 and then 0.7 to get to 10.7. That's adding 8.8 altogether.

e.g., Problem A: It's about 4 equal groups and the total amount, so you have to add or multiply.

e.g., You know the whole, 1.2 kg, and a part of it (0.35 kg + 0.29 kg) and you want to know the other part.

e.g., All of them involve subtracting 0.9 from 2.1.

e.g., A is a “how much more do you need?” problem, B is a take-away problem, and C is a comparing problem.

6. a) Create 2 subtraction problems about this picture.

b) Create a problem about the picture that is not a subtraction problem.

7. a) Finish the sentence to create a subtraction problem.

A large bowl holds 6.5 L of punch

b) Finish the sentence to create a different type of subtraction problem.

A large bowl holds 6.5 L of punch

8. What clues can you use to decide if a problem can be solved using subtraction?

6.5 L

Recognizing the different types of subtraction problems when decimals and whole numbers are involved will help you recognize subtraction problems when other types of numbers, such as fractions, are involved.

FYI



Problem 1: How much taller is Nivra than Janis? Problem 2: How much does Nivra have to grow to be

as tall as Andreas?

e.g., What is the average height of the three friends?

and 4.9 L was used at a party. How much is left?

and you pour it into a

larger bowl that holds 10.0 L. How much more punch will you need

to add to fill the 10.0 L bowl?

e.g., If you know the whole and you know a part of it and you want to

know the amount of the other part, it's probably subtraction.

e.g.,

e.g.,

Recognizing Division Situations 1

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