Chapter 8 of 8

All 8 chapters
  1. 1Multiplying and dividing in your head, readunderstood
  2. 2Short multiplication, readunderstood
  3. 3Long multiplication, readunderstood
  4. 4Short division, readunderstood
  5. 5Long division, readunderstood
  6. 6Order of operations, readunderstood
  7. 7Estimate, then check, readunderstood
  8. 8Choosing what to do, readunderstood

M3.2.8 Chapter 8

Choosing what to do

How do I know whether to multiply or divide?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Solve a problem with several steps using all four operations
  • Draw a bar model to decide whether to multiply or divide

Joseph reads: "6 boxes hold 24 pencils each. How many pencils are there?" Should he multiply or divide? Many people guess. A drawing can decide for you.

A bar model is a bar cut into parts. The whole bar is the total. Write what you know on the parts. Write a ? where the answer goes.

Draw it first

A bar model for 6 boxes of 24 pencils

  1. A bar cut into 6 equal parts, each labelled 24. A bracket over the bar shows the total. The total is first ?, and then 144.242424242424

    1 Draw 6 equal parts for the 6 boxes. Write 24 in each part, because each box holds 24 pencils.

  2. A bar cut into 6 equal parts, each labelled 24. A bracket over the bar shows the total. The total is first ?, and then 144.242424242424?

    2 The whole bar is the total number of pencils. We do not know it yet, so write a question mark over the bar.

  3. A bar cut into 6 equal parts, each labelled 24. A bracket over the bar shows the total. The total is first ?, and then 144.242424242424144

    3 The parts are equal and we want the total, so multiply. 6 × 24 = 144 pencils.

Now turn the problem round. 144 pencils are shared between 6 boxes. How many go in each box? The total is known. The parts are what we want.

A bar cut into 6 equal parts, each marked ?. A bracket over the bar says 144.??????144
The total is known and each part is wanted, so divide: 144 ÷ 6 = 24.

Here is the rule. Equal parts: from the parts to the total, multiply. From the total to one part, or to the number of parts, divide. Parts that are not equal: add them for the total. Take away to find a missing part. The bar shows which one you have.

Worked example

Mei has 3 bags with 24 beads in each, and 18 loose beads. How many beads does she have?

  1. Draw a bar with 3 parts of 24 and one part of 18. The whole bar is the total, so write a question mark over it.
  2. The bags: 3 × 24 = 72.
  3. Add the loose beads. 72 + 18 = 90.

Answer: 90

See it three ways

With things

Lay out 3 bags of 24 beads in a row, and then 18 loose beads at the end.

As a picture

A bar with four parts, 24, 24, 24 and 18. A bracket over the whole bar has ?.24242418?

In symbols

3 × 24 + 18 = 72 + 18 = 90.

Worked example

Mei shares her 90 beads equally between 6 friends. How many beads does each friend get?

  1. The total is known, and so is the number of parts. We want the size of one part, so divide.
  2. Use short division. 6 goes into 9 once, leaving 3. 6 goes into 30 five times.
  3. 90 ÷ 6 = 15.
  4. Check by multiplying back. 6 × 15 = 90.

Answer: 15

Many problems have several steps. Draw the bar for the first step, then the next. Each step uses one operation, which is adding, taking away, multiplying or dividing. You may need all four.

A bar for 250 stickers with four parts: 40, 40, 40 and a part marked ?. A bracket over the bar says 250.404040?250
Hana gives 40 to each of 3 friends. The part with the question mark is what is left.

Worked example

Hana has 250 stickers. She gives 40 to each of 3 friends. She shares the rest equally between 5 cousins. Each cousin then finds 4 more stickers. How many stickers does each cousin have now?

  1. Draw a bar for the 250 stickers. Three parts of 40 are given away. The last part is what is left.
  2. Stickers given away: 3 × 40 = 120.
  3. Stickers left: 250 − 120 = 130.
  4. Each cousin gets 130 ÷ 5 = 26.
  5. Each cousin finds 4 more. 26 + 4 = 30.

Answer: 30

Try it

Draw a bar for 84 eggs

You need: paper and a pencil, a ruler, if you have one.

  1. Draw a long bar and write 84 over it. A farmer packs 84 eggs into trays of 12.
  2. Mark off parts of 12 along the bar, one after another, until the bar is full.
  3. Count the parts. That is the number of trays.

What you will see: You make 7 parts of 12. So the farmer fills 7 trays. The total is known and the number of parts is wanted, so the calculation is 84 ÷ 12 = 7.

If you cannot try it, this is what happens.
A bar cut into 7 equal parts, each labelled 12. A bracket over the bar says 84.1212121212121284

A wrong idea many people have

The word "each" always means multiply.

Why it feels right: In "4 boxes with 6 pencils in each", the word each goes with multiplying.

The test that breaks it: Try "12 pencils are shared between 3 boxes. How many go in each box?" The word each is there, but the calculation is 12 ÷ 3 = 4.

What is true: Do not hunt for words. Draw the bar. With equal parts, parts to total means multiply, and total to a part means divide. With unequal parts, add or take away.

Key words

bar model
a bar cut into parts that shows what you know and what you want to find.
operation
one of adding, taking away, multiplying and dividing.

Back to your guess

The answer is B: Draw a bar model. With equal parts, multiply to find the total and divide to find one part. With unequal parts, add and take away.

A drawing shows what you know and what you want. Equal parts: from the parts to the total, multiply. From the total to one part, divide. Unequal parts: add them for the total, and take away to find a missing part. The word "each" can go with either one.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. The bar shows 144 pencils shared in 6 equal parts. Which calculation finds one part?

    A bar cut into 6 equal parts, each marked ?. A bracket over the bar says 144.??????144
    Tap your answer.

    1. Not this one. Adding makes the bar longer. We know the whole bar and want one part, so divide. Go to M3.2.8. Try another.

    2. Not this one. Multiplying makes a number bigger than 144, but one part cannot be bigger than the whole bar. Go to M3.2.8. Try another.

    3. Not this one. Taking away 6 does not cut the bar into 6 equal parts. Equal parts mean divide. Go to M3.2.8. Try another.

    4. Yes, that is right. The total, 144, is known, and the bar is cut into 6 equal parts. From the total to one equal part, divide. 144 ÷ 6 = 24.

    See the answer

    144 ÷ 6 = 24

    The total, 144, is known, and the bar is cut into 6 equal parts. From the total to one equal part, divide. 144 ÷ 6 = 24.

    • If you chose A: Adding makes the bar longer. We know the whole bar and want one part, so divide. Go to M3.2.8.
    • If you chose B: Multiplying makes a number bigger than 144, but one part cannot be bigger than the whole bar. Go to M3.2.8.
    • If you chose C: Taking away 6 does not cut the bar into 6 equal parts. Equal parts mean divide. Go to M3.2.8.

  2. The bar shows 6 equal parts of 24 pencils each. Which calculation finds the total?

    A bar cut into 6 equal parts, each labelled 24. A bracket over the bar has ?.242424242424?
    Tap your answer.

    1. Not this one. That adds the number of parts to the size of one part. The whole bar has 6 parts of 24, so multiply. Go to M3.2.8. Try another.

    2. Not this one. That cuts one part of 24 into 6 smaller pieces. The question asks for the whole bar. Go to M3.2.8. Try another.

    3. Yes, that is right. The 6 equal parts are known and the total is wanted. Parts to total means multiply. 6 × 24 = 144.

    See the answer

    6 × 24 = 144

    The 6 equal parts are known and the total is wanted. Parts to total means multiply. 6 × 24 = 144.

    • If you chose A: That adds the number of parts to the size of one part. The whole bar has 6 parts of 24, so multiply. Go to M3.2.8.
    • If you chose B: That cuts one part of 24 into 6 smaller pieces. The question asks for the whole bar. Go to M3.2.8.

  3. Mei has 3 bags of 24 beads and 18 loose beads. She shares all the beads equally between 6 friends. How many beads does each friend get?

    Tap your answer.

    1. Yes, that is right. First find all the beads. 3 × 24 = 72, and 72 + 18 = 90. Then share them. 90 ÷ 6 = 15.

    2. Not this one. You left out the 18 loose beads. All the beads are shared, so add them to the 72 first. Go to M3.2.8. Try another.

    3. Not this one. You divided only the 18 loose beads by 6. All the beads must be shared, so add first, then divide. Go to M3.2.8. Try another.

    4. Not this one. You multiplied the total by 6. Sharing between 6 friends means dividing the total by 6. Go to M3.2.8. Try another.

    See the answer

    15

    First find all the beads. 3 × 24 = 72, and 72 + 18 = 90. Then share them. 90 ÷ 6 = 15.

    • If you chose B: You left out the 18 loose beads. All the beads are shared, so add them to the 72 first. Go to M3.2.8.
    • If you chose C: You divided only the 18 loose beads by 6. All the beads must be shared, so add first, then divide. Go to M3.2.8.
    • If you chose D: You multiplied the total by 6. Sharing between 6 friends means dividing the total by 6. Go to M3.2.8.

  4. A rope is 156 m long. Ravi cuts it into pieces of 12 m. How many pieces does he get?

    Tap your answer.

    1. Not this one. Multiplying gives a number far bigger than the rope. Each piece is 12 m long, so there are fewer pieces than metres. Go to M3.2.8. Try another.

    2. Yes, that is right. The total length, 156 m, is known, and each part is 12 m. The number of parts is wanted, so divide. 156 ÷ 12 = 13.

    3. Not this one. Taking 12 m off the rope gives the rope that is left after one piece. It does not count the pieces. Go to M3.2.8. Try another.

    4. Not this one. Adding makes the rope longer. Cutting it into equal pieces means dividing. Go to M3.2.8. Try another.

    See the answer

    13

    The total length, 156 m, is known, and each part is 12 m. The number of parts is wanted, so divide. 156 ÷ 12 = 13.

    • If you chose A: Multiplying gives a number far bigger than the rope. Each piece is 12 m long, so there are fewer pieces than metres. Go to M3.2.8.
    • If you chose C: Taking 12 m off the rope gives the rope that is left after one piece. It does not count the pieces. Go to M3.2.8.
    • If you chose D: Adding makes the rope longer. Cutting it into equal pieces means dividing. Go to M3.2.8.

  5. Ana has 3 times as many stickers as Tariq. Tariq has 15 stickers. The bars show this. How many stickers does Ana have?

    Two bars of the same scale. The bar for Tariq has one part, 15. The bar for Ana has three parts, 15, 15 and 15. A bracket over the bar for Ana has ?.Tariq15Ana151515?
    Tap your answer.

    1. Not this one. Taking away makes the number smaller. Ana has more stickers than Tariq, so the answer is more than 15. Go to M3.2.8. Try another.

    2. Not this one. Dividing finds a third of 15. "3 times as many" means Ana has 3 parts of 15, so multiply. Go to M3.2.8. Try another.

    3. Not this one. Adding 3 would give her only 3 more. "3 times as many" means 3 equal parts of 15. Go to M3.2.8. Try another.

    4. Yes, that is right. Ana has 3 equal parts of 15. Parts to total means multiply. 3 × 15 = 45.

    See the answer

    45

    Ana has 3 equal parts of 15. Parts to total means multiply. 3 × 15 = 45.

    • If you chose A: Taking away makes the number smaller. Ana has more stickers than Tariq, so the answer is more than 15. Go to M3.2.8.
    • If you chose B: Dividing finds a third of 15. "3 times as many" means Ana has 3 parts of 15, so multiply. Go to M3.2.8.
    • If you chose C: Adding 3 would give her only 3 more. "3 times as many" means 3 equal parts of 15. Go to M3.2.8.

  6. Mei has 4 bags of 25 beads and 36 loose beads. She shares all the beads equally between 8 friends. How many beads does each friend get?

    See the answer

    17

    First find all the beads. 4 × 25 = 100, and 100 + 36 = 136. Then share them. 136 ÷ 8 = 17.

  7. Hana has 250 stickers. She gives 40 to each of 3 friends. She shares the rest equally between 5 cousins. How many stickers does each cousin get?

    See the answer

    26

    Given away: 3 × 40 = 120. Left: 250 − 120 = 130. Each cousin: 130 ÷ 5 = 26.

  8. From an earlier chapter

    Round each number to one digit to estimate 487 × 31. Give the estimate.

    See the answer

    15 000

    487 rounds to 500 and 31 rounds to 30. 5 × 3 = 15, and three zeros go back. The estimate is 15 000.

Keep it

Copy this into your exercise book.

Draw a bar model. Equal parts: multiply for the total, divide for one part. Unequal parts: add, or take away a part. Do one operation at a time, and check the answer makes sense.

Draw: Draw 6 equal parts with 24 in each, and a bracket over them. Write 6 × 24 = 144.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (792 KB). It opens on any phone.