4

1. Number Sentences

In this chapter you will learn about:• Number puzzles• Writing number sentences for word problems • Writing word problems• Solving number sentences using strip diagrams and

balances

1.1 Number PuzzlesIn this section you will solve number puzzles.

Exercise 1.1

Warm Up

1 Copy and complete in your notebook.

a Add 10 and 3. The answer is ____?____ .

b 12 take away 5 is ____?____ .

c 3 plus 18 is ____?____ .

d 15 minus 4 equals ____?____ .

4

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Yes!

Khalid

Faisal

13 24 25 6 17

2 Use the numbers in the cloud. Answer each secret number puzzle.

a ‘Add 2 to my secret number and you get 8.’

b ‘Subtract 5 from my secret number and you get 12.’

c ‘Add 10 to my secret number and you get 23.’

d ‘Take away 12 from my secret number and you get 12.’

e ‘Add 9 to my secret number and you get 34.’

I have a secret number. Take away 3 from my secret number

and you get 12.

Is your number 15?

Khalid is playing a game with Faisal.

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Main Exercise

3 Faisal says, ‘Add 7 to my secret number and you get 35’. Find Faisal’s secret number.

Hint Use a number line.

4 a Jasem sent invitations two weeks before his wedding. Jasem’s wedding is on September 24. When did Jasem send the invitations?

b How is this question like the secret number game?

5 Salma and Wafaa play the secret number game.

a Solve Wafaa’s secret number puzzle.

b Solve Salma’s secret number puzzle.

Word fact: Solve means ‘find the secret number’.

–?

? 30 35

–5

Wafaa

Salma Divide my secret number by 2 and you get 21.

Multiply my secret number by 3 and you

get 15.

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6 Khalid and Faisal play the secret number game again.

a Solve Khalid’s secret number puzzle.

b Solve Faisal’s secret number puzzle.

7 Play the secret number game with your partner. Take turns making a puzzle for your partner to solve. Choose a secret number between 1 and 20 each time.

8 Hamdan and Rashid are playing the game. Hamdan chooses a secret number. He does not say a sentence to Rashid. Instead, he writes this:

❑ + 20 = 84

a What do you think this means?

b How can Rashid find Hamdan’s number?

Khalid

Faisal

Multiply my secret number by 2. Then subtract 4. You get 24.

Divide my secret number by 3. Then add 4. You get

10.

TALKABOUT IT

TALKABOUT IT

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9 Solve these number sentences.

a ❑ + 3 = 12

c ❑ - 12 = 15

e 8 = ❑ ÷ 5

10 Solve these number sentences.

a 2 × ▲ + 3 = 13

b ❑ - 5 = 6 + 2

c ★ × 2 = 36 ÷ 2

d 16 - 4 = 3 × ?

e ❑ ÷ 3 = 15 + 10

11 Making decisions Copy and complete these number sentences.

Choose your own numbers to fill the __ spaces, so that the answer for ★ will be a whole number.

a ★ ÷ 2 = ____?____

b ____?____ × ★ = 12

c 20 - ★ = ____?____

d ★ - ____?____ = 6

e 43 + ★ = ____?____

f ★ × 2 = ____?____

b ? × 2 = 40

d 17 + ▼ = 27

f 15 - ? = 4

Hint Try a number in the ____?____ space. If it doesn’t work, try a different number. Think about why some numbers work and some don’t.

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12 Defining problems Write three of your own number sentences using a shape like ❑ or ▼. Make sure the answer for ❑ or ▼ is a whole number.

Hint Here are two ways to make ❑ a whole number:

1 Write a correct number sentence with whole numbers e.g.

2 × 3 – 4 = 2. Then replace one number with ❑.

For example, 2 × ❑ – 4 = 2

2. Write a number sentence e.g. 2 × ❑ = 7. Solve it: ❑ = 7 ÷ 2.

If the answer is not a whole number, change the numbers in the number sentence. Solve it again. Do this until the answer is a whole number.

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13 Organizing information Ahmed starts to write this number sentence:

▼ ÷ 2 + 1 = …

He wants ▼ to be a whole number.

He uses Excel to help him find numbers he can use to finish the number sentence.

a Open a spreadsheet on a computer.

• In cell A1, enter ‘triangle number’

• In cell A2, enter ‘Triangle / 2 + 1’

• In cells B1-F1, enter the numbers 1, 2, 3, 4, 5

• In cell B2 enter ‘B1/2+1’

• Use the fill tool to extend this formula to cells C2-F2. Drag across the row to F5

b Which numbers can Ahmed use to complete his number sentence? Write the two number sentences you have found with whole number answers.

Hint

The numbers in Row 2 give the numbers he can use to complete his sentence.

Fill tool

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Math facts: 10 is more than 3. 7 is less than 12. If you double 4 you get 8. If you halve 8 you get 4. The difference between 13 and 9 is 4.

1.2 Writing Number Sentences for Word Problems

Exercise 1.2

Warm Up

1 Making decisions Copy and complete in your notebook.

a If you ____?____ 8 you get 16.

b 8 is ____?____ than 11.

c 17 is ____?____ than 9.

d The ____?____ between 18 and 21 is 3.

e If you ____?____ 12 you get 6.

2 Copy and complete in your notebook. Choose your own numbers.

a ____?____ is less than ____?____ .

b If you double ____?____ you get ____?____ .

c ____?____ is more than ____?____ .

d The difference between ____?____ and ____?____ is ____?____ .

e If you halve ____?____ you get ____?____ .

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Main Exercise

Example

Faisal said, ‘Add 7 to my secret number and you get 35’.

We can write this as a number sentence.

Use ❑ for the secret number. ❑ + 7 = 35

3 Write number sentences for these number puzzles. Use ❑ for each secret number.

a ‘Add 3 to my secret number and you get 8.’

b ‘Subtract 9 from my secret number and you get 12.’

c ‘Add 10 to my secret number and you get 34.’

d ‘Take away 12 from my secret number and you get 15.’

e ‘Add 9 to my secret number and you get 28.’

To solve word problems, there are three things to remember:

1 Write a number sentence. Use a shape like ❑, ▼ , ★ for the secret number.

2 Write the answer as ❑ = … (or ▼ = … or ★ = …)

3 Write the answer in words. Don’t only write a number!

E.g. The time was 10 minutes.

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Example

Ahmed has some sweets. He eats 14. Then he has 23 left. How many sweets did Ahmed have at the beginning?

Answer

Step 1 Write a number sentence. ❑ - 14 = 23

Step 2 Solve the number sentence. ❑ = 37

Step 3 Write the answer in words. Ahmed had 37 sweets at the beginning.

4 Khalid has some counters. Hassan gives him 8 more counters. Khalid now has 20 counters.

a Write a number sentence. Use ❑ for the number of counters Khalid started with.

b Solve the number sentence.

c Copy and complete this sentence.

Khalid started with ____?____ counters.

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5 Saif visits Ferrari World in Abu Dhabi. He spends 200 AED on tickets. He has 350 AED left.

a Write a number sentence. Use ★ for the amount of money Saif took with him.

b Solve the number sentence.

c Copy and complete this sentence.

Saif took ____?____ AED with him.

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HANDS ONACtiVITY

6

You will need:

• multilink cubes

Start with 20 multilink cubes on your desk.

• Ask your partner to turn away. • Hide some cubes under your notebook. • Ask your partner to look again and count the cubes left on the table. • Help your partner to write and solve a number sentence to work out

how many cubes are under your notebook.• Check your partner’s answer by counting the cubes under your

notebook.

Example

10 football shirts and 1 goalkeeper’s shirt cost 1200 AED in total. The goalkeeper’s shirt costs 200 AED. How much does a football shirt cost?

Answer

Step 1 Write a number sentence, using ❑ for the cost of a football shirt.

10 × ❑ + 200 = 1200

Step 2 Solve the number sentence.

10 × ❑ must be 1000, so ❑ = 100.

Step 3 Give the answer in words.

One football shirt costs 100 AED.

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7 Mubarak has a tablet computer. He deletes 5 apps, then buys 8 more. Mubarak now has 14 apps. How many apps did he have at the beginning?

a Write a number sentence.

b Solve your number sentence.

c Write your answer in words.

8 Wafaa’s mother has some ribbon. She gives half of it to Wafaa. Wafaa buys 5 m more ribbon. Wafaa now has 9 m of ribbon altogether. How much ribbon did Wafaa’s mother have at the start?'

Hint Use ❑ for the amount of ribbon Wafaa’s mother gave her.

Half of this will be ❑ ÷ 2.

a Write a number sentence.

b Solve your number sentence.

c Write your answer in words.

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9 An airplane flies 163 m above the control tower at Abu Dhabi International Airport.

The airplane is also 300 m above sea level.

The airport is 27 m above sea level.

a Copy and label the diagram.

b How tall is the control tower at Abu Dhabi International Airport?

Write and solve a number sentence. Remember to give your final answer in words.

Hint Start your number sentence with 27 + ❑ + ____?____ .

airplane

controltower

sea level27 m

300 m

?

____ m

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10 Umm Salem gives the same amount of money to each of her 4 children. She then gives 30 AED to her nephew.

Altogether, she gave out 230 AED.

How much did she give each of her 4 children?

11 Hamdan has some oranges.

His brother eats half of them.

Then Hamdan’s mother gives him 3 more.

Now Hamdan has 10 oranges.

How many oranges did Hamdan have in the beginning?

Write and solve a number sentence.

Hint

Half of the oranges can be written as ❑ + 2

12 Ibrahim plays on his computer for the same number of hours each school day.

At the weekend he spends 4 hours on his computer altogether.

Ibrahim plays on his computer for 19 hours each week.

How long does Ibrahim spend on his computer on a school day?

Write a number sentence and solve it. Give your answer in words.

Hint

How many school days are in one week?

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13 When Salem goes to bed he can see some boats harbored in Dubai Creek.

When Salem wakes up, he sees most of the boats have gone. Only

one third are left. While he eats breakfast, he sees another 12 boats leave the creek. There are now only 2 boats left. How many boats were on the creek when Salem went to bed? Write and solve a number sentence. Give your answer in words.

Hint Use ❑ for the number of boats when Salem goes to bed. When

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HANDS ONACtiVITY

1.3 Writing Word ProblemsIn this section you will write word problems to match number sentences.

Get Ready

You will need:

• multilink cubes

Exercise 1.3

Warm Up

1 Work with a partner. You will need 24 multilink cubes.

Put them into groups of two:

Count the groups.

There are 12 groups of 2.

We can write: 12 × = 24 or 12 groups of 2 = 24.

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Repeat this for:

a groups of 3

b groups of 4

c groups of 6

d groups of 8

e groups of 12

2 Use your answers from Q1 to find ❑ in these number sentences.

Hint Write your answer as ❑ = …

a 12 × ❑ = 24

b 8 × ❑ = 24

c 4 × ❑ = 24

d 6 × ❑ = 24

e 3 × ❑ = 24

f 2 × ❑ = 24

Hint

8 ×

= 24

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3 Now start with 18 multilink cubes.

Count them, then put them into blocks of 2.

Copy and complete: ____?____ blocks of 2 = 18

Repeat this for:

b groups of 3

c groups of 6

d groups of 9

Use your answers to solve:

e 9 × ❑ = 18 f 6 × ❑ = 18

g 3 × ❑ = 18 h 2 × ❑ = 18

Hint Write your answers as ❑ = …

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20 - ❑ = 4

❑ + 4 = 20

❑ × 4 = 20

20 ÷ ❑ = 4

4 Match each word problem to a number sentence.

a Maryam has 20 books. She gives some to her son. She has 4 left. How many books did Maryam

give to her son?

b Eman has 20 books. She shares them equally among her children. Each child gets 4 books. How many children does Eman have?

c Rashid has some books. His father gives him 4 more. Now Rashid has 20 books. How many books did Rashid have at the start?

d Abdullah has some children. He gives each child 4 books. Altogether he gives out 20 books. How many children does Abdullah have?

Example

Use the number sentence 3 + ❑ = 5 to complete the word problem below.

Answer

I have ____?____ sweets. I have 3 sweets.

My friend gives me some more. My friend gives me some more.

I now have ____?____ sweets altogether. I now have 5 altogether.

How many did my friend give me? How many did my friend give me?

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5 Copy and complete the word problem so it matches the number sentence.

8 + ❑ = 17

Reem has ____?____ pencils.

She buys some more.

She now has ____?____ pencils altogether.

How many pencils did Reem buy?

6 Copy and complete the word problems so they both match the number sentence.

24 - ❑ = 14

a Salem has a box of ____?____ dates.

He eats some of the dates.

There are 14 dates left.

How many dates did Salem eat?

b One day on Earth is 24 hours.

A day on Jupiter is shorter than this.

The difference between the length of a day on Earth and on Jupiter is ____?____ hours.

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7 Use this number sentence: 3 × ❑ = 300

Copy this word problem into your notebook. Complete the question to match the number sentence.

The top speed of a Jaguar F-type V8 is 300 km/h.

This is three times the top speed of a bus in Abu Dhabi.

What is ____?____ ?

8 Here is a number sentence: ❑ + 15 = 27

a Copy and complete this word problem to match the number sentence.

Mansour reads 15 books in August.

He read ____?____ books altogether in July and August.

How many books did Mansour read in July?

b Copy and complete this word problem to match the same number sentence. Write your own question.

Abdulla’s brother is 15 years older than him.

Abdulla’s brother is ____?____ years old.

__________________?__________________ ?

c This is the start of another word problem that matches the same number sentence. Copy and complete the word problem.

Saif buys a pen. He also buys a magazine that costs 15 AED. ..

Word fact: The top speed is the fastest speed something can go.

Math fact: We can make many different word problems

for the same number sentence by using different real-life situations.

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Hint You could also use

another situation such as cars, people, money, days.

TALKABOUT IT

Example

Write a word problem for this number sentence: 3 + ❑ = 5

Step 1 Understand the number sentence.

We start with 3 of something.

We add or get some more.

Then there are 5 altogether.

Step 2 Choose a real-life situation.

E.g. A trip to the mall.

Step 3 Write the word problem.

I go to the mall.

I spend 3 hours shopping.

Then I go to a movie.

Altogether I spend 5 hours at the mall.

How long was the movie?

9 Defining problems Work with your partner. Write word problems that match these number sentences:

a ▼ + 5 = 12

b 17 = 20 - ❑

c 2 × ? = 14

d ▲ ÷ 2 = 15

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Example

Write a word problem to match this number sentence: 2 × ❑ - 1 = 17

Step 1 Understand the number sentence.

We start with 2 groups of the same number.

Then we take 1 away.

There are 17 left.

Step 2 Choose a real-life situation.

E.g. Food

Step 3 Write the word problem.

I have 2 equal bags of cucumbers.

I use 1 cucumber in my salad.

I have 17 cucumbers left altogether.

How many cucumbers were in each bag?

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10 Defining problems Write a word problem to match the number sentence.

5 + 2 × ▼ = 21

Hint You could use 5 bananas; 2 equal bags of apples,

and 21 pieces of fruit altogether.

11 Defining problems Write word problems to match the number sentences:

a ❑ ÷ 2 - 5 = 15

Hint For example – Faisal has some pens.

He gives half of them to his brother …

b 3 × ▼ - 1 = 17

c ❑ ÷ 3 + 4 = 9

d 100 + 6 × ▲ = 400

e ? ÷ 4 + 15 = 30

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1.4 Using Strip Diagrams and Balances

In this section you will learn about solving problems with strip diagrams, and using balancing.

Exercise 1.4

Warm Up

1 How many balance one ?

Hint Cross out pieces that are the same on each side of the balance.

2 How many balance one ?

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3 How many balance one ?

4 The top and bottom strips are the same length. How long is the green part?

5 A ribbon is 20 cm long.

It is cut into 4 equal parts. How long is each small part?

10 cm

14 cm

?

20 cm

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Hint Remember the strip diagram means ● + ● + ● = 24.

Main Exercise

Example

Saif and Ali play a game. These are the rules:

Player 1: Write a number sentence. Use a counter to cover a number.

Player 2: Work out the covered number.

Player 1: Saif begins. He writes a number sentence. He uses a counter to cover a number.

5 + ● = 16

Player 2: Ali draws a strip diagram. This helps him understand the number sentence.

5 ●16

At the top is 5 + ●. This is equal to 16. So the counter covers 11, because 5 + 11 = 16.

6 Ali writes a number sentence for Saif to solve: ● × 3 = 24.

a Copy and complete Saif’s strip diagram.

● ● ●?

b Write down the covered number.

Word fact : A strip diagram is a way to show a number sentence. The two strips are the same length because the two sides of the number sentence are equal

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7 Ali writes another number sentence: 5 × ● = 45

a Draw a strip diagram to help Ali.

b What is the covered number?

8 Saif and Ali draw these strip diagrams. Write the number sentences and work out the covered number for each strip diagram.

a 11

31

b

48

c

21

d35

50

9 Work with your partner. Play the same game as Saif and Ali. You will need a counter.

Player 1: Write a number sentence. Use a counter to cover a number.

Player 2: Draw a strip diagram. Work out the covered number.

HANDS ONACtiVITY

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10 Ali writes a number sentence. Then he covers the operation with a blue counter. He covers the number with a red counter.

6 ● ● = 30

Draw two possible strip diagrams.

For each strip diagram, find the covered number.

Example

Ali writes this number sentence for Saif to solve:

× 5 + 6 = 26

Saif draws this strip diagram:

6

26

Next Saif thinks about cutting the strips. He draws another diagram, partitioning the 26 into 20 + 6.

6

20 6

Then he draws this diagram:

20

He thinks about cutting the strips into 5 equal sections. Saif says = 4

Check: 4 × 5 + 6 = 26. Saif’s answer is correct.

THINKABOUT IT

Word fact: +, -, × and ÷ are all operations.

This shows + + + + + 6 on the left side of the number sentence.

This shows 26

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11 Here is a number sentence: 4 × ▼ + 10 = 38.

a What number should be in this pink box?

▼ ▼ ▼ ▼ 10

?

b What number should be in this pink box?

▼ ▼ ▼ ▼ 10

? 10

c What is the number ▼?

12 Copy and complete the strip diagrams for each number sentence.

a ❑ × 2 + 1 = 25

Step 1

❑ ❑ 1

?

Step 2

❑ ❑ 1

? 1

Step 3

❑ = ____?____

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b ▼ × 5 + 9 = 44

Step 1

▼ ▼ ▼ ▼ ▼ 9

?

Step 2

▼ ▼ ▼ ▼ ▼ 9

? 9

Step 3

▼ = ?

c 3 × ❍ + 20 = 53

Step 1

❍ ❍ ❍ ?

?

Step 2

❍ ❍ ❍ ?

? ?

Step 3

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13 Match each number sentence i-iii to a strip diagram a-c.

i 3 × ▼ + 6 = 30 ii ▼ × 6 + 6 = 30

iii 3 × 6 + ▼ = 30

a▼ ▼ ▼ ▼ ▼ ▼ 6

30

b6 6 6 ▼

30

c▼ ▼ ▼ 6

30

14 Generating solutions Write and then solve three different number sentences for this strip diagram.

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ?

?

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15 Noura and Jameela look at this set of scales. The scales are balanced.

Noura says, ‘ I think one box weighs 20 g’.

Jameela says, ‘I think one box weighs 200 g’.

Who is correct? Explain.

16 Rashid looks at this set of scales. The scales are balanced.

Rashid says, ‘I can solve this the same way as I did with strip diagrams. I just need to take a cube off each side because the scales will still balance. Then I know that 4 cubes are the same as 2 cylinders...’.

a Discuss Rashid’s strategy. Can you see how it works?

b If a cylinder weighs 60 g. How heavy is a cube? Explain how you worked it out.

Word fact: When scales are balanced, the weight on each side is equal.

600 g

TALKABOUT IT

TALKABOUT IT

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17 For each diagram, work out the weight of one box.

a

Hint Use Rashid’s strategy from Q16.

b

18 a How many spheres balance one cylinder?

Word fact: Cylinders and spheres are 3D solids. cylinder

sphere

2 kg

300 g

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Math fact: Each coat hanger must balance. We can draw simple diagrams to show the balances, and solve problems.

b How many triangles balance one cube?

19 Work in groups.

You will need: • Coat hangers • Clothes pegs • plastic bags • multilink cubes

Use two coat hangers, some pegs and plastic bags to build a coat hanger balance, as shown in the picture. Ask your teacher where you may hang your balance.

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Example

In this balance = 2. Find the value of and .

First look at the smaller balance on the right side.

Take a off each side.

We know each is 2.

So must be 4.

Put the numbers on the smaller balance.

4 + 2 = 6 2 + 2 + 2 = 6

The total for the smaller balance on the right is 6 + 6 = 12.

4 2

2 22N

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Now look at the whole balance.

We have + on the left.

We have 12 on the smaller balance on the right.

So + balances 12.

So must be 10.

20 In this balance = 3.

a Look at the smaller balance on the right side. What is ?

Hint

You can take a and a off each side and it will still balance.

b What is the total of the smaller balance on the right?

c Now look at the whole balance and find the value of .Not f

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21 This balance works the same way.

In this balance = 6. Find the value of and .

Hint

Remember to look at the smaller balance on the left side first.

22 = 4.

a Find the value of and .

b Explaining Explain how you solved the problem.

Hint First look at the smaller balance on the left.

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Summary

Number sentences use a symbol like ?, ❑, ♦ or ★ for a number we don’t know. An example of a number sentence is:

3 + ❑ = 5

Solve means ‘find the number we don’t know’.

Number sentences can match word problems. For example:

To solve word problems, there are three things to remember:

1 Write a number sentence. Use a shape like ❑, ♦, ★ … for the number you don’t know.

2 Write the answer as ❑ = … (use your shape.)

3 Write the answer in words. Don’t only write a number.

Strip diagrams also match number sentences. For example:

▼ ▼ ▼ 4

10

3 × ▼ + 4 = 10

Khalid has some counters.

Hassan gives him 15 more counters.

Khalid now has 34 counters.

How many counters did Khalid start with?

♦ is the number of counters Khalid started with.

♦ + 15 = 34

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We can solve this strip diagram by cutting off 4 from each strip.

▼ ▼ ▼ 4

6 4

▼ ▼ ▼

6

Then cut again:

▼ ▼ ▼

2 2 2

So ▼ = 2.

This is an example of a balancing problem:

The objects on each side of each balance are equal.

On the smaller balance on the left, and equals and . This means equals .

On the right side of the whole balance, equals . This means equals and together.

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Review

1 Solve these number sentences:

a 12 + ★ = 27

b ▼ × 3 = 600

c ❑ _ 31 = 11

d ▼ ÷ 4 = 16

2 Solve these number sentences:

a 3 × ★ _ 4 = 14

b ❑ _ 12 = 19 + 12

c 10 _ 4 = ❑ ÷ 3

d ▼ × 2 = 18 ÷ 3

3 Copy and complete these number sentences. Make sure the answer is a whole number:

a ▼ + 6 = ____?____

b ❑ _ ____?____ = 42

c ★ ÷ 3 = ____?____

d 4 × ▼ + ____?____ = 10Not f

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4 Laila gives her sister 5 pencils.

Laila now has 18 pencils.

How many pencils did Laila have at the beginning ?

Write and solve a number sentence.

5 Write a word problem to match each number sentence.

a ▼ _ 30 = 31

b ★ ÷ 8 = 50

c 2 × ▼ = 16

d 19 + ❑ = 37

Hint Try using pens, pencils, pizza slices, sweets or oranges. Use the

names of your family or friends.

6 Draw strip diagrams for these number sentences.

Find the missing numbers.

a 11 + ❑ = 19 b ❑ × 5 = 45

7 Draw strip diagrams for these number sentences. Solve the number sentences:

a 3 × ★ + 4 = 19 b ▼ × 4 + 7 = 23 Not f

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8 How many balance one ?

9 = 6. Find and . Check your answer.

I can statements

What can you do?

• I can solve number sentences.

• I can write a number sentence where the answer is a whole number.

• I can write a number sentence and use it to solve a word problem.

• I can match word problems to number sentences.

• I can make a word problem that matches a number sentence.

• I can use strip diagrams to help solve number sentences.

• I can solve balancing problems.

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