1. Number Sentences

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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 Puzzles
In 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 ____?____.
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Khalid is playing a game with Faisal.
I have a secret number.
Take away 3 from my secret number
and you get 12.
Is your number 15?
Khalid
Yes!
Faisal
2 Use the numbers in the cloud.
Answer each secret number puzzle.
a ‘Add 2 to my secret number and you get 8.’
13 24
25 6 17
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.’
5
Main Exercise
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3 Faisal says, ‘Add 7 to my secret number and you get 35’.
Find Faisal’s secret number.
Hint
Use a number line.
–5
–?
?
30
35
4aJasem 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 mea
ns
‘find the se
cret
number’.
Multiply my secret
number by 3 and you
get 15.
Salma
Divide my secret number
by 2 and you get 21.
Wafaa
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6 Khalid and Faisal play the secret number game again.
Multiply my secret number by 2.
Then subtract 4. You get 24.
Khalid
Divide my secret number
by 3. Then add 4. You get
10.
Faisal
a
Solve Khalid’s secret number puzzle.
b
Solve Faisal’s secret number puzzle.
7Play 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.
TALTKIT
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ABO
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:
K
TAOL
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❑ + 20 = 84
a What do you think this means?
b How can Rashid find Hamdan’s number?
7
9 Solve these number sentences.
b
? × 2 = 40
c
❑ - 12 = 15
d 17 + ▼ = 27
e
8=❑÷5
f
15 - ? = 4
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a
❑ + 3 = 12
10Solve 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 = ____?____
Hint
b
____?____ × ★ = 12
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.
c
20 - ★ = ____?____
d
★ - ____?____ = 6
e
43 + ★ = ____?____
f
★ × 2 = ____?____
8
12Defining problems Write three of your own number sentences
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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.
9
13Organizing information Ahmed starts to write this number
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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
Fill tool
b
Which numbers can Ahmed use to complete his number
sentence? Write the two number sentences you have found with
whole number answers.
int
H
The numbers in Row 2 give the numbers he
can use to complete his sentence.
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1.2 Writing Number Sentences
for Word Problems
Exercise 1.2
Warm Up
1Making 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.
Math fact
s
:
10 is more
than 3.
7 is less th
an 12.
If you doub
le 4 you ge
t 8.
If you halv
e 8 you
get 4
The differe .
nce betwe
en
13 and 9 is
4.
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
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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 2Solve the number sentence.
❑ = 37
Step 3Write 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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5Saif visits Ferrari World in Abu Dhabi. He spends 200 AED on tickets.
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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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6
You will need:
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DS O HAANCtiVITY
• 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 1Write a number sentence, using ❑ for the cost
of a football shirt.
10 × ❑ + 200 = 1200
Step 2Solve the number sentence.
10 × ❑ must be 1000, so ❑ = 100.
Step 3Give 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.
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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
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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.
airplane
____ m
300 m
control
tower
?
27 m
sea level
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 + ❑ + ____?____.
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10Umm Salem gives the same amount of money to each of her 4
How much did she give each of her 4 children?
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children. She then gives 30 AED to her nephew.
Altogether, she gave out 230 AED.
11Hamdan 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
12Ibrahim 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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13When Salem goes to bed he can see some boats harbored in Dubai
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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
Salem wakes up there are ❑ ÷ 3 boats left.
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1.3 Writing Word Problems
In 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.
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DS O Put them into groups of two:
HAANCtiVITY
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:
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a groups of 3
b groups of 4
c groups of 6
Hint
8×
= 24
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 ❑ = …
a12 × ❑ = 24
b 8 × ❑ = 24
c
4 × ❑ = 24
d 6 × ❑ = 24
e
3 × ❑ = 24
f
2 × ❑ = 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:
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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 × ❑ = 18h
2 × ❑ = 18
Hint
Write your answers as ❑ = …
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4 Match each word problem to a number sentence.
20 - ❑ = 4
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a Maryam has 20 books.
She gives some to her son.
She has 4 left.
How many books did Maryam
give to her son?
❑ + 4 = 20
❑ × 4 = 20
20 ÷ ❑ = 4
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
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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.
How long is a day on Jupiter?
24
Word fact
7 Use this number sentence: 3 × ❑ = 300
:
The top sp
eed
is the faste
st
speed som
ething
can go.
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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.
Math
f
act:
We ca
n
m
ake
differe
nt wor many
d pr
for the
same oblems
num
senten
ce by u ber
si
differe
nt real ng
l
ife
situati
ons.
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. ..
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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.
Hint
You could also use
another situation such
as cars, people,
money, days.
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
TALKIT
UT
ABO
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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10Defining problems Write a word problem to match the number
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sentence.
5 + 2 × ▼ = 21
Hint
You could use 5 bananas; 2 equal bags of apples,
and 21 pieces of fruit altogether.
11Defining 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
?
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balance one
4The top and bottom strips are the same length. How long is the
green part?
10 cm
?
14 cm
5 A ribbon is 20 cm long.
20 cm
It is cut into 4 equal parts. How long is each small part?
30
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Main Exercise
Example
Saif and Ali play a game. These are the rules:
Player 1: W
rite 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
Word fact
●
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.
Hint
Remember the strip diagram means ● + ● + ● = 24.
31
:
A strip dia
gram is a
way to sho
w a numbe
r
sentence.
The two st
r
ips
are the sam
e length
because th
e
two sides o
f the
number se
ntence are
equal
7 Ali writes another number sentence: 5 × ● = 45
a Draw a strip diagram to help Ali.
b What is the covered number?
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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
d
35
50
9 Work with your partner. Play the same game as Saif and Ali. You will
N need a counter.
DS O
HAANCtiVITY
Player 1: Write a number sentence. Use a counter to cover a
number.
Player 2: Draw a strip diagram. Work out the covered number.
32
10Ali writes a number sentence. Then he covers the operation with a
blue counter. He covers the number with a red counter.
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K
THIN IT
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Word fact
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:
This shows
+
+
+
+
+ 6 on the left side of the number sentence.
6
26
This shows 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.
33
:
+, -, × and
÷ are
all operatio
ns.
11Here is a number sentence: 4 × ▼ + 10 = 38.
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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 ▼? 12Copy and complete the strip diagrams for each number sentence.
a ❑ × 2 + 1 = 25
Step 1
❑
1
❑
?
Step 2
❑
❑
?
1
1
Step 3
❑ = ____?____
34
b ▼ × 5 + 9 = 44
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Step 1
▼
▼
▼
▼
▼
9
▼
▼
9
?
Step 2
▼
▼
▼
?
9
Step 3
▼=?
c
3 × ❍ + 20 = 53
Step 1
❍
❍
?
❍
?
Step 2
❍
❍
?
❍
?
?
Step 3
❍ = ?
35
13Match each number sentence i-iii to a strip diagram a-c.
i 3 × ▼ + 6 = 30 ii ▼ × 6 + 6 = 30 iii
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3 × 6 + ▼ = 30
a
▼
▼
▼
▼
▼
6
▼
30
b
6
6
6
▼
▼
6
30
c
▼
▼
30
14Generating solutions Write and then solve three different number
sentences for this strip diagram.
▼
▼
▼
▼
▼
▼
?
36
▼
▼
▼
?
15Noura and Jameela look at this set of scales.
The scales are balanced.
Word fact
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TALTKIT
:
U
When scale
s are
balanced,
the
weight on
each side
is equal.
ABO
600 g
Noura says, ‘ I think one box weighs 20 g’.
Jameela says, ‘I think one box weighs 200 g’.
Who is correct? Explain.
16Rashid looks at this set of scales. The scales are balanced.
K
TAOL
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AB
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.
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17For each diagram, work out the weight of one box.
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a
2 kg
Hint
Use Rashid’s strategy from Q16.
b
300 g
18a How many spheres balance one cylinder?
Word fact
:
Cylinders
and
spheres a
re 3D solid
s.
cylinder
sphere
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b
How many triangles
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balance one cube?
19Work in groups.
You will need:
• Coat hangers
• Clothes pegs
•
• multilink cubes
plastic bags
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.
Math fact
:
Each coat
hanger mu
st
balance. W
e can draw
simple
diagrams t
o show the
balances, a
nd solve
problems.
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Example
= 2. Find the value of
In this balance
and
.
First look at the smaller balance on the right side.
Take a
off each side.
We know each
So
is 2.
must be 4.
Put the numbers on the smaller balance.
4
2
4+2=6
2
2
2
2+2+2=6
The total for the smaller balance on the right is 6 + 6 = 12.
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Now look at the whole balance.
+
on the left.
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We have
We have 12 on the smaller balance on the right.
So
+
So
must be 10.
balances 12.
20In this balance
= 3.
a
Look at the smaller balance on the right side. What is
?
Hint
You can take a
balance.
and a
off each side and it will still
b What is the total of the smaller balance on the right?
c
Now look at the whole balance and find the value of
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.
21This balance works the same way.
In this balance
= 6. Find the value of
and
.
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Hint
Remember to look at the smaller balance on the left side first.
22
= 4.
Hint
First look at the
smaller balance on
the left.
a
Find the value of
and
.
bExplaining Explain how you solved the problem.
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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:
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
To solve word problems, there are three things to remember:
1Write a number sentence. Use a shape like ❑, ♦, ★ … for the
number you don’t know.
2Write the answer as
❑ = … (use your shape.)
3Write the answer in words. Don’t only write a number.
Strip diagrams also match number sentences. For example:
▼
▼
4
▼
10
3 × ▼ + 4 = 10
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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,
equals .
means
and
On the right side of the whole balance,
equals
and
together.
This means
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equals
equals
and
. This
.
Review
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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 × ▼ + ____?____ = 10
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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.
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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 D
raw 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
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balance one
?
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8 How many
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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