🦘 Kangaroo Grade 3 · session 2 of 24 · 60 minutes · free

Balance scales and missing numbers

A balanced scale is an equation you can see. Learn to take the same away from both sides, swap one thing for another and work backwards, and many Kangaroo puzzles become two-line answers.

🎯 Practice mode 🖨 Print a worksheet

A complete lesson · 60 minutes

Teach it, see it, practise it, all on this page.

Open this page with your learner and work down it, or press Teach this lesson to go through it one step at a time on a big screen. Every section has something to do.

See the teacher's plan

Warm-up4 min
Missing numbers8 min
Balance scales9 min
Swapping7 min
Working backwards7 min
Activity 17 min
Activity 27 min
Activity 36 min
Quiz4 min
Set the project1 min

By the end you can

Find missing numbers with the opposite operation
Solve balance-scale puzzles by taking the same from both sides
Swap things that weigh the same
Work backwards through a number machine

Before you start

Times tables to 10
Adding and subtracting to 100

Have ready

Paper and pencil
Coins or counters to act out the scales

Grade 3 · ages 8 to 9 · contest level 3–4

Warm-up · think first

A cat weighs the same as 3 rabbits. A rabbit weighs the same as 2 hamsters. How many hamsters weigh the same as the cat?

Show the answer

6 hamsters. Swap each of the 3 rabbits for 2 hamsters: 3 × 2 = 6.

Teacher tip: Draw the cat on one side and the rabbits on the other, then swap one rabbit at a time.

More warm-ups for every class →

Part 1 of 4 · learn, see, practise

Missing numbers

To find a missing number, do the opposite. Adding is undone by subtracting, and multiplying is undone by dividing.

☐ + 27 = 63 means 63 − 27 = ☐. Always check by putting your answer back in the box.

Example 1

☐ + 27 = 63

Undo the + 27: 63 − 27 = 36.
Check: 36 + 27 = 63.

Answer: 36

Example 2

45 − ☐ = 18

45 take away something leaves 18, so the something is 45 − 18.
45 − 18 = 27. Check: 45 − 27 = 18.

Answer: 27

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up1☐ + 15 = 40

Warm-up2☐ − 8 = 17

Core36 × ☐ = 42

Core4☐ ÷ 4 = 9

Stretch 🔥5☐ + ☐ + 7 = 31 (the same number in both boxes)

Stretch 🔥63 × ☐ − 5 = 25

Answers (for the teacher)

Part 2 of 4 · learn, see, practise

Balance scales

When a scale balances, both sides weigh the same. You can take the same weight off both sides and it still balances.

Two apples and 3 kg balancing 11 kg means two apples weigh 8 kg, so one apple weighs 4 kg.

Example 1

Two equal apples and a 3 kg weight balance 11 kg. How heavy is one apple?

Take 3 kg off both sides: two apples balance 8 kg.
One apple is half of that: 4 kg.

Answer: 4 kg

Example 2

An apple and a pear together balance 10 kg, and the apple is 2 kg heavier than the pear. How heavy is each?

Take the extra 2 kg away: two pears' worth is 8 kg, so the pear is 4 kg.
The apple is 4 + 2 = 6 kg. Check: 6 + 4 = 10.

Answer: apple 6 kg, pear 4 kg

Balance puzzles

Give each fruit a weight with the − and + buttons. A puzzle is solved when every scale balances.

Solve puzzle 1.
Solve puzzle 2.
Solve puzzle 3 (the hardest).

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up1A box and a 3 kg weight balance 10 kg. How many kg is the box?

Warm-up2Two equal bags balance 18 kg. How many kg is one bag?

Core33 equal balls and a 2 kg weight balance 14 kg. How many kg is one ball?

Core4A melon balances 2 apples and a 1 kg weight. An apple weighs 2 kg. How many kg is the melon?

Stretch 🔥5A cube and a ball weigh 10 g together. Two cubes and a ball weigh 16 g. How many grams is the ball?

Stretch 🔥63 pears weigh the same as 2 apples. An apple and a pear weigh 15 g together. How many grams is an apple?

Answers (for the teacher)

Part 3 of 4 · learn, see, practise

Swapping things that weigh the same

If two things balance, you can swap one for the other anywhere. This turns a hard puzzle into one with a single kind of thing.

Swap step by step and write each new balance down.

Example 1

1 dog weighs as much as 2 cats, and 1 cat as much as 3 mice. How many mice weigh as much as 2 dogs?

2 dogs = 4 cats.
4 cats = 4 × 3 = 12 mice.

Answer: 12

Example 2

🔺 + 🔺 = ⭐ and ⭐ + 🔺 = 18. What is 🔺?

Swap ⭐ for 🔺 + 🔺 in the second line: 🔺 + 🔺 + 🔺 = 18.
So 🔺 = 18 ÷ 3 = 6.

Answer: 6

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up11 big box weighs as much as 4 small boxes. How many small boxes balance 3 big boxes?

Warm-up2🔵 + 🔵 = 10. What is 🔵?

Core3🔺 + 🔺 + 🔺 = 🟦 and 🟦 + 🔺 = 24. What is 🔺?

Core41 apple weighs as much as 2 plums, and 1 plum as much as 3 cherries. How many cherries weigh as much as 2 apples?

Stretch 🔥5🔺 + 🟦 = 13, 🟦 + 🟢 = 15 and 🔺 + 🟢 = 10. What is 🟦?

Stretch 🔥6A bottle and a glass hold as much as a jug. A bottle holds as much as a glass and a cup. Two jugs hold as much as three cups. How many glasses fill the bottle?

Answers (for the teacher)

Part 4 of 4 · learn, see, practise

Working backwards

When you know the end of a story, undo each step in reverse order: the last thing done is undone first.

"Add 7, then double, get 30" is undone by "halve 30, then take away 7".

Example 1

I think of a number, add 7 and double it. I get 30. What was my number?

Undo the doubling: 30 ÷ 2 = 15.
Undo the + 7: 15 − 7 = 8. Check: (8 + 7) × 2 = 30.

Answer: 8

Example 2

A machine multiplies by 3 and then takes away 4. 23 comes out. What went in?

Undo the − 4: 23 + 4 = 27.
Undo the × 3: 27 ÷ 3 = 9.

Answer: 9

Practise it

Start with the warm-up and go as far as you can. The stretch questions are meant to be hard.

Warm-up1I add 5 to a number and get 20. What was the number?

Warm-up2I double a number and get 18. What was the number?

Core3I take away 6, then multiply by 4, and get 20. What was the number?

Core4I halve a number, then add 9, and get 15. What was the number?

Stretch 🔥5I multiply a number by 3, add 4, then halve it and get 11. What was the number?

Stretch 🔥6Kanga had some sweets. She gave half of them to Roo, then 3 to Joey, and had 5 left. How many did she have at the start?

Answers (for the teacher)

Watch out

Common mistakes

Undoing in the wrong order

Undo the last step first. For "add 7 then double", halve first, then subtract 7.

Taking weight off one side only

Whatever you take off one side you must take off the other, or the scale stops balancing.

Not checking

Put the answer back into the puzzle. It takes ten seconds and catches most mistakes.

Class activity 1 of 3

Fruit puzzle chain

Each answer helps with the next one. 🍌 + 🍌 + 🍌 = 12, then 🍓 + 🍓 = 🍌 + 6, then 🍒 + 🍌 + 🍓 = 20, then 🥝 + 🥝 = 🍒 + 3.

Find 🍌 from the first line.
Put it into the second line to find 🍓.
Keep going down the chain.
Check every line at the end.

1🍌

2🍓

3🍒

4🥝

Answers (for the teacher)

Class activity 2 of 3

Animal see-saw

A dog and a cat weigh 14 kg together. The dog and a rabbit weigh 11 kg. The cat and the rabbit weigh 5 kg.

Add all three facts: each animal is counted twice.
Halve to get the weight of all three animals.
Take away each pair to find the third animal.

1All three animals together (kg)

2The dog (kg)

3The cat (kg)

4The rabbit (kg)

Answers (for the teacher)

Class activity 3 of 3

Number machine factory

Machine A doubles a number and then adds 3. Machine B adds 4 and then multiplies by 3.

Run the machines forwards for the first and third questions.
Run them backwards for the second and fourth.
For the last one, try small numbers until the input and output match.

15 goes into machine A. What comes out?

221 comes out of machine A. What went in?

32 goes into machine B. What comes out?

430 comes out of machine B. What went in?

5A machine doubles and then takes away 5. Which number comes out the same as it went in?

Answers (for the teacher)

After class · project

Build a balance puzzle

Invent three balance puzzles with pictures, each with exactly one answer, and test them on someone at home.

Choose secret weights for two or three objects.
Draw two or three scales that balance with those weights.
Check that only your weights make every scale balance.
Give the puzzle to someone and watch how they solve it.

What to make: Three drawn puzzles with an answer page.

Stretch: Make a puzzle with three objects and three scales, like the 🔺 🟦 🟢 question.

How it is marked

Look for	Done well when
Balanced	Every scale balances with the secret weights
One answer	No other weights work
Clear	Someone else can read and solve it

Project template

Copy it onto paper or print the practice sheet.

Objects and secret weights: ______ = ___   ______ = ___   ______ = ___
Scale 1: left ____________ | right ____________
Scale 2: left ____________ | right ____________
Scale 3: left ____________ | right ____________
Who solved it, and how: ____________________________

Homework

Practise at home

Try each on paper, then tap to see the answer and working. Or print the practice sheet.

☐ × 7 = 56
8
Two equal parcels and a 4 kg weight balance 20 kg. How heavy is one parcel?
8 kg
1 orange weighs as much as 3 plums. How many plums balance 4 oranges?
12
I multiply a number by 5 and take away 3. I get 32. What was the number?
7 (32 + 3 = 35, 35 ÷ 5 = 7)
A pen and a pencil cost ₹30. The pen costs ₹20 more than the pencil. How much is the pencil?
₹5 (not ₹10: then the pen would be ₹30 and the total ₹40)

🔥 Challenge round · finished early?

The hardest questions of the lesson, no notes

Cover the notes and try these from memory. They are meant to make you think hard: draw, try small cases, and check your answer.

Stretch 🔥1☐ + ☐ + 7 = 31 (the same number in both boxes)

Stretch 🔥23 × ☐ − 5 = 25

Stretch 🔥3A cube and a ball weigh 10 g together. Two cubes and a ball weigh 16 g. How many grams is the ball?

Stretch 🔥43 pears weigh the same as 2 apples. An apple and a pear weigh 15 g together. How many grams is an apple?

Stretch 🔥5🔺 + 🟦 = 13, 🟦 + 🟢 = 15 and 🔺 + 🟢 = 10. What is 🟦?

Stretch 🔥6A bottle and a glass hold as much as a jug. A bottle holds as much as a glass and a cup. Two jugs hold as much as three cups. How many glasses fill the bottle?

Stretch 🔥7I multiply a number by 3, add 4, then halve it and get 11. What was the number?

Stretch 🔥8Kanga had some sweets. She gave half of them to Roo, then 3 to Joey, and had 5 left. How many did she have at the start?

Answers (for the teacher)

🧠 Think-hard question

🔺 + 🟦 + 🟢 = 20, 🔺 + 🔺 = 🟦 and 🟦 + 🟦 = 🟢. Find all three.

Talk it through or use the drawing board. There is more than one way to get there.

🎯 Practice mode: one question at a time 🧩 Brain warm-ups 🎮 Maths and logic games

Quiz · 8 questions

Final quiz: check what you learned

Pick an answer to see if it is right and why. Your best score is saved on this device.

1☐ + 38 = 75. What is ☐?

Show the answer

37. 75 − 38 = 37.

24 equal weights balance 20 kg. How heavy is each weight?

Show the answer

5 kg. 20 ÷ 4 = 5.

3🔺 + 🔺 + 🔺 = 21. What is 🔺 + 10?

Show the answer

17. 🔺 = 7, so 🔺 + 10 = 17.

41 bear weighs as much as 3 dogs and 1 dog as much as 2 cats. How many cats balance 2 bears?

Show the answer

12. 2 bears = 6 dogs = 12 cats.

5I add 9 to a number and halve the result. I get 11. What was the number?

Show the answer

13. 11 × 2 = 22, and 22 − 9 = 13.

6A ball and a cube weigh 12 g together. The ball is 2 g heavier than the cube. How heavy is the cube?

Show the answer

5 g. Without the extra 2 g, two cubes weigh 10 g, so a cube is 5 g.

7🟦 + 🔺 = 9 and 🟦 + 🟦 + 🔺 = 14. What is 🔺?

Show the answer

4. The second line has one more 🟦 and is 5 more, so 🟦 = 5 and 🔺 = 4.

8Three cats and a dog weigh 22 kg. Two cats and the dog weigh 18 kg. How heavy is the dog?

Show the answer

10 kg. The difference is one cat: 4 kg. Then the dog is 18 − 8 = 10 kg.

For parents and teachers

The 60-minute lesson plan

Use it as it is, or stretch any part over two sessions. Each row links to the part of the page you need.

Time	Part	The teacher	The learner
0–4 min	Warm-up	Act the warm-up out with coins before answering.	Guess, then explain.
4–12 min	Missing numbers	Explain, reveal the examples one step at a time, then the practice.	Practise up to the stretch question.
12–21 min	Balance scales	Explain, use the interactive part, then the practice.	Practise; try the stretch.
21–28 min	Swapping	Explain and show the examples.	Practise.
28–35 min	Working backwards	Explain and show the examples.	Practise; the stretch can be homework.
35–42 min	Activity 1	Let the learner work alone; hint only after a wrong check.	Solve and check.
42–49 min	Activity 2	Do the first item together, then let them finish.	Finish the activity.
49–55 min	Activity 3	Set as homework if time is short.	Solve the puzzles.
55–59 min	Quiz	Run the quiz like the contest: no calculator, pick one of five answers.	Answer and read why.
59–60 min	Set the project	Show the template and agree a due date.	Start the project.

Questions to ask

What can you do to both sides of a scale and keep it balanced?
Why do we undo the last step first?
How does swapping help?

If it feels hard

Act out every scale with coins or counters and remove the same from both sides with your hands.

If it feels easy

🔺 + 🟦 + 🟢 = 20, 🔺 + 🔺 = 🟦 and 🟦 + 🟦 = 🟢. Find all three.

FAQ