Grade by Grade
C. Algebra · Patterns, Variables, Coding and Modelling

Balancing Act: Making Two Sides Equal

C2.2About 40 minutes
Learning goals
  • I can find the value of each side to see if two expressions are equal.
  • I can figure out how much to add to one side to make the two sides equal.
  • I can figure out how much to subtract from one side to make the two sides equal.
  • I can use a balance, a number line or counters to show my thinking.

Words to know

expression
Numbers joined by + or −, with no equal sign, like 8 + 4 or 15 − 6.
equivalent
Having the same value. 8 + 4 and 10 + 2 are equivalent, because both are 12.
equal sign (=)
Means is the same amount as.
not equal sign (≠)
Means is not the same amount as.
balance
A scale that is level when both sides are the same amount.

Remember from Grade 1

The equal sign means is the same amount as. A balance is level when both sides are equal, and it tips when they are not.

6 + 1 = 3 + 4, because both sides are 7.

This year you will be a balance fixer. When two sides are not equal, you will figure out exactly how much to add or take away to make them equal.

What is an expression?

An expression is a group of numbers joined by + or −. It has no equal sign.

  • 8 + 4 is an expression. Its value is 12.
  • 15 − 6 is an expression. Its value is 9.

Two expressions are equivalent when they have the same value.

8 + 410 + 5
8 + 4 is 12. 10 + 5 is 15. The right side is heavier, so the balance tips to the right. They are not equivalent: 8 + 4 ≠ 10 + 5.
Worked example

How much to add?

8 + 4 and 10 + 5 are not equal. What could you add to the left side to make them equal?

  1. Find the value of each side. Left: 8 + 4 = 12. Right: 10 + 5 = 15.
  2. Which side is smaller? The left side (12).
  3. How far is it from 12 to 15? Count on: 13, 14, 15. That is 3.
  4. Add 3 to the left side: 8 + 4 + 3 = 15.
  5. Check: 15 and 15. Now they balance!

Add 3 to the left side: 8 + 4 + 3 = 10 + 5.

10111213141516+3
From 12 to 15 is a jump of 3.
Worked example

How much to subtract?

Use the same two sides: 8 + 4 and 10 + 5. This time, what could you subtract from the right side to make them equal?

  1. Left is 12. Right is 15.
  2. The right side is bigger by 3.
  3. Take 3 away from the right side: 10 + 5 − 3 = 12.
  4. Check: 12 and 12. Balanced!

Subtract 3 from the right side: 8 + 4 = 10 + 5 − 3.

Two ways to fix it

If one side is smaller, you can add the difference to the smaller side, or you can subtract the difference from the bigger side. Both work!

  1. Find the value of each side.
  2. Find the difference (how far apart they are).
  3. Add it to the smaller side or take it away from the bigger side.
13 − 54 + 6
13 − 5 is 8. 4 + 6 is 10. The right side is 2 more.
Worked example

With subtraction expressions

Make 13 − 5 and 4 + 6 equal. Give both ways.

  1. Left: 13 − 5 = 8. Right: 4 + 6 = 10.
  2. The difference: from 8 to 10 is 2.
  3. Way 1: add 2 to the smaller (left) side: 13 − 5 + 2 = 10. ✔
  4. Way 2: subtract 2 from the bigger (right) side: 4 + 6 − 2 = 8. ✔

13 − 5 + 2 = 4 + 6 or 13 − 5 = 4 + 6 − 2.

🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪🍪
Top row: 8 cookies on one plate. Bottom row: 10 cookies on the other plate. Move or add cookies to make the plates fair: give the first plate 2 more, or take 2 off the second plate.
Worked example

Bigger numbers

What must be added to 30 + 20 to make it equal to 40 + 25?

  1. Left: 30 + 20 = 50. Right: 40 + 25 = 65.
  2. From 50 to 65: 50 + 10 = 60, then 60 + 5 = 65. That is 15.
  3. Add 15 to the left: 30 + 20 + 15 = 65. ✔

Add 15: 30 + 20 + 15 = 40 + 25.

Just for fun

Why is a balance scale so good at keeping secrets?

It never tips anyone off! ⚖️

Your turn

1

Are 5 + 4 and 3 + 6 equal?

2

What do you add to the left side to make them equal? 6 + 2 ☐ 5 + 5

3

What do you subtract from the right side to make them equal? 7 + 3 ☐ 9 + 4

4

Make them equal by adding to one side: 12 − 2 and 6 + 6

5

Which side is bigger, and by how much? 20 + 20 and 30 + 15

6

Fill in the box to make it true: 9 + 1 + ☐ = 8 + 7

Hands-on activity

Hanger balance

You need: a clothes hanger, 2 cups or small bags, string or tape, and lots of the same small object (pennies, washers or Lego bricks that are the same size).

  1. Hang a cup on each end of the hanger. Hang the hanger from a doorknob.
  2. Put 7 pennies in one cup and 4 in the other. Which side goes down?
  3. Ask: How many pennies do I need to add to the light side to make it level? Predict, then test.
  4. Reset to 7 and 4. Now ask: How many do I take away from the heavy side? Predict, then test.
  5. Write a number sentence for each, like 4 + 3 = 7 and 7 − 3 = 4.

What to look for: your child finds the difference (3) before testing.

Math talk

Say: The left side is 12. The right side is 15. The right side is 3 more. So I can add 3 to the left or take 3 from the right.

More practice: warm-up

1

Equal or not equal? 4 + 4 and 6 + 2

2

What do you add to 5 to make it equal to 9?

3

What do you subtract from 10 to make it equal to 7?

4

Which side is bigger? 3 + 3 or 2 + 5

5

What does equivalent mean?

More practice: on your own

1

What do you add to the left to make them equal? 15 − 5 ☐ 8 + 6

2

What do you subtract from the left to make them equal? 9 + 9 ☐ 10 + 5

3

Give two ways to make 7 + 5 and 6 + 9 equal.

4

Fill in the box: 50 + 10 = 40 + 10 + ☐

5

Noor has 6 + 7 marbles in two bags. Ravi has 20 − 4 marbles. How many more marbles does Noor need to have the same as Ravi?

Challenge

1

Fill in the box: 25 + 25 − ☐ = 30 + 10

2

Jay says: 8 + 6 and 10 + 7 are 3 apart, so I add 3 to the right side. Is Jay right? Explain.

3

A team has 34 + 20 points. The other team has 60 + 9 points. How many points does the first team need to tie?

Check your understanding

Quiz

1.

What is an expression?

2.

What do you add to the left side to make them equal? 5 + 3 ☐ 6 + 6

3.

What do you subtract from the right side? 10 + 1 ☐ 9 + 7

4.

Which pair is equivalent?

5.

A balance has 9 + 2 on one side and 15 on the other. Which side goes down?

6.

Fill in the box: 20 + 5 + ☐ = 30

For parents and teachers

This lesson builds a strong meaning of the equal sign: both sides have the same value. Children find how much to add to, or subtract from, one side to make two expressions equivalent.

Common mistakes

  • Adding the difference to the bigger side. Ask: Which side needs more?
  • Computing only one side. Encourage finding the value of both sides first.
  • Thinking = means write the answer next. Show sentences like 10 + 5 = 9 + 6 and 15 = 8 + 7.

How to help

  • Make a hanger balance, or share snacks onto two plates and make them fair.
  • Ask How many more? and How many fewer? in daily life (You have 8 grapes and I have 11. How many more do you need?).
  • Use a number line to count up from the smaller value to the bigger one.