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Year 5 Maths Number Sentences & Order of Operations

Equivalent number sentences

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Theory

Two calculations are equivalent when they have the same value, so they can be joined with an equals sign: \(7 \times 6 = 21 \times 2\) because both are \(42\). This Year 5 maths page shows how to test whether a sentence is true, and how to find a missing number by keeping the sentence balanced, using compensation and doubling and halving.

A number sentence is true when both sides have the same value. \(34 + 19 = 40 + 13\) is true: both sides are \(53\).

Compensation (adding): if one number goes up, the other must go down by the same amount. \(34\) went up \(6\) to \(40\), so \(19\) goes down \(6\) to \(13\).

Compensation (subtracting): if both numbers go up by the same amount, the difference stays the same. \(45 - 18 = 50 - 23\) (both went up \(5\)).

Doubling and halving (multiplying): if one factor is doubled, halve the other. \(16 \times 5 = 8 \times 10\).

If you change one side, change the other side the same way to keep the balance: add \(5\) to both sides and the sentence stays true.

A level balance with 34 plus 19 and 40 plus 13 A level balance. The left pan holds 34 plus 19 and the right pan holds 40 plus 13. Both are 53, so the balance is level. 34 + 19 40 + 13 =
Both sides are \(53\), so \(34 + 19 = 40 + 13\).
Array of 3 rows of 8 split in half An array of 24 dots in 3 rows of 8, split by a dashed red line into a blue half and an orange half of 3 rows of 4 each. Moving the orange half under the blue half gives 6 rows of 4, the same 24 dots.
\(3 \times 8 = 6 \times 4\): double the rows, halve the row length.

Ways to keep a sentence balanced:

\[ 34 + 19 = (34 + 6) + (19 - 6) = 40 + 13 \]
34+19=40+13
\[ 45 - 18 = (45 + 5) - (18 + 5) = 50 - 23 \]
45−18=50−23
\[ 16 \times 5 = 8 \times 10 \]
16×5=8×10
Equals means same value. It is not a signal to write the answer. Look at both sides.
Look for the change. Compare the numbers on each side. How did one change? The other must change to make up for it.

How to test a number sentence

  1. Work out the value of each side.
  2. Same value? The sentence is true.

How to find a missing number

  1. Work out the side with no gap.
  2. Find the number that gives the other side the same value, or use compensation.
  3. Check both sides.
Example 1 — Value of each side
Is \(7 \times 6 = 21 \times 2\) true?
Solution
\(7 \times 6\)\(=\)\(42\)
\(21 \times 2\)\(=\)\(42\)
7×6=21×2=42

True; both sides are 42.

Example 2 — Compensation
Find the missing number: \(34 + 19 = 40 + \Box\).
Solution
\(34 + 6\)\(=\)\(40\)
\(19 - 6\)\(=\)\(13\)
34+19=40+13

\(13\). Check: both sides are \(53\).

Example 3 — Same difference
Find the missing number: \(45 - 18 = 50 - \Box\).
Solution
\(45 + 5\)\(=\)\(50\)
\(18 + 5\)\(=\)\(23\)
45−18=50−23

\(23\). Both sides are \(27\).

Example 4 — Double and halve
Find each missing number.
(i) \(16 \times 5 = 8 \times \Box\)
(ii) \(14 \times 4 = 28 \times \Box\)
Solution
\(16 \times 5\)\(=\)\(80 = 8 \times 10\)
\(14 \times 4\)\(=\)\(56 = 28 \times 2\)
16×5=8×10,14×4=28×2

(i) \(10\)
(ii) \(2\)

Example 5 — Equivalent arrangements
Hall A has \(3\) rows of \(20\) chairs. Hall B has \(5\) rows of \(12\). Do they have the same number of chairs?
Solution
\(3 \times 20\)\(=\)\(60\)
\(5 \times 12\)\(=\)\(60\)
3×20=5×12=60

Yes, both have 60 chairs.

Example 6 — Keep the balance
\(18 + 7 = 20 + 5\) is true. If \(4\) is added to the left side, what must happen to the right side?
Solution
\(18 + 7 + 4\)\(=\)\(29\)
\(20 + 5 + 4\)\(=\)\(29\)
18+7+4=20+5+4

Add \(4\) to the right side too.

Example 7 — Find the mistake
To work out \(6 + 9 + 4\), Kai wrote \(6 + 9 = 15 + 4 = 19\). Explain what is wrong.
Solution
\(6 + 9\)\(=\)\(15\)
\(15 + 4\)\(=\)\(19\)
6+9+4=19

His line says \(6 + 9\) equals \(19\), which is false. Write two separate sentences, or \(6 + 9 + 4 = 19\).

Common pitfalls

Running sentences together. Writing a chain like \(6 + 9 = 15 + 4 = 19\) says \(15 = 19\). Use separate lines.
Copying a number across. In \(18 + 7 = 20 + \Box\), the answer is not \(7\): \(18\) went up \(2\), so \(7\) goes down \(2\) to \(5\).
Changing both numbers the same way when adding. For addition, one goes up and the other goes down. For subtraction, both move the same way.
Doubling both factors. \(16 \times 5\) is not \(32 \times 10\). Double one factor and halve the other.

Frequently asked questions

What makes a number sentence true?

Both sides have the same value. \(7 \times 6 = 21 \times 2\) is true because both are \(42\).

What is compensation?

Making up for a change. If you add \(6\) to one number in a sum, take \(6\) from the other: \(34 + 19 = 40 + 13\).

Does compensation work the same for subtraction?

No. Move both numbers the same way: \(45 - 18 = 50 - 23\).

How does doubling and halving work?

Double one factor and halve the other; the product stays the same: \(16 \times 5 = 8 \times 10\).

Why can you not write 6 + 9 = 15 + 4 = 19?

It claims \(6 + 9\) is \(19\). Each equals sign must join two things with the same value.

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