From Missing Numbers to Simple Equations
By NeurofiED Editorial Team · Reviewed by NeurofiED Learning Science Team
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Help a Year 5 child move from missing-number arithmetic to simple equations by representing unknowns, preserving equality and checking every solution.
A simple equation is a statement that two expressions have the same value, with an unknown standing for a number we need to find. The bridge from arithmetic to algebra is shorter than it looks: replace a familiar empty box with a letter, keep both sides equal, undo operations carefully, then substitute the answer into the original equation to check it.
For example, `□ + 7 = 19` and `a + 7 = 19` ask the same mathematical question. The letter does not make the arithmetic harder. It gives the unknown a name, so the relationship can be written, discussed and reused.
England’s Year 6 mathematics programme says pupils should express missing-number problems algebraically and find pairs of numbers that satisfy an equation with two unknowns.[2] Its non-statutory guidance introduces letters and symbols through situations pupils already understand, including missing numbers and equivalent expressions.[2] For a Year 5 pupil preparing for later work, the useful foundation is not advanced algebra: it is secure reasoning about unknowns, equality and checking.
Contents
From a blank box to a letter
Children often meet unknowns long before anyone calls the work algebra:
Each blank represents a number that makes the statement true. Replacing the blank with a letter changes the notation, not the underlying task:
In `6n`, writing the number beside the letter means multiplication: `6 × n`. At this stage, say that aloud rather than expecting the notation to explain itself.
A letter can play different roles in mathematics, but simple equations usually use it for a particular unknown value. In `n + 9 = 16`, the job is to find the value of `n`. That differs from a formula such as `p = 4s`, where the letters can take different values while preserving a relationship. Keeping the first meaning clear prevents “a letter is secretly one fixed number” from becoming a later misconception.
Start by moving between forms. Ask a child to rewrite `□ + 14 = 31` as `x + 14 = 31`, then explain what stayed the same. The answer should be the relationship: an unknown number plus 14 has value 31.
What the equals sign really says
The equals sign does not mean “the answer comes next”. It means has the same value as. Both sides must balance.
Consider:
`8 + 4 = 7 + 5`
There is no single answer sitting on the right. Instead, both expressions have value 12. This understanding matters when the unknown appears in an unfamiliar position:
`24 = y + 9`
A child who reads `=` as “now calculate” may hesitate because the total comes first. A child who reads it as “is equal in value to” can see that this says the same thing as `y + 9 = 24`.
A simple balance drawing can help: place one expression on each side of a level beam. Whatever change is made to one side must be matched by an equal change to the other if the balance is to remain level. The Education Endowment Foundation advises that manipulatives and representations should have a clear mathematical purpose, reveal structure and help pupils move towards using mathematics independently.[3] A balance is useful when it makes equality visible; it should then be connected directly to the symbols and gradually removed.
Try true-or-false statements before solving:
Ask the child to calculate both sides. This turns equality into something they test, not a symbol they pass over.
A reliable solve-and-check routine
Use the same four questions each time:
“Inverse” means an operation that reverses another: subtraction can undo addition, addition can undo subtraction, division can undo multiplication, and multiplication can undo division. This is stronger than telling a child to “move a number across and change the sign”, because the latter hides why the equation remains true.
For a one-step equation, a known arithmetic fact may be fastest. For a less familiar equation, write the balancing step explicitly. Both routes should lead to a check.
Worked example: an unknown addend
Solve:
`x + 17 = 45`
The unknown has 17 added to it. Subtract 17 from both sides:
`x + 17 − 17 = 45 − 17`
On the left, `+17` and `−17` cancel:
`x = 28`
Now substitute 28 into the original equation:
`28 + 17 = 45`
The left side is 45 and the right side is 45, so the equation is true. Therefore `x = 28`.
A child may instead reason from a missing-addend fact: “What must be added to 17 to make 45?” That is valid. Ask them to connect it to subtraction: `45 − 17 = 28`. The bridge to algebra comes from explaining why the same value satisfies the written equation.
Worked example: multiplication
Solve:
`5m = 65`
Read this as “five multiplied by `m` equals 65”. Multiplication by 5 is undone by division by 5. Divide both sides:
`5m ÷ 5 = 65 ÷ 5`
Therefore:
`m = 13`
Check by substitution:
`5 × 13 = 65`
Both sides equal 65, so `m = 13` is correct.
Notice why subtracting 5 would not solve the equation. `5m` is not `m + 5`; it is five lots of `m`. Reading notation accurately is part of the reasoning.
Worked example: a two-step equation
Solve:
`3n + 8 = 29`
There are two operations around the unknown: first multiply by 3, then add 8. Undo them in reverse order.
First subtract 8 from both sides:
`3n + 8 − 8 = 29 − 8`
So:
`3n = 21`
Then divide both sides by 3:
`n = 7`
Check in the original equation:
`3 × 7 + 8 = 21 + 8 = 29`
The original left side becomes 29, matching the right side. The solution is `n = 7`.
If a child divides 29 by 3 first, return to the operation order in the expression. The equation builds `3n + 8` by multiplying and then adding. Solving dismantles that construction: subtract, then divide.
When two unknowns have several answers
Some equations do not have one unique answer. Consider:
`a + b = 18`
If positive whole numbers are allowed, `(1, 17)`, `(2, 16)`, `(3, 15)` and `(9, 9)` all work, along with their reversed pairs. One equation with two unknowns usually needs more information to select a single pair.
A tidy way to enumerate possibilities is to hold the total constant and change one value systematically:
| `a` | `b` | Check | |---:|---:|---| | 1 | 17 | `1 + 17 = 18` | | 2 | 16 | `2 + 16 = 18` | | 3 | 15 | `3 + 15 = 18` | | 4 | 14 | `4 + 14 = 18` |
Continue the pattern rather than guessing. If the question adds `a > b`, only pairs meeting that condition remain. If it adds `a = 2b`, the pair must satisfy both statements. This is still early equation reasoning: understand the conditions, generate candidates and check each one.
Common mistakes and useful adjustments
Treating the letter as decoration
A child calculates `3 + 8 = 11` in `3n + 8 = 29` and ignores `n`. Ask them to read `3n` aloud and replace `n` with a blank box: `3 × □ + 8 = 29`.
Reading equals as “write the answer”
A child says `7 + 5 = 9 + 3` looks wrong because there is calculation on the right. Use two quick calculations to show both sides are 12. Then try equations with the total first, such as `20 = q + 6`.
Using an inverse on only part of the equation
A child changes `x + 17 = 45` to `x = 45` by removing 17 from the left without changing the right. Draw a balance and remove the same amount from each side. Rewrite the full line `x + 17 − 17 = 45 − 17` before shortening it.
Confusing `4p` with `p + 4`
Use a concrete value. If `p = 6`, then `4p = 24`, while `p + 4 = 10`. Ask the child to sketch four equal groups of six.
Stopping without checking
An arithmetic slip can leave a neat but false answer. Make substitution part of the solution, not an optional extra. The check must use the original equation, because checking only a rearranged line may repeat the same mistake.
The notation is taking over
If every letter causes hesitation, return briefly to a box, counters or a balance sketch. Once the relationship is secure, put the letter back. If the representation is doing all the thinking, remove it gradually: first keep a small sketch, then ask for symbols plus an oral explanation, then symbols alone.
A parent action sequence
Keep a session short enough for explanations to stay careful.
If the child is accurate but slow, keep the structure and vary the numbers. If they can calculate but cannot explain equality, compare both sides of true and false statements. If they choose operations at random, return to the story of what was done to the unknown and undo it in reverse.
Keep this practice focused on representing an unknown and maintaining equality once an equation has been formed. Choosing operations from a word problem is a separate reasoning step.
Short FAQ
Must the unknown be `x`?
No. Any clear symbol can represent the unknown. Switching among `a`, `m`, `n` and a blank helps show that the reasoning does not depend on one special letter.
Can the unknown appear on the right?
Yes. `31 = k + 12` is equivalent to `k + 12 = 31`. Equality is symmetric: if the left has the same value as the right, the right has the same value as the left.
Is guessing allowed?
A sensible trial can reveal structure, but every candidate must be checked. For one-step and two-step equations, inverse operations usually give a more reliable route than unsystematic guessing.
Why check by substitution?
Substitution tests the proposed value against the exact condition in the original equation. If both sides calculate to the same value, the candidate satisfies that equation.
Is this already algebra?
Yes. Representing an unknown with a symbol and reasoning from equality are algebraic actions. The numbers and operations may be familiar arithmetic; algebra makes the relationship explicit.
Next step
Choose one missing-number statement, rewrite it with a letter, solve it while preserving equality, and check the value in the original equation. Then change one number and explain what remains the same about the method.
Sources
[2] https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study — Department for Education, National curriculum in England: mathematics programmes of study.
[3] https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks-2-3 — Education Endowment Foundation, Improving Mathematics in Key Stages 2 and 3.
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