A child who can do 3 + 4 can solve x + 4 = 7. What stops them is not the arithmetic. It is that nobody has told them what the equals sign means, or what the letter is. Explain those two things with a picture, and equations become one of the easier parts of school maths. This page gives the picture, the order to teach in, and the mistakes to expect.
The three ideas a child needs first
1. The equals sign means "the same as". After years of sums written as 3 + 4 = ▢, most children believe = means "now write the answer". Test it: ask your child what goes in the box in 8 + 4 = ▢ + 5. A child who says 12 has the wrong idea of equals; the answer is 7, because both sides must be worth the same. Fix this before anything else, with sentences such as 5 + 5 = 6 + ▢ and 10 = 7 + ▢.
2. A letter is a number we do not know yet. Not an object, not a label. Start with a box or a question mark, and only later call it x.
3. Every operation has an opposite. Adding undoes subtracting; multiplying undoes dividing. A child should be able to say, quickly, "the opposite of adding 6 is taking away 6".
The balance picture
Draw a see-saw balance. Say: an equation is a balance that is level. Whatever is on the left weighs the same as whatever is on the right.
Now put x + 4 = 7 on it. On the left pan, a closed box (the hidden number) and four counters. On the right pan, seven counters.
The one rule: you may do anything you like, as long as you do the same thing to both pans. Then it stays level.
Take four counters off the left. The balance tips. Take four off the right as well: level again. What is left? The box on one side, three counters on the other. The box holds 3.
Do this with real things: a paper bag with counters hidden in it, coins, building blocks. Ten minutes with a bag and some coins does more than a page of written examples.
The order to teach in
| Step | Example | What the child does | Usual grade |
|---|---|---|---|
| 1. Missing numbers | 3 + ▢ = 7 | Finds the number by thinking or counting | Grades 1 to 3 |
| 2. Equal sides | 8 + 4 = ▢ + 5 | Learns that = means the same as | Grades 2 to 4 |
| 3. A box on the balance | ▢ + 4 = 7 | Takes the same from both pans | Grades 4 to 5 |
| 4. A letter for the box | x + 4 = 7 | Writes "− 4" under both sides | Grades 5 to 6 |
| 5. All four one-step types | x − 5 = 9, 3x = 12, x ÷ 2 = 6 | Uses the opposite operation | Grade 6 |
| 6. Two steps | 2x + 3 = 11 | Undoes in reverse order | Grade 7 |
| 7. Letters on both sides | 5x − 3 = 2x + 12 | Collects the x's on one side first | Grades 7 to 8 |
Do not skip to step 4. Most trouble with equations in Grade 7 is steps 1 to 3 that were never done.
One-step equations, written properly
Teach one layout and insist on it: the equals signs in a column, one step per line, the same operation written on both sides.
x + 4 = 7 Take 4 from both sides. x + 4 − 4 = 7 − 4 x = 3
3x = 12 ("three boxes weigh 12") Divide both sides by 3. x = 4
x − 5 = 9 Add 5 to both sides. x = 14
x ÷ 2 = 6 ("half the box is 6") Multiply both sides by 2. x = 12
Then the habit that makes a child independent: check. Put the answer back in the first line. Is 3 + 4 equal to 7? Yes. A child who checks never needs to ask "is this right?"
Two-step equations
2x + 3 = 11. On the balance: two boxes and three counters on the left, eleven counters on the right.
Ask: what happened to x? It was doubled, then 3 was added. Undo it in reverse, like taking off shoes before socks.
- Take 3 from both sides: 2x = 8
- Halve both sides: x = 4
- Check: 2 × 4 + 3 = 11. Level.
A story helps: "I think of a number, double it and add 3. I get 11. What was my number?" Children solve these in their heads by working backwards, which is exactly the method. Then show them that the equation is the same story written short.
The five mistakes to expect
| What the child writes | What it tells you | What to do |
|---|---|---|
| x + 4 = 7 = 3 | Reads = as "the answer comes next" | Back to equal-sides sentences; one equation per line |
| x + 4 = 7, so x = 11 | Did the operation instead of its opposite | Ask "what was done to x? How do we undo it?" |
| 2x + 3 = 11, so x + 3 = 5.5 | Divided only part of a side | On the balance: halving means halving everything on the pan |
| 3x = 12, so x = 9 | Thinks 3x means 3 + x | Say "three boxes"; write 3 × x for a week |
| Right answers, no working | Doing it by inspection | Fine for now, but it will fail at step 7; ask for one written step |
A common shortcut, "move it to the other side and change the sign", gives right answers for a while and then fails with division and with negatives. If your child has been taught it, do not ban it; ask them to show the same step on the balance, so that the rule has a reason.
Games that teach it
- The bag game. Hide counters in a paper bag, put the bag and some loose counters on one side of a table, counters on the other, and say "these sides are equal". The child works out what is in the bag, then opens it to check.
- Think of a number. Take turns. "I think of a number, add 5, and get 12." Then two steps.
- Is it true? Write 6 + 2 = 4 + 4, 9 = 5 + 3, 7 = 7. True or false? This is the fastest cure for the equals-sign problem.
- Make it level. Give one side, 12 + 3, and ask for five different other sides.
When to get help
If a child in Grade 7 or above is still stuck on one-step equations after a few weeks of this, the cause is usually underneath: negative numbers, fractions, or the order of operations. Those are found quickly by watching the child work, line by line; a short check for the common gaps is in why am I bad at algebra. In a 1-on-1 online pre-algebra class the tutor watches every line the child writes on a shared whiteboard and fixes the step where it goes wrong. The free 30-minute first class is a real lesson on this week's homework.
Questions parents ask
1How do you explain equations to a child?
2What is an equation, in simple words?
3At what age can a child learn equations?
4Why does my child find equations hard?
5How do I teach my child to solve two-step equations?
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Shobha
Founder and lead tutor, Science with Shobha
Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.