The part you cannot see

Before you read anything

Draw it before you answer it. A tank holds 18 litres. There are 11 litres in it.

Do not answer yet. Draw the bar, and put a box where the missing number goes.

Put a box in the picture

The same bar with one part hidden. Read down and it is an addition; read up and it is a subtraction. Both find the same box.

Same picture as the last two lessons. The only change is that one of the two parts is not written down.

Two ways to say the same thing

$6 + \square = 9$ and $9 - 6 = \square$ are not two questions. They are one picture, read down and read up.

6 + \square = 9 \quad\Longleftrightarrow\quad 9 - 6 = \square

The box is a part. Take it off the whole and you have found it.

So "how many more" and "how many are missing" are the same sum, and you never have to decide which one a question wants. Draw the bar. The bar decides.

Add or subtract? It turns on one thing: whether the whole is what you do not know. Missing whole, add. Missing part, subtract. Nothing else in the wording matters.

The word that lies to you

"Altogether" means add, "left" means subtract. It usually does, which is exactly what makes it dangerous: it works often enough to become a habit, then fails on the ones that count.

Amina sold 7 mangoes and had 5 left. The word is "left". The answer is $7 + 5 = 12$, because what is missing here is the whole.

Amina

sold 7, has 5 left. How many did she start with?

Musa

saw "left" and wrote $7 - 5$

What the bar still cannot do

It tells you which operation. It does not make you quick. You can read $8 + \square = 15$ off the picture and still pause over the answer — and a pause every time is what stops you thinking about the problem.

So: what would have to be true for that pause to disappear?

Check yourself

Three to try

Draw the bar first, every time, even when you can see the answer. In two units' time these numbers get big enough that you cannot.

1

Amina sold 7 mangoes and had 5 left. How many did she start with?

2

There are 16 pupils in a class. 9 are girls. How many are boys?

3

Otim has 8 goats. Musa has 14. Which calculation says how many more Musa has?

Show that you have it

The box, and then the letter

The last one has a letter in the box instead of a shape. Nothing else about it changes, which is the point — you have been solving equations since you were six.

1

A bar shows a whole of 23. One part is 9 and the other is a box. What goes in the box?

2

$\square + 16 = 41$. What goes in the box?

3

Same bar, same box, a letter instead of a shape: $x + 16 = 41$. What is $x$?

Next: The bit that sticks out The whole thing on one page