Why the Formula Works

Four rules most of us learned without a reason, rebuilt with paper: long multiplication as a rectangle, dividing by a fraction, cross-multiplying and completing the square.

Spark & Whetstone · sparkwhetstone.org/learn/why-the-formula-works · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. Multiplication is a rectangle

Draw 23 × 14 as a rectangle 23 wide and 14 tall. Cut the width into 20 and 3 and the height into 10 and 4. You now have four smaller rectangles.

Their areas are 20 × 10 = 200, 3 × 10 = 30, 20 × 4 = 80 and 3 × 4 = 12. Added up they make 322, the whole rectangle.

The long multiplication you learned at school does exactly these four multiplications; the columns just hide the rectangle.

Question. Cut 23 × 14 into tens and ones. Which four areas make up the rectangle?

200, 30, 80 and 12
200 and 12 only
230 and 140

Say it back. In your own words: where is the rectangle hiding in long multiplication?

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Lesson 2. Dividing by a fraction

"Turn it upside down and multiply" is a rule without a reason. The reason is a question: how many of these fit into that?

3 ÷ ½ asks how many halves fit into 3. Each whole holds two halves, so three wholes hold six. Dividing by a half is the same as multiplying by two.

The same picture works for any fraction. 2 ÷ ¼ asks how many quarters fit into 2: four in each whole, so eight.

Question. Using "how many fit?", what is 2 ÷ ¼?

Eight quarters fit, so the answer is eight
Half
Six

Say it back. Say it back: what question does dividing by a fraction really ask?

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Lesson 3. Cross-multiplying

If 3/4 = x/20, the school rule says multiply across: 3 × 20 = 4 × x. The reason is that you are multiplying both sides by the same thing, 4 × 20, which keeps them equal and clears both fractions.

Then 60 = 4x, so x = 15. The two fractions are the same amount cut into different numbers of pieces: 3 out of 4 is 15 out of 20.

Question. If 3/4 = x/20, what is x?

x is fifteen
x is twelve
x is sixteen

Say it back. In your own words: why is it fair to multiply across?

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Lesson 4. Completing the square

Draw x² as a square with sides x. Draw 6x as a strip x long and 6 wide. Cut the strip in half lengthways and put one half on the right of the square and one half along the bottom.

You now have an almost-square with a missing corner. The corner is 3 by 3, so adding 9 completes it: x² + 6x + 9 is a square with sides x + 3.

That is the whole method. Half the strip goes on each side, and the missing corner is that half, squared.

Question. Which piece completes the square for x² + 8x?

A corner of 16
A corner of 8
A corner of 64

Say it back. Say it back: where does the number that completes the square come from?

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At the table: The scissors proof

Players: a grown-up and a teenager. You'll need: squared paper, scissors, a pencil.

The grown-up picks one rule from school they never understood.
Together, find the drawing in this course that explains it and cut it out of squared paper.
The teenager explains the drawing back to the grown-up in their own words, without the formula.

Answers

200, 30, 80 and 12 — 20 and 3 across, 10 and 4 down: 20×10, 3×10, 20×4 and 3×4. They add up to 322.
Eight quarters fit, so the answer is eight — Four quarters fit into each whole, and there are two wholes.
x is fifteen — 20 is five times 4, so x is five times 3. Cross-multiplying says the same thing: 3 × 20 = 4x, so 60 = 4x.
A corner of 16 — Half of 8 is 4, so the corner is 4 by 4. Then x² + 8x + 16 = (x + 4)².