7 × 13 isn't on most times tables. But 7 × 10 and 7 × 3 are — and somehow adding them gives exactly the right answer. That's not a trick. It's a seam hiding inside a rectangle, and once you see it, a(b + c) = ab + ac stops being a rule to memorize.
Step 1 of 8
Drag across the rectangle to move the seam. Wherever you cut, the two pieces add back up to the whole — they have to, because nothing was added or thrown away.
Tip: cut at a round number — 10, 20 — and both pieces become easy.
One rectangle. One cut. Nothing else.
7 × 13 is 7 rows of 13 unit squares. The answer — the product — is simply how many squares fit inside: the rectangle's area.
Write 13 as 10 + 3 and cut there. You get two rectangles, and here's the key: both are still 7 rows tall. The cut went through every row, so every row is in both pieces. That's why the 7 multiplies the 10 and the 3.
Cutting doesn't create or destroy squares, so 70 + 21 has to equal the full 91. The numbers were never special — any height, any split, same picture.
That's the distributive property: the multiplier outside the parentheses gets handed out — distributed — to every part inside. Read right-to-left, it's factoring: ab + ac = a(b + c), gluing two rectangles of the same height back into one.
A very common mistake: 7 × (10 + 3) = 7 × 10 + 3
In the picture, that means the gold piece only got one row. The dashed 18 squares just vanished — the answer comes out 73 instead of 91.
The rectangle won't let you do that. Every part inside the parentheses has to be multiplied, because every part spans all 7 rows.
With minus signs and square roots, this slip gets sneakier. Try the practice traps ↓
Nothing stops you from cutting the height too. Split 12 into 10 + 2 and 13 into 10 + 3, and the rectangle falls into four easy pieces:
100 + 30 + 20 + 6 = 156
That's exactly what the long-multiplication you learned in school is doing, column by column. And later in algebra, (x + 2)(x + 3) is the very same four-piece rectangle — just with an x where the 10 was.
Eleven problems built around the mistakes people actually make — lost minus signs, half-distributed roots, and places where distributing doesn't work at all. Pick an answer, then tap the wrong ones too: each one hides a specific slip.
First-try score: 0 / 0