Concept 3 of 7
Arrays & the Turn-Around Trick
Today's big idea
A multiplication is a rectangle of dots, so turning it on its side proves a × b = b × a.
3 rows of 8 and 8 rows of 3 hold the same 24 dots.
How it works
👀 What's going on?
Line up 3 rows of 8 dots and count them: 24. Now turn the whole grid on its side and it becomes 8 rows of 3. Count again and it is still 24. The dots never moved — you just looked at them a different way.
💡 Here's the trick
Look at the array above. A multiplication is a rectangle of dots: rows times columns. Because you can turn any rectangle on its side without adding or removing a dot, 3 × 8 and 8 × 3 give the exact same answer. This is the turn-around (commutative) trick.
🪄 How to do it
- 1.Read the multiplication as a rectangle: this many rows of that many.
- 2.If one order is hard to picture, turn it around to the easier one.
- 3.The product is the same either way, so use whichever you know.
✍️ Try it
You know 10 × 4 = 40. Use the turn-around trick to find 4 × 10 without counting.
4 × 10 is just 10 × 4 turned around, so it is also 40. The rectangle is the same; only the label swapped.
⚠️ Watch out
The turn-around trick works for multiplication, not division. 12 ÷ 3 is 4, but 3 ÷ 12 is not — division cares which number goes first.
🌟 Why this matters
Turning a hard fact into an easy one halves how many tables you must learn, since every fact you know hands you its twin for free.
Exam tip: Stuck on 7 × 3? Flip it to 3 × 7 and count up in 7s.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“You know 8 × 4 = 32. What is 4 × 8?”
- 1
Same rectangle, turned
4 × 8 is 8 × 4 turned on its side.
- 2
Rotating changes nothing
No dot is added or removed.
- 3
Answer ✓
4 × 8 = 32
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
THE PROBLEM
26 × 47
Picture a rectangle 26 tall × 47 wide — its AREA is the answer.
47
26
Big multiplications feel hard because your brain juggles too many things at once. The Area Model fixes that — we'll slice this rectangle into 4 easy rooms.
Quick check
✅ VERIFICATION
You know 6 × 9 = 54. What is 9 × 6?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Find the Rule
3→15, 4→20, so the rule is × 5.