Difference of Squares
Two squares become one multiplication.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: a squared minus b squared equals (a plus b) times (a minus b) — two squares become one easy product.
★ · Difference of Squares Identity
Compute 51 squared minus 49 squared without a calculator.
L0 · Border = Sum of Sides
A 4-by-4 quilt of patches grows by one extra row and one extra column, becoming a 5-by-5 quilt. How many new patches were added?
L1 · Two Squares, One Multiplication
A bigger quilt is 11-by-11 patches. A smaller quilt is 9-by-9 patches. How many more patches does the bigger quilt use?
L2 · Sum Times Difference
Two square quilt sides add up to 30 patches. The bigger side is 6 patches longer than the smaller side. How many more patches does the bigger quilt use than the smaller one?
L3 · Reverse the Identity
A bigger quilt uses 192 more patches than a smaller quilt. The bigger side is 4 patches longer than the smaller side. What is the bigger side length?
L4 · Factor Pairs of the Difference
A community centre wants two square quilt sizes so the bigger quilt uses exactly 693 more patches than the smaller. The bigger side must be less than 50 patches. How many different pairs of whole-number side lengths are possible?
L5 · Factor Pairs, Bounded Window
Two square quilts have side lengths whose area difference is exactly 3465 patches. The bigger side must be less than 100 patches. The smaller side must be at least 10 patches. How many different pairs of whole-number side lengths are possible?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · The Two-Squares Trick
Peek the trick — Two squares, one multiplication
Any squared minus squared splits into the sum of the two numbers times their difference. Two squares collapse into one easy product.
L0 · The Border Trick
Peek the trick — Border = sum of sides
When a square grows by one row and one column, the L-shaped border has exactly old-side plus new-side patches. No multiplication needed.
L1 · Two Quilts
Peek the trick — Sum times difference of the sides
When two square sizes are given, never square either one. Add the sides, subtract the sides, multiply once.
L2 · Sum × Difference
Peek the trick — Multiply the sum and the difference directly
When only the sum and difference of two sides are given, multiply them. The actual side lengths are never needed.
L3 · Working Backwards
Peek the trick — Divide the area gap by the side gap
Knowing the area difference and the side gap, divide to recover sum of sides, then average up and down to recover each side.
L4 · Count the Pairs
Peek the trick — Factor the area gap and filter
Every factor pair of the area gap with matching parity gives one side-length pair. Convert each pair, then keep only those passing the constraint.
L5 · Fit the Window
Peek the trick — Factor, then filter twice
Same factor-pair recipe, but now intersect two constraints. Convert each factor pair to sides and keep only those whose bigger side fits the upper bound and whose smaller side meets the lower bound.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
