Tangyuan MathLesson 7 · 15 min Stage 3 · Power strategies
Break it apart
Nobody handed us a rule for 7 × 8. So we build it — from two facts we already own. This is the most powerful idea in the whole course.
In this lesson your child will cut a hard array into two friendly pieces and add the parts — 7 × 8 = 7 × 5 + 7 × 3 = 35 + 21 = 56 — and learn to choose good places to cut.
Talk it through
"You now own ×1, ×2, ×5, ×10, the nines, and the swap. But what about a fact like 7 × 8 — no rule, no shortcut? Here's the master move: if you can't do it whole, cut it in half."
Open the demo: 7 rows of 8. Slide the cutter to 5, so the array splits into 7 × 5 and 7 × 3. "Look — it's just two arrays glued together. The left part you know: 7 × 5 = 35. The right part you know: 7 × 3 = 21. Glue them back: 35 + 21 = 56."
"The recipe: cut one of the two numbers into friendly pieces — fives, tens, and doubles are your friends — multiply each piece, add the answers. You just figured out a fact you never memorized. That's allowed. That's what mathematicians do."
Two more together, out loud: 6 × 7 = 6 × 5 + 6 × 2 = 30 + 12 = 42. And a big one beyond the table — 12 × 4 = 10 × 4 + 2 × 4 = 48. Slide the demo around to check different cuts on the same array: 7 × 8 as 7 × 4 + 7 × 4 works too — 28 + 28.
Try it together
Your child drives the cutter. Challenge: find a cut where both halves are facts you own. (5+3? 4+4? Both fine — many roads, one answer.)
Quick check
Common mix-ups
Cutting into unfriendly pieces. 7 × 8 as 7 × 6 + 7 × 2 trades one hard fact for another. Good cuts land on 5s, 10s, 2s and doubles — numbers with rules.- Adding where you should multiply. The pieces are 7 × 5 and 7 × 3 — multiply each piece first, then add the two answers. Writing 7 + 5 + 7 + 3 is a different (wrong) road.
- Thinking there's one right cut. 7 × 8 = 7×5 + 7×3 = 7×4 + 7×4 = 7×6 + 7×2… every honest cut lands on 56. Your child's favorite cut is the correct one.
Play it now
The meaty part of the table: both numbers 3–9. Breaking apart out loud is encouraged — this round is about figuring, not recalling.