Multiplication and Division of Multi-Digit Numbers
Warm-up
Show 6x4=24 on an area model (6 wide, 4 tall, 24 squares). Now show 6x40: same height, ten times wider, 240 squares. Extend: 6x43 = 6x40 + 6x3 = 240+18 = 258. Students can see the breakdown happening visually.
Explore
Area model station: students solve 4 two-digit-by-one-digit problems using area models on grid paper, colouring each partial product a different colour. Then solve the same problems using mental math. Are answers the same? Which method is clearer?
Formalize
Camping trip problem: 5 families share the cost of a camping permit (37.
Multiplication and Division of Multi-Digit Numbers
Connect multiplication and division: if 37 x 5 = 185, then 185 / 5 = 37. The multiplication fact confirms the division answer. Teaching students to verify division with multiplication is both a metacognitive strategy and a demonstration of inverse operations.
Practice
Students solve 4 two-digit-by-one-digit and 2 three-digit-by-one-digit problems using area models, plus 4 division problems using chunking. Exit ticket: use the area model to solve 8x47.
Exit ticket
Students solve 4 two-digit-by-one-digit and 2 three-digit-by-one-digit problems using area models, plus 4 division problems using chunking. Exit ticket: use the area model to solve 8x47.
Step 1: Estimate: 34 ≈ 35, and 35 × 6 ≈ 210 (or 30 × 6 = 180 as a floor). Expect a bit over 200.
Step 2: Draw the area model: a rectangle 6 tall, split into widths 30 and 4. Two rooms: 6 × 30 = 180 and 6 × 4 = 24.
Step 3: Add the rooms: 180 + 24 = 204.
Step 4: Now run the compact algorithm beside it: 6 × 4 = 24 → write 4 carry 2; 6 × 3(tens) = 18 tens, +2 = 20 tens → 204. Match each mark to a room: the carried 2 is the 20 from 24; the "18 tens" IS the 180 room.
Step 5: Keep both alive deliberately: the algorithm is fast; the model explains and scales (it will draw 2-digit × 2-digit next month and (x+3)(x+4) in Grade 10). Never let the shortcut orphan the picture.
Check against the estimate: 204 vs ≈210 ✓.
Step 1: Build 78 as 7 rods and 8 units. The job: share into 3 equal piles.
Step 2: Share the rods first: 7 rods into 3 piles → 2 rods each, 1 rod stuck (can't split a rod across piles).
Step 3: Trade the stuck rod for 10 units: now 18 loose units. Share: 18 ÷ 3 = 6 each.
Step 4: Read each pile: 2 rods + 6 units = 26. So 78 ÷ 3 = 26.
Step 5: Reveal the algorithm as the diary of what just happened: "3 into 7 goes 2" (2 rods each), "remainder 1" (the stuck rod), "bring down the 8" (trade it in with the loose units → 18), "3 into 18 goes 6." Every mysterious move in long division is a share-or-trade you can do with your hands.
Do two more with materials BEFORE ever writing the bracket. The notation should feel like shorthand for a familiar act, not an incantation.
The problem: 86 students, each van seats 8. How many vans?
Step 1: Divide: 86 ÷ 8 = 10 remainder 6 (8 × 10 = 80, six students left on the curb).
Step 2: Interrogate the remainder — the step that separates arithmetic from problem solving: those 6 students are PEOPLE. They need a van. Round UP: 11 vans.
Step 3: Contrast with sibling problems sharing the same division: "86 cookies shared by 8 kids — how many EACH?" → 10 (the remainder crumbles into extras; answer rounds DOWN). "How many cookies left over?" → 6 (the remainder IS the answer).
Step 4: Post the three fates of a remainder: forces rounding UP (vans, buses, boxes needed), gets DROPPED (full bags only), or IS the answer (leftovers). The division is identical; the STORY chooses.
Exit ticket: write one story for each fate of 50 ÷ 7. Students who can author all three own the concept.