Multiplication and Division Concepts
Warm-up
I have 4 groups of 5 linking cubes. How many cubes total? Count by 5s: 5, 10, 15, 20. Write the addition: 5+5+5+5 = 20. Write the multiplication: 4 x 5 = 20. All three say the same thing. Now rearrange into an array: 4 rows of 5. Does it still equal 20?
Explore
Fish drying scenario: a community is preparing for winter. Each drying rack holds 6 fish. How many fish on 7 racks? Model with cubes. Write the multiplication. Now reverse: 42 fish, 6 per rack. How many racks? Write the division. Discuss: how are these two problems related?
Formalize
Hundred chart patterns: colour all multiples of 3. What pattern do you see? (Diagonal stripes.) Now colour multiples of 4. Different pattern. Overlay: which squares are multiples of both 3 and 4? (Multiples of 12.) This is a preview of common multiples and connects multiplication to visual pattern.
Multiplication and Division Concepts
Commutativity through the array: build a 3x7 array. How many? (21.) Now rotate the array 90 degrees: it becomes a 7x3 array. How many? (Still 21.) 3 x 7 = 7 x 3. This is the commutative property of multiplication, proved visually with a single array turn.
Practice
Students solve 4 multiplication and 4 division problems in context (equal groups, array, sharing, grouping), showing concrete or pictorial models for each. Exit ticket: write a division story for 36 / 6 = 6 using the fish-drying context.
Exit ticket
Students solve 4 multiplication and 4 division problems in context (equal groups, array, sharing, grouping), showing concrete or pictorial models for each. Exit ticket: write a division story for 36 / 6 = 6 using the fish-drying context.
Step 1: Build a 4-by-6 array of counters (4 rows, 6 in each row). Count it however you like — skip-count rows: 6, 12, 18, 24.
Step 2: Read the array four ways, writing each fact as you go: 4 × 6 = 24 (4 rows of 6). 6 × 4 = 24 (turn your head — 6 columns of 4). 24 ÷ 4 = 6 (24 shared into 4 rows → 6 per row). 24 ÷ 6 = 4 (24 dealt into groups of 6 → 4 groups).
Step 3: Say the relationship plainly: multiplication builds the rectangle from its sides; division recovers a side from the whole. They are the same picture read in different directions.
Step 4: Fact-family triangle for the week: 24 on top, 4 and 6 at the base — quiz all four readings until any one instantly summons the others.
Watch for: students who treat ÷ as a brand-new operation to memorize. It isn't — it's the array asking "what's my missing side?"
Story A (sharing): 12 cookies shared fairly among 3 friends. Deal them out like cards — one for you, one for you, one for you, around and around. Each friend ends with 4. Here 3 is the NUMBER OF GROUPS and the answer is the size of each.
Story B (grouping): 12 cookies packed into bags of 3. Scoop 3 at a time: one bag, two, three, four bags. Here 3 is the SIZE of each group and the answer is how many groups.
Step 3: Compare the two acts physically — dealing versus scooping. Different motions, different questions, same equation: 12 ÷ 3 = 4.
Step 4: Why teachers must know the difference even though the arithmetic matches: word problems come in both flavours, and students who only ever "dealt" will misread a grouping story (and vice versa). Give one of each flavour daily and have students name which kind it is before solving.
Bonus connection: grouping division is literally repeated subtraction — 12 − 3 − 3 − 3 − 3 = 0, four subtractions.
The claim: "5 times zero is 5, because you have 5 and then… nothing happens." The student is treating × 0 like + 0.
Step 1: Return to the meaning of multiplication: 5 × 0 means 5 GROUPS OF 0 — five empty plates. Count the cookies on five empty plates: 0.
Step 2: Contrast with the addition fact they're borrowing: 5 + 0 = 5 says "five cookies, then nothing added" — a real plate of 5 stays 5. The two rules feel similar but describe different acts.
Step 3: Descend the pattern so the answer arrives by structure, not decree: 5 × 3 = 15, 5 × 2 = 10, 5 × 1 = 5, 5 × 0 = … each step drops by 5, so 0. Patterns persuade where rules bounce off.
Step 4: Check the flip side too: 0 × 5 (zero groups of five — no plates at all) is also 0.
Exit question: "Which is bigger, 8 × 0 or 8 × 1?" — and WHY. (8 × 1 is eight; 8 × 0 is none.)