Multiplication as equal groups, arrays and area, and as a shortcut for repeated addition; division as sharing and as grouping, and as a multiplication with a missing factor. Fact families, so one fact gives you four. The order, break-apart and grouping properties, and the strategies that make the facts to 100 easier: skip counting, doubling, tens minus one. Times 0, times 1, and why you cannot divide by 0. Two-step word problems. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
Three groups. Four in each group. The groups have to be equal — that is the whole rule.
Multiplying is equal groups. Dividing is the same picture read backwards: you know the total and you are finding how many groups, or how big each group is.
Two things people get wrong
⚠️People often think…
20 cookies shared by 4 friends is 20 − 4.
Sharing is dividing: 20 ÷ 4 = 5 cookies each. Subtracting takes 4 cookies away and leaves 16 — which is what would happen if somebody ate them, not if you shared them.
“Each” is the word that means divide.
⚠️People often think…
7 × 0 is 7, because multiplying by nothing leaves it alone.
7 × 0 is seven groups with nothing in them — zero. It is times ONE that leaves a number alone. Times zero erases it.
Times one keeps it. Times zero erases it.
How many groups, how big each
1Watch one
3 × 4.
Three groups of four.
4 + 4 + 4.
That is 12.
12 — not 7. Multiplying is not adding the two numbers.
2Do one with me
20 cookies, shared evenly by 4 friends.
Sharing evenly means you
20 ÷ 4 =
So each friend gets
💬One sentence, then you move on
Why is sharing a division and not a subtraction?
3Try one
7 × 0 = ?
I want a hint first
Seven groups with zero in each one. How much is that altogether?
Where this goes
Where this lives
Splitting a bill, packing equal boxes, figuring out how many packs you need for a whole class.
What this feeds
Next unit is multi-digit multiplication — the same equal groups, with bigger numbers.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 A word problem — the four questions
Are the groups EQUAL? (If not, add.) → Do I KNOW the total? (No → multiply: groups × in each. Yes → divide: total ÷ groups, or total ÷ in each.) → Write the sentence with the numbers in: 4 × 6 = 24 or 24 ÷ 4 = 6 → Check by running the fact family backwards: 6 × 4 = 24.
Equal? Total known? Write it. Check it.
💡 The whole unit in one idea
Multiplication and division are ONE picture read two ways. An array of 4 rows with 6 in each row has 24 dots: 4 × 6 = 24. Cover the 6 and ask how many in each row — that is 24 ÷ 4 = 6. Cover the 4 and ask how many rows — that is 24 ÷ 6 = 4. Every fact you learn is really four facts, and every strategy (break apart, double, tens minus one) is just a different way of counting the same dots.
⚠️ Traps the test loves
Adding the two numbers: 3 × 4 as 7. Three groups of 4 is 4 + 4 + 4 = 12.
Subtracting when the problem shares: 20 cookies for 4 friends as 20 − 4. Sharing is 20 ÷ 4 = 5.
Swapping a division: 24 ÷ 3 is 8, but 3 ÷ 24 is not. Only multiplication swaps.
Times 0 as the number itself: 7 × 0 is 0, not 7. Times 1 keeps the number; times 0 erases it.
Losing or adding a zero: 4 × 30 is 120, not 12 and not 1200. Do 4 × 3, then put ONE zero back.
Stopping after step one of a two-step problem. 3 × 8 = 24 is the total; the question asked what was LEFT after giving 5 away: 19.
🗂️ The strategies — the chart
Strategy
How it works
Example
Skip count
count by the second factor, first-factor times
4 × 5: 5, 10, 15, 20
Order (commutative)
swap the factors — same product
8 × 3 = 3 × 8 = 24
Break apart (distributive)
split one factor into 5 + something or 2 + something
7 × 8 = 5 × 8 + 2 × 8 = 40 + 16 = 56
Grouping (associative)
in a chain of three, multiply the easy pair first
2 × 5 × 7 = 10 × 7 = 70
Doubling
× 4 is double × 2; × 8 is double × 4
6 × 2 = 12, 6 × 4 = 24, 6 × 8 = 48
Nines
ten groups minus one group
9 × 7 = 70 − 7 = 63
Missing factor
turn the division into “what times?”
42 ÷ 6: 6 × 7 = 42, so 7
🎯 How the test will ask
A picture of equal groups or an array — “write a multiplication sentence.” Groups × in each.
A short story — “which equation matches?” Decide: total unknown (×) or total known (÷).
An equation with a blank: 6 × ? = 42, or ? ÷ 4 = 5. Use the fact family.
“Which shows another way to find 7 × 8?” Look for 5 × 8 + 2 × 8 — break apart.
A rectangle with its side lengths — “find the area,” or area and one side — “find the other side.”
A two-step story: multiply to get a total, then add or subtract. The answer to step 1 is NOT the final answer.
“Which is true?” about 0 and 1: 5 × 1 = 5, 5 × 0 = 0, 0 ÷ 5 = 0, and 5 ÷ 0 has no answer.
✅ Before the test, can you…
Draw an array for 3 × 6 and write all four facts in its family?
Tell a sharing story and a grouping story for 12 ÷ 4?
Find 7 × 8 by breaking apart, and 9 × 6 with the nines trick?
Solve 5 × ? = 40 and ? ÷ 6 = 3 and check each answer?
Multiply 4 × 30 and 6 × 50 by counting tens?
Find the area of a 5-by-7 rectangle, and the missing side when the area is 35 and one side is 5?
Solve a two-step problem and show both steps?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like a mathematician: sort real situations into equal groups, arrays and area; label a dot array and read both a multiplication and a division out of it; put a break-apart strategy for 7 × 8 in order; then settle an argument about whether 3 × 8 and 8 × 3 are really the same. Every activity checks itself, and hints are free.
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Your practice record — saved on this device
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