Place value first: multiplying by 10, 100 and 1,000, and by 30 or 400 without losing a zero. Then one problem, 23 × 14, worked four ways — breaking a factor apart, the area model with its four boxes, partial products, and the standard algorithm with its two rows — so you can see they are the same rectangle. Regrouping, and why the second row starts with a placeholder zero. Estimating before you multiply, so an answer with the wrong number of digits gets caught. Two-digit by two-digit, three-digit by two-digit, and two-step word problems. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
A rectangle, 23 across and 14 down. How much it covers IS the answer to 23 × 14.
Every box has to get filled. In the stacked method, that second row is 23 × 10 — which is exactly why it ends in a zero. The zero is not decoration; it is a whole box.
Two things people get wrong
⚠️People often think…
23 × 14 is 92 + 23 = 115.
The second row is not 23 × 1 — it is 23 × 10 = 230. That is where the placeholder zero comes from. 92 + 230 = 322, and 115 is not even close.
That 1 is a ten. The zero proves you knew it.
⚠️People often think…
36 × 7 is 2,142 — 21 written next to 42.
Partial answers get ADDED, never glued side by side. 7 × 6 = 42, so write the 2 and carry the 4; 7 × 3 tens = 21 tens, plus the 4 carried = 25. The answer is 252.
Estimate first: 36 × 7 is about 250, not 2,000.
Four boxes, then add
1Watch one
23 × 14, using the four boxes.
20 × 10 = 200.
20 × 4 = 80, and 3 × 10 = 30.
3 × 4 = 12.
200 + 80 + 30 + 12 = 322.
2Do one with me
14 × 12. Split them into 10 + 4 and 10 + 2.
The big box: 10 × 10 =
The two side boxes: 10 × 2 plus 4 × 10 =
The corner box: 4 × 2 =
💬One sentence, then you move on
Why does the second row of the stacked method end in a zero?
3Try one
Before working out 48 × 52, estimate it by rounding both to 50. What do you get?
I want a hint first
50 × 50. Fifty fifties is five thousand halved — or just 5 × 5 with two zeros put back.
Where this goes
Where this lives
Square footage for flooring, how many chairs fit in a room, the cost of 24 of something that costs $15.
What this feeds
Next unit is fractions — pieces of one whole instead of many whole groups.
Estimate something you would need to buy a lot of, then say about how much.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🧭 The standard algorithm — five moves
Estimate first (23 × 14 ≈ 20 × 15 = 300) → Row 1: multiply the top number by the ONES digit, regrouping as you go (23 × 4 = 92) → Row 2: write a 0 in the ones place, then multiply the top number by the TENS digit (23 × 1 ten = 230) → Add the rows (92 + 230 = 322) → Compare with the estimate: 322 is close to 300, so it is reasonable.
Estimate, ones row, zero, tens row, add, compare.
💡 The whole unit in one idea
Every method is the same rectangle. 23 × 14 is a rectangle 23 wide and 14 tall. Cut it at the place values — 20 | 3 across, 10 / 4 down — and you get four boxes: 200, 30, 80 and 12. Breaking apart names the boxes. Partial products lists them. The standard algorithm folds them into two rows (80 + 12 = 92, then 200 + 30 = 230) and adds. The placeholder zero is simply the tens row remembering that it is made of tens.
⚠️ Traps the test loves
Leaving out the placeholder zero: 23 × 14 as 92 + 23 = 115. The second row is 23 × 10 = 230, so the answer is 322.
Forgetting to add the carry: 27 × 3 as 61. It is 3 × 2 tens = 6 tens PLUS the 2 tens carried = 81.
Writing partial products side by side: 36 × 7 as 2,142 (21 next to 42). They are added and carried, not glued: 252.
Only three boxes in a four-box area model. 23 × 14 needs 20 × 10, 20 × 4, 3 × 10 AND 3 × 4.
Losing or adding a zero with tens and hundreds: 23 × 400 is 9,200, not 920 and not 92,000.
Skipping the estimate — and handing in 496 for 48 × 52 when 50 × 50 = 2,500 says it must have four digits.
A bare problem: 46 × 32 or 123 × 45. Two rows, placeholder zero, add. Estimate before you start.
An area model with empty boxes — “fill in the partial products” or “what is the product?”
“Which expression is equal to 6 × 23?” Look for 6 × 20 + 6 × 3 — tens and ones, both multiplied.
Someone’s wrong work — “what mistake did they make?” Check the carry and the placeholder zero first.
“Which is the best estimate?” Round EACH factor to its nearest ten (or hundred), then multiply.
“Which answer is NOT reasonable?” Count the digits against the estimate.
A story with boxes, rows or buses — multiply for the total, then maybe add or subtract for step two.
✅ Before the test, can you…
Multiply by 10, 100 and 1,000, and by 30, 400 and 5,000, without losing a zero?
Draw the four-box area model for 23 × 14 and fill in all four boxes?
Do 27 × 3 and 36 × 7 in columns, carrying correctly?
Do 46 × 32 with two rows and explain why the second row starts with 0?
Estimate 61 × 39 before working it, and say whether 2,379 is reasonable?
Do 123 × 45 and 4 × 235?
Solve a two-step story 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: fill in the four boxes of an area model for 23 × 14, put the standard algorithm’s moves in order, argue for the placeholder zero with real numbers, then sort estimates into reasonable and not. Every activity checks itself, and hints are free.
Your practice never leaves this device. There is no account and no sign-in. Your work is saved in this browser only, and you can erase it whenever you want.
Your practice record — saved on this device
This is your record of the module on screen — it stays here and goes nowhere. Independent means you got it right on the first tap; supported means you got it after the explain-and-retry, or marked ‘I had it’ on a revealed answer. Both count, and neither is a grade. If your teacher asks, copy the row or show them this screen.
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The answer key is for a teacher: it prints only from here, for the unit on screen. Print the study packet prints the study pages and a blank quiz — never the answers.