Blog

Teaching Multiplication So It Sticks

2026-10-18 · 9 min · English

Multiplication is where a lot of homeschool years quietly go wrong. Not because it is hard, but because of what gets taught in what order.

The usual sequence is: hand over a times-table chart, drill it, test it, get frustrated. Six weeks later the student can produce 6 × 7 under no pressure and freezes on it in a word problem, which tells you the fact was memorized and the idea was not.

Here is the order that works instead, drawn from the two sequences we publish — the US Common Core K–5 progression and the Singapore MOE syllabus — which disagree about plenty but agree completely about this part.

Meaning before speed

A student who has not physically built multiplication will eventually guess at it. The classic error, and we see it often enough that it has its own entry in our grade 3 lesson notes, is this:

Wrong: 3 × 4 means 3 + 4, so the answer is 7.

You cannot drill that away, because it is not a memory failure. The fix is a minute with cups and beans: three cups, four beans in each, count them, twelve. Multiplication means groups of. Say "groups of" out loud every single time you write ×, for weeks, until the student says it instead of you.

Both sequences build the idea the same way, from the concrete up:

That array turn is worth more than it looks. When a student rotates the beans and the total does not change, they have discovered the switch-around property themselves, and the number of facts to learn drops by nearly half. Tell them that explicitly. It is the best news in the subject.

In the Common Core sequence this ground is 3.OA.A, "Meaning of multiplication and division" — equal groups, arrays, unknown factors — and it comes before 3.OA.C, "Multiplication facts within 100." The order is not decorative.

Division at the same time, not later

The temptation is to finish multiplication, then start division. Resist it. Division taught as a separate later topic becomes a second thing to be afraid of.

Teach both meanings of division alongside the multiplication:

• Sharing: 12 beans dealt one at a time into 3 cups. How many in each? 12 ÷ 3 = 4.
• Grouping: 12 beans made into groups of 4. How many groups? 12 ÷ 4 = 3.

Then write the fact family — 3 × 4, 4 × 3, 12 ÷ 3, 12 ÷ 4 — and say the thing that makes division manageable: division is a multiplication question asked backwards. For 42 ÷ 6, the student asks "6 times what is 42?" That single reframing converts every division fact they are dreading into a multiplication fact they already have.

About a dozen facts, not a hundred

A 10 × 10 table has 100 cells, which is why a chart feels like a wall. Strip out what is free and the wall becomes a short list.

• The switch-arounds halve it.
• The 0s, 1s, 2s, 5s and 10s come almost free from counting.
• Doubling handles the 4s and 8s: 4s are double 2s (4 × 7 is 14 doubled, 28), and 8s are double 4s (8 × 6 is 24 doubled, 48). The Singapore track uses the same move for 6s, since 6 is double 3.
• The 9s are one group short of a 10: 9 × 7 = 70 − 7 = 63.
• Add a group handles the neighbours: if 5 × 6 = 30, then 6 × 6 is one more six, 36.
• Break apart handles the stubborn 7s: 7 × 8 = (5 × 8) + (2 × 8) = 40 + 16 = 56.

What is left after all that is a short list of genuinely tricky facts — for most students a handful in the 6, 7 and 8 range. Write them on paper, put it on the fridge, and work on three or four at a time. That is a weekend's worth of problem, not a term's.

Two strategy errors show up so reliably that we wrote them into the lessons as things to watch for:

• The nines trick, applied to the wrong number. A student computes 9 × 6 as 60 − 9 = 51. The fix is to draw ten boxes of 6 and cross out one whole box: you remove one group of 6, not 9, so 60 − 6 = 54. The check is the digit pattern — in the nines, the digits of the product add to 9, and 5 + 4 = 9 while 5 + 1 does not.
• Zero. "Any number times 0 is that number, so 7 × 0 = 7." Seven empty cups. Count the beans. Zero groups, or groups of zero, always make zero.

Where the two sequences differ

If you are choosing a math track, this topic is a fair place to look at the difference.

The Common Core progression puts arrays and repeated addition in grade 2 (2.OA), the meaning and the facts in grade 3 (3.OA), and factors and multiples in grade 4 (4.OA.B). It leans on strategy talk — name the strategy, explain the choice.

The Singapore MOE syllabus splits the tables across two years on purpose: the 2, 3, 4, 5 and 10 tables in Primary 2, the 6, 7, 8 and 9 tables in Primary 3, each introduced through doubling and known facts, with the bar model used to make the structure visible. Drawing ten bars of 7 and crossing one out is why the nines trick stops being a trick and becomes a picture.

Neither is better. One is more explicit about reasoning out loud; one is more systematic about visual models and pacing. We publish both as full K–5 tracks and expect you to pick one and stay with it — see the curriculum page for what each covers topic by topic.

A week that actually works

1. Monday. Build it. Cups, beans, arrays, the equation written after the building. No speed.
2. Tuesday. Fact families and the division link. Four facts from every three numbers.
3. Wednesday. One strategy — doubling, say — applied to a whole table. The student says the answer and how they got it.
4. Thursday. A card game. Flip two cards, multiply, bigger product takes the pair. When they stall, ask "which strategy could help?" rather than giving the answer.
5. Friday. A one-minute speed round of ten facts, scored, with the score written down to beat next time. Then add any fact they missed to the fridge list.

Then five minutes a day, in the car or at dinner, on three tricky facts. Not twenty minutes on Sunday.

What our materials give you for this

Every topic has a Quick Check before you teach, so you find out whether the array stage is already solid rather than guessing. The lesson itself is scripted — what to say, the worked examples, the misconceptions above with the fix for each, and a parent tip. Then guided practice with the answers printed after each question, independent practice, and a Show What You Know at the end with an answer key that explains the reasoning rather than just marking it right. The guide walks through how a topic runs start to finish.

A grade 3 math workbook is $9.99 and covers the whole year; the full K–5 math subject is $39. Both tracks, Common Core and Singapore, are in the store.

If multiplication is the specific wall you have hit, do not buy anything yet — the first topic of every grade in every subject is free, and in grade 3 math that first topic is the meaning of multiplication. Teach it this week with cups and beans and see whether the explanation lands. If it does, the rest is on the store.