Four jars of eight marbles and eight jars of four hold the same 32. Turning the jars sideways never changes the count.
So a child who owns the 3 times table already owns every 3 on the chart. You only have to spot it the other way round.
Step 1 of 4the fact we want
9 × 3
That makes the chart smaller than it looks. The 2 to 12 tables hold 121 facts, and turning them around leaves 66 to learn, as we counted on the multiplication tables check page.
This move works for times only. 12 ÷ 3 and 3 ÷ 12 are different questions.
Double it, split it, or round it up
Doubling covers the 2s, 4s, and 8s. One double for a 2, two for a 4, three for an 8.
Step 1 of 4the fact we want
4 × 6
A split breaks the number into two easy pieces. A dozen is 10 and 2, a 7 is 5 and 2, and a 3 is 2 and 1.
Step 1 of 5the fact we want
12 × 8
Rounding up covers the 9s. We take ten of the number, one group too many, and give that number back.
Step 1 of 5the fact we want
9 × 6
The fact next door
The fifth move borrows from the fact beside the one you want. If 6 × 6 is yours, 6 × 7 is one more 6.
Step 1 of 4the fact we want
6 × 7
The squares make the best neighbors. 6 × 6, 7 × 7, and 8 × 8 stick in a child's head, and a square has no second number to pick, so the fact next door is its only other road.
Two roads, one answer
Here is the part worksheets leave out. Three of these five moves take 8 × 9, so let's run all three.
double the 9 three times9 → 18 → 36 → 72
ten eights, give one back80 − 8 = 72
9 × 9, give a 9 back81 − 9 = 72
When two roads disagree, one of them went wrong. That is the sign to start again.
And when they agree, the answer has passed a check. No answer key, nobody to ask.
Practice: nine mixed facts
Five friendly ones first. Get one wrong and the box names the move.
11 × 3 =
Same rules as above. You've got this.
4 × 7 =
Same rules as above. You've got this.
6 × 8 =
Same rules as above. You've got this.
8 × 7 =
Same rules as above. You've got this.
9 × 7 =
Same rules as above. You've got this.
Now the four that take the most practice. Run each one twice, by two moves, and check the answers agree.
Turn it around first. Almost every fact has a twin, so learning one gives you both.
Pick the move from the number. Doubles for the 2s, 4s, and 8s, splits for the 3s, 6s, 7s, 11s, and 12s, and a round-up for the 9s.
Run a second road. Two moves landing on the same answer give you a check you can run alone.
Which move will you reach for first?
Double
0%
Split
0%
Round
0%
Flip
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FAQ
Five of them carry the whole chart. Turn the fact around, because 9 × 3 and 3 × 9 share an answer. Double it, for the 2, 4, and 8 times tables. Split the number into friendly pieces, so a dozen becomes 10 and 2. Round up to ten and give one group back, which is the whole 9 times table. And borrow from the fact next door, so 6 × 7 is 6 × 6 plus one more 6. Teachers call these commutativity, doubling, partitioning, compensation, and derived facts.
Pick it from the number you are multiplying by. Doubling handles the 2s, 4s, and 8s. A split handles the 3s, 6s, 7s, 11s, and 12s. Rounding up to ten handles the 9s. The 10s put a zero on the end, and the 5s are half of that: ten 7s make 70, so five 7s make 35. Most facts give you two numbers to choose from, and when the two match, like 7 × 7, the fact next door is the second road.
Turning the fact around, because it costs nothing to learn. England teaches it in year 2: multiplication of two numbers can be done in any order. A child who owns the 3 times table already owns every 3 in the chart, including 9 × 3 and 12 × 3, and only has to recognize those facts written the other way.
Run a second strategy on the same fact. 8 × 9 can be reached by doubling the 9 three times, by taking ten 8s and giving one back, or by starting at 9 × 9 and giving a 9 back. All three land on 72. When two roads disagree, one of them went wrong and the fact is worth starting again.
They are how the facts get learned, and recall is still the finish line. England expects year 4 pupils to recall multiplication and division facts for tables up to 12 × 12, and its year 4 check gives six seconds a question, which is too little time to work anything out. A strategy is what a child uses on the way to knowing a fact cold.
