Multiplication
Multiplication is equal groups. The times tables are worth knowing by heart, but a child who only has them memorised and cannot draw what 6 times 7 means will stall as soon as the numbers get bigger.
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Seeing a fact as an array
6 x 7
- Draw 6 rows of 7 dots, or 7 rows of 6 — either way round gives the same total.
- Split it somewhere easy. Six rows of 5 is 30, and six rows of 2 is 12.
- Add the parts: 30 + 12 is 42.
- So 6 x 7 is 42.
Worth knowing. Splitting a hard fact into two easy ones is the same idea as the area model used later for long multiplication. It is worth the time now.
Two-digit by one-digit
34 x 6
- Multiply the ones: 4 x 6 is 24. Write 4 and carry the 2.
- Multiply the tens: 3 x 6 is 18, plus the 2 you carried makes 20.
- Write 20 in front of the 4.
- The answer is 204.
Worth knowing. The carried 2 here is two tens, and it is added after the multiplication, not before. Multiplying the carried digit is one of the most common mistakes in this topic.
Where it usually goes wrong
- Adding the carried digit before multiplying instead of after.
- Forgetting the place-holder zero when multiplying by the tens digit in a two-digit by two-digit problem.
- Treating 6 x 7 and 7 x 6 as separate facts to memorise rather than the same one.
The Professor still gets 7 times 8 wrong sometimes. He works it out from 7 times 4 and doubles it, and nobody minds.
Practise it
All level pathsQuestions teachers and parents ask
Which times tables are worth learning first?
The 2s, 5s and 10s come almost free from counting. The squares (3 x 3, 4 x 4 and so on) anchor the middle. That leaves a handful of genuinely hard facts — 6 x 7, 6 x 8, 7 x 8 and 8 x 9 — which are worth naming as hard and practising on their own.
When should times tables be fluent?
Common Core asks for fluent multiplication and division within 100 by the end of grade 3, which is the 10 x 10 table. The 11s and 12s are a British and traditional addition rather than a Common Core requirement.
Do the tables have to be memorised?
Enough of them that long multiplication does not stall. A child working out 7 x 8 from 7 x 4 doubled is doing real mathematics, but if every single fact needs working out, multi-digit problems become too slow to finish and confidence goes with it.
Where this shows up
Standards
- 3.OA · Operations and algebraic thinking
- 3.NBT · Number and operations in base ten
- 4.OA · Operations and algebraic thinking
- 4.NBT · Number and operations in base ten
- 5.NBT · Number and operations in base ten
- CCSS.MATH.CONTENT.2.OA.C.4 — Use repeated addition to find the total number of objects in a rectangular array.
- CCSS.MATH.CONTENT.3.OA.A.1 — Interpret products of whole numbers as equal groups.
- CCSS.MATH.CONTENT.3.OA.A.3 — Use multiplication and division within 100 to solve word problems.
- CCSS.MATH.CONTENT.3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation.
- CCSS.MATH.CONTENT.3.OA.B.5 — Apply properties of operations as strategies to multiply and divide.
- CCSS.MATH.CONTENT.3.OA.C.7 — Fluently multiply and divide within 100; know all products of two one-digit numbers from memory.
- CCSS.MATH.CONTENT.3.NBT.A.3 — Multiply one-digit whole numbers by multiples of 10 from 10 to 90.
- CCSS.MATH.CONTENT.4.OA.A.1 — Interpret a multiplication equation as a comparison.
- CCSS.MATH.CONTENT.4.OA.A.2 — Multiply or divide to solve word problems involving multiplicative comparison.
- CCSS.MATH.CONTENT.4.NBT.B.5 — Multiply a whole number of up to four digits by a one-digit number, and two two-digit numbers.
- CCSS.MATH.CONTENT.5.NBT.B.5 — Fluently multiply multi-digit whole numbers using the standard algorithm.
- CCSS.MATH.CONTENT.5.NBT.B.7 — Add, subtract, multiply and divide decimals to hundredths.