Understanding multiplication (PDF, 31 KB)
Multiplicative Strategies progression, 3rd step;
Number Facts progression, 3rd step
The purpose of the activity
In this activity, the learners move from a ‘repeated addition’ model of multiplication to one where they use multiplication facts. They identify ‘known’ and ‘unknown’ multiplication facts and use already known facts to develop quick recall of unknown facts.
The teaching points
- Using multiplication facts gives the same result as repeated addition but is more efficient
- compare 6 x 9 = 54 with 9 + 9 + 9 + 9 + 9 + 9 = 54.
- Learners find out which multiplication facts they already know and derive unknown facts from known facts.
- Learners understand that 3 x 4 represents three groups of 4 and 4 x 3 represents four groups of 3 and that they both equal 12 (the commutative property of multiplication).
- Knowing the commutative property of multiplication reduces the number of multiplication facts to learn.
- Discuss with learners the occasions when they have derived facts from those they already know.
- Objects – paper clips, screws, etc.
- A set of multiplication fact cards for each learner (small cards with a fact on the front and the answer on the back).
The guided teaching and learning sequence
1. Write 3 x 4 on the board and ask the learners to represent this problem with objects. Possible representations are three groups of four and four groups of three. Ask the learners to share their representations with each other and notice any differences. Write 4 x 3 on the board and discuss the fact that it represents four groups of 3 while 3 x 4 represents three groups of 4.
2. Ask the learners to find out the total number of objects and share how they did it. Listen for counting 1, 2, 3, 4, etc. (counting all), 4 + 4 + 4 (repeated addition) and knowing 3 x 4 or 4 x 3 = 12 (basic facts). Ask the learners to consider which method is quickest – considering larger numbers may make the point clearer (6 x 9 = 54).
3. Encourage the learners to notice that while 3 x 4 and 4 x 3 mean different things, the total number of objects is the same.
4. Give each learner a set of multiplication fact cards and ask them to sort out the facts into two piles; one containing the facts they can recall the answer to quickly (in under 3 seconds) and the other containing the facts they can’t. (Alternatively, the learners could work in pairs, taking turns to ‘test’ one another.)
5. Ask the learners to spread out the unknown fact cards and find pairs that have the same answer (for example 6 x 9 and 9 x 6). One of each pair can be removed from the ‘unknown’ fact pile.
6. Ask the learners to take an ‘unknown’ fact and share how they might work it out from a known fact.
If 4 x 7 is unknown but 2 x 7 = 14 is known, then 4 x 7 = 28 because it is 2 x 2 x 7.
If 6 x 9 is unknown but 6 x 10 = 60 is known then 6 x 9 = 56 because it is 6 x 10 – 6 x 1.
Explain to the learners that the aim is to eventually have all facts in their known pile. Have the learners store their two piles of cards in two envelopes so that they can be used in the “Deriving multiplication and division facts” activity to follow.
Ask the learners to find a known fact card from which they can derive the answer to an unknown fact card for each of their unknown facts. Ask them to explain how they solved the problem to another learner.