Multiplication and division facts (PDF, 49 KB)
Number Facts progression, 3rd step
The purpose of the activity
In this activity, the learners learn strategies that will help them remember and recall the basic multiplication and division facts. The learners focus on learning those facts they are not able to recall quickly.
The teaching points
- Basic facts for multiplication are the combinations where both factors are less than 10.
- Recalling basic facts is often referred to as ‘mastery’ and means that a learner can give a quick response (within about 3 seconds) without having to work out the fact by a method such as counting.
- Number relationships are the foundation for strategies that help learners remember basic facts.
- All of the multiplication basic facts are conceptually related, which means you can figure out new or unknown facts from those that are already known. For example, if you know that 5 x 6 = 30, you can work out that 6 x 6 = 36 by adding one more lot of 6 to get 36.
- Division facts correspond to multiplication facts. For example, 4 x 5 = 20, 5 x 4 = 20, 20 ÷ 5 = 4, 20 ÷ 4 = 5. The exception to this relates to the facts involving 0. While 0 ÷ 5 = 0, there is no real number or meaningful answer to 5 ÷ 0. As an example, if you have $20 to share equally between 5 people, then each person receives $4. You can use this to illustrate the problem of dividing by 0.
- Let’s say you have $20 to share between 0 people and ask how much does each ‘person’ get? An attempt to calculate this is meaningless because the question itself is meaningless since there are simply no people to share anything with in the first place.
- Traditionally, drill has been the most popular approach used in schools for students to learn to recall their basic facts. However, the very fact that many learners are unable to recall their multiplication facts indicates that drill alone does not work for many people.
- It is important that the learners are not drilled in a basic fact until they at least have an efficient strategy for working it out. For example, if the learner has to skip-count or count in ones to work out 7 x 8 = 56, they are not ready to practise it for quick recall. Once they can work it out quickly, for example, by working from 6 x 8 = 48, then they could use drill or practice activities to develop mastery of the basic facts.
- There are hundreds of software programmes and websites that offer drill of basic facts. While these can help learners increase their speed of recall, you need to ensure that the learners first have efficient strategies for the facts included in the drill.
- Discuss with the learners the pros and cons of different ways of learning multiplication facts and any barriers they may have to learning them.
- A set of multiplication and division fact cards, preferably one set for each learner (masters available from the nzmaths website).
- Multiplication facts chart.
The guided teaching and learning sequence
1. First you need to develop an inventory of the known and unknown facts for each learner.
a) This can be done by ‘testing’ each learner on the multiplication basic facts, one at a time, using flash cards. If the learner responds quickly (within 3 seconds) and without obviously skip-counting or counting in ones to solve the fact, place it in their ‘known’ pile. Continue with all the facts until they are sorted into two piles – those that are known and those that need to be learnt. (Note: This approach is also described in the “Deriving multiplication and division facts” activity.)
b) Alternatively ask the learners to sort the facts into two piles and place in the ‘unknown’ pile those facts they are unsure about. Ask them to include any facts they solve by counting.
2. Give the learners a multiplication facts grid and show them how to record facts from their ‘known’ pile on the grid.
Recording the known facts on the grid allows the learner to see the facts they know and the ones they need to learn. The focus should be on developing strategies to learn the unknown facts. The remainder of this sequence is presented as a series of ideas or approaches to help the learners fill the specific gaps in their quick recall of the multiplication basic facts. Rather than working through each idea, choose the ones that best suit the learner’s gaps. As the learner builds on their mastery of multiplication facts, add these to the chart and to the ‘known’ pile of facts.
It is very important the learners understand the commutative property, for example, that 4 x 3 gives the same result as 3 x 4. Although some facts seem easier to master than others, it is important that the ‘turn-around’ facts are learnt together as this halves the number of facts that need to be learnt.
Facts that have 2 as a factor are equivalent to the addition doubles and should already be known by the learners. For example, if a learner knows 7 + 7 = 14 they should also know it as 2 x 7 = 14 and 7 x 2 = 14.
Zeros and ones
It is interesting to note that 36 of the 81 facts have either 0 or 1 as one of their factors. While these should be reasonably easy to learn, many learners seem to get confused by the ‘rules’ that are often associated with 0 and 1 facts. These facts are straightforward when learnt in a context. For example: if I have one bag of 6 apples, I have 6 apples in total.
Skip-counting in fives gives the learners practice in gaining a ‘feel’ for the numbers that are products of 5. The learners may also be able to connect to the fives facts if they think about them as being half of the tens facts.
The nines facts can be some of the easiest facts to learn as there is an interesting and helpful pattern in the products.
- The digits in the product sum to 9.
- The digit in the tens place is one less than the number of nines you are finding. For example, 6 x 9 = 54 (where 5 is one less than 6).
Although the nines pattern helps some learners, others find it confusing. The nines can be also worked out by taking one group away from the tens facts. For example, 9 x 6 is 6 less than 10 x 6 = 60, giving 9 x 6 = 54.
The left-over facts
After learning the 0, 1, 2, 5 and 9 facts, there are just 25 left to learn, and 10 of these are turn-around facts. In the chart below, the 10 turn-around facts have been left unshaded. The 15 highlighted ‘leftover’ facts can be learnt by deriving them from a closely connected fact. For example 3 x 3 is 3 more than 2 x 3. This approach is explained in detail in the “Deriving multiplication and division facts”.
Division facts linked to multiplication facts
Division facts tend to be more difficult to recall than the multiplication facts. Encourage the learners to learn the related division facts with the multiplication facts.
There are hundreds of websites offering practice activities for basic facts. Here are two websites where the user can select the numbers and the operation(s) practised in the activity.
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