Whenever there is a back‐to‐basics movement in education, the issue of learning ‘times tables’ (and other basic facts) arises. This seems to happen very frequently in England, most recently in 2014 with the introduction of a new curriculum. To a large extent this argument about rote learning times table facts is irrelevant for dyslexics and probably many other students, too. In our combined experience of over 50 years of teaching mathematics to dyslexics, we have found that the rote learning of times tables is a frustrating exercise for both learner and teacher (see also Miles 1983; Pritchard et al., 1989; Chinn, 1995a; Turner Ellis et al., 1996; Threlfall and Frobisher, 1999; Geary 2003). Yet still there are unrealistic expectations which should not be applied to all children, dyslexic or not.
It is one of our hopes for the future that, as neuroscience becomes increasingly sophisticated, there will be neurological evidence as to why this task is so difficult for so many children, and what solutions are possible.
Rote Learning Strategies
We believe that there is an effective alternative solution to this problem. Although within this chapter we suggest a highly effective rote learning technique, we believe that strategies, or derived fact strategies (Dowker, 2005) based on patterns and the interrelationships of numbers are effective in learning how to work out times table facts, a principle stated some 18 years ago in the Primary School Mathematics Curriculum Document for Ireland: Teacher Guidelines (1999):
All children can gain from using strategies for number facts making them, in effect, number combinations. They can learn the ‘easy’ number facts first (×1, ×2, ×5, ×10) and use these to build up the others using doubles, near‐doubles and patterns of odd and even. These strategies are of particular help to children with memory problems, but can also develop a more secure knowledge of these facts in all learners.
These strategies give the learner routes to an answer, as opposed to him or her relying on only using memory, which will give no possibility of obtaining a correct answer when he or she forgets the fact (Chinn, 1994) or a wrong answer inhibits the access to the correct fact. Again, our experience is that many children already use strategies, as others have found (Gray and Tall, 1994; Bierhoff, 1996), which they have devised for themselves, though often these strategies are neither consistent nor organised mathematically.
Siegler (1991) writes that: ‘Children often know and use many strategies for solving a class of problems. Knowing diverse strategies adds to the children’s flexibility in solving problems.’ and: ‘Children’s strategy choices may be less subject to conscious, rational control than often thought.’ Our experience is that strategies need to be taught and organised, through working with what the child already knows and uses. We have tried to use strategies which are developmental, for example the strategy used to work out 7 × 8 by breaking down the single step to 5 × 8 plus 2 × 8 will be used for products such as 23 × 54 and later for algebraic expressions (Wigley, 1995). We are mindful of the Bransford et al. (2000) Key Finding 2:
To develop competence in an area of enquiry, students must:
have a deep foundation of factual knowledge,
understand facts and ideas in the context of a conceptual framework, and
organise knowledge in ways that facilitate retrieval and application.
It is the importance of (b) and (c) in supporting (a) that we are demonstrating in this (and other) chapter(s).
McCloskey et al. (reported in Macaruso and Sokol, 1998) hypothesise that the process involved in the retrieval of arithmetic facts is separate from those involved in the execution of calculation procedures. This is further support for the principle of not holding a pupil back in maths just because he cannot retrieve basic facts from memory.
Hattie’s (2009) meta‐analysis of education research found that meta‐cognitive strategies and Piagetian programmes were among the most effective in influencing learning.
Use of music
There are CDs of times tables set to music. The rhythm and the tune help some to learn the tables, but in our experience, it is not the panacea.
Use of ‘fun’ games
These are rarely fun for very long and still rely on rote learning. If it works then use it, but remember that the ability to retrieve something from memory does not guarantee that you understand it.
The ARROW technique
If the child is to be encouraged to learn by rote, then this technique is powerful, but, as ever, it will not be successful for every child. It does not claim to be ‘fun’ other than giving some learners the rewards of success.
The learner can use the ARROW technique to rote learn these facts (Lane and Chinn, 1986; Lane, 1990; 2012). This is a multisensory method using the learner’s own voice which, in the initial trials in the 1980s, was recorded on audiotape. Now it is possible to record the data onto computer and use visual input via the screen to provide a multisensory presentation.
The child types into the computer the table facts he wishes to learn.
He records them onto the computer, in his own voice, leaving a 3–5‐second gap between each fact.
He puts on headphones and listens to and sees just one fact.
He listens again to the fact several times.
If the learner sub‐vocalises the fact as he hears it this can be a powerful aid to learning. This can be a very effective method for many people, but as with so many interventions, not for all. The process should be repeated several days in a row for the same set of facts. The learner will probably find that 5–10 facts per session are enough, but success has a great motivating effect, so more may be possible. The goal is to achieve long‐term retention.
Learning by Understanding
There are many advantages in learning times table facts by understanding. The methods we advocate provide memory ‘hooks’ on which to hang several connected facts, and some of them are introductions to procedures used later on in mathematics, such as in estimation, long multiplication and algebra. This is part of the developmental aspect of this approach to teaching maths. The strategies suggested here encourage the learner to look for patterns and interrelationships between numbers and operations; they help develop a facility with numbers and an understanding of algorithms. They may also enhance pupil confidence.
It is our experience that the basic structure for the strategy approach links with the times table square, even though initial work is with separate tables. The square gives an overview of the task, encourages interrelating facts, also presents facts as division and can be used to illustrate gains in an encouraging way. Also a student can learn how to fill in a blank table square (partially), a useful technique which makes good use of some of the extra time which may have been allocated for an examination.
There are 121 facts in the table square (Figure 7.1). The size of this task can be reduced quite quickly and easily. This progress can be readily shown to the learner and contrasts with the normal approach of, ‘Which times tables facts do you know?’ It is worth noting that for some children this mass of information, presented at one time may be overwhelming: too much information. You (the teacher) should ask the child to look at the table with you, to see several helpful things.
There are patterns, some easier than others, for example the column and row for the ten times facts is 10 20 30 40 50 60 70 80 90 100, the numbers from 1 to 10 with an extra digit, a 0, at the end (see also place value). If information can be seen to be in patterns or if it can be organised in patterns, it is easier to learn. There is also a pattern in the sound of the words for the ten times facts which links to the numbers one to nine: ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety – one of us (SC) sometimes uses ‘tenty’ as well as one hundred to reinforce the verbal pattern, to emphasise the place value need for a special word for 100 and to refer to when in subtraction a hundred is renamed as ten lots of ten, ‘tenty’.
There are other patterns in the square that the child can look at later. At this stage you are providing an overview. You must use your professional judgement to see how far to go at this stage without becoming counter productive. It is meant to be a supportive and motivating activity, putting a positive perspective on the number of facts that they do know.
Numbers which do not appear
Not all the numbers between 0 and 100 appear, for example 43 and 71. This does not mean they are not important. They are just not part of this collection of numbers.
Limiting the task
The numbers on the square have a lowest value of 0, and a highest value of 100. This gives some limits for the task, making it appear possible and, with a little understanding of how numbers relate to each other, even more possible.
Currently (2015) children in England are expected to learn the 11× and 12× facts. This seems, in our metric world, to be a superfluous requirement. It is worth noting that the 12× table was found to be the most difficult for primary children in Smith and Teague’s (2014) study. The use of strategies, particularly combining 10× and 2× facts for 12× facts, makes this an unnecessary rote task.
Remember that each time the child learns a set of facts, shading in known facts on the square shows that the task remaining is getting smaller. Furthermore, when he learns a fact from say the five times table, for example 7 × 5 = 35, he also learns 5 × 7 = 35, two facts for the price of one. This commutative property can be introduced quite early in the work and, like all interventions, revisited and reviewed frequently until it is thoroughly internalised by the student.
The order in which to learn the facts
It seems sensible to learn first the facts which lead to the quickest and most secure gains and therefore encourage confidence. You may wish to change the order given, but we suggest that the first five remain set as shown. Our order is based on:
The facts that a dyslexic or dyscalculic learner is most likely to know (0×, 1×, 2×, 5× and 10×)
The type of strategy advocated
By the time the child has learnt the times tables facts for 0, 1, 2, 5 and 10, he has reduced his task from 121 facts to learn down to 36 (and this can be almost halved to acknowledge the commutative property of ab = ba). These first 85 facts are the easiest to learn and you can demonstrate how the child can start to make significant gains.
Check‐backs/reviews
Constant reviews are important. You are often dealing with severe long‐term memory deficits and incorrect previous learning. It helps to revise and review material with the child quite often and, as with all skills, without practice the skill level decreases. This is especially so with dyslexics and dyscalculics. We maintain that those learning‐check charts you see with headings ‘Taught, Revised, Learnt’ should also have a fourth column, ‘Forgotten’. When S.C. lectures, and it is the same response in many countries, he often asks teachers and parents, ‘What are the biggest problems you face with children learning maths?’ One of the most frequent responses is, ‘He learns it that evening, but by the next morning he has forgotten it again.’
The Commutative Property
The commutative property is expressed algebraically as:
It can be introduced to a child as a way of getting double value for most of the times table facts that he learns. Obviously this is not so for the squares such as 4 × 4 and 9 × 9.
One of the models or images used for ‘a times b’ is area. Base ten (Dienes) blocks, Cuisenaire rods, square counters and squared paper are useful to illustrate this model. To illustrate the commutative property a learner can draw a rectangle of 4 × 10, oriented to have the side of 10 units horizontal, then he can draw a second rectangle, 10 × 4, with the side of 10 units vertical. These areas could be representing rooms or carpets. If it is not obvious that the two areas are the same, then the learner can cut out the two rectangles and place them on top of one another to show they are the same size (Figure 7.2).
Another illustration of this property can be achieved with Cuisenaire rods. So for 3 × 5 and 5 × 3, three five rods (yellows) can be put down to make a rectangle and then five three rods (light green) can be placed next to them to show that 3 × 5 and 5 × 3 cover the same area (and that three lots of five and five lots of three are the same) (Figure 7.3). Area is a powerful model for the developmental aspect of this work, leading ultimately to algebraic expressions.
Another effective demonstration which focuses on the ‘lots of’ version of ‘times’ is to use square counters in rows and columns. This additional ‘picture’ reinforces and develops further understanding of the concept of multiplication. For example, twelve counters can be placed down as three rows of four and a second set of twelve counters placed down as four rows of three (Figure 7.4).
Each of these demonstrations looks at a different facet of multiplication and each has future currency. This suggests that all three should be used to demonstrate and reinforce the concept. Examples of future currency for these demonstrations are using area to provide a picture of multiplications such as (a + b) (a + 3b), and extending 5 × 8 from five lots of eight to six lots of eight.
You have then demonstrated that 4 × 10 is exactly the same as 10 × 4, that 5 × 3 = 3 × 5, that 3 × 4 = 4 × 3 and so on. Each fact the child learns can have the order changed round, giving him another fact – free. You may wish to digress to discuss squares, such as 4 × 4 – as ever, judge the readiness of the individual.
Learning the Table Square
Zero: 0
Zero is an important concept, so time should be spent establishing that the child has some understanding of zero; zero, nought, nothing – as ever, the language should be varied.
In later numeracy work the child will meet examples like 304 × 23 or 406 ÷ 2, where the process of multiplying a zero, multiplying by zero or dividing into zero is used. The introduction of a zero into any computation increases the number of errors. Zero is not well understood.
You can start by explaining the meaning of 3 × 0 and so on: 3 × 0 means:
3 times 0
or three lots of zero gives the answer zero.
0 × 3 is the same as 3 × 0, zero lots of three is also zero.
This is another example of the need to use varied language to present a comprehensive image of the concept.
Two suggested teaching models
Talk about having nothing in one pocket, nothing in two pockets and so on.
Use empty plastic cups and discuss how much in one empty cup, two empty cups and so on.
The child should then realise that any number times zero equals zero and zero times any number equals zero. So:
1 × 0 = 0
and
0 × 1 = 0
2 × 0 = 0
0 × 2 = 0
3 × 0 = 0
0 × 3 = 0
4 × 0 = 0
0 × 4 = 0
5 × 0 = 0
0 × 5 = 0
6 × 0 = 0
0 × 6 = 0
7 × 0 = 0
0 × 7 = 0
8 × 0 = 0
0 × 8 = 0
9 × 0 = 0
0 × 9 = 0
10 × 0 = 0
0 × 10 = 0
Children like enormous examples such as ‘a billion lots of zero’ or ‘zero lots of a billion’ – it impresses much more than ‘zero lots of two’ even if the result is the same.
Now you can tell the child to look at the table square.
‘You will see a row of 0 s across the top, and a column of 0 s down the left hand side. You have just learnt your first 21 facts.’
Progress check
If you want the child to keep a check on his progress use the table square in Figure 7.1. Copy one and hand it to the child to act as his record of progress. Tell him to shade in all the zero facts – the top row and the first column. You will probably find the child needs a second table square to keep as a ‘clean’ copy.
One: 1
One is the basic unit.
So, 1 × 4 has the same answer as 4 × 1.
Any number times 1 equals that number.
One times any number equals that number.
Multiplying a number by 1 does not change its value. (This pedantic statement is important for work on renaming a fraction, where the fraction’s value does not change, but its appearance will.).
Counters are quite a good manipulative aid for demonstration. They can also be used on an interactive screen or via an animated slide with PowerPoint (e.g. as shown in www.mathsexplained.co.uk), or for the child to use to understand ‘one lot of n’ or ‘n lots of one’.
The concept you are introducing here is summed up by the equations:
Again, tell the child to look back at the table square and see that the one times table facts appear twice, first written across, second row down; then written down, second column in.
0 × 1 = 0
1 × 0 = 0
1 × 1 = 1
1 × 1 = 1
2 × 1 = 2
1 × 2 = 2
3 × 1 = 3
1 × 3 = 3
4 × 1 = 4
1 × 4 = 4
5 × 1 = 5
1 × 5 = 5
6 × 1 = 6
1 × 6 = 6
7 × 1 = 7
1 × 7 = 7
8 × 1 = 8
1 × 8 = 8
9 × 1 = 9
1 × 9 = 9
10 × 1 = 10
1 × 10 = 10
Again, explain and demonstrate the important fact that the number you multiply by 1 does not change in value. When 1 multiplies a number it leaves the number with the same value as it had before.
The child has now learnt 19 new facts (he had already learnt 0 × 1 and 1 × 0), making a total so far of 40 out of 121 – almost a third.
Progress check
The child can now shade in the 1× facts. He can shade in the second row and the second column. These are the numbers 1–10 across and down.
Ten: 10
1, 2, 3, 4, 5, 6, 7, 8, 9 are single digits. 10 has two digits, a 1 followed by a 0. The 0 means zero ones, and the 1 means 1 ten. It is necessary that the child has retained earlier work on place value from Chapter 5. A moment’s reinforcement will check this. Because maths is a developmental subject there are many opportunities to review, revisit and revise previous topics.
Ten is a key number in this chapter (and, indeed throughout the book). The ten times table facts will be extended to teach the child how to work out the five times table facts and the nine times table facts (and can also be extended to access the 11×, 12×, 15× and 20× facts). Thus, it is well worth reviewing the child’s understanding of ten and place value.
So explain that 20 has a 0 digit for 0 ones and a 2 digit for 2 tens. 2 × 10 means:
2 times 10 equals 20
2 lots of 10 are 20
There is an easy pattern to show this and the commutative facts:
0 × 10 = 0
10 × 0 = 0
1 × 10 = 10
10 × 1 = 10
2 × 10 = 20
10 × 2 = 20
3 × 10 = 30
10 × 3 = 30
4 × 10 = 40
10 × 4 = 40
5 × 10 = 50
10 × 5 = 50
6 × 10 = 60
10 × 6 = 60
7 × 10 = 70
10 × 7 = 70
8 × 10 = 80
10 × 8 = 80
9 × 10 = 90
10 × 9 = 90
10 × 10 = 100
10 × 10 = 100
Get the child to listen to the pattern as he says the ten times table and hears the oral connection, for example:
Four tens are forty
Six tens are sixty
Nine tens are ninety
Even two tens are twenty gives a two‐letter clue. We find that sometimes a brief digression to ‘twoten’ to ‘twoty’, ‘threeten’ to ‘threety’ and ‘fiveten’ to ‘fivety’ reinforces rather than confuses.
It is worth noting that the sounds of twelve and twenty are frequently confused, as are thirteen and thirty.
The auditory and visual clues to each answer within the ten times tables enable the student to access an answer without having to count from 1 × 10 to the required answer (which many children do to access the answers for the 2× table facts).
This pattern can be practised with trading money, always remembering to have the child say as he trades one 1p coin for one 10p coin, ‘one times ten is ten’. He then trades two 1p coins for two 10p coins, and says ‘two times ten is twenty’.
And so on, till he trades 10 × 10p coins for ten 10p coins, and says ‘ten times ten is tenten or tenty’; there are no such words of course and a special word is used instead – hundred. A hundred, 100, has three digits, the only number with three digits in the table square. A hundred pence has its own coin, a pound. So 10 × 10p = 100p = 1 pound. All this reinforces the special importance of 100.
There are other ways to practise the ones/tens relationship.
Single straws, and bundles of ten straws:
Each time ‘ten times bigger’ means exchanging a ten‐straw bundle for a single straw.
Base ten (Dienes) blocks, a one hundred bead string or a metre rule can be used to add to the development of the idea of the ten times table. Base ten blocks reinforce the place value aspect.
Remember that some materials are proportional in size to their value, for example Cuisenaire rods; some are proportional in quantity, for example bundles of straws; some are proportional by volume and area, for example base ten blocks; some are proportional by length, for example a metre rule; some are representative of value, for example 1p and 10p coins. Using a mixture of these and the numbers/digits ensures development from concrete to symbolic understanding.
Some ‘everyday’ examples may be used to provide reinforcement:
How many legs on ten cows?
How many wheels on ten bikes?
How many pence in ten 5p coins?
How many legs on ten spiders?
How many sides on ten 50p coins?
Progress check
If the child is comfortable in his ability to retrieve the ten times facts from memory then he can shade in the end column and the bottom row of his table square, the ten times facts. Filling in the tens column and the tens row should remind the child that for each times fact he can write the numbers in either order, so, for example, 2 × 10 = 10 × 2. This means that, if he remembers that 10 × 3 = 30, then he knows 3 × 10 = 30, one fact from the three times table and one fact from the ten times table – one again, two for the price of learning one. The commutative property should be reinforced frequently.
The goal for retrieving these core facts is automaticity.
So far the child has learnt 57 facts, almost half. He has 64 to learn.
Two: 2
First, as for each number, the lessons should look at the concept of the number and its interrelationships with other numbers. There can be some demonstrations, discussions and some information on two, such as:
Two is one more than one.
It is twice as big as one.
It is an even number.
Even numbers are numbers that share/divide into two equal parts.
Some examples:
Eight divides (shares) into two lots of four:
Twenty divides into two lots of ten:
Each child can try equal sharing with a random pile of pennies, sharing them out, one at a time, into two piles. If the two piles are equal, then he started with an even number. If there is one penny left over, then he started with an odd number.
The even numbers from 1 to 10 are 2, 4, 6, 8, 10
The odd numbers from 1 to 10 are 1, 3, 5, 7, 9
A useful extra fact, actually a generalisation, here is that any even number ends in 2, 4, 6, 8 or 0, and any odd number ends in 1, 3, 5, 7 or 9. Some review/revision questions can be used, such as:
Which of these numbers is even? 2341, 4522, 57399, 34, 70986, 11112, 335792
Which of them divide evenly by two?
If the pattern for even numbers is:
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Continue the pattern to 102.
The 2× table and the commutative facts:
0 × 2 = 0
2 × 0 = 0
1 × 2 = 2
2 × 1 = 2
2 × 2 = 4
2 × 2 = 4
3 × 2 = 6
2 × 3 = 6
4 × 2 = 8
2 × 4 = 8
5 × 2 = 10
2 × 5 = 10
6 × 2 = 12
2 × 6 = 12
7 × 2 = 14
2 × 7 = 14
8 × 2 = 16
2 × 8 = 16
9 × 2 = 18
2 × 9 = 18
10 × 2 = 20
2 × 10 = 20
Note that the ones digit pattern repeats: 2, 4, 6, 8, 0.
The first four facts can be learnt as a chant:
‘Two, four, six, eight, who do we appreciate?’
This chant brings the child to almost halfway in learning the two times table.
Often it is useful to have reference points in calculations. The child already has a start reference point, 1 × 2 = 2, and an end reference point, 10 × 2 = 20. The middle reference point has its value on the child’s hands – two hands, each with five fingers, two lots of five fingers is ten fingers, 2 × 5 = 10 or 5 × 2 = 10. It also is illustrated by 5 × 2p = 10p, a trading operation, where five 2p coins are traded for one 10p coin. So 5 × 2 = 10 is the middle reference point on which to build the remaining facts 6 × 2 to 10 × 2.
The answers for 6 × 2 to 9 × 2 have the same last digits as the first four facts, 12, 14, 16, 18 – show the child the pattern. The numbers have the same digit pattern because 6 × 2 is one more two than 5 × 2 and 7 × 2 is two more twos than 5 × 2, etc., and because 5 × 2 has zero in its ones digit column. This is a use of the strategy of a middle reference point, which in this case combines with the strategy of breaking down numbers to build up on known facts (6 = 5 + 1, 6 = 5 + 2 and so on).
So if the child can remember the reference value 5 × 2 = 10, he can quickly work out, say, 7 × 2.
Seven is 5 + 2
7 × 2 is five lots of two plus two lots of two:
There are three useful and regularly occurring strategies here:
Breaking down a number into key/core numbers (1, 2, 5, 10) as with breaking down seven into five and two, so that these facts are used and extended.
The use of a reference point in the middle of the task. Many children will claim to ‘know’ the two times table; however, it is often the case that when asked for 7 × 2 they begin at 2 and work up 2, 4, 6, 8, 10, 12, 14. A middle reference point means that the child can start at 10.
The use of ‘lots of’ for times leads to six lots of two being seen as one more lot (of two) than five lots of two. This image of multiplication as repeated addition of the same number and addition of ‘chunks’ of that number will be used again later.
However effective they may be eventually, strategies still need practice and reinforcement.
Some practical work can be built around coins and trading, using a 10p coin for tens and 2p coins for two. The learner trades five lots of 2p for a 10p coin to reinforce the middle reference point and the repeating 2, 4, 6, 8 pattern. An example with 8 × 2 is to take eight 2p coins, take out five of these and trade them for a 10p coin. This leaves one 10p coin and three 2p coins, which combine as 10 and 3 × 2 to make 16. This reinforces the image of 8 × 2 breaking into clusters of 5 × 2 and 3 × 2.
Trading is a procedure that will be used in addition and subtraction. The issue of consistency arises again here for this procedure, with the range of words in use for trading, including renaming and decomposing.
Four: 4
As always, it is worth starting with an overview, looking at the properties of four and relating four to other numbers. The most useful of these relationships are:
Four is two times two. 4 = 2 × 2
Four is twice two (be careful as some children interpret ‘twice’ as ‘add two’).
Four is an even number.
Numbers which are divisible by four can be divided by two twice.
Four is four ones added together 1+ 1 + 1 + 1.
Four times facts are accessed by doubling the two times facts. This also has the benefit of revisiting the two times facts. This strategy also introduces the procedure of two‐step multiplications. So 4 is used here as 2 × 2. Later, learners will multiply by numbers such as twenty by using two stages, ×2 and then × 10 (or vice versa). This × 2 × 2 method is building on knowledge the child has already learnt and makes use of the interrelationships of numbers. The child is taught to double the two times table. You have to establish the strategy, using methods such as 2p coins set out to show the two times table alongside the four times table, for example:
2 × 2 compared with 2 × 4
3 × 2 compared with 3 × 4
The two times table is shown with single piles of 2p coins and the four times table is shown with double piles of 2p coins. The child can be shown, for example, the piles for ‘three lots of’ and it should be possible to convince him that he ends up with twice as much from piles with 4p in as he does from three piles with 2p in. (This is a similar strategy to comparing the five times and ten times tables.)
Once the idea of the strategy is established, you can move on to comparing the answers to the 2× and 4× tables in the same way that the 5× and 10× tables will be compared.
1 × 2 = 2
4 = 1 × 4
2 × 2 = 4
8 = 2 × 4
3 × 2 = 6
12 = 3 × 4
4 × 2 = 8
16 = 4 × 4
5 × 2 = 10
20 = 5 × 4
6 × 2 = 12
24 = 6 × 4
7 × 2 = 14
28 = 7 × 4
8 × 2 = 16
32 = 8 × 4
9 × 2 = 18
36 = 9 × 4
10 × 2 = 20
40 = 10 × 4
It is worth reminding the learner that he already knows 0 × 4, 1 × 4, 2 × 4 and 10 × 4 from the tables he has learned previously.
1 × 4 to 5 × 4 are achieved by doubling within the range of the two times table, for example the learner can manage 4 × 4 as 2 × 4 = 8 and 2 × 8 = 16 and thus 4 × 4 = 16. Some practice to reinforce this ‘known’ pattern may be needed.
6 × 4 and 7 × 4 are relatively easy since there is no carrying to complicate the second doubling:
6 × 2 = 12
12 × 2 = 24
7 × 2 = 14
14 × 2 = 28
The second doubling for 8 × 4 and 9 × 4 can be done via breakdown strategies, using 8 as 5 + 3 (2 × 10 + 2 × 6) and 9 as 5 + 4 (2 × 10 + 2 × 8) or 10 – 1 (2 × 20 – 2 × 2). Alternatively, 9 × 4 can be done, later, as 4 × 9 from the nine times table.
It may be good practice for the learner to give you the first step in practice sessions so that 7 × 4 is delivered in two stages: 14 then 28.
When the four times facts are shaded in on the table square the learner has just 36 facts to learn.
(Note: The eight times facts can be accessed by a triple multiplication of 2× 2× 2×. Although this is a long strategy it does take the learner to an answer instead of them feeling helpless when they have no recall from memory available for this fact.)
Five: 5
As with all the times tables, the first step is to establish a basic understanding of the number.
Some information about five
A key fact is that five is halfway from zero to ten. The learner can be reminded how five was used as a halfway reference point in the two times table.
Five is half of ten.
Ten divided by two is five. 10 ÷ 2 = 5.
Five is an odd number.
Even numbers multiplied by five have zero in the ones place.
Odd numbers multiplied by five have five in the ones place.
Five can look like: 5 or l l l l l or l l l l or V or 10/2 or 10 ÷ 2 or .
The five times table and the commutative facts
0 × 5 = 0
5 × 0 = 0
1 × 5 = 5
5 × 1 = 5
2 × 5 = 10
5 × 2 = 10
3 × 5 = 15
5 × 3 = 15
4 × 5 = 20
5 × 4 = 20
5 × 5 = 25
5 × 5 = 25
6 × 5 = 30
5 × 6 = 30
7 × 5 = 35
5 × 7 = 35
8 × 5 = 40
5 × 8 = 40
9 × 5 = 45
5 × 9 = 45
10 × 5 = 50
5 × 10 = 50
Note:
The child knows the start reference point 1 × 5 = 5, and the end reference point 10 × 5 = 50.
There is a repeating pattern in the ones digits: 5, 0, 5, 0, 5, 0, 5, 0, 5, 0.
This means another pattern/generalisation: an odd number times five gives an answer that ends in five and an even number times five gives an answer which ends in zero, thus providing another example of the concept of odd and even numbers.
It is useful to set up a comparison of the 10× and 5× tables by writing the answers side by side. Looking at the answers illustrates the relationship between them, that is, each 5× answer is half of each 10× answer, for example 6 × 5 = 30 and 6 × 10 = 60, and 30 is half of 60. It is possible to work out the fives by taking the tens and halving the answers. So, for 8 × 5, start at 8 × 10 = 80 and half of 80 is 40. As a check, eight is an even number, so the answer ends in zero. Again, for 5 × 5: 5 × 10 = 50 and half of 50 is 25. As a partial check, five is odd, so the answer ends in five.
This strategy will be useful later when working out key number percentages.
The strategy of looking at the ones digit also helps to reinforce the child’s attention on reviewing answers to check their validity.
Some practical work
The learner can practise halving tens by trading 10p and 5p coins; for each 10p trade one 5p. Each time you can help the child rehearse the process:
‘Seven times five. Start at seven times ten. Half of seventy is thirty‐plus five, that is, thirty‐five. Seven was odd. The answer ends in a five. That checks.’
This can be reinforced by taking seven 5p coins and explaining that each coin is worth half as much as a 10p coin and that collectively they are worth half as much as seven 10p coins.
If the child has difficulty in dividing 30, 50, 70 and 90 by 2, remind the child how to break numbers down, for example 50 is 40 + 10. Halve 40 (answer 20) and halve 10 (answer 5), so that 50 + 2 = 25.
Again, you may have to remind the child how sometimes it is easier to use two small, quick steps than to struggle with one difficult step.
Other materials may be used to reinforce this relationship between five and ten; these include Cuisenaire rods, and patterns of dots. The digits can also demonstrate the ‘lots of’ aspect of multiplication, used when extending knowledge of, say five ‘lots of’ to six ‘lots of’ or seven ‘lots of’:
Six lots of five
5 + 5 + 5 + 5 + 5 + 5
Seven lots of five
5 + 5 + 5 + 5 + 5 + 5 + 5
As before, the goal is for the learner to be able to recall a five times table fact from memory, or work out an answer quickly and to get as close to automaticity as is possible. The strategy is designed to give security and consistency in the answers and inhibit the influence of incorrect answers. Starting from 1 × 5 and counting up to the required answer is not the goal. When the learner can remember or work out the five times facts, then he can shade in the five row and column on his table square. The times table task is reduced to 25 facts.
Three: 3, six: 6 and seven: 7
The strategy used for these times table facts is the same and is very much a part of the developmental nature of this programme. The strategy is to break down a ‘difficult’ number into two ‘easier’ numbers. So 3 becomes 2 + 1, 6 is 5 + 1 and 7 is 5 + 2. This procedure is used in long multiplications such as 35 × 78, where the 35 is broken down to 30 + 5 and the two partial products, 30 × 78 and 5 × 78, are then recombined for 35 × 78. Whilst this example is the procedure advocated by many texts, most people do not need to use it for easier numbers such as 3, 6 or 7. We have extrapolated the method back to help with retrieving basic facts. This also serves as an introduction to the area model for partial products in multiplication and, it is hoped, sets up this concept for future work in other multiplications such as fractions and quadratic equations. The Figures 7.5 to 7.7 illustrate the model and its developmental property.
The language of multiplication can be quite abstract. ‘Three times four’ or ‘three fours’ requires the student to know the code. ‘Three lots of four’ is more closely related to multiplication, hinting at repeated addition. This latter wording also lends itself to the concept of 3n = 2n + n again setting the foundations for the development of further maths skills and concepts. The same concept applies for 6n = 5n + n and 7n = 5n + 2n.
So the 3× table is calculated by taking a 2× table fact and adding on one more, for example:
and the 6× table takes a 5× table fact and adds on one more multiplicand, for example:
and the 7× table takes a 5× table fact and adds on the appropriate 2× table fact, for example:
Nine: 9
The key fact is that nine is one less than ten.
There is an easy method to work out the 9 times facts using fingers. If we were being rigidly principled, we might not mention a method which is radically different from the other methods and strategies mentioned in this book. However, working with dyslexics can make you very pragmatic and eclectic, on the basis that the gains in the child’s self‐confidence may outweigh any doubts about the academic validity of a particular technique.
So, if you want to know the answer to 4 × 9, for example, put the fingers of both hands down on a surface and tuck back the fourth (4) finger from the left (Figure 7.8). The answer lies each side of this fourth finger, the tens to the left, three fingers means thirty, and the units to the right, six fingers, giving an answer of thirty‐six. Often parents like this method.
Figure 7.8 Using fingers to find the nine times facts.
However, we prefer a strategy with potential for further use, a developmental strategy. Therefore, the strategy we advocate is based on estimation, the particularly useful estimation of ten for nine, and the subsequent refinement of this estimation. The strategy could also be perceived as a breakdown/partial products method, with a subtraction of the partial products rather than the addition we have used so far.
The first step is to establish the principle of the method, that is, that nine is one less than ten. This is helped by examining nine.
Nine is nine units.
Nine is one less than ten.
9 = 10 − 1 and 10 = 9 + 1.
The closeness in value of nine and ten can be demonstrated by showing the child a pile of ten 1p coins and asking him to say, without counting them, if there are nine or ten. It does not matter which the child guesses. It is the uncertainty that is important; the nearness of nine and ten makes it hard to give an answer with certainty. The demonstration can move on to Cuisenaire rods. A ten rod (orange) and a nine rod (blue) are placed side by side. A one rod (white) is added to the nine rod to show that the difference is one. This is presented in numerals as:
9 + 1 = 10
10 − 1 = 9
This demonstration is now extended to show how to estimate and refine from the 10× table facts to the 9× table facts.
Two nine rods are placed on a flat surface. Two ten rods are placed alongside and two whites are added to the nine rods to show that the difference in value is two. The process is repeated to develop the pattern that n nine rods are n ones less than n ten rods insert (See Figure 7.9). In numbers:
Thus any nine times fact can be worked out from a ten times fact, for example 8 × 9 is worked out as:
8 × 10 = 80
80 − 8 = 72
8 × 9 = 72
This is verbalised as:
‘Eight times nine is eight less than eight times ten. Eight times ten is eighty, so, eight times nine must be seventy something.’
The ‘something’, the unit digit, can be found by subtracting eight from eighty, or eight from ten, using number bonds/combinations for ten (another example of revisiting the key facts). It can be found by counting backwards from 80, though this is a very difficult task for some dyslexics, or a further pattern can be used:
1 × 9 = 9
2 × 9 = 18
3 × 9 = 27
4 × 9 = 36
5 × 9 = 45
6 × 9 = 54
7 × 9 = 63
8 × 9 = 72
9 × 9 = 81
10 × 9 = 90
Notice:
The units column digits go from 9 to 0, whilst the tens column digits go from 0 to 9. This results in the sum of the two digits in each answer always being nine, for example for 63, 6 + 3 = 9.
So the child can work through the following process for, say, 4 × 9.
4 × 10 = 40;
4 × 9 is smaller and must be ‘thirty something’;
The ‘something’ must be the number which adds on to 3 to make 9, that is 6. So the answer is 36. For 8 × 9, again:
The child may think that this is a long process, but with regular practice it becomes quicker. Also, as the child becomes more adept, he starts to short‐circuit the process and use it to top off a half‐known answer. In other words, the strategy provides a memory hook for the child so that he is not left floundering when faced with an ‘impossible’ question.
When the child has grasped this strategy, he may shade in the nine times column and row. He now has only one fact left to attack.
Eight: 8
8 × 8 can be accessed by repeated multiplication, that is:
Final notes
The squares, 3 × 3, 4 × 4, 5 × 5, 6 × 6, 7 × 7 and 8 × 8 are connected to the products of the numbers ‘each side’ of them, that is 2 × 4, 3 × 5, 4 × 6, 5 × 7, 6 × 8 and 7 × 9 respectively by:
n2 = (n − 1)(n + 1) + 1
3 × 3 = (2 × 4) + 1
3 × 3 = 9
2 × 4 = 8
4 × 4 = (3 × 5) + 1
4 × 4 = 16
3 × 5 = 15
5 × 5 = (4 × 6) + 1
5 × 5 = 25
4 × 6 = 24
6 × 6 = (5 × 7) + 1
6 × 6 = 36
5 × 7 = 35
7 × 7 = (6 × 8) + 1
7 × 7 = 49
6 × 8 = 48
8 × 8 = (7 × 9) + 1
8 × 8 = 64
7 × 9 = 63
Developmental aspects
Multiplication facts are, of course also division facts. For example, when students are factorising equations in algebra or reducing fractions to the simplest form they will need to use the times table facts in division format. The maths vocabulary used is pertinent to the process, for example:
For 30 ÷ 5
How many fives in thirty? … ‘five’ comes before ‘thirty’
How many times does five go into thirty? … ‘five’ comes before ‘thirty’
What is thirty divided by five? … ‘thirty’ comes before ‘five’
Share thirty dollars between five people. … ‘thirty’ comes before ‘five’
Five people share thirty dollars equally. … ‘five’ comes before ‘thirty’
Division facts should be linked to multiplication facts and children need to know that whilst multiplication is repeated addition, division is repeated subtraction (of the same number). The multiplication square can be used for division. The student may need guidance in doing this as directional issues are involved.
There are some simple clues as to divisibility and factors for some of the factors. Some are obvious, such as the rules for a number being divisible by two (and thus four and eight) or by ten or by five. The rule for divisibility by nine has already been mentioned, that is, that all the digits will add up to nine, though this may take more than one step. For example, 4914 adds up to 18 and then 1 + 8 = 9.
The rule for divisibility by three is that the digits will add to three, six or nine. If the number is also an even number then it will divide by six.
These simple rules allow the learner to deal with divisibility by 2, 4, 5, 6, 8, 9 and 10.
Times table facts and exams
If a candidate is not allowed to take a times table square into a maths examination it is likely that he will be able to have a sheet of squared paper on which he can make a 12 × 12 grid (needed for the 0× to 10× facts). A student can be taught how to fill in a blank square very quickly, especially if he chooses not to fill in every blank, say filling in only the core facts for 0×, 1×, 2×, 5× and 10×, leaving the others until needed for a specific question. It is good practice and good revision of the facts, and how they are related, to do this exercise reasonably frequently, especially in the run up to an exam. The square, of course, gives factors, too, and equivalent fractions. The square gives security.
Summary
In this chapter we have introduced the idea of teaching strategies to learn/work out the times tables facts, treating them as ‘number combinations’. We believe that this approach is pragmatic, since few dyslexics or dyscalculics can rote learn this information. It has the added bonus of introducing and teaching several useful mathematical processes and concepts, which include: estimation, factors, that number values and operations are interrelated, that multiplication is repeated addition and partial products, all via the strategy of breaking down numbers into convenient and appropriate parts. We hope that a child may, by using these strategies, learn to produce quick answers for the times table facts, whilst having a back‐up strategy for those occasions when the mind goes blank or is uncertain which of a number of possible answers stored in the brain is the right one. We have also tried to introduce some flexibility in the methods described, ever mindful of our basic premise that not all children learn in the same way.
Finally, it is worth repeating the cautionary note concerning division facts. We think that so many children perceive the multiplication square as the times table facts that they forget that they are also the basic division facts.
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