In this chapter we begin with presenting the concept of multiples with the beads, then common multiples with beads and working with common multiples on paper, to extend the knowledge of factors up to one hundred. The pegs and board are used to find the lowest common multiples and on paper the children use Charts, Chart C gives an intuition into factors, before factors are explicitly explored and Prime Factors found by the child of 9-12 years.
Material Description:
The first two exercises can be done with any of the two short bead chains, of none-hierarchical colours and tickets. Select none sequential bars and ones over five so there is enough segments of bars to give examples.
2 sets of boxes of bead bars
Method:
Small group presentation
Concept of multiples
Ask the child to skip count the short chain of six.
Lay out the tickets vertically at the side
Say, ‘Today I will tell you something new about these numbers, six is a multiple of six, twelve is a multiple of six, eighteen is a multiple of six, twenty-four is a multiple of six, thirty is a multiple of six, thirty-six is a multiple of six, so what is a multiple? A multiple is a number which contains another number an exact multiple of times, what does that mean? It means there is nothing left over.
So, we said six is a multiple of six, how any times does six contain six? (once) How many times does twelve contain six? (2) What about eighteen?’, Continue to thirty-six, stressing that nothing is left over, there is no remainder.
Ask the child to lay out the tickets.
‘What have we been talking about, we have been saying that six, twelve, eighteen, twenty-four, thirty, thirty-six, are all multiples of six.’
Repeat, asking the child to lay out the beads and tickets, ask the child, ‘Why is X a multiple of Y?
Mix the tickets from the chains, ask the child, ‘Is X a multiple of six or nine? (or both) Lets place it.’
Ask the child to continue laying out the chain, placing the tickets and reading the multiples and to explain that they are all multiples, because there is nothing left over
Aim:
Introduce the language of multiples
Notes:
This work is essentially sensorial and language giving.
Casually mention ideas like, that is the first multiple and that some multiples belong to more than one chain (common Multiples.
Do this as simply as possible without over complicating it, or over simplifying it (to do this we use longer short chains
A multiple is a number that contains another number an exact number of times. (The first multiple is always the number itself)
A common multiple is a number that contains more than one number an exact number of times.
The lowest common multiple is the first number we reach that contains all our numbers exactly.
A factor is a number that divides into another number exactly.
A prime factor is any factor of a number which is a prime number.
The highest common factor is, in a set of number, the highest number which will divide evenly into all of them.
When to give the lesson:
This is early work, which can be given once the children know their multiplications tables.
Lots of work is done with the concept of multiplication before factors are introduced.
After the lesson:
The children can continue with the long chains, maintaing their own tickets.
Follow up work
The children with common multiples
Common multiples
Material Description:
Two boxes of bead bars
Method:
Summarise earlier work, decide which box to use for the multiples and which to use for the product.
Take a bar of two from the top bead box and place it horizontally, saying, ‘We are going to take two once’. From the bottom box of beads take out one bar of two and lay out a vertical bar of two, beneath the multiple of two. ‘Two taken twice.’ Lay out two bars of two, horizontally one below the other and a bar of four under the beads. ‘Two taken three times’ Take three bars of two, place them horizontally, vertically below each other and their product, the bar of six, directly beneath them. ‘Two taken four times’, place four bars of two and a bar of eight beneath.
Continue placing the beads until seven
‘Look at what we have already, two taken once, is two, two taken twice if four so four is a multiple of two, two taken three times is six, so six is a multiple of two…’
Continue placing the beads until ten
These are some of the multiples of two, two, four, six, eight …there are some of the multiples of two.
Now let’s do the multiples of three, it is not necessary to line up the next set of multiples directly under the previous ones
Read through the multiples of two and three, ask the child if they notice anything. Give examples of common multiples and define them, it contains more than one number an exact number of times, six is a common multiple of two and three, so is twelve and eighteen. Ask, ‘What did we discover about eighteen, six and twelve? We found that they are common multiples.’
Tidy up by taking the bars of common products first
After the lesson:
The children can continue making multiples of four, five and six, and so on
Multiples of Numbers (papers)
Material Description:
Paper squares with arrangements of numbers, coloured pencils
Method:
With multiples of four, ask, ‘What is the first multiple of four, the first multiple is always the number itself’. Show the child how to make an attractive shape around the first multiple, e.g. a triangle or circle and draw around 4. Ask, ‘What comes next?, What is the next multiple?’ highlighting them up to 100, occasionally breaking off to summarise the numbers circled so far, stressing the idea that these are all the multiples of four between one and a hundred.
When it is complete look at the pattern formed, they might like to connect them, joining the multiples.
With Common Multiples
The children do the multiples of two numbers, e.g. 3 and 5 on the same sheet.
Have different colours for different multiples, and form different shapes, e.g. 3, blue in a circle and 5, green squares. When you read a common multiple, e.g. 15, 30, 45 draw both shapes.
At the end read through all the common multiples.
Title the page, ‘Common multiples of the numbers three and five’
Notes:
This is early work, the children can count if necessary.
The exercises reinforce the language of multiplication and common multiples and extend the work beyond chains while limiting it to one hundred.
It is an indirect preparation for prime numbers
Tables A, B and C
Material Description:
Tables A, B and C and a control
Method:
Table A
Sitting with a group of children who have tables of their own work through Table A. Begin by saying, ‘Here we have some tables, we can use them to do more work with multiples. What is a multiple?’ (a number which contains another number an exact number of times). While filling out the first vertical column say, ‘Two taken once is two, two is a multiple of two, two taken twice is four, four is a multiple of two, two taken three times is six, six is a multiple of two…’ Continue systematically with the multiples of two, three, four to ten.
The children must complete a table and put their name on it, because they will need them later, finishing at least a column at a sitting.
After the child can check it with a control. They can also read it out loud and use a social control of error.
They can continue to work with other blank pages working however they like.
Table B
Once Table A is complete and you are sitting with a group of children, who have tables of their own work through Table B. Say, ‘Where did we finish on Table A?’ (10 x 5 =50), ‘Now we continue with Table B, two taken twenty-six times is fifty-two, two taken twenty-seven times is fifty-four, and what are these numbers?’ (multiples of two). Continue systematically with the multiples of two, three, four to ten.
The children must complete a table and put their name on it, because they will need them later, finishing at least a column at a sitting.
After the child can check it with a control. They can also read it out loud and use a social control of error.
They can continue to work with other blank pages working however they like.
Table C
Once Table A and B are complete they can be used by the children to complete Table C. Say, ‘We can also find multiples with Table C. You can see that on the Table C we have the numbers 1-50, like in Table A and then the numbers 51- 100 just like in Table B. Looking at Table C, the first number we see is ‘1’, do we have a product of 1 on Table A?’ (no) Leave this. ‘Let’s look at the next line on Table C, it says ‘2’. Do we have the number ‘2’ on Table A? (yes) ‘We do we have ‘2’ taken once, let’s make an agreement when we see a number taken once we will ignore it because it really doesn’t tell us anything more about the number. I am going to put a red line under the number ‘2’ on Table C. Do we have a number ‘3’ on Table 1’ (yes) ‘Now, remember our agreement, I am going to ignore this because it is taken once, so I will just draw a red line under ‘3’. Now, do we have an answer of 4 on Table A?’ (yes) ‘We have two taken twice is four, I will write the multiplication ‘2×2 on Table C, Do I have the answer ‘4’ elsewhere?’, (yes) ‘but it is taken once so I will ignore it. Now let’s look at 5, it is only taken once so I ignore it and write a red line, Now, let’s look at ‘6’, Can you see a product of 6,?’ (yes), ‘Two taken three times is six so I can write this multiplication’, Write 2×3 ‘I can also see that three taken twice is six so I record that’, Write 3×2 and six taken once, but I don’t write this.’
Continue underlining prime numbers and writing the factors. If possible try to complete the chart in one sitting, if the children are restless complete one column and come back to it later.
Unless the child asks before hand, review the chart, saying, ‘Let’s look at the numbers with the red line under them, 2, 5, 7, 11, 13… these numbers cannot be written in any other way, apart from a multiple divided by one, so they are called Prime Numbers’.
Aim:
To introduce the child to the language and concept of prime numbers and to indirectly prepare for factors.
Notes:
If the children aren’t sure they can continue adding when completing Table A and B, only when the children are comfortable with their tables begin Table C.
The children may ask about prime numbers when working with the papers with multiples earlier, after using Chart C bring the children’s attention to the red line which indicates a prime number and give the terminology before introducing Prime Factors.
Lowest Common Multiplies: Pegs and board
Material Description:
Peg board, box of hierarchically coloured pegs, little sticks, tickets or mathematical alphabet, squared paper
Method:
First example with consecutive numbers
Recap, what is a multiple and what is a common multiple? Say, ‘We are going to look for common multiples, we are going to look for two, three and four’. Place tickets with ‘2’, ‘3’ and ‘4’ horizontally along the top of the peg board frame and place that amount of green pegs vertically beneath them. Say, ‘The rules is you must add to the shortest line first’ Place a stick beneath the lowest pegs under each number, ask, ‘What is my shortest like?’ (2) add two green pegs and place the stick. Then ask, ‘Now what is my shortest line?’, (3) place three pegs and a stick. Then ask, ‘What is my shortest line?’ (2 and 4) ‘Which would you like me to add to first?’ Place two under the ‘2’ and four under the ‘4’. ‘Now what is my shortest line?’ (3) add to this. Continue.
Say, ‘Look, they are all in a horizontal line, when we get a horizontal line we have reached the common multiple, what is the common multiple?’ (12) Say, ‘Mathematicians have a way of talking about these, they call it the lowest common multiple, now thats a lot to say every time you want to talk about it so Mathematicians take the first letters, L for Lowest and a full stop, C for common and a full top and M for multiple and a full stop. Write, ‘L.C.M. 2, 3, 4 is 12’
Notes:
This is a simple, sensorial introduction to lowest common multiples
Use 2,3, and 4 because you reach the lowest common multiple quickly and without running out of space in the units.
Second example, without consecutive numbers
Say, ‘We want to find the common multiple of six and twelve’, put out the tickets, ‘6’ and ’8’, six and eight unit beads below and place the stick. Say, ‘So, what is our rule, our rule is to add to the shortest line’. Put six beads beneath the six and say, ‘I think I could do some changing here, let’s exchange ten units for one ten,’ Do this, counting the pegs from the bottom upwards, put the ten in the column left of the units, and the stick at the end of the unit line. Say, ‘Now I must add to the shortest line, but I need to remember the numbers, what is the lowest number now?’ (8) Add eight beads and then exchange. Ask the child, ‘Which line has the lowest value?’ and continue.
Summarise, saying, ‘We have come to a straight line, the lowest common multiple of six and eight is twenty-four. Let’s record that’. Write ‘L.C. M. 6, 8 = 24’
After completing an example say, ‘What its the lowest common multiple? It is the lowest number which contains all out numbers evenly, the smallest number to contain these numbers evenly’.
Aim:
To provide concept, language and notation for work with the lowest common multiple.
Notes:
Working with the change is less mechanical, the children have to perceive the value of the beads without seeing them.
Choose numbers to present which will not go on for two long
When to give the lesson:
After all the previous work on multiples
Factors (concept): Pegs and Board
Material Description:
Pegs and board, squared paper and pencil and green bowl
Method:
‘Today we are going to look at some numbers, we are going to look at eighteen, I am going to write this on squared paper’. Count out 18 unit pegs into the bowl. ‘I am going to see how many ways I can make eighteen, can I make groups of two with eighteen?’ Place the pegs in the bowl in pairs, arranging them in a pattern, beginning high on the board. ‘Yes, I can make groups of two with 18, I will record that’. Record ‘2 yes’ on the paper below the number 18 ‘Now I will go again, can I make groups of three with eighteen’, count out another eighteen pegs into the bowl and place the pegs on the board in a different pattern to separate them from the groups of two. Continue with four. ‘I cannot make groups of four with eighteen, I will record that on paper and remove the beads, I only leave out the beads that can be divided.’ Now try five, once it is obvious to the child that groups of five are impossible stop, put the beads back and record the result, ‘no’, try to make groups of six, record the positive result and leave the beads out, continue trying seven and all the numbers till nine, even if the children can see you can’t make the groups.
‘Ask the child to look at the paper and the peg board and ask her if we can make groups of ten or more (no), We have groups of 2,3, 6 and 9. We can make groups of 2,3,6 and 9, these have a special name, these are factors of nine, which comes from the Latin ‘facere’ which means, ‘to make’, each of these divides into eighteen exactly.’
Give another example systematically searching for groups in twenty-one, this time title the page, ‘Factors of 21’.
Key:
Introduces concept and to finding the factors of a number
Notes:
We search systematically, we are interested in making groups, not how many groups.
The factors of a number are always lower or equal to it, the multiples of a number are the number itself and numbers higher.
Prime numbers are those which have only that number and 1 for factors
If the child asks about dividing numbers into 1, we say, one is a factor, but it doesn’t tell us anything new about a number so we don’t do it here.
When to give the lesson:
After Chart C
After the lesson:
Encourage a lot of exploration, the children can take examples from Chart C and discover the link with prime numbers.
Prime Factors
Material Description:
Chart C, the child’s or classroom control
Method:
‘Remember what a factor is? It is a number which divides into another number evenly? I am going to look at eighteen again. Let’s find eighteen on Chart C to record all the ways we can make eighteen, do this on squared paper. These are the way we can make eighteen. Using Table C we are going to see if there is any other way to make eighteen’. Look at the chart, searching for the fist digit in the multiplication 2, ask, ‘Is there another way to make two?’ (no), ‘Is there another way to make nine?’, (yes) write 3×3. Look at the next multiplication, ask, ‘Is there another way to make three?’ (no), ‘Is there another way two write 6?’ (yes) Change it for 2×3. Continue for the other two multiplication
Say, ‘We know that 2 and 3 are prime numbers, these are prime factors of eighteen’.
Check by multiplying the prime factors.
Give another example, with 24. When the child breaks down 2X12 to 2x2x6 ask for a different way to make six and record 6 as 2×3 immediately, discuss six but do not record it. We try to get to 2 or 3 every time if it is possible. Continue to use Chart C.
Notes:
If you have not done so already discuss the word prime numbers with the red lines on Chart C before starting
When to give the lesson:
After a great deal of exploration with multiples, Chart C – which gives a link to Prime Numbers and factors this is given.
The commutative law is assumed here, take the first way to form a number you come across, for 6, that is 2×3, don’t consider 3×2.
After the lesson:
Children continue to explore, they realise that some numbers have prime numbers, some have many, some have a few others have none, but 12, 36 and 96 have many and these are always interesting to explore
Lowest Common Multiple (using prime factors)
Material Description:
Pegs and board, blank tickets, long black strips instead of short sticks
Method:
Recap work with multiples, factors, prime factors.
‘Today we want to see if we can find the lowest common multiple of twelve, twenty-four and thirty-six, we are going to use Prime Factors when we do this’
Write the tickets and put out the tens and unit pegs to represent the numbers in the same way that was done for finding the Lowest Common Multiple. This time use the black line vertically instead of the sticks.
‘We are going to use Prime Numbers, going through each number one at a time, let’s start with twelve. Remember I said I would use prime numbers? I will try to use my lowest prime number first, which is two, can I divide twelve by 2? (yes) Write ‘2’ on a ticket and place it at the top by the twelve beads, ‘If I divide twelve by two, what do I get?’ (6) Place six pegs alongside the black line. ‘Can I divide my six by two?’ (yes) Write a ticket for ‘2’ and place it by the six pegs and take two pegs. Ask, ‘Can I divide 2 by 2?’ (yes) write a ticket for ‘2’ and place one peg beneath. Say, ‘When we have only one peg we have finished’
‘Now we will do twenty-four, can twenty-four be divided by two?’ (yes) ‘What will I get?’ (12) Write the ticket of ‘2’ and place twelve pegs along the line. ‘Can I divide this by 2?’ (yes) Place a ticket of ‘2’ and place six pegs. Ask, ‘Can I use my lowest prime still?’ (yes) ‘If I divide 6 by 2 what will I get?’, (3) Write a ticket and place three pegs. Ask, ‘Can I divide 3 by 2?’ (no) ‘So I will try my next lowest multiple which is 3, can I divide three by three?’ (yes) Now put a ticket ‘3’ and take 1 bead.
With 36, ‘Can I use my lowest prime number?’ (yes) Write a ticket for ‘2’ and place 18 beads below, ‘Can I divide thirty-six this by 2?’ (yes, it gives 9) Place another ticket for ‘2’ and 18 pegs,. Ask, ‘Can I divide this by 2’ (yes) Place ticket of ‘2’ and nine pegs, ‘Can I divide this by 2 ?’ (no) ‘by 3?, the next lowest prime number?’ (yes) Write a ticket for ‘3’ and take three pegs, Say, ‘Three divided by 3 is one, there is no more space, I can put it on the frame, I have finished.’
Say, ‘We were looking for the lowest common multiple of 12, 23 and 36, now we will write them’ Record all the prime factors (written on the tickets) for the first number which is 12, which is 2x2x3, then look at the factors of 24, this time only record additional numbers, so while there is 2x2x3x2 we only record one more 2. Then look at the factors of 36, there are 2x2x2x3x3, the 3 is the only additional number, so we record only ‘x3’.
L.C. M. 12,24, 36 = 2x2x3x2x3 =72
‘So the lowest common multiple of 12,24 and 36 is 72’
Aim:
Preparation for working in the abstract, the children now have another way for finding Lowest Common Multiples.
Notes:
The lowest common multiple must contain the other numbers evenly.
Show three numbers because it allows for more exploration, the children can work with less or more.
The limitations of the space afforded by the material move the children on to work with paper and in abstraction.
After the lesson:
The children can use many numbers
Highest Common Factor
Material Description:
Pegs and board, blank tickets, long black strips instead of short sticks
Method:
Lay the board out as for finding the lowest common multiple.
Say, ‘I will find the highest common factors of 24 and 36’. Place the tickets for ’24’ and ’36’ at the top of the frame, put out the black strips and pegs. Work the same with the pegs.
‘We use the lowest prime number till we can’t use it anymore. Can I divide 24 by 2?, What will I get?’ Place a ticket ‘2’ and put out 12 pegs. ‘Can I use two again? And what will I get?’ (6) Place ticket ‘2’ and six pegs. ‘Can I use two again? And what will I get?’ (3) Place ticket ‘2’ and three pegs? ‘Can I use two again?’ (no) ‘Can I divide it by 3?’ (yes) Place the ticket ‘3’, say ‘What will I get?’ (1) place ticket ‘3’ and one pegs. Say, ‘Now I have one, I am finished’.
‘Now we use the lowest prime number to divide 36. Can I divide 36 by 2?, What will I get?’ (18) Place a ticket ‘2’ and put out 18 pegs. ‘Can I use two again? and what will I get?’ (9) place ticket ‘2’ and nine pegs? ‘Can I use two again?’ (no) ‘Can I divide it by 3?’ (yes) ‘And what will I get?’ (9) Place ticket ‘3’ and nine pegs. ‘Can I divide this by 3?’ (yes) ‘And what will I get?’ (3) place ticket ‘3’ and three pegs. ‘Can I divide this by 3?’ (yes) ‘And what will I get?’ (1) place ticket ‘3’ and one peg. Say, ‘Now I have one, I am finished.’
‘Now we have the prime factors of 24 and 36, there is something different we want to find today, that’s the highest common factor, so we are going to record in a different way.’
‘What are the prime factors of 24? I am going to write them down.’
Write 2 x 2 x 2 x 3
Say, ‘And the prime factors of 36?’ Write 2 x 2 x 3 x 3
‘What I am looking for is the highest common factor’, write Highest Common factor, it take s a long time to write, so we write H.C.F. 24, 36
‘I have to look at what is common to both groups’, only write the numbers both groups have, so write ‘2 x 2 x 3’, say, ‘These are the only ones which are common to 24 and 36. Now we multiply’, 2 x 2 x 3, that is 12, write ‘=12’ and asa control ask, ‘is 12 contained in 24 and 36? (yes)
Another example with 27 and 15.
Lay out the tickets and board, this time we find that not all numbers can be divided by 2 and 3.
‘Now we have the prime factors of 27 and 15,
‘What are the prime factors of 27? I am going to write them down.’
Write 3 x 3 x 3
Say, ‘And the prime factors of 15?’ Write 3 x 5
Write H.C.F. 27, 15 = 3
Aim:
To give the concept, language and technique to find the highest common factor
Notes:
When taking the quotient count it into your hand before placing it.
When to give the lesson:
After the child has worked with
The lowest common multiple and Highest Common Factor: Paper
Material Description:
Squared paper and pencil
Method:
‘What I am looking for now is the lowest common multiple of 12, 30 and 36’. Underline the numbers and draw a vertical lines beneath them.’
‘So what have we got, 12, I am going to find the lowest common factors, first, what it the lowest prime number (2) can I divide 12 by 2, (yes) I have six, can I divide it by 2 again, (yes,3). can I divide it by 2? (no) three’ (yes, I have one).
Do the same for 30 and 36
| 12 | 30 | 36 |
| 12 2 | 30 2 | 36 2 |
| 6 2 | 15 3 | 18 2 |
| 3 3 | 5 5 | 9 3 |
| 1 | 1 | 3 3 |
| 1 |
Remember what am I looking for, it’s the lowest common multiple of 12, 30 and 36
We take all of the first number 2 x 2 x3 and I taken from the second number those numbers I haven’t taken already, which is x5, in the third number there is an extra 2, so I take that
Write L.C.M. 12, 30 and 36 = 2x2x3x5 x3
We already know how to do exponential notation. So we can rewrite that, 2 appears twice so we can write it as 22, how many times does 3 appear? 9twice) so I can write 32 and five (once) so we multiply it by 5
Write = 22 x32 x5
We know that 22 is 4, 32 is 9 and then we times it by 5
=4 x 9 x5 =180
To find the highest common factor we list what is common to all the groups H. C. F 12, 30 and 36 = 2×3 = 6
Another example with 13, 30 and 21.
Find the common factors as before
L.C. M. 13, 30, 21 = 13 x 2 x 3 x 5 x7
= 2,730
H. C. F 13, 30 and 21 = none. Say, ‘No number fits into all of these’
Aim:
Children discover that every number have a lowest common multiple but not every group will have a highest common factor.
Notes:
Odd numbers are less likely to give a highest common factor
We lay out the work in this way to replicate the peg board
In schools this may be show an a factor tree
Note on whole chapter:
This is an early exercise are for six and seven year olds, the Charts A, B and C are for seven and a half to eight year olds, the lowest common multiple with pegs, factors and prime factors are for eight to nine year olds. Introduce the Highest Common Factor at nine.Lowest common multiple and highest common factor on paper is for children aged nine and above.
This chapter does not deal with a single operation, but the finding of patterns in numbers, to be used in relation to other chapters, particularly when trying to reduce fractions.
Any even number except two can be expressed as the sum of two prime numbers.
When to give the lesson:
After exponential notation
Then give the children word problems using multiples and factors