Racks and Tubes
Material Description:
Test tubes from units to millions in 7 colour-coded stands
Box with green, blue, red skittles
3 division boards
Squared/graph paper for recording
Pencil
Notes:
The quotient is always what the unit gets, this is given in the Casa in the group games.
The first few examples with the unit division board show what the children already know before you work further.
There are stages in recording
- The final quotient; single digit, double digits and triple digits
- Intermediate remainders; single digit, double digits and triple digits
- Conventional notation; single digit, double digits and triple digits
Each time a new concept is shown go back to using a single digit.
This work is very important because the sensorial aspect helps the child to understand the distributive aspect of long division.
The work is done in stages, beginning in late Casa until the children are eight, assuming they have all the pre-requisites.
Abstraction does not immediately proceed abstraction.
When to give the lesson:
After long multiplication, especially the large bead frame, the Laws of Multiplication have been introduced. The children need to be familiar with the concept of division – as sharing – given in the Casa with the golden bead material and the stamps.
Quotient, no remainders
Material Description:
one units board, squared paper, pencil, rubber, ruler
Method:
Say, ‘Today we are going to do some division with the racks and tubes, we are going to divide 7565 by 5. The racks are going to represent our divided and the skittles and boards our divisor’.
Take out the racks first and then the bowls you will need and put in the dividend, training the children to check that there are ten in each tube, to remove them carefully and to check what is in the bowl. Write the problem on squared paper. Then remove the unit board and the five green skittles, place them along the top.
Ask, ‘Which category do we begin with?’, (thousands). Place the thousands bowl above the board, share the beads among the skittles until you cannot share them equally.
Ask the children what each unit skittle got and record (one) ‘‘My answer is always what one unit gets, so what is that?’ (one thousand) so you record above the thousands.
Clear the board. Ask the children, ‘What category should we share next? (hundreds) bring the hundreds bowl and transfer the remaining thousand beads, so there are 2500s, replace the thousands bowl and turn in over.
Say, ‘There is a rule, you can only share the beads which are the same colour of the bowl, share the hundred beads amongst the skittles, then change a thousand bead for ten-hundreds. Ask the children what each unit skittle got and record (five) ‘Five of what?’ (five hundred) so you record above the hundreds. Clear the board, take the tens bowl and continue.
Read out the problem and quotient.
After the lesson:
The children continue work, suggest examples so that they don’t come across remainders just yet.
Quotient, with remainder
Say, ‘Today we are going to do something different with division with the racks and tubes, we are going to divide 6,427 by 3. The racks are going to represent our divided and the skittles and boards our divisor’.
Take out the racks first and then the bowls you will need and put in the dividend, training the children to check that there are ten in each tube, to remove them carefully and to check what is in the bowl. Then remove the unit board and the three green skittles, place them along the top.
Ask, ‘Which category do we begin with?’, (thousands). Place the thousands bowl above the board, share the beads among the skittles until you cannot share them equally. Ask the children what each unit skittle got and record (two) ‘My answer is always what one unit gets, so what is that?’ (two thousand) so you record above the thousands.
Continuing to give the language as before, clear the board. Ask the children, ‘What category should we share next? (hundreds) bring the hundreds bowl, replace the thousands bowl and turn in over. Share the hundreds and record one over the hundreds and replace the bowl. Clear the board, share the tens, making the necessary exchange, clear the board. Take the units bowl and share them equally, recording the beads that have been shared look back at the bowl, ‘We have one left, we will record it, to remember it is a remainder we write a small ‘r’ and the number 1 because we have one left.
Read out the problem and quotient.
After the lesson:
The children continue to work with their own examples with divisors in the units.
Final quotient with double digits
Material Description:
1 child uses one board for tens the adult uses the units board
Method:
Divide 39,464 by 32, express the dividend in beads in bowls. Check. Write the problem on squared paper. Beginning with the two highest categories share the beads onto the units and tens board. ‘You are going to start with the ten thousands on the tens board, I will take the thousands, which are ten times less’. When sharing, the highest category dictates, continue asking, ‘Can you continue to share?’, before sharing at the same time one bead per unit, only changing when necessary. Share them until the highest category can no longer be shared, equally, if the person with the highest category can continue but the person with the lower category cannot ask, ‘If you were to give me that ten I could share again?’, take one of the higher category beads and exchange it for a tube, counting that there are ten.
Record the quotient, ‘We always record the share of the units, what is the share of the unit? which category is it? Where shall I record?’, (the ten-thousands on the units board). Read from the vertical digit. If one category has a remainder follow the ‘units of the bowl’ rule, ‘Units can only be shared if they are the same colour as the bowl they are in’ share the units that are the same colour of the bowl first and then exchange the ones of the higher category for ten which match the bowl.
Clear the board and overturn the tens thousand bowl, pass the thousands to the child and take the hundreds, (if there were to be any remaining thousands they move with their bowl to the child, remainders of the highest category are transferred from the high category bowl to the next highest when the child receives it.) Say what each board gets You are going to continue with the thousands on the tens board, I will take the hundreds, which are ten times less’. Share these beads and continue to share the other categories. When all the categories have been worked through ask,‘Is there a remainder?’
Read through the problem and quotient and remainder 1,233 r8
Notes:
Lower categories can borrow form higher categories, but not the other way round.
When to give the lesson:
After sufficient practice with the units
After the lesson:
The children continue to work with their own examples with divisors in the tens, later the strongest child operates the unit board.
The final quotient and triple digits
Material Description:
2 children use two boards the adult operates the unit board, have the hundreds, tens and units divisor
Method:
Divide 6,438 by 234, express the dividend in beads in bowls (this example has many changes). Check. Write the problem on squared paper. Beginning with the two highest categories share the beads onto the hundreds and tens board. Say, (to the person on the hundreds board) ‘You are going to start with the tens of thousands on the hundreds board’ (to the person on the tens board) ‘you will put the thousands on the tens board, these are ten times less, and I will put the hundreds on the unit board, the hundreds which are ten times less again‘ Share the beads until the highest category can no longer be shared. Record the quotient, ‘We always record the share of the units, what is the share of the unit? Which category is it? Where do you record?’, (the thousands on the units board).
Clear the board and overturn the thousand bowl, move the bowls up, exchanging the beads as necessary, following the pattern used with double digits.
Read through the problem and quotient and remainder of 27 r120
When to give the lesson:
After sufficient practice with the units and tens
After the lesson:
The children continue to work with their own examples with divisors in the tens, later the strongest child operates the unit board.
Intermediate remainders; single digit
Method:
As before, the difference here is recording the intermediate remainder.
23,784 divided by 7
As 20,000 does not divide by 7, exchange one ten-thousand bead for 10 thousands and share.
The intermediate quotient is three thousand, it is recorded on the paper above the thousands place.
To record the remainder say,‘What we are going to do differently is when we record, we will record what is left over in the bowl. What is it? (one) One what? (one thousand). I am going to record it here, under the thousands’.
Clear the board, On your working, bring down the next category (7 hundreds) under the hundreds and adjacent to the one remainder. When you do this bring forward the beads representing the seven hundred to show the child the connection. You are now dividing 1,700 by 7.
Share as before, remembering ‘the rule of the bowl’, and record that each unit gets 2. Record what is left in the bowl (3 hundred remainders) under the hundreds, adjacent to the remainder. Clear the board. Bring down the next category, 8 tens on paper and with the beads.
Share, changing when necessary, remember ‘the rule of the bowl’. Record how much each unit gets, (five tens) and what is left in the bowl, (three tens, the remainder). Clear the board. Bring down the next category, four units, by recording on paper and moving the units bowl forward.
Continue sharing out the units, record the units (four). record the remainder under the units, then as there is nothing to divide further record the remainder to the right of the quotient r.
Read through the problem and quotient and remainder of 325 r.6
When to give the lesson:
After sufficient practice with the units
After the lesson:
The children continue to work with their own examples with divisors in the tens, later the strongest child operates the unit board.
Intermediate remainders; double digits
Material Description:
1 child uses one board for tens the adult uses the units board
Method:
As before, the difference here is recording the intermediate remainder.
92,998 divided by 64
Continue as before with two boards. When all the beads have been shared, record the quotients and remainders, clear the boards, pass the bowls to the left, with the remainders following the ‘rule of the board’. Change as necessary ensuring that there are enough beads for the highest category and making any changes for the second category that are obvious.
Continue, record the units and the remainder.
Read through the problem and quotient and remainder of 1453 r.6
When to give the lesson:
After sufficient practice with the units
After the lesson:
The children continue to work with their own examples with divisors in the tens, later the strongest child operates the unit board.
Intermediate remainders; triple digits
Material Description:
2 children use two boards the adult operates the unit board, have the hundreds, tens and units divisor
Method:
Divide 64,382 by 234
Continue as before,
Read through the problem and quotient and remainder
Notes:
Test our examples before you give them, let the children choose their own examples later
When to give the lesson:
After sufficient practice with the units
After the lesson:
The children continue to work with their own examples with divisors in the tens, later the strongest child operates the unit board.
Conventional notation; single digit
Material Description:
One units board
Method:
Divide 7,687 by 5 using the beads, bored and paper as before. Share the thousands bead and record the thousands quotient on the paper.
Say,‘What we are going to do differently we will record what we have used. What have we used up? (point to the beads to encourage her to multiply the divisor by the quotient, or she can count write a minus and the figure below, checking the category with the child, e.g. -5 below the thousands) Then ask, ‘What have we got left?’ (draw a line beneath the figure representing the number of beads used, do the subtraction on paper, writing only the meaningful ‘0‘s then check category by category with what is left in the bowls)
Clear the board. Bring the hundreds down on paper – adjacent to the 2 remainder and the hundreds beads forwards.
Divide the 26 hundred, record the hundred quotient, the hundreds dividend and remainder, asking the questions as before.
Clear the board and bring the tens down on paper and in beads, record as before, clear the board and bring the units down.
Aim:
Prepare for work in the abstract by the sensorial work now.
Notes:
To find the answer to the question ‘what have I used?’ we relate the process of multiplication to division but we do not say this explicitly at this stage.
It might be necessary to show the child how to exchange the numbers for subtraction while they do there own work, encourage them to ‘hold the numbers in their head’ rather than write a line through, do not use the term borrow, it is an exchange. Saying, ‘I’m going to take exchange by taking one from here, now I have X, so I can subtract…now what is this number? so Y from X’
Conventional notation; double digits
Material Description:
1 child uses one board for tens the adult uses the units board
Method:
Divide 8, 827 by 24 using the beads, bored and paper as before. Share the thousands and hundreds bead, ask, ‘can you share again?’ exchanging where possible as before, and record the hundreds quotient, three on the paper.
Say,‘We will record what we have used. What have I used up?’ (add from the lowest category, your board) ‘I’ve used up four taken three times, that’s twelve. I am going to record the ‘2’ under my hundreds and keep the ‘1’ in my head. What have you used up? (the child counts six) (point to the child to encourage her to multiply the dividend by the quotient), ‘That’s six thousand, plus the one in my head, that’s seven thousand, I am going to record that under the thousands. Then ask, ‘What have we got left?’ (draw a line beneath the figure representing the number of beads used, do the subtraction on paper, then check category by category with what is left in the bowls)
Clear the board. Move the bowls. Bring the hundreds down on paper – adjacent to the 2 remainder and the hundreds beads forwards.
Divide the hundreds and ten beads, record the quotient – six
Say, ‘Let’s see what I have used, I used four taken six time, thats twenty four, record four, I will write that, and keep the two in my head. What have you used up? two taken six times (12) plus the two in my head, (14)I will write that under the hundreds’.
Clear the board and bring the tens down on paper and in beads, record as before, clear the board and bring the units down.
Say, What have I used up? four taken seven times (28) record the 8, keep the 2 in my head, what have you got? two taken seven times, (14) plus the two in my head. Record 16
Notes:
After a number of times the children begin to see that division is a series of repeated subtraction.
Instead of counting the children should begin to calculate what has been used up by multiplication, as is modelled in the presentation
Conventional notation; triple digits
Material Description:
2 children use two boards the adult operates the unit board, have the hundreds, tens and units divisor.
Method:
As above, using three boards.
64, 382 by 234
Special cases
A ‘0’ in the tens
Material Description:
2 children use two boards the adult operates the unit board, have the hundreds, tens and units divisor. There is a great deal of changing, the child with the tens board will need to have patience.
Method:
Divide 51,252 by 202
The child with the tens board receives no skittles. The child with ten thousand must exchange and then the thousand child must exchange to give you enough beads to begin. The ten board get nothing, otherwise the procedure is the same.
Record at the level of the final quotient, no working below the written sum.
The answer is 247 r.123
Notes:
Children do not need to wait till the end of the work to discover these, they will come across examples when they work with their own numbers.
They can record at whatever level they are working at.
Special cases are always concerned with ‘0’ in the divisor, it is simpler to begin with ‘0’ in the tens place and record at the level of final quotient, this leaves the children able to concentrate on the new part.
A ‘0’ in the units
Material Description:
2 children use two boards the adult operates the unit board, have the hundreds, tens and units divisor.
Method:
Divide 19,293 by 370
The adult with the units board receives no skittles. The child with ten thousand must exchange and then the thousand child must to begin. The ten board get units, otherwise the procedure is the same.
Record the units even though no units will be on the board. Asking, ‘We record what the unit gets, the unit gets ten times less than the tens, what did the tens skittles get, how much do you think the unit skittle would have?
Notes:
As well as being careful of the empty category the children also have to work out the quotient for an unrepresented unit.
A ‘0’ in the units and the tens
Material Description:
2 children use two boards the adult operates the unit board, have the hundreds, tens and units divisor.
Method:
Divide 6,756 by 500
The adult with the units board receives no skittles, nor does the child with the tens. The child with thousands shares the beads amongst her skittles. The ten board get units, otherwise the procedure is the same.
Record the units even though no units will be on the board. Asking, ‘We record what the unit gets, What have the hundred skittles got (1000) What would the ten skittles get?, they are ten times less than the hundreds? (100) What would the unit skittles get, they are ten times less than the tens, (10), so we record 10.
Clear the board, move the bowls, share and record for the tens, units and remainders.
Group Division: Stamp Game
Material Description:
Stamp game
Units, tens, hundreds, thousands
Pencil and eraser for recording
Paper strip to represent division line
Neutral colour disc (not red, blue, or green)
Small coloured square of paper (1cm x 1cm) for remainder
Small white paper (2cm x 2cm) to record products
When to give the lesson:
After the racks and tubes, but don’t necessarily wait till after the child can write conventionally.
Method:
Unit divisor – in the units
Divide 9 by 3
Write the problem on a strip of paper and put the problem at the top of the mat
Say, ‘Today we are going to use stamps, I am going to take nine units of stamps, my dividend and lay them out in an orderly way here’. Place a long white strip horizontally towards the top of the mat. Take nine stamps place them vertically, counting them.
Say, ‘I am going to take my divisor’, Write it on a ticket and place it to the left of the dividend. ‘Let’s see if I can make groups of three’. Move the divisor down to the level below the line and group the units into three threes horizontally. ‘So how many units was I able to make?’ (3) ‘So that is my quotient, I write it on a ticket and put it above the line’.
Example with tens and units
‘This time lets take 16 and divided it by four’
Write the problem and place the line, Take a dividend of 16 stamps (one ten stamp and six units, placing them as in the stamp game and write the divisor. ‘The question is, can I make groups of four with sixteen? What should I do?’ (change) ‘How many groups of four did I make? (4) I have to record that and put it above the line, so four groups of four, that is my quotient’.
Show many examples at this level
Example with categories including 1,000, with a remainder in 1 category
‘Now we are going to go on to do some really interesting examples’
Divide 3,936 by 32
‘We are going to lay out our dividend in the usual way’…and write the problem…am going to need a disk this time.
‘Can I make groups of 32 with 3?’ (no) [This is a remainder]. Move the disk over to the hundred stamps. “Can I make groups of 32 with 39?’ (yes) place the 32 stamps – three thousands and two hundreds – horizontally. ‘I have made one group of 32, the disk shows me how to record it’, Record ‘1’ on a slip over the hundreds on the problem slip. ‘Can I continue?’(no). Place a slip with a ‘1’ over the problem.
Continue by moving the disk to the next category, ask, it is possible to make groups of 32 with the hundreds and tens. Form these and place a slip with the number of groups as before. Then move the disk to make groups with the tens and units.
‘How many groups of 32 did I make with 3,936? Read the slips placed over the line; ‘123’. There is no need to record at this stage.
To check multiply the quotient by the divisor to produce the dividend
Example with a remainder in 2 categories – with a single digit divisor
Divide 5,648 by 245
Proceed in the usual way, asking the children, ‘Can I divide 245 by 5? (no) 245 by 56? (no) 245 by 564? (yes) as you move along the categories bring the disk along the problem till it is possible to make groups – 2 groups of 245, ‘I used thousands, hundred s and tens so I record ‘2’ over the tens’.
‘I move over my disks and bring down my units. What am I working with? (758), I bring down my divisor, What do I have? (758 divided by 245) Do you think I can make groups of 245 with 758?’ (yes)
‘Arrange the group horizontally, can I keep going?’ (yes if I change – first remainder) Make the change, continue to make two more groups.
‘Can we go any further?’ (no) [second remainder] ‘I have a remainder and I must record it’, write r.13 on a slip and place it. So what have we got? We have 5,648 divided by 245 giving us 23 r.13.
There is no need to record at this stage.
To check multiply the groups by the intermediate quotients and add them.
Example with a remainder in 2 categories – with a two digit divisor
Divide 2,164 by 52
Proceed in the usual way, asking the children if we can make groups and moving the disk along the problem till it is possible to make groups. Make 4 groups of 52 horizontally and write the problem. Move the divisor and dividend down, form the units and remainder.
So what have we got? We have 2,164 divided by 52 giving us 41 r.32
There is no need to record at this stage. To check multiply the groups by the intermediate quotients and add them.
Example with a remainder in 2 categories – with a zero in the quotient
Divide 4,848 by 48
Proceed in the usual way, asking the children if we can make groups and moving the disk along the problem till it is possible to make groups. Make the group of 48 with 480. I can’t make a group of 48 with 40, record a 0 for the tens and 1 for the units.
Group Division: Abstraction
Material Description:
Squared paper and material for stamp game
Method:
Unit divisor – in the units
Divide 612 by 5.
‘This morning we are going to do something new’
Write the problem on squared paper, to conventionally record the division
Set up the line, dividend and divisor, using the same verbalisation, ‘Can I make groups of five with six?’…I was able to make one group of five’. Record in the usual way above the line and record the intermediate quotient on paper. ‘And what did I use up? This time find out on paper, not by looking at the stampsI used five taken once’, write -5 and do the subtraction saying, ‘If I took five once I used five’ and write it. ‘I am left with one in the abstract, lets see if I have one left here, I do’. Bring down the tens on paper and with the stamps.
‘Can I make groups of five with eleven?…What must I do?’ (change). Make two groups and record ‘2’ with slips above the unit and on the squared paper. ‘I have made two groups of five. I record on my square paper what I have used up, I have used five taken twice’. Write 2 ‘What’s that?’ (10) Write – 10 and calculate the subtraction.
‘I can check with my material…and I move the units down. Can I make groups of five with twelve? (yes)… I have two groups and I have a reminder of 2’.
Make the units and the remainder with the slips and on paper.
With an example in the tens
Divide 7,967 by 34.
Continue as above, make 2 groups of 34 with 79 in the hundreds, recording on slips and squared paper recording what has been used and subtracting it. Bring down the tens and make 3 groups in the tens, recording the intermediate quotient and what has been used as before, make the subtraction, saying, ‘If I took thirty four twice I used sixty eight’ and write this. Bring down the units. Make 4 groups of 34 with the units and a remainder of 11.
To check go from the abstract to the material. multiplying the amount of groups made with each intermediate quotients and adding, starting in the units. Do this in your head.
Aim:
To lead to abstraction, specifically by combine what the children know about notation from the test tube material and verbalisation from their work with the stamps.
Notes:
The focus remains on the groups
Bring down in the material before on paper
This is an almost unknowing passage to abstraction, aided by the verbalisation and with checks made between the material and paper calculations.
Do not make too many crutches for the children.
The work with the stamps gives the technique of group division.
Our emphasis is on how many groups are in the dividend, and secondly the multiplication of the divisor by each digit of the quotient.
Reinforce the idea that whatever was used up has to be subtracted or taken away.
The use of the disk emphasised the number of categories (digits in the divisor and the number of digits in the dividend, used up at any one time).
When to give the lesson:
This immediately proceeds abstraction, give at around age eight to nine.