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Division


Created on 06 January 2014 by ncetm_administrator
Updated on 30 June 2014 by ncetm_administrator

Division

This suite of videos includes consideration of the structures of division in terms of grouping and sharing at Key Stage 1; moving into a written algorithm at Key Stage 2 and demonstration of fluency at Key Stage 3.

You may find it useful to download the videos to your own device, using the links given beneath each video. If you encounter any problems, please contact us.

 

The structure of division - Key Stage 1

Woodberry Y2 - sharing and grouping

Sharing and grouping - whole class

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

Sharing and grouping - in pairs

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

 

Division - Key Stage 2

Springfield Y4 - place value counters for division

Moving to a written algorithm for division

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

Representing division with place value counters

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

Using place value counters and recording division

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

Division with remainders

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

Division with exchange

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

 

Division - Key Stage 3

St Marylebone - group working on problems

Problems involving division

Download the accompanying PowerPoint slides.

This video can also be watched on Vimeo, the video hosting site, where it is also possible to download a copy for your own use.

 
 

 


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Comments

 


25 January 2018 07:23
Really clearly shows what division is and how to manipulate numbers. Worked so well with my Year 3/4 class!
By davidtrace11
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06 June 2017 20:53
Where can I purchase magnet place value counters like this for demonstrating with? I can't seem to find them anywhere online...
By annette.s4
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14 November 2015 20:18
Really useful page of videos for modelling division progression. Thank you!
By gvickers
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03 March 2015 19:41
I am a him, but with a name like Bernie who is to know? :-)
By bjwestacott
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29 December 2014 09:37
Nicola - you might want to contact Bernie about this: you can click her username, then click the blue Send Message button.
28 December 2014 17:13
I have found this discussion thread very interesting. I was wondering if you have had the online conference you mentioned or if this is still to be organised? Thanks. Nicola.
By MissTinnion
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22 June 2014 12:51
I had another look at the video 'Representing Division with Place Value Counters'. You can hear the teacher referring to the divisor as the number OF groups (1 m 41s) and then switching to it being the number IN a group (2m 01s), then number OF groups (2m 04s). and then back to the number IN a group. This is easily done but it can confuse pupils. When doing divison, as either sharing or grouping, it is important that teachers distinguish clearly between the number IN a group and the number OF groups when speaking to pupils.When sharing, the divisor is the number OF groups and the quotient is the number IN a group; when grouping, the divisor is the number IN a group, and the quotient is the number OF groups. The Y2 video showing grouping and sharing with hoops and blocks was much more consistent on this point and it was easy for pupils to grasp the difference between the sharing and the grouping. Once we move to number discs, the language needs to be equally precise. PS These comments are not intended to be critical of the video but are the sort of points I make when working with teachers and my intention is to be helpful in getting this style of teaching into UK classrooms. Bernie Westacott
By bjwestacott
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15 June 2014 11:04
Thanks, Debbie. I will get in touch, hopefully later this week, and we could think about an online conference as that sounds a great idea. All the best, Bernie
By bjwestacott
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05 June 2014 08:59
Hi Bernie

Thankyou for sharing your thoughts, I do agree with you that children need to understand division in terms of equal grouping. There are two structures for division in the curriculum grouping and sharing and I am sure that you would agree that they need to understand both. The Year 2 video on division shows both. These Year 4 pupils have already been through that stage. There are a few key points in my thinking that I would like to share:

Using manipulatives and recording abstract mathematics at the same time can be very powerful. It allows the abstract symbols to take on meaning. We do have a tradition in this country of engaging in the concrete first before moving into the abstract. There is evidence from research that this often results in a gap between the concrete and the abstract and that children don't transfer their understanding from the concrete, resulting in an insufficiently deep understanding of the abstract. In Singapore they are more likely to introduce the two at the same time. They introduce the abstract earlier but keep the concrete going for longer. Thus the abstract symbols have the opportunity to take on meaning and learning is deeper and more sustainable. The practice of working in pairs is something we have sought to exemplify in several of the videos where one child uses the manipulatives and the other does the recording and then they swap over. What they are recording is what they see going on with the manipulatives so a strong link is made between the concrete and the abstract. I would direct you to the last video for Year 2 fractions, this is a good example where the abstract symbols of fractions take on real meaning for the children due to the manipulation of the fraction cards. I do think this practice is contributing to the high achievement of all the children in the class. Abstract and concrete are not in opposition. Mathematics is an abstract subject , the role of the concrete is to expose the structures and relationships embedded in the mathematics and for pupils to have sufficient depth of understanding to apply abstract mathematics to concrete contexts. They need to be able to move fluently and flexibly between the two.

Another reflection I would like to share is how we apply and think about place value in a standard algorithm. The 3 place value counters in the hundreds coumn assist children in thinking of the number not as 300 ones but instead as 3 ONE hundreds - they are 3 units of a hundred and with these 3 units of a hundred I can only make 1 group of 3. I can only deal with units of a hundred in the hundreds column. If there were 4 counters I would still only be able to make one group of 3, even though my left over counter has a value of one hundred ones. I have to exchange it for 10 tens and move it into the next column, then I can continue to make groups of 3's. The idea of place value and unitizing is a very powerful idea, it allows us to work with the mathematics in easier and more efficient ways. The five year old child who proudly says I know that 300 + 200 = 500 has begun to grasp this.

I could continue, but now getting too long, if you want to talk more, happy to do so over the phone, or perhaps we could do some sort of online conference where other could join us and share ideas. Children do find division hard and the new curriculum is a good opportunity to try and crack it. Thankyou Bernie for taking the time to share your thoughts.
05 June 2014 07:34
Thanks for your reply Debbie.i feel that the choice of division into equal groups is an easier introduction to division for children using manipulatives as the outcome is concrete and does not rely upon abstract recording for meaning, thus it has meaning for pupils of all abilities, whether they are ready to move to the abstract or are still at the concrete (maniupulatives) stage. Once the children move to abstract recording, there is no difference in either method and both rely upon a sound knowledge of tables. In the example you quoted, if you divide a number into groups of 11 you need to know how many 11s in that number; if you divide it into 11 equal groups then this still requires that you know how many 11s there are as if you put 1 in each group that is 11, if you put 2 in each group that is 2 lots of 11, etc. thus by the time the pupils are ready to move to abstract the two methods become about knowing and applying the 11 times tables. The vital thing is to make the initial learning experience as concrete as possible, i.e. it needs to be possible for pupils to carry out division only with maniupulatives and with no recording for it to have meaning. This division into equal groups approach has proved very successful starting with children from Year 1 tackling division with manipulatives (we start very slowly, focusing on mastery of pre-arithmetic concepts and skills in EYFS, prior to moving onto arithmetic in KS1) progressing through to them mastering long division by the end of Year 4. Bernie
By bjwestacott
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04 June 2014 19:43
Thank you for your comment. There is a choice to be made as to whether we focus on a sharing structure (as you suggest where the dividend is divided into three equal groups) or a grouping structure as in the video ( where the dividend is divided into groups determined by the divisor) The sharing structure is effective where the the divisor is small but becomes difficult where the numbers are larger. For example where we might be dividing by 11 without the use of place value counters. It is quite difficult to apply an effective strategy to work out how many would be in each of the 11 groups. Asking how many groups of 11 there are and to use knowledge of multiplication tables is more effective. The teacher is introducing the algorithm in a way that will work for any number.
01 June 2014 15:02
In these videos the teacher is using manipulatives to model division as dividing the numbers into an equal number IN each group - this results in a rather abstract result such as: 1 with the 100s, 1 with the 10s, and 2 in the 1s, thus the answer of 1 hundred, 1 ten, and 2 ones only makes sense in terms of the abstract written recording. Had the division been done with dividing the numbers so that the number OF groups is 3, then the pupils would have ended up with discs with 112 in each of the 3 groups - a far more concrete outcome that does not need the abstract, written version for it to be visually clear - it does, however, lead easily to abtract, written recording and leads on to the formal written algorithm for long division. Bernie Westacott
By bjwestacott
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