Venn diagrams without probability

Venn diagrams...one of those topics sorely missed when removed from the curriculum, and a topic whose reintroduction to the GCSE was welcomed by many maths teachers. Of course those of us that have taught A-Level stats have been keeping ourselves nicely abreast of set notation and its applications to Venn diagrams; but mainly to do with application to probability. Something that the SAMs have made clear is that Venn diagrams will be used more widely than just with probability at GCSE, and so I thought I would share a couple of questions I have developed to support using Venn diagrams.

1: Prime factorisation

To be fair this is an approach that some people will already be using; the application of Venn diagrams to find HCF and LCM from the prime factorisations of two numbers. This question however uses it in reverse, gives the prime factors in the Venn diagram without telling the numbers

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2: Venn diagrams and expressions/equations

Something similar to this is employed in the SAMs, but to my recollection it again involved probability. To be fair this questions could, but I have left it slightly short of that (this question is to be used with Year 7 upper set). It still does bring in some nice expression writing and equation solving.

Both of these questions are in the Venn Diagrams Word document in editable form on my TES resource area, which is at the link here: https://www.tes.co.uk/teaching-resource/venn-diagrams-11082977 so feel free to use, adapt etc; I hope it serves as a source of inspiration for developing resources for these new topics and that tie different areas of the curriculum together.

10ticks Level 9/10 or the new GCSE?

A while since my last post I know - a well earned break following the end of the school year (which for Leicestershire was the 10th of July) and I have now re-turned my attention to making sure we are well resourced for next year. It was while doing this I came across a great little find...

I think every maths teacher has at some point used 10ticks. Rarely an entire sheet (although I have known some teachers use a whole sheet for homework or a follow-up lesson) but certainly stealing parts of it. When I first started my career the school I worked at only had access to Levels 3 up to 7/8, and like many new teachers (as I was at the time) I had trained in the era when Level 9 or 10 were no longer really talked about, even though the original National Curriculum did cast its gaze to those lofty heights. When I encountered the Level 9/10 worksheets in my second school, I had a cursory glance, and haven't had much cause to glance that way again, save for the odd top-end trig or volume resource. Imagine my delight then when going back through I read the contents list again for 10ticks Level 9/10.

Honestly I wouldn't be surprised if the people designing the new KS4 programme of study didn't have one eye on this document when they were writing the new content - so richly is it found here. A brief summary if I may:

Pack 1 - Accuracy of Measurement, Variation, Indices (negative and fractional), compound growth and decay, Surds and irrationals (including in trigonometry), approximating root 2 by iteration.

Pack 2 - Trigonometry, Tree diagrams, solving quadratics by factorising, quadratic sequences, completing the square and the quadratic formula.

Pack 3 - Ratio, Scale Factors and Similar triangles, Congruence, Distributions, Histograms.

Pack 4 - Graphs and Equations (including circle equations), Volume and Surface area of curved shapes, Density, Algebraic fractions, Transforming formulae, transforming graphs.

Pack 5 - Vectors, Proof, Distance/Velocity - Time graphs including curves, Perpendiculars.

Pack 6 - Set notation, Venn Diagrams (after lots of matrices stuff).

Now if I were looking for a list of things that fit somewhere in the 3 part Venn diagram of Content new in Foundation tier, Content new in Higher tier, Content crucial for further study beyond GCSE I think this would be quite a reasonable list, and I think many maths teachers would agree with me. I haven't had chance to look through all of the individual pages yet, but undoubtedly some gems await.

So if you are preparing for the new GCSE over the summer, or even over the course of teaching next year, don't neglect an old faithful resource like 10ticks; it might just surprise you.

Practical Cumulative Frequency - Some thoughts

Another lesson based in the hall this week; this time cumulative frequency. Went quite well considering it was bottom set Year 9! I can see this working better with pupils  who have a reasonable attention span (my Year 9 second set would have been the proof had my intern not been taking them for the cumulative frequency). Equipment today was:

1) 6 metre rulers
2) A big ball of wool
3) Sellotape
4) Scissors
5) Blu-tack
6) Printed axes numbers (I had the numbers 1 to 30 and heights from 140cm to 190 cm)
7) Bean bags (one per person).
8) Flipchart paper

We set up two rulers taped to the wall one on top of each other to create a 2 metre high line, one at each end of the hall, and ran the wool across between them at 10 cm intervals (starting at 140 as no one in the class was below 130 cm tall) to create a way of seeing how many people were in each group, and also set up a big axes on the floor by blu-tacking the printed numbers down at intervals using the final two metre sticks to measure the intervales (I used 20 cm going up and 1 metre going across for the heights, which seemed OK). 

Once the measuring station and axes were set up, they lined up in height order between the two rulers and numbered off (there were 23 in my bottom set, so numbered 1 to 23). We then draw a frequency table (flipchart paper) by looking at how many people were between the lines, or below the lowest line (or above the highest line). What was great was that I could then throw out things like, "Raise your hand if you are below the 170 cm line" and then after a pause, "How could we tell this from the table we just drew?" From here the concept of cumulative frequency was born.

Once that was clear I had each person in turn, from 1 to 23, come and place their bean bags at the correct point on the axes, creating what was actually quite a good cumulative frequency curve (Year 9 bottom set remember!). We talked about the shape, explaining why it appeared the way it did, and even got as far as looking at median and quartiles; first doing it from the line, with everybody raising their hands and then putting them down in pairs, leaving one person, and then breaking the line in half leaving that one person on their own, before doing the same with each half to create the quartiles; and then seeing how we could see that from the graph by walking along from those peoples numbers to where they had placed their bean bag, and then down to read off their height. I am hoping that following this up with a classroom based lesson tomorrow, starting with the same table, that we will begin to really understand how cumulative frequency works and how it is represented.

Any practical lesson like this has it dangers and pitfalls, and there were times that some kids were standing around whilst other did bits (my TA was invaluable with supporting the creation of the measuring station and axes in particular), but on the whole I would say the lesson was quite successful and would certainly run it again.

Teaching Mean - a follow up.

Last week I was chatting with a couple of tweeps (Mr Bayew and Mr Reddy) about my blog http://educatingmrmattock.blogspot.co.uk/2015/05/mean-average-dont-add-them-all-up-and.html  and discussing ways of teaching mean - I talked about the approach I would take and thought I would outline it here.

The biggest misconception I have come across with mean calculations is the blind 'add them all up and divide by how many there are' process, with no appreciation for the information this is designed to convey. Therefore the primary goal of my approach is to reinforce the need for a total that is shared, and so my approach looks at other ways of finding the total before looking at addition. You will see kids in my class lying down in a line being measured to see how long the line is, so we can see the total length (i.e. height) of all of the pupils. Weighing is another good one; putting lots of items on a scale to find their total weight without the need to add. In true 'mastery' style we do these practically before looking at images like these:

(images courtesy of mathszone.co.uk and teachitprimary.co.uk)

On these we can talk about mean average weight of each apple or lollipop. Once I am confident that pupils understand the idea of mean, we can then look at it in reverse, i.e. if a bag of 6 apples have a mean weight of 42 grams, how much would the bag weigh? Questions like this reinforce the sharing nature of the mean average, and highlight division as the primary operation rather than addition. 

Once the concept is clear, finding totals from addition just become another type of problem, rather than an integral part of the process; we can pose problems like find the mean of 5 objects of lengths 12 cm, 17 cm, 9 cm, 6 cm and 11 cm in the knowledge that pupils are already secure that the sharing of the total is the important thing, and that the adding is just there to find the total. I also like to throw in questions like "what is the mean of a group of 10 objects, 4 of which weigh 30 grams, 3 of which weigh 32 grams, 2 of which weigh 35 grams and 1 which weighs 36 grams. As a pre-cursor to mean from tables this sort of question is really nice as it shows the use of multiplication with addition to find totals, and importantly stops kids getting bogged down in lots of adding.

So do your kids a favour, teach them what the mean means!