## Semi-circles on a hypotenuse

Today’s puzzle is from Catriona Shearer (@cshearer41). It started out when I saw this tweet:

I looked at this then put my phone down and thought about it while doing something else. Only I didn’t, I misremembered entirely what the question was. This is what I thought it was:

Find the hypotenuse.

While thinking on it I though, “hang on, if the diameter of the circle is 6 then the radius of the circle is 3 which by similar triangles means the hypotenuse is 12.”

Like this.

I then thought, “that’s a bit simple for a Catriona puzzle, I wonder what it really said.”

So I looked and realised it was an entirely different puzzle. I also noticed this tweet underneath it:

I followed that link and found the puzzle as intended:

The difference being the light green dotted line. I think perhaps my misremembering and the knowledge this dotted line makes the puzzle possible rather than impossible probably influenced my solution. Firstly I sketched it out:

I realised I could reflect triangle CBD in the line CD and the other vertex would be at point E. I could then reflect this new triangle in the line EC. As ED is the size of the radius of each semi-circle and AC is a tangent it follows that the angle BCD must be one third of the angle BCA, or 1/3 of 90, or 30. Which means the angle labelled x must be 60. Although I over complicated this slightly when I wrote it out:

I know in a right angled triangle with a 60 degree angle that the hypotenuse is always double the side adjacent to the angle. So from here I knew that it was going to be the triangle I had assumed earlier. But I wanted to show it logistically.

I considered the trapezium EFCB, I could make a smaller triangle EGB that was similar to the large one. I wrote the side lengths in, but realised I still hadn’t justified them.

I felt happier once I had justified it, although now I look at it I’m a little annoyed I used x again to mean something different.

I know that this “x” is the radius of each semi circle, and that the semi circles are congruent, hence the area is x^2 pi. Or 9pi in this case.

I enjoyed this puzzle, but I wonder if I would have taken a different approach had I not know the importance of the dotted line or had not wrongly assumed the correct diameter in the first instance. How did you approach it? I’d love to hear your solutions in the comments or on social media.

## Polygons, Area and some generalisations

On Tuesday the maths teaching world recieved the terrible news that the great Don Steward has sadly passed away. I didnt know Don personally, but friends I have who did speak of him very highly. I do know his blog, median. It is an amazing source of lesson resources and some insightful musings that have often got me thinking and have helped me plan many lessons. Without knowing it, Don has helped me (and I imagine many others) become a better maths teacher and a better mathematical thinker with his posts, and I will be forever grateful to him.The news prompted Chris Smith (@aap03102) to share this:I thought that if Don had found this puzzle interesting enough to engage in email chat about it then it was definitely something worth looking at.I drew a sketch and sid some preliminary workings:I know the areas are the same. And I’ve got a load of trapeziums so that’s easy enough to work out:I can then set areas equal to each other:This brings about the answer that it will produce 3 sections with the same area every time k is 2 bigger than n. I thought about what this looks like, theres the simplest case of course:Which gives us 3 rectangles. This pair of solutions is probably the simplest to actually work out. But I think lots of students would not even consider it as it doesn’t look like the picture in the question. I also wondered about the examples where n was bigger than 2? I.e. if n = 3 then k = 5 and this would look inverted. Would people discount this also because it didn’t look like the diagram in the drawing?I then considered this one.For some reason, while I was looking at this one another route to the solution jumped out at me. If I consider one of the top trapezia, a or b in the first diagram:Then as the area of the square is 36, each section is 12 so the solution falls out from knowing that the sum of k and 6-n is 8 (from area of a trapezium formula.k + 6 – n = 8Simplifies to:k = n + 2A nicer solution.I considered the case where n = 0 as I thought it was quite nice:My thoughts turned to the general case offered in the question. Following my original working it falls out this way:So k need to be one third of the side length of the square larger than n.We could have just considered that the trapezium again:(1/2)(k + s – n)(s/2) = (s^2)/3Which reduces tok + s – n = 4s/3So againk = n + s/3Interestingly I think equating areas is a simpler solution in the general case, even though the area of the trapezium seemed simpler when we had numbers.All this, and the thoughts on how it looked earlier made me wonder what would need to happen if we wanted the angles at the “middle” vertex were all equal (so 120).I did some preliminary sketches:I know some ratios for right angled triangles with a 30 degree angle, looking first at s = 6:Then the general:Lots of interesting maths coming out of this puzzle. I wondered whether other angles would produce other ratios. I wondered what would happen if rather than equal areas we assigned the areas other ratios. There are plenty more directions to go in, but it’s getting late so they will have to be for another post and another day.

## A nice money puzzle

Here is an interesting money puzzle I came across on twitter. It’s from @puzzlecritic and I came across it via Ed Southall (@edsouthall).

My first thought was that a barn owl shopping for books is a bit strange. But then I got over that and started to think about the puzzle.

It appeared to me that there were actually more than one way to get to an answer here. But knowing Ed, I figured only one would be correct. I assumed that some would give monetary values that are impossible (I.e. answers with part pennies involved).

I thought about the various ways. We have 3 books listed in order of price, a,b and c. The first way to get a solution is that b = 3a, and c = 8b = 24a. Thus the total price 10.50 divided by 28 would give the price of the cheapest book. This would give an answer of 37.5p so I discounted it.

The next was would be that b=8a and c=3b=24a. This would lead to 10.50/33 which is a shade less than 32p. Again, not a whole number.

The third way was where b=3a and c=8a. This would lead to 10.50/12, which would be 87.5p again, not a whole number.

The fourth way would be where c=3b and c=8a. This is a little less straightforward to solve. I set a=3x for ease meaning c=24x and b =8x. This leads to x=10.50/35 which is 0.3, meaning the cheapest book, a, cost 90p.

In hindsight, I should have known that it would be the final way that generated the solution!

*I enjoyed this little puzzle, it got me thinking about ratios and algebra and it strikes me as a very good puzzle to show to my students. It made me wonder if there were other methods to solve it, if there was a way to do it without trialling different solutions. I can’t see any at the minute but will continue to think about it. I also wondered about the thought process behind setting it up, how easy would it be to create a question like this that gives answers that don’t work the other ways. I’m not sure, but I do intend to ponder it further. How did you solve it? Do you have any other questions or puzzles like this? I’d love for you to let me know in the comments or on social media.*

## Pandemic Graphs

People who know me, or who have read any of my other posts, will know that I’m quite a fan of mathematics. I enjoy doing it, I enjoy reading about it, writing about it and teaching it. I tend to see maths in most places, and a couple of things I really like finding is hideous statistical deception, or just plain bad graphs, and great examples of real life usages of maths.

During the pandemic there have been plenty of things flying around that fall into both of these categories. So I thought I’d share some of them here:

**Horrific graphs**

This one is from Fox 31, whatever that is and someone sent me to me in whatsapp. Its hideous isn’t it? It looks ok to start with, but then you notice the y-axis! The numbers are equidistant but the values start in 30s, some go up in 10s and some in 50s! I’m not sure why this would be drawn like that. I’m thinking there was something that happened on march 25th or 29th that they wanted to show an impact for but I’m not sure. It could just be someone has no idea how graphs work.

Another I was sent via WhatsApp and it comes from Australia presumably, as I can’t think what NSW would be if it isn’t New South Wales. I also saw this on Twitter a few times. I think it gets worse the longer you look at it. At first glance it’s a harmless bar chart, but then you notice, the 190s are different heights, the gaps between 190, 186 and 174 are all similar, 91 and 190 are almost the same. ARGH. I can’t fathom a reason why you would draw it like this. Initially I thought it must be done to give the impression of falling infection rates, but then the 116 jumps up to higher than 127 which comes earlier. It’s just mind boggling. I can only imagine they downloaded a stock bar chart and put their number on the bars as they were.

This one was sent to me via Facebook. I don’t know the source, and it’s a little different to the other 2 as I think (hope) it’s a work of satire. It’s too far removed from reality to be real, surely? But it certainly serves to show the sort of thing that does get issued by people who are hoping to serve a purpose, and hoping people just look at the visual and not the actual data.

**Logarithmic scales **

I’ve seen a lot of people posting about logarithmic scales. Lots of folk seem to have an issue with them, and in some circumstances I feel they are being used deceptively without proper signposting.

These 2 graphs show the same data, the 1st is a logarithmic scale and the second is a linear scale. Arguably the logarithmic scale gives a much more easily compared picture of the death rates between a number of the countries affected by the pandemic and it much more useful for those looking at the different approaches and analysing what is happening in an attempt to respond to the crisis better.

However, I have seen in lots if places logarithmic scales graphs like this being shared in an attempt to suggest the curve is flattening, when the reality is it’s not. And this misuse of the logarithmic scale is rather worrying and needs calling out.

I think these graphs give us a good example to use in lessons, and will certainly help with the teaching of logarithmic scales and exponential modelling in future.

**A final thought:**

Here are 2 amazingly accurate graphs that people have sent me recently:

*Apologies that the graphs here aren’t referenced properly, all of them have been sent to me via social media, some multiple times, if you happen to be, or know, the source do let me know in the comments or on social media. Also, I’d love to hear about and see any other graphs about the pandemic, or any other maths that you’ve come across.*

## Constructions

One topics I have never been a fan of teaching is constructions. I think that this is due to a few factors. Firstly, there is the practical nature of the lesson, you are making sure all students in the class have, essentially, a sharp tool that could be used to stab someone. I remember when I was at school a pair of compasses being used to stab a friend of mines leg and this is something I’m always wary of.

Secondly, the skill of constructing is one that I struggled to master myself. I was terrible at art, to the point where an art teacher kept me back after class in year 8 to ask why I was spoken about in the staffroom as the top of everyone else’s class but was firmly at the bottom of his. I explained that I just couldn’t do it, although it was something I really wished I could do. He was a lovely man and a good teacher and he offered to allow me to stay back every Monday after our lesson and have some one to one sessions. I was keen and did it, this lasted all through year 8 and although my art work never improved my homework grades did, as he now knew I was genuinely trying to get better. I have always assumed the reason I am poor at art is some unknown issue with my hand to eye coordination, and I have always blamed this same unknown reason for struggling sometimes with the technical skills involved in constructions. Since coming into teaching I have worked hard to improve at these skills, and I am certainly a lot lot better than I used to be, but I still feel I have a way to go to improve.

For these reasons I chose to go to Ed Southall’s (@solvemymaths) session “Yes, but constructions” at the recent #mathsconf19. Ed had some good advice about preparation and planning, but most of that was what I would already do:

*Ensure you have plenty of paper, enough equipment that is in good working order, a visualiser etc.*

*Plan plenty of time for students to become fluent with using a pair of compasses before moving on.*

He then moved on to showing us some geometric patterns he gets students to construct while becoming familiar with using the equipment. Some of these were ones I’d not considered and he showed us good talking points to pick out and some interesting polygons that arise. The one I liked best looked like this:

*This is my attempt at it, I used different coloured bic pens in order to outline some of the shapes under the visualiser.*

The lesson was successful, the class can now all use a pair of compasses and we managed to have some great discussions about how we knew that the shapes we had made were regular and other facts about them.

Next week we need to move on to looking at angle bisectors, perpendicular bisectors, equilateral triangles, and the such. I hope to get them constructing circumcircles of triangles, in circles of triangles and circles inscribed by squares etc.

Here are some more of my attempts at construction:

“Constructing an incircle” – I actually did this one in Ed’s session!

“A circumcircle” – I drew the triangle too big and the circle goes off the page. Interesting to note the centre is outside the triangle for this one.

“A circle inscribed within a square” – this is difficult. Constructing a square is difficult and that is only half way there if that. This is the closest I have got so far and two sides are not quite tangent.

“A flower” – nice practice using a pair of compasses and this flower took some bisectors too.

*If you have any ideas for cool things I can construct, and that I can get my students to construct, please let me know in the comments or on social media.*

## Late tiering decisions

Last week year 11 sat their mocks. Some did really well, others did really poorly. It’s the latter group that has me purplexed. Students sitting the higher tier paper but only scoring single digits per paper, or even earlt teens per paper. What to do with them?

Some of them asked if they could move to foundation, I think its best for them. 1 student got 32 marks over 3 higher papers, did the 3 foundation and was well over 100. 1 student got 40 marks over 3 higher papers spent 30 mins in a foundation paper and got 60 marks. The grade 5s they want seem more achievable on foundation.

My issue lies with a few students desperate to do higher and try for 6s. Scoring around 50 marks over 3 higher papers it seems a risk. But having taught them both i feel that it’s within their capabilities. But from November to march they have made only tiny gains in marks. On the ine hand, foundation means they cant get a 6 and for at least one of them means rethinking post 16 choices, but on the other hand sitting higher means they might end up with only a 4 or less and thst would mean rethinking post 16 again. It’s tricky, any thoughts are welcomed.

## Dividing by fractions

For a while I’ve been pondering on dividing by fractions. It started out with thinking about methods of teaching dividing by fraction. I’ve always taught to multiply by the reciprocal and built from the idea that if you divide something, say 5 by a fraction, say 1/3, then you are looking to find (in this example) how many 1/3s go into 5. As there are 3 thirds in 1 then you can get to your answer by multiplying 5 by 3, which is the reciprocal of 1/3. Then examples can follow using none unit fractions there are 3 thirds in a whole, so there are 3/2 2/3s in a whole and so on. I think it’s a method that builds on understanding well, but I find that most students just ignore the reasons it works and boil it down to a method, many have already heard it referred to as KFC and just stick to that mnemonic and don’t try to understand. It was this lack of understanding on why the method works that had me thinking about whether it is the best method, especially when dividing a fraction by a fraction.

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