## Reverse percentages and compound interest

The other day a discussion arose in my year 10 class that I found rather interesting. There was a question on interest which incorporated compound interest and reverse percentages. One student was telling the other how to find the answer to the reverse part, “you need to divide it, because it was that amount times by the multiplier to get this amount and divide is the inverse of times.” All good so far, then they discussed how to complete it if it was a reverse of more than one year, “so in that case it’s the new amount dived by the multiplier to the power of how many years.” I was pleased at the discussion so I didn’t really interject.

Then one of them aid, “if I’m looking for two years ago, can’t I just times it by the multiplier to the power -2? Wouldn’t that work.” I thought this was an excellent thought process. The other student disagreed though, sating “no, it has to be divide.” So I thought at this point I’d better interject a little.

“Does it give you the same answer?” I asked. They both thought about it and tried it and discussed it and said yes. So I asked “does it ALWAYS give the same number?” they tried a number of scenarios using different amounts, different interest rates and different numbers of years. Eventually they had convinced themselves. “Yes, yes it is always the same.”

“So is it a valid method then?” I probed. Some more discussion, then one ventured “yes. It must be.”

“Why does it work?”, I then asked. And left them discussing it.

When I came back to the pair I asked if they could explain why it works and one of them said, “we think that it’s because multiplying by a negative power is the same as dividing by the positive version.”

## Dodgy Microsoft Graphics

So my new laptop arrived today and I quickly set about using it. It’s a Windows 10 laptop and as such has all the usual Microsoft stuff preloaded in it. I was going to set chrome as the default browser when it suggested I try Microsoft edge as it’s apparently faster and made for Windows 10. When I opened it it showed me this graphic:

Immediately I called shinanegans. The 5% difference between the green and the blue looked far too big. Initially I thought it was just down to the scale starting from 25000 and the size, but looking deeper there are also 4 extra sets if 5 notches on the blue which further add a to the illusion.

All in all a terrible diagram. Poor form Microsoft. Poor form.

## Infuriatingly impossible exam questions

Today I was working on some Vectors exam questions with my Y13 mechanics class and I came across this question:

A student had answered it and had gotten part d wrong. What he had done was this:

*I have recreated is incorrect working.*

Obviously he had found out when the ship was at the lighthouse, instead of 10km away. I explained this to him and started to explain how he should have tackled this when a sudden realisation angered me.

Now for those if you that didn’t work through the question, here is the actual answer:

Can you see what had me infuriated?

This is an impossible answer! If the lighthouse is on the trajectory of the ship and it will hit said lighthouse at t=3 then that would stop the ship! At the very least it would slow it down!!!! In reality it would have to avoid the lighthouse and change trajectory. Meaning the second answer, T=5, would not happen under any circumstances!

My initial thought was: *“are they expecting students to spot this and discount the second answer? That’s a bit harsh.”*

So I checked the markscheme:

Nope, they are looking for both answers. Argh! I can understand using a real life context in mechanics, I really can. But why not check for this sort of thing!

*What do you guys think? Is this infuriating or am I just getting get up over nothing? I’d love to hear your views in the comments or via social media.*

## A new term, and a new school

I’ve started a new job this year at a new school. This is the second time I’ve moved schools and I have to say it has been a much smoother transition than it was the last time.

This school is very close geographically to my last school and as such has a similarge make up of students.

I’ve now got to the point where I can remember most of the names of my students and we are working hard to put some real progress in the classroom.

It’s been a time of change all round really, my daughter started school this year too, and as such I have now become the parent of a school kid. That’s been weird all round but she’s enjoying it and I think we picked a really good school. My wife and I have been invited to attend a meeting there next week when the will tell us how they teach English and maths in reception. I’m interested to see what they say about it, particularly in mathematics!

Also this summer I finished my MA and I’m awaiting results for the dissertation. I think it went OK, but I won’t know until the brown envelope arrives with my feedback and grade. The dissertation was entitled “Investigating problem solving as a means to improving understanding in A level mathematics” – catchy I know. I enjoyed writing it and I may share a summary on here at a later date.

All in all its currently a time of change and that brings with it excitement and challenges.

*How has your start to the new year been? How are your new classes? Have you started a new job? I’d love to hear about it in the comments or via social media.*

## Circles and Triangles

Regular readers will know that I love a good puzzle. I love all maths problems, but ones which make me think and get me stuck a bit are by far my favourite. The other day Ed Southall (@solvemymaths) shared this little beauty that did just that:

I thought “Circles and a 3 4 5 triangle – what an awesome puzzle”, I reached for a pen an paper and drew out the puzzle.

I was at a bit of a loss to start with. I did some pythag to work some things out:

Eliminated y and did some algebra:

Wrote out what I knew:

And drew a diagram that didn’t help much:

I then added some additional lines to my original diagram:

Which made me see what I needed to do!

I redrew the important bits (using the knowledge that radii meet tangents at 90 degrees and that the line was 3.2 away from c but the center of the large circle was 2.5 away):

Then considered the left bit first:

Used Pythagoras’s theorem:

Then solved for x:

Then briefly git annoyed at myself because I’d already used x for something else.

I did the same with the other side to find the final radius.

A lovely puzzle using mainly Pythagoras’s theorem, circle theorems and algebra so one that is, in theory at least, accessible to GCSE students.

I hope you enjoyed this one as much as I did!

## End of term emotions

What an emotional few weeks. This time of year is always emotional, but this year that has been ramped up to a whole new level. There is all the usual emotion of Y11 and Y13 classes finishing the year, and this year that has been compounded by the fact that I am leaving my current school at the end of term.

I’m sad that I won’t work. With some of my colleagues anymore and I’m sad that I won’t get to teach some of my classes next year. On the flip side, I’m excited by the challenge that lays ahead and I’m excited by the fact I’m going to be working with some former colleagues and friends again.

Then I’m devastated by the referendum result. I thinks it’s a disaster for the country for so many reasons. The economy will suffer, the rich diverse culture that we have in Britain will suffer, it will affect touring musicians which may mean many UK based ones will give it up and less overseas stars grace our shores.

Then there’s the rise in hate crime. In the first week after the referendum there were 300 reported hate crimes against non brits. Up from 60 on a normal week. I find both those figures abhorrent, but the larger one particularly so. To me it shows that the racist and xenophobic underbelly of our society now feel they have been legitimised. It was always going to happen they way Nigel Farage and his cronies have spent the last two decades selling the EU debate as “we want our country back”.

## Angle problem

Today has been quite a geometric based day for me. I spent a couple of hours solving non-RAT trigonometry problems with year 10 and then a while with year 11 looking at various algebra angle problems. Then I went on Twitter and saw this from Ed Southall (@solvemymaths):

A couple of nice parallel lines questions that I might grow at y11 tomorrow.

Both are fairly straight forward to solve. I looked at the first one, imagines a third parallel line through the join if x and saw x must be the sum of 40 and 60 hence 100.

The second I saw an alternate angle to the 50 in the top triangle and used angle sum of a triangle is 180 to spot that x is a right angle. I glanced down at the responses and saw the vast majority had the same answers as me. That would probably have been the end of it but then I noticed this response:

The same thought process for the first one, but a significantly different approach to the second.

It made me wonder what approach others would take, and which approach my students would take. I wondered if the first problem had led this respondent into this solutions the second, and if so why it hasn’t had the same effect as me.

I don’t know if either approach is better, I just thought the differences were interesting. I’d love to hear your thoughts on it and how you would approach it.

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