## Area Puzzle 21

I came across the following area puzzle on Ed’s (@solvemymaths) site.

I found it quite an interesting idea and had a little go at solving it.

First I considered an equilateral triangle, all the angles are 60 degrees so we can see the area would be (x^2 sin (60))/2. As sin 60 = 3^(1/2)/2 it was easy enough to solve.

I then realised I could have used Heron’s Formula, so did it that way too. Luckily I got the same answer:

The square was a tough one:

Then I considered a pentagon, I wasn’t sure how to approach it at first, so I reverted to my favourite shape, the triangle. Split the pentagon into 5 congruent isosceles triangles and solved with trig.

Wolfram alpha gave me the lovely exact answer:

I then considered the hexagonal case, which is really just 6 equilateral triangles.

I approached the decagon in the same manner I had the Pentagon.

And got an equally lovely exact answer:

I enjoyed working through these, and thought it would make a nice lesson to build resilience and cognitive activation. I also thought about what else could be done. What does the sequence of side lengths generated look like? Could an equation be formed to describe it? What if we were looking at the areas of regular polygons with side length six, what would the sequence look like then? All these would make nice investigations.

## 100% Chance

*“100% Chance of getting a safety car.”*

That line was repeated numerous times in the build up to today’s signapore grand prix and in the early stages of the race. It was repeated by various pundits and comentators and it has made my blog boil.

**THERE IS NOT A 100% CHANCE OF A SAFETY CAR**

That should be enough, to be honest. It’s not certain that it will happen, there is a chance that it might be avoided. The problem stems from the confusion between relative frequency and probability.

Relative frequency IS a good proxy when it comes to probability but it’s isn’t always exactly the same thing. The relative frequently of a safety car being needed at Singapore was, still is, 100% because There has been one at every race ever held here, which gives us a relative frequency of 100%. But the sample size is tiny (9 races) and this isn’t big enough in this case. Please sky sports, sort your maths out.

## Bicycle Puzzle

This week’s puzzle from Chris Smith is a nice contextualised simultaneous equations puzzle I intend to use next week.

It boils down to:

2X + B + T = 135

2X + 2B + 3T = 269

X + B + T = 118

Via elimination using 1 and 3 we can see that X (number of tandems) is 17.

That gives:

34 + B + T = 135

34 + 2B + 3T = 269

17 + B + T = 118

1 and 3 rearrange to the same leaving:

B + T = 101

2B + 3T = 235

Elimination gives T (number of Tricycles) to be 33 which leave B (number of Bicycles) to be 68.

A nice little problem.

## Carnival of Mathematics

Hello, and welcome to the 126th edition of the Carnival of Mathematics. *For those of you who are unaware, a “blog carnival” is a periodic post that travels from blog to blog and has a collection of posts on a certain topic. This is one of two Maths Carnivals, the other being Math(s) Teachers at Play, the current edition can be found here.*

Some interesting facts about the number 126: it’s a pentatope number, decagonal number and a pentagonal pyrimadal number. It’s is also palendromic in quinary (1001).

So a carnival with such an interesting number must surely include some interesting posts? Well there are some classics, but first this from one of my pupils: “Sir, I know why you wear glasses….. They help you with di-vision! ”

Now the posts:

Shecky Rieman send us this post on Oliver Sacks, saying every Carnival this month should make some mention of Dr. Sacks, but not sure how many math posts were done on him, since math wasn’t really his genre.

Ben Orlin has given us this post which uses “The smartest dumb error in the state of colorado” to discus the history of multiplication.

Bit Player have this exploration of a prime generating sequence.

Marginal revolution looks at an interesting dilemma for drug dealers.

Diane Jolie sends us this post looking at Ada Lovelace.

Carnival protectors **The Aperiodical** have plenty of posts since the last carnival my favourite are This on game theory in the new York times, This on pentagonal tilings, and, of course, puzzlebomb!

Denise at Let’s play math has this review on an interesting looking puzzle book.

Ed at solvemymaths.com has given us these Maths Mr Men, and this awesome area puzzle, amongst other stuff.

Colin at flying colours maths has loads of great posts including this from the mathematical pirate on circles and this from the mathematical ninja on why it works.

Tom at mathematics and coding has been extremely busy recently and my favourite post is this on that damp quotient rule.

Emily at I love maths games has been setting up her new classroom, including this great maths meme wall.

John at Math hombre has this on MacMahon squares.

Dave, the stats king, over at reflective maths has this lovely collection of maths puzzles. (I have my own here.)

Kris Boulton has posted this great piece on the identity symbol and it’s importance.

Sam Shah has This on bliermoins, cyclic quadrilaterals and cross chords.

Manan at Math misery talks about developing mathematical fluency.

**Maths in the news**: Katie has sent this from the BBC discussing Alex Bellos new eliptical pool table and many more interesting things about conic sections. Handy as I’m teaching them at the moment so I dug out this old post.

## Lessons to learn

*This post was originally published on 28th August 2015 on Labour Teachers and can be viewedÂ here.*

Keiron Cunningham is perhaps the greatest hooker to have ever played the sport of rugby league, and how he coaches his former club St Helens. On Thursday they were on the telly and in the pre game build up it showed some of his comments about his team in the previous few weeks. The team have been on a poor run of form recently and in the press he has publically chastised them. Much of the build up was focused around whether this was the best was to improve the performance or if the effect would be detrimental on performance.

This made me think about the rhetoric we have seen in education from the government in recent years, and what potential effects they might have. There has been a range insults thrown towards the profession. They language of war has been prevalent in the majority of speeches and the effect has been to lower morale – something the sky sports commentators feared Cunninghams comments may do to the St Helens team.

I saw Mick Waters speak recently on the subject of government rhetoric and he was in agreement that the use of military language and the constant blows (“enemies of promise”, “dealers in despair” etc) do nothing positive for the profession. Teachers are left feeling threatened and under pressure. Surely a better tactic would be to focus on positives and target the criticisms in a much more constructive and supportive way.

Yes, there are plenty of schools and teachers that need to improve, we all should be striving to improve day on day because we can all be better than we are, but battering the profession verbally does nothing for morale and nothing for retention rates. Let’s hope the new year brings a new philosophy.