# Working with angles in Polygons

There are loads of different ways to approach the problem of “How do I remember that the total of the angles in a hexagon is 720?” (and all the other polygons for that matter!). Which one suits YOU is pretty dependent on your learning style, and on how much Maths you do week on week….

Memorisation (and a cheat!)

This may work for you, to some extent. You need to KNOW that the total of angles in a triangle is 180 degrees. If you really struggle with this, perhaps it can help you to dig out your protractor from your pencil case. The biggest number on it is 180 and that’s the number you need to remember!

Again most people can remember that a RIGHT ANGLE is 90 degrees so the TOTAL of the angles in a rectangle is 360. (4×90).

If you are really good at memorising, then it may be helpful for you to remember that, every time you add one more side to your polygon, you add another 180 degrees. That means that a 5 sided shape (pentagon) has 540, 6 sided has 720, 7 sided has 900, 8 sided has 1080, etc etc…..

One single angle in a regular polygon

For example, an 8 sided regular octagon has a total of 1080 degrees. Each internal angle must be 1080/8 = 135 degrees.

Another approach…. less to memorise.

This may suit you if you really have a flaky memory…. it IS important that you know The total of the angles in a triangle is 180 degrees. Then you can sketch the shape…. split it into triangles….. see the video below.

I have taught this the “traditional” way where you break the octagon up into 4 triangles. It’s cool. Mathematically very clever. But VERY hard for some of my students to actually remember… so this (less elegant) way is what I tend to teach more often.

Working Backwards is almost as easy….

Sometimes you are given the interior angle and told “This is from a regular polygon. How many sides has it got? Here’s how to do that kind of question….

# Problem Solving in Maths Lessons

I have been a member of the Association of Teachers of Mathematics  for years now, and very much enjoyed the 2011 conference – a brilliant opportunity for a Freelance Maths Tutor to do some networking and catch up with CPD. So I was very pleasantly surprised recently to find myself a member of the ATM Facebook Group, where more networking but at less cost, suddenly becomes a possibility. On February the 23rd Mike Ollerton is doing a session in Leicester entitled

I’d love to come but can’t, so I have had to content myself with holding this question in my mind for the past fortnight as I teach my Primary School groups.

It is SO tempting just to TEACH Maths, and it can be quite effective with some pupils, especially in the short term. I have  bunch of tricks I can serve up and they are simple, powerful, they go down well. But I have taken a break from that recently and given the kids some actual problems to solve… The results have not been quite as I expected…

# How much does one Strawberry Pencil weigh?

I picked up a packet of 12 strange-looking sweets on the way into school recently, and put this question on the board at the start of each of the group sessions. The extension group, who decided to run this as a boys-v-girls challenge, sailed into a process with ease, using the “obvious” method, which was to read from the packaging the total weight of the sweets and divide by 12 using a bus stop division. It was so quick that they had time to work with another pack of sweets as well….

The interesting outcome for this group was they found three different answers. The division was 75/12, and I was offered 6.3, 6.6 and 6.25. We’ve done a lot of work recently on team work and collaboration, and so I offered this back to the group and asked them to resolve the differences. Pretty soon 6.6 was withdrawn (Oh, I made a mistake), but two remained for consideration. I was pretty chuffed when one of the girls (Kate) admitted to the group that she couldn’t do the division, and she knew her answer (6.3) was wrong. This took a lot of courage and I think was a complement to the level of trust she now feels in the group’s process.

She wrote her workings on the board and, predictably, several hands shot up accompanied by “Oooh”s and much enthusiasm. It would have been easy to get one of them (or several) to teach Kate how to do division when a remainder is not acceptable as an answer. However, I was skeptical whether she would get it, or remember, so I said to the group that we were ALL hugely tempted just to teach Kate, but I was going to NOT teach her, she was going to work it out herself. The atmosphere crackled – the hands reached higher and dislocated shoulders threatened… this was not at all what the audience wanted… one wannabe teacher appeared to be on the verge of exploding, so I offered her the chance to go and briefly run around outside. To the huge amusement of the group, she kangarooed theatrically up and down outside the window then returned…

Back to Kate…

I began to gently engage her with finding her own solution. I got her to rework the bits of the sum that she was confident with and identify the “unstuck” moment (which was the arrival at “remainder three”). She put “point three” into the answer area.. but looked worried. More hands reaching up. This time the exploding pupil was a boy but was sent out with instructions to kangaroo OUT OF SIGHT so the focus could remain on Kate. I continued to ask Kate questions about where she might put the three, where did she normally put remainders, just opening her to her own solution-finding strengths, and then the magic moment, the penny dropped and was missed by several of the group whose desire to teach had become almost unbearable – but Kate’s face as she inserted the required noughts, put the remainder 3 in the correct place and finished the sum correctly, was a picture of triumph. 100% agreement now on 6.25 being the answer.

We finished the lesson with a quick bout of the Inverse Function Dance to release the required steam.

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I think sequels are important. The long term effects of an experimental approach matter and it is important not to make judgements too quickly. So…

# Ten days later

Kate assures me that she is still 100% confident with divisions that push on into the decimal world in this way. The group’s cohesion, focus and mutual trust continues to grow and they have spent two one-hour lessons now exploring “What is the largest (area) shape you can draw with a perimeter of 30cm”, they now believe it’s a circle but are very keen to spend next lesson getting to grips with HOW to best find the area of a circle – they are becoming dissatisfied with counting squares, and have found the investigation a genuine struggle in places, but seem to be experiencing a delight in their learning which is motivating them to go deeper.

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A Quote..(from ATM)

The power to learn rests with the learner. Teaching has a subordinate role. The teacher has a duty to seek out ways to engage the power of the learner