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….

Measuring Angles

Dear M,

There are a lot of quite nice resources online that help to build the skills of estimating and measuring angles. Using a protractor is about 3 times more confusing than using a ruler, so it’s something that needs a whole lot of practice. If I was Prime Minister, then angles on exam papers would always be TRUE angles so you could use REAL estimation skills to work them out, but they aren’t. Oh, and I’m not Prime Minister…. As we discussed today in our lesson, it doesn’t matter if you photocopy an angle larger or smaller, it stays the same size (which is pretty weird when you put it like that….. most things DO get bigger when they get bigger, don’t they!)

I have ordered the builders’ angle measurer that we looked at online  and chatted to the plumbers who are in my house today who don’t use one of those but thought it sounded incredibly useful!

Link to the online game of measuring angles on screen….

Brilliant “make an angle game” that we played, and you were so good at! The link takes you to the start screen – we skipped the intro then chose “make the angles game”. To practice full-on measuring, where you have to place the protractor without any help from the computer, the option to pick is “up to 180 degrees in ones”.

Build your own printable worksheet (to use with a real protractor) – gave you one to do, here is a link to it just in case, feel free to make some more worksheets if you want more practice.

I wonder how you will get on, measuring angles over 180 degrees, when your own protractor only goes up to 180…. It’s a genuine mathematical problem, and I think that if you give it 5 minutes of careful thought, you may come up with something. It isn’t impossible, but it isn’t obvious either. As always, if you can figure out your own solution, you are MUCH more likely to remember it!

If your mind goes a blank when you try to use your 180 degree protractor, I would recommend this video because the teacher is making an effort to not be boring, is quite funny, but rather quiet, so you’ll have to listen carefully!

Rosalind

The power of practice

Dear M,

You are making great progress with algebra, I think. The crucial link between the written symbols and what they actually MEAN is not easy to make, but you are putting the hours of practice that you need, in order to become confident.

Here are some more questions. I know we discussed the idea that I could build an interactive game for you that would check each piece of your working, and your answers, but on reflection, I think you are getting sufficiently fluent with algebra now, that this level of feedback would actually be unhelpful, like bolting stabilizers back onto a bicycle after you have begun to balance it by yourself.

You are already in the (excellent) habit of keying the question into your calculator, keying in the answer, and comparing the two, What I would like you to focus on this week, is actually writing down the “easy version” and, if the question and answer DON’T match, then key in these workings out as well to narrow down the problem.

The huge advantage of you and the calculator (rather than the computer) doing the checking, is that you are able to use your calculator throughout the IGCSE exam, so you are practicing really useful skills, not using a tool that will be taken away from you.

After working on the last worksheet with you, we both realised that the sums were too densely packed on the page, and a bit dazzly. I hope this version is easier. I have left gaps for you to write the “easy version” out directly below the sum.

As you work, it may be worth being aware of the commonest errors you were making in the lesson (which are errors most people make, you are not alone 🙂  )

• If there was a minus sign in front of the first bracket, you were not always “seeing it” clearly when you keyed expressions into your calculator
• If there was a minus in front of the bracket as well as one inside it, you sometimes made the wrong decision (although you had pretty much stopped doing that by the end of the lesson, errors have a nasty habit of creeping back in a few days later!)
• You made more errors when you did everything mentally, and fewer when you jotted the “easy version” as a working step.

Jargon-wise, what we are doing is called “multiplying out brackets and simplifying”. That’s the terms the examiner would use (instead of “writing down the easy version and writing down the answer”.

So here are three more worksheets:

multiply out and simplify 1

multiply out and simplify 2

multiply out and simplify 3