Category Archives: algebra

NCETM article on Functions – new spec GCSE

A short post, this.  Just a link to a useful NCETM article on functions. I haven’t taught this yet, but I remember getting pretty upset by the notation f(x), fg(x), etc. when I learned it at school.  One way of dealing with this might be to say that they have already encountered this function notation when doing trigonometry, after all sin(x) means sin of x, not sin times x.  But then don’t we also say “of means times“.  Arghh!

Anyway, it’s a great article and includes some ideas about linking this to:

  1. area and scale factor:

screen-shot-2016-11-01-at-15-43-26

2. angles

Screen Shot 2016-11-01 at 15.45.55.png

3. graphs

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https://www.ncetm.org.uk/resources/49564

Sum of Consecutive Numbers, a multi-layered investigtion

One of those classic investigations that gets forgotten about all too easily.  So much scope for generalising at different levels.

The fact that all odd numbers can be expressed as sum of two consecutive numbers is probably the first thing that will be established.  But why is this the case? And can students express this as a generalisation, first in the form of concise words and then algebraically?

The beauty of this is that there are then many other layers of things to discover, right down to a generalisation explaining which numbers cannot be expressed as a sum of consecutive numbers.  And maybe even a proof.

This nRich page gives away some of the answers.

Thanks to Alan Parr for reminding me about it with this excellent blog post:

The All I Can Throwers – Sessions with Den and Jenna. #1 – Consecutive Numbers

The paper napkin trigonometry trick with a smattering of Pythagorean triples. 

Take a piece of paper and do the following:

  1. Make it into a square (interesting discussion on best way to do this).
  2. Fold in half then unfold so you have created crease along a vertical line of symmetry
  3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
  4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??

IMG_20160612_190015IMG_20160612_190022

Fun with Fibonacci

This is an old one but fun, and a good way to use algebra to show why a trick works.  It’s a similar to showing how Magic Squares work.  It’s not a formal proof as such, but I think it’s a good way to introduce the topic.

Screen Shot 2016-06-03 at 16.13.14

Once students have grasped the basic concept of a Fibonacci Series (something which, in my experience they often see at Primary School even if they can’t remember what it is called), then you are ready to start the trick.

Fibonacci series don’t have to start with a 1 and a 1 as in the diagram above.  You start by asking students which two numbers they want to start with.

Then they get ready to be wowed with your powers of mental arithmetic. Tell them that you will be able to add up the first 10 digits of this sequence in your head faster than they can on calculators. Get one student up to the board to write down the numbers one by one.  TOP TIP here: make sure you have the numbers 1-10 written in a vertical column and that the chosen student writes down each term in the sequence against the numbers.  You should end up with something like this on your board:

Screen Shot 2016-06-03 at 16.22.39

As soon as term 7 goes up on the board, you start calculating.  You should be able to find the sum of the first 10 terms before they even get to term 10 and this is why:

Screen Shot 2016-06-03 at 16.20.56
I quite like doing the calculation on a miniwhiteboard, then writing the answer face down on a students’ desk and then walking to the other side of the room.  Once they have finally totalled the column of numbers on their calculator, you ask a student to have a look under the whiteboard.

And like all good magicians, you DO then go on to reveal the secrets of your trick!

Going off piste with the difference of two squares

Sometimes it’s worth taking a risk and changing course of a lesson halfway through. And sometimes it pays off.

Today’s lesson was supposed to be about algebraic proof and we started with these nice questions from Don Steward. I thought they looked like good practice for multiplying out double brackets at the same time as introducing algebraic proof.

Picture2

By the way question 3 is particularly tricky.

Following on from the previous lesson, lots of them started by trying values for n.  Great to then have the discussion on what makes a proof vs. an example.

We got into a good discussion on Question 5 and I wanted to know if they were familiar with the difference of two squares.  This is where the lesson changed course completely.  None of them could tell me what it was called but I got the sense that they had seen it before. So next, I wrote these questions on the board:

Factorise:

  1. n² – 9
  2. 4n² – 25
  3. 81 – n²
  4. 100 – 81

They raced through these because they had all spotted the short cut.  So next, I put this up, again from Don Steward:

Picture2

Which followed on nicely from the final of the 4 questions I had put on the board.  Many of them initially struggled to see the link immediately or see the pattern in the numbers.  But with a bit of time and just the right amount of help (i.e. not much actual help, just encouragement!) they started to find others.  I heard them forming statements like: “I need to number that multiply to give 2016, then I need another two numbers that add to give one of those number and subtract to give the other.”

About 5 minutes before it was time to pack away there was a great buzz in the room as 3 students found some other numbers and then the race was on. Lots of solutions started coming, but nobody got all 12 (including me!).

I went away and built a spreadsheet to investigate further, but I still didn’t find all 12.  Can you help???