Category Archives: algebra

Interleaving Algebra and Perimeter

Over the last couple of years we have organised our Year 7 curriculum so we do some introductory algebra early on.  Forming expressions from words and collecting like terms, would be topics that I would put in the introductory bucket. The benefit of this is that it can be interleaved into various other topics to extend thinking and promote generalising.  Perimeter is an example of this, when we can give side lengths letters instead of numerical values.

An alternative to presenting a bunch of text-book type questions is to investigate a simple 4 piece Tangram, as described in this task from Mike Ollerton.

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(Click on the image to access the full Word document)

The task as presented is primarily an exercise in shape, but I might use a slightly modified version of the Tangram to focus purely on perimeter.

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Before asking students to cut the triangles out of the shape, we might agree on some labels for the side lengths.  If we focus just the shorter sides, we could call Triangle A’s short lengths a, Triangle B’s short lengths b, and Triangle C’s short lengths c, so we end up with something like this.

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Again, before getting to work with the scissors we could discuss how we might describe the hypotenuse lengths in terms of a, b and c.  And in fact, if we really need c at all…

Using this notation for the side lengths, we can then cut up the 4 shapes and generate many other shapes and find their perimeters.  Using 1 shape, 2 shapes, 3 shapes, all 4 shapes: what is the shape with the longest perimeter? The shortest perimeter? What is the difference between longest and shortest?  What shapes are different but have the same perimeter – can we prove this using algebra?

It hopefully presents a need for “collecting like terms” as well as some introductory practice in using the technique.

9-pin Geoboard

A second task that links algebra to perimeter uses a 9-pin geoboard.  This sounds fancier than it is. You don’t need the actual boards, students can create their own in their books or you can give them some dotty paper. First, we tell students that we are going to make triangles using only these lines a, b, c. This is a key image that we will need to refer to either on the board or on a handout.

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The next task is to construct triangles using various combinations of these lengths. Each triangle must fit within the 3×3 array.  Depending on the class and your objectives for the lesson, at some point you show them that there are only 8 “different” triangles. An opportunity here for a discussion on what we mean by different and what congruency is.

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Once we have these, we can go through the process of calculating the perimeter for each one using the a, b, c notation we introduced earlier.

There are some options from here.  Mike’s suggestion is that we order these from smallest to largest.  We could do that by just looking at the shapes and having a guess. It’s pretty obvious that C is smallest although some of the in-between ones are harder to see. We could establish that a<b<c in the first picture (again by looking).  Then we would also need to decide, for example which is bigger: 2a or c.  With an older group, you might even use Pythagoras and express c and b in terms of a using surd form.

By looking at the differences between each shape’s perimeter, we start dealing with negative quantities of a, b, and c.  If we then sum up all those differences, we should end up with an expression the same as the difference between the smallest and largest, with the c’s cancelling out. Which is quite satisfying and obvious if you think about the expressions lined up on a number line.

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There is a fair degree of flexibility within tasks like these and I believe that as teachers we need to select carefully what routes we expect to go down in a lesson. There is a danger that we try to encompass too many different topics in one go and if all of these topics are new to a class then they (and you!) are likely to lose track of what they are actually meant to be learning in this lesson. However, if you are confident that the learners in your class are secure with certain concepts (in this case collecting like terms with negative coefficients) then it is a good way to consolidate and practice this knowledge whilst pushing into new territory.

 

 

 

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:

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2. angles

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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??

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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.

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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:

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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:

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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!