Four operations of fractions by folding paper

Another idea from Mike Ollerton’s workshops.  This file gives a comprehensive explanation of the activity, which starts like this:

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I have used a similar task before but I realised that I had missed a key step which is to label each fraction after folding:

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The file then goes on to describe how to use this for demonstrating all four operations: add, subtract, multiply, divide. It’s a lovely way of reinforcing the concept of equivalent fractions at every stage.

My only reservation with this task is that doing the folding in the first place might be a barrier for some learners.  Especially folding something into thirds – it’s not straightforward.

I have added some Powerpoint printables that provide guidelines along which to fold. Note these are set up as A4, so print them 2 to a page and then cut.

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In a sense, I can see that this might detract from the notion of “folding in half” because it becomes “fold along that line”.  I haven’t had enough experience of which is the “better” way to do this – I’d be very happy if anyone wanted to share their thoughts!

 

Just follow these steps and you’ll be OK at Enlargements

Here’s a statement which I don’t think will be too controversial – I would have thought maths teachers the world over would agree with this:

I want my students to gain a deep understanding of the mathematics, not just follow a procedure to get the right answer.

This is our aim. We don’t always get there. We have different ways of getting there. I have recently re-read this seminal paper by Skemp from 1976.

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In it he talks about Relational Understanding, which I have generally thought of as understanding of concepts, and Instrumental Understanding, which I think of as understanding of procedures.  In my teaching I have been inclined to build conceptual understanding first, and then see what methods make sense from there.  However, I’m starting to think that it’s not that simple.  There are some situations where some instrumental understanding might come first and act as a foundation on which to build relational understanding.  Ultimately we want both, but the order in which we achieve this is not always the same.  We should not dismiss a didactic approach that provides a clear sequence of steps and worked examples as a part of the journey to a deeper mathematical understanding.

I recently observed a colleague teaching the topic of Enlargements to a Year 11 revision class, who are entered for the Foundation paper.  He used a method which I don’t think I had thought of before.  It was heavy on the Instrumental Understanding, but it worked and the students were doing some more tricky fractional and negative enlargements with centres of enlargement not at the origin.  About as hard as it gets for these types of exam questions (thanks for Maths Genie for these examples)

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So, what was this wonderful method?  Well, it might be nothing new to you dear reader, but it used vectors.  It relied on students being secure with describing the translation between two points using vector notation.  Given that this is how Translations are described and that we typically teach Translations (along with Reflections and Rotations) before Enlargements in a topic called Transformations, this should be a build on / consolidating of what was learned a few lessons ago.

The steps go something like this.

  1. Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
  2. Circle the centre of enlargement, CoL (helpful to distinguish it from a vertex later)

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3. Find the vector that moves you from the CoL to each vertex.
4. Multiply each vector by the scale factor

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5. Now use those vectors to plot the new points, starting from the CoL again. Connect the vertices to form the shape.

6. Finally draw in some ray lines to convince yourself that you have not made any mistakes.

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I think that’s quite a neat method that enables us to go straight to what might be seen as the most difficult example. The only real difficulty here, however is multiplying a (pretty simple) negative fraction by an integer.  Something which should be secure by the time this topic is being taught. And if it’s not secure, well here is an opportunity to practise it without detracting too much from the main objective.

Another benefit, is that it might be easier for students to plot points by counting squares rather than draw accurate, extended ray lines, as pointed out by Mr Blachford on Twitter.

Once students have done a few examples, we can draw attention to some things, for example:

  • If the scale factor is >1, it gets bigger, if it’s <1 it gets smaller
  • The scale factor applies to each side length of the shape (but not the area…)
  • Negative scale factors always place the image the “other side of the CoL”
  • All ray lines must go through the CoL. This is how it is constructed.

I would use Geogebra (as in fact I did to create these images) to examine what happens when we move the CoL. (Note: Geogebra uses the US terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

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And from there we can go onto questions that show the enlargement and ask for a description.  I would use Geogebra for that bit as well as it is very easy to create images that can then be used to ask learners to describe what they see.

Just for fun, I had a go at a 3D version in Geogebra.  I’l leave you to decide whether or not it adds anything. You should be able to access it here. This is what it looks like:

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This is an example.  It’s one way of doing it. For other topics, for example fractions, I would prefer to spend a decent chunk of time on building conceptual understanding before focussing on algorithms to get right answers. But that’s for another post…

 

 

 

 

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.