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.

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)

So, what was this wonderful method?  Well, it might be nothing new, 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)

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

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.

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: If you are the in the default US language setting, Geogebra uses the terminology Dilation rather than Enlargement – which actually is more descriptive of what we are doing, isn’t it?!)

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:

This is a specific pedagogy for this topic.  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…

 

 

 

 

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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.

(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.

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.

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.

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.

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.

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.

 

 

 

Concentric Equilateral Triangles

The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.

What is the ratio of the height of the red triangle to the height of the pink triangle?

Can you solve using trigonometry or only using Pythagoras?

The border is now 2cm, whilst the side length of the red triangle remains 4cm.  What is the ratio of heights now?

Explore what happens for other border widths.  Can you generalise for any border width w?

Geogebra file here.

Spoiler here.

I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.

 

 

Growing a cube – an introduction to 3D coordinates

I built this on Geogebra.  It’s pretty simple but might be a good way in to 3D coordinates and more generally explaining the concept of dimensions.

You can download and open the original Geogebra file here which gives more control than just playing the gif.

1. Start with all sliders at zero.
2. As you increase slider a, talk about the first dimension.  Any point in the 1D world can be described by a single number which shows how far along the line you are. Every object in a 1D world is just a line. Long or short. It can be described by a single number which we can call length.
3. Once a has reached 1, talk about the second dimension. This is now like a floor, or the surface of the earth. We call this a plane.
4. Increase the slider b.
5. The world in 2D contains two dimensions, which we can call length and width. There are other words: e.g. breadth, depth.
6. Every point in a 2D world can be described by 2 coordinates. These are the x and y coordinates.  It’s important to notice that the x-direction (i.e. the x-axis) and the y-directions (the y-axis) are a right-angles to each other or orthogonal. Why is this?
7. Once b has reached 1, what shape to we have? How many vertices does this shape have? How many edges? What are the coordinates of its vertices? Do we need 3 figure or 2 figure coordinates for a 2D shape?
8. Now we can bring things into the real world in which we live, the 3D world where shapes also have height.
9. Increase the slider c to grow the cube upwards.
10. When c has reached 1, what shape do we have? How many edges, vertices, faces does it have? What are the coordinates of the vertices of this shape?

From there, you could always say:

Why stop at 3 Dimensions?

 

 

Origami for the end of term

Origami is one of those things that I think I would love to spend more time exploring but rarely do.  I have used the Origami Player with my students  (it works really well as an App within Chrome), which gives excellent visual instructions on making things. The timings have been well thought out and it gives a little timer prompt so you know how long you have got to do that fold before looking up at the screen for the next one.  It’s been an end of term, easy lesson. Nothing wrong with that.

The first session on the nRich Teacher Inspiration Day  last week where we looked at some of the activities here got me thinking about how I could make it a slightly more meaningful learning experience.  

Still not highly mathematical, but at least it gets students working together and struggling with something.  To build resilience in our students they need see the struggle as a positive and not something to be avoided at all costs.  It was a bit of a metaphor for all learning. A discussion that can be had with students when reflecting on this task might be along the lines of:

• Did you need help from someone at some point? (yes, good)
• Did you help someone else at some point? (yes, good)
• Did you struggle at some point? (yes, good)
• Did you give up? (hopefully no, good)
• Did you achieve something you didn’t think you could do before?

This type of discussion can be a powerful motivator and more useful than vague questions like “Did you enjoy it?” or “Did you have fun?”

There are lots of Origami ideas on this page of nRich’s new Wild Maths site. I really like the idea of modular Origami, i.e. each person makes a module and then they come together to create something beautiful.  I have an end-of-term cross curricular session with Year 8 to plan. My Origami paper arrived this morning.  You can use A4 paper and fold down the corner to make a square, but proper origami paper is really lovely and this pack was only £8 for 500 sheets.  It looks great so time to start practising!

I will post an update on this as planning progresses and share some pictures of the final event.  In the meantime, if anyone has an ideas for how to make Origami more mathematical (without spoiling the enjoyment of it!) then let me know.

 

Great problem solving tasks from FMSP

My Year 10 class did these 3 this week:

They have their GCSE RS and Science exams next week, so I needed to find something a bit “fun” but still wanted it to be “mathsy”.  They are a strong group with enough keen beans amongst them for me to feel confident that something like this would work.  I was impressed with their teamwork and tenacity.  At first it seems hard because you are faced with a blank sheet (their words). It took the quickest group 20 mins, 1 out of the five didn’t finish after 30 minutes.

Something I always bear in mind when doing any sort of team activity (see my earlier blog post) is that everyone should have a clear role to play and something to contribute. In this task everyone has their own set of clues, that they are not allowed to show to the others. I encouraged them to make sure they were the experts in their own clues so that when proposals were made they could say if this proposal broke any of their rules.

There is no need to cut up these sheets into “cards” as the resource suggests.  Quicker and better if you just cut them into strips:

That way you can give a strip to each student. They are less likely to drop a card on the floor which kind of ruins it! If you cut some horizontally and some vertically as shown you can cater for groups of 4 or 5 students.

We finished with 10 minutes to spare and had a pretty good discussion about the task.  Questions I asked them:

How did they start the task? (Talking or just reading)

Did it help to group the clues in any way?

Were they tempted to share the clues? (I told them I’d give them 30s time penalty if they were caught sharing.) Or were they happy having their own? (most were)

How did you work well as a team? What did you think went well?

Was it hard? What made it hard?

 

From squares to Platonic Solids

Some questioning as a way to get to Platonic Solids:

• What is special about a cube? (compared to a cuboid)
• What is special about squares? (the sides on a square)
• What do we call 2D shapes where all the sides are the same?
• So, can we generalise, what is special about a cube?
• What other regular 2D shapes do you know. Draw them now.
• What other 3D shapes can you picture that can be constructed only with regular 2D shapes? These are called Platonic Solids.

Have a go.  See if you can find some dusty old 2 shapes in a cupboard somewhere.

These are quite cheap or you could make your own from card using this template.

So, here they are in all their glory. The wikipedia page has some nice animations.

The Greeks discovered these 5 and mathematicians since have proved that there are definitely no more, whichever polygons we use.

Next you could complete this table.

Once you have done that, can you spot any patterns in a relationship between F, E and V?  You might even discover Euler’s formula, which holds true for any convex polyhedra, not just the platonic solids.

V – E + F = 2

Or use (a lot!) of 2d shapes to make nets of these shapes.

This page gives all the answers in a nice table!

Do try this at home!

I liked the look of this Nrich task thinking that my Year 7s, who have shown some appetite for investigative tasks, would enjoy it.

It’s a great task, but like so many investigations, you really need to have a go at it first yourself.  I did it with my daughter (Yr7) and we took nearly an hour to find the 3×3. And I was trying pretty hard!  Interestingly the 4×4 was much easier.  Here are our combined efforts!

   It’s going to be critical how I introduce / explain the task, so I will create a notebook file to help with this.

I have also created a worksheet as I feel that my class will need a bit more scaffolding on this task.  I now know that I will encourage them to move onto the 4×4 if they get fed up with the 3×3.

I’ll update this post tomorrow once I have taught the lesson.

Perimeter of Regular Polygons

Based on a task from nrich but I felt that the context was a bit confusing.  And I thought some colour would help!

Make a square with one cut

This is one of those amazing sheets which keep them going for quite a while!  I’ve tried this with a lower set Yr10, an extension class in Yr8 and even a few teachers!  It’s hugely differentiated all on one sheet and the way it’s laid out encourages students to wander to another shape if they get stuck – some of these are really hard.

I quite often offer scissors but most people quickly realise that they prefer just to do it with a pencil and visualising it – which of course is the whole point!

Make a square with one cut