What other questions could be asked here?

Powerpoint file, with solutions here.

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Powerpoint file used to create this here.

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Maybe this is an important thing to establish before talking about irrational numbers. With irrational numbers, we are effectively trying to convince students that there is a separate class of numbers on the number line that can’t be expressed as a division of two integers.

There is a proof for why ∏ is irrational but it’s not pretty. I’m taking Peter Mitchell’s word for it on that who presented on this topic at the recent MEI conference. He has a proof here, but in his own words “it’s really, really tedious!” So maybe surds are a better place to look for an example of a proof that irrational numbers exist. Although this is an A level topic, I think with the right class this could be used at KS4.

This is a proof by contradiction, which in itself is a bit strange. But the logic is sound: if I assume something to be true and then work through it to show that there is something inherent within it that is false, then I have proved that thing cannot be true therefore it must be false.

In this case, we are going assume that √2 * is* a rational number, prove that that is

If √2 is a rational number, then we can write it √2 = *a/b* where *a*, *b* are whole numbers, *b* not zero.

We additionally assume that this *a/b* is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for *a/b* to be in simplest terms, both of *a* and *b* cannot be even. One or both must be odd. Otherwise, we could simplify *a/b* further.

Going back to our first statement:

√2 = *a/b*

we can square both sides to get:

2 = *a*^{2}/*b*^{2}

or

*a*^{2} = 2*b*^{2}.

So the square of *a* must be an even number since it is two times something. If *a²* is even then *a* itself must also be even. Any odd number time an odd number creates an odd number (some more of these here).

Okay, if *a* itself is an even number, then *a* is 2 times some other whole number. In symbols, *a* = 2k where k is this other number.

If we substitute *a* = 2k into the original equation 2 = *a*^{2}/b^{2}:

2 = (2k)^{2}/b^{2}

2 = 4k^{2}/b^{2}

2b^{2} = 4k^{2}

b^{2} = 2k^{2}

Again, because b is 2 times something, b **must** be an even number.

We have shown that *a* and *b* are both even numbers, but we started saying that *a/b* was a fraction in its simplest form.

I might want students to explore what happens with √4 in this same proof, i.e. prove why √4 is **not** irrational. From there we could go on to look at √3. It’s a bit harder, but only really requires that all odd numbers can be written in the form 2n+1. Here is a spoiler if you are stuck.

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What is the general formula for the surface of *a* cuboid of width 1, depth 1 and height h?

What is the general formula for a cuboid of width 1, depth *d* and height *h*?

What is the general formula for a cuboid of width *w*, depth *d* and height *h*?

What other 3D shapes can you find the general formula for the surface area? Try:

- A tetrahedron, side length
*a* - A square based pyramid, base length
*a*, height,*h* - A cylinder radius
*r*, length*l*

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So, having given up trying to get down with the kids, here is another approach which involves them generating data so they feel they have some ownership of it. It’s quick, they can do it in their seats, they get a bit competitive and it’s interesting.

How many dots?

Display this for about 2 seconds. Tell them what they are going to see and make really clear that they are not to discuss it but write down their estimate. Then go round the class capturing those estimates on a spreadsheet. I might use Geogebra for this as it is great a creating box plots, but Excel would be equally good.

You must plead with your students not to cheat and change their estimates from the one they wrote down. Tell them they will get a second chance. Do it all again to get a second second set of data. You now have two sets of randomly generated data that you can use to compare using averages, box plots, standard deviation, etc. It should be a great example of regression to the mean. Also “The wisdom of the crowd” – always interesting to see how wise your crowd actually is!

Oh, and how many dots? 46. The bare bones of a Powerpoint is here.

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This approach builds on the presentation that Mike Ollerton gave at the recent Mixed Attainment Maths conference in Sheffield (keep an eye on the site for details of the next conference in November!)

I didn’t actually attend Mike’s presentation – I was too busy giving my own – but Mike has kindly shared his ideas and I have been thinking about how I might use Geogebra as a tool to aid learning.

As with most of my use of Geogebra, I am using it as an exposition tool to structure whole-class questioning and discussion around. In an ideal world, I might get students to do this themselves, but that is not practical in my classroom. I feel that starting Geogebra from a blank sheet can be nearly as powerful as them doing it themselves and is likely to be a much more efficient use of lesson time.

The basic principle used is that of rotating a fixed line segment, a “spinner” if you like, around a point. We are aiming to explore the co-ordinates of the point at the end of the line as the angle increases from 0º to 90º (and beyond) in a table.

I must say that from this point onwards this post is **not** Mike’s recommended approach (which is here) – but my interpretation of it using Geogebra.

So, first step is to form the spinner by by plotting points at (0,0) and (1,0), zooming in and connecting the points with a line segment. There is something in observing what happens to the scale on the axes as we zoom in.

Next, we create the angle by using the Angle with Given Size tool. As the prompt says when you hover, the tool: select leg point, then vertex. Rather than fix the angle, I want to make a slider so I can easily change it. I set the angle “a” making sure to leave the degree symbol in place (otherwise you get radians). The slider then needs tweaking by double-clicking to set the max, min and increment.

Next, I need to make the line segment a bit bolder by right clicking on it…

..and change the properties of the point so it shows the coordinates to 2 decimal places, also using right click.

We now have a tool that can tell us the co-ordinates. Before using this, however, I think I would want pupils to do some work on paper, using Mike’s handout to get a feel for the numbers and get their own results. To fill in this table:

I feel that it’s useful that they have the opportunity to correct any measurement inaccuracies before the next step and this is where the computer helps.

As per the worksheet, a series of questions can be posed before using the calculator’s Sin and Cos functions to complete the following:

From there we can easily start exploring what happens when the side length is not 1 and use the ideas of scaling and similar triangles.

And then, of course change the angle again.

Once this is all set up, it’s easy to display / hide the coordinates and maybe so some miniwhiteboard work to assess how well the class has grasped the use of the Sin and Cos functions. And then keep the coordinates, but hide the angle to demonstrate inverse Sin and Cos.

With some practise and familiarity with Geogebra you are spending less than 3 minutes on the computer. If you like the idea of “Geogebra from a blank sheet”, click on the Geogebra category link at the top to see posts on using the same idea for other topics.

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I expect students to think of translating the graphs along x and y so that the turning point ends up at the origin. Then they can test whether y = x² . What other methods might you use? What other prompts might you give to learners that are stuck? What additional questions could you ask?

Here is the above image as a pdf. I created it in Excel here as a scatter graph with a smoothed curve, so can tweak the values if you wish.

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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:

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.

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!

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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 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.

- Label the vertices of the shape you’ve been given (say A,B,C,D, something like that)
**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: Geogebra uses the US 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 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…

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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.

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.

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