Rational and Irrational numbers

I only recently properly considered why all fractions are either terminating or recurring decimals.  Fundamentally, this is because there are a finite number of options for the remainder, which is a maximum of 1 less than the divisor.  This is most easily seen when dividing by 7.  All six potential remainders are used and the 7th division goes back to the beginning of the sequence as shown on this task from Don Steward.

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.

Proof that √2 is irrational

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 false, thereby proving that √2 must be irrational.

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 = a²/b²

or

a² = 2b².

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 = a²/b²:

2 = (2k)²/b²

2 = 4k²/b²

2b² = 4k²

b² = 2k²

 

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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Generalising Surface area

What is the surface area of a cube of side length 1?
If we then cut this cube in half, and throw one of the halves away, what is surface area of the remaining cuboid?
Repeat the process, cut the shape in half along the same plane. What pattern can you see?

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

 

Student generated data

I know many teachers find teaching Statistics at KS3 & 4 a bit dry.  One way to make it a bit more interesting is to make the data somehow relevant to the students.  I’m not talking about football scores or download charts here. I’ve gone to great lengths to source that kind of data and create beautiful resources with it only to find that, while it engages some students, it has the opposite effect on others if it is something they are definitely not into.

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.

Trigonometry, another way

Trigonometry falls firmly into the camp of one of those areas which I don’t feel I have cracked yet.  If “cracked” means finding a bomb-proof way to introduce it to any class I encounter, then maybe I never will!  Either way, I am always interested in different approaches to teach this topic which many students seem to struggle with at first.

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.

Is this a graph of a quadratic function?

Here’s an idea that came to me in Anne Watson’s plenary today at the MEI conference in Keele.  I think it’s quite challenging but I haven’t used it with any classes yet, so have no idea really!

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.