Times tables grid

A simple grid to stick in the back of the books of students.  I tend to give to everyone in Year 7, and then observe carefully who uses it and who doesn’t.  Once it becomes easier to look inside their memory than flip to the back of the book, I know they are secure in their times tables!

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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.

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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?

 

 

Why stop at 3 dimensions? 

A friend of mine (who is not a maths teacher) recently sent me this article that Marcus Du Sautoy wrote in 2013 when he was president of the MA. He makes the case for a “Mathematical Literature” GCSE to sit alongside the utilitarian curriculum a bit like English Literature sits alongside English Language GCSE. It should aim to develop a student’s love of mathematics through doing maths for enjoyment to foster an appreciation of our rich and varied mathematical heritage.

At the end of the article Du Sautoy he says he is a mathematician not an educationalist. But I think the example he gives in here is a great one. There is definitely a lesson in it, and not just for higher attaining classes. Once the basics of plotting coordinates in a Cartesian system are secure, I think this would be accessible to any KS2/3 class upwards. But this comes with a health warning: I haven’t tried it yet,  so (as with anything you find on the Internet) proceed at your own risk!

The lesson plan

Prerequisites:

  • Pupils need to know how to plot coordinates in 2D (1st quadrant only)
  • They need to know what a square is and be able to reason why the basic square has coordinates (0,0), (0,1), (1,0), (1,1).  You could maybe start with something like this from Don Steward, although this is already maybe more than is required.Picture1.png
  • They need a systematic way of finding arrangements of things.  e.g. how many ways are there of arranging the letters ABC? ABCD? etc.

And that is about it.

So next, we can introduce the 3D coordinate system.  Visualising things in 3d can be hard; often it is not closely correlated to mathematical attainment. But there are a wealth of options now to show and “spin” a 3D object on a screen. Here is one I created on Geogebra:

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I would spend some time on this, looking at systematic ways to make sure we have the coordinates of all 8 vertices.  You might want to take the approach I wrote about here.  I would try to convince students that they could have done this systematically with having had to see the cube in 3D space.  Because what they are going to do next is find the coordinates of an object that they definitely can’t see – a 4D cube.

Although it’s impossible to see in our 3D world, we can use maths to work out the coordinates of each vertex of a 4D cube.  Again, if this is done systematically students will hopefully soon see that the number of vertices doubles each time we add another dimension. Here is 2D, 3D and 4D:

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The next part of the article is where it really gets interesting. Because as well as this being an important piece of maths in its own right, it also has an application in computing.  It is used in error correction when sending digital signals. The rules are reasonably straightforward and the article provides an example of a piece of code that contains an error.  Can your students apply the rule to find the incorrect bit?

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Do you really understand addition and subtraction?

This was the question Mark McCourt was getting us to ponder in the first session of Maths Teacher Network.  He started off by running through the classic 1089 “trick” which I have written about before here.

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These types of activity are such a powerful way to get students to carry out repeated practice to build fluency. If you handed out a sheet with 20 column subtractions and 20 column additions, you would get groans and do no favours for the general popularity of maths as a subject.  If those repeated calculations have a purpose as they do here, the dynamic and energy in the classroom is completely different.

I’d used this before but had never given it the time that it deserves, nor had I extended it:

  • Try it with 4 digits, 5 digits, etc.
  • Go back to three digits and try it with a, b, c. In other words prove it algebraically.
  • Now try in a different base, e.g. Base 7!

I actually got a bit obsessed with this when I got home.  I do love a good spreadsheet challenge, so I attempted to build something that would provide an algebraic proof of different numbers of digits and bases. Here is the result…

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My reflection from this session was that even if I have “seen” something before, have I really investigated it deeply and could I use it more effectively with my classes? To which my answers were no and yes!

How many zeros?

A conversation in the Maths department this week about how to extend a particularly strong Year 7 student whilst teaching multiplying and dividing by 10, 100, 1000 led to this idea:

If you multiply all the numbers from 1 to 100 how many zeros are at the end?

I like this question.  It’s actually really quite hard. Have a go. I had previously used is as, “How many zeros are the end of 100! ?” But phrasing it this way gets around the need to explain factorial notation in the middle of a lesson that is not about factorial notation and therefore is can be used as an extension challenge.  Another less challenging, but similar(ish) question:

How many numbers with repeated digits (e.g. 55, 101, 155) are there between 1 and 201?

It is sometimes hard when teaching “basic” topics like decimal place value to find suitable challenge.

Before thinking about finding hard questions, I think it’s worth putting ourselves in the shoes of some of our higher attaining Year 7s and their experience of our mixed attainment classes in the first few months of secondary school.

Primary school experiences vary a lot. Many are taught in sets, especially in larger 2 and 3 form entry schools which are common in London.  My hunch is that the propensity to set children for Maths increases the closer they get to Year 6 although I don’t have any data to back that up (if you know of such data, please share!)  Most of the primaries that I am working with this year that are implementing mastery approaches and so are moving away from setting to mixed attainment teaching, but starting in the lower years.

In many cases, children who felt they were “good at maths” in Primary, developed that self-perception by completing calculations and getting correct answers. After all, that’s what maths is, right?  In some cases they will have established rules in their own minds schemas, etc. An example might be “to multiply by 10, simply add a zero on the end”. And, of course, for integers this works.  Which is why it is so important to highlight misconceptions e.g. 3.4 x 10 = 3.40 as soon as possible before these “false friends” become established.  This is why Year 7 can be such a challenging year for pupils and their teachers.  There is often a need to undo and deconstruct some false ideas that have become well-established before building back up the correct concepts.

So, back to the problem.  Maybe it’s worth saying that there are 24 zeros at the end of that huge number.  That’s the answer.  But as we all know, the answer is just the beginning.

Geometric proof

A quick post to share a useful site that I learned about today.  Thanks to  who shared this during this evening’s #mathscpdchat on Twitter.

I’ve always struggled teaching students how to explain their reasoning when solving angle problems to a sufficient level to gain full marks on a GCSE question.  Here is a nice site that scaffolds this for students by presenting a series of mathematically concise statements that students need to chain together into a coherent and complete reasoning.

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I would get students to write these in their books so they get a good sense of what is required in a full and coherent proof.

There are lots of good examples on this site.

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It was obviously built a while ago, and sadly it doesn’t cover circle theorems, but I have e-mailed the author, so I’ll post any updates here!

 

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