## Teaching and Learning Framework for a lesson in Mathematics

Here’s a flexible ‘framework’ for the types of things I look for when observing a maths lesson. One of my goals was to make sure that it fit onto one A4 page (back-to-back) which I just about managed. Before I discuss this and amend with my department, does anyone have anything to add or subtract?

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## Dodgeball Dash

On Monday, a colleague of mine found an interesting problem (I don’t know where) and brought it over to me to take a look at. It was presented in the textbook in one of those “fake-real world type scenarios” so I thought I could do better with the context. Originally it was about finding the distance from the corner of a garden to a tap in the middle of the garden. Here it is now – it’s not exactly an “any questions?” type task because the question is obvious I think: “Will John get to the dodgeball before Sarah?”

After a brief discussion on the lack of information given within the problem, and the modelling assumptions about constant speed vs acceleration, present the students with more information to tackle the problem:

Given that the court is a perfect rectangle,  if John runs at 8 m/s and Sarah runs at 5 m/s, will John get to the dodgeball first?

Happy Problem Solving!

Question/Extension Exploration: Can anyone solve this without the use of Pythagoras?

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## Mathematical Reflections after the SGIS Education Conference

A few hand-picked highlights and reflections from the SGIS Conference 2016. Just to qualify some of the points made here, Rob Eastaway delivered sessions to both maths and non-maths specialists in a fun and dynamic way with some great ideas. There is no way that he could have, or should have, delved deeply into some of the points I am making.

1. The Creative Side of Maths (Rob Eastaway)

Some lovey ideas from Rob and some of my own reflection points in this fun packed opening keynote.

• I loved how he used jumping up stairs to teach his children about prime numbers (e.g. 13 stairs: Can you get to the top in equal jumps? What about with 12 stairs?)
• “How to measure the height of a tree with a walkers crisp packet?” was a lovely open problem with many different solution strategies. Some great hands on maths there.
• Another maths teacher said at one point during this talk that it is okay to make mistakes. I think this is somewhat dangerous if we aren’t clear on the types of mistakes being made. I’ve found it helpful with my classes to refer to sloppy mistakes and stretch mistakes. Sloppy mistakes aren’t great. They tend to refer to things like numerical errors, not reading the question properly or issues with algebraic manipulation. Stretch mistakes are great. They can happen when someone tries a new strategy or provides a strategy for the class to analyse. Mathematics is both creative and rigorous after-all!
• Rob said that we should be encouraging any solution strategy so as not to make students believe there is only one way to solve problems. I completely agree, but I also agree with the Japanese philosophy of ensuring that the class reflects on the most efficient strategy and/or the strategy that could be applied to a larger proportion of different problems – we are ultimately also trying to help students become effective mathematicians.
• When Rob referred to Pythagoras as being about lengths, I understand why he would say that, ultimately we use the idea of the triangle inequality (and Pythagoras) when walking around everyday. However, I do love how it is a beautiful theorem about area and it therefore can be applied to calculating lengths (sorry for being pedantic here – I just can’t help it – see one of my previous posts).

2. An Inquiry into the Brain (Dr Robert Greenleaf)

• Manipulatives and visuals are only good if students are engaged and focused. Otherwise they can be a distraction and learning can become confused [I guess just like any form of teaching and learning].
• Never define anything that you can get students to define themselves. One example that came to mind here for me is showing lots of trapezia and asking students to develop the simplest and shortest definition possible. This makes the students active participants in the defining process.
• Whenever possible, get students to analyse and collaborate on the similarities and differences between things. I believe that he claimed this to have the highest understanding and retention effect size over any other learning strategy. A classic mathematical example here would be in ascertaining the odd one out from a group of quadratics.

3. Maths, Magic and Mind-Reading (Rob Eastaway)

Lots of lovely mathematical magic tricks in this workshop to astound students and then get them working out the ‘how’ of the trick. I won’t share the magic tricks but I think Rob’s point about introducing algebra with mathematical magic really does engage students in understanding what a variable is. This is one he shared but also one I have used in the past during initial algebra units.

E.g. Think of a number -> Multiply it by 2 -> Add 10 -> Divide your current answer by two -> subtract the original number you thought of. The answer is always 5 – why?

4. The Importance of Puzzles in Maths (Rob Eastaway)

Again, so many great ideas for ‘low floor, high ceiling” tasks here. I decided to take one of them and extend it into a more meaty investigation.

Imagine a football field with a length of 100 m. Two teams are playing a cup final and a team assistant is asked to run and grab some bunting before the end of the game. Since the bunting needs to pass across the entire length of the pitch (100 m), but be pulled upwards in the centre so that players can fit underneath it, the manager asks the assistant to get more than 100m of bunting. The assistant comes back with 101 m which appears to be way too short.

1. Can the players fit under the bunting?
2. What would be the minimum length of bunting needed for the players to fit underneath? (given to an appropriate accuracy and assuming an averaged height person)
3. Can you find a general formula for the length of bunting needed for any length football pitch and for players of any height?

(This isn’t at all intuitive. It would make a nice 3 ACT task for anyone good enough to make the ACT 1 animation)

Summary

If you haven’t attended one of Rob Eastaway’s presentations before then I’d highly recommend that you do! If not, check out some of his books. I personally loved “Why do buses come in threes?” – see previous blog post on this.

## GeoGebra Pedagogy Workshop

Annoyingly I’ve never been to a GeoGebra workshop; I’ve only presented on GeoGebra and so I don’t know what other presenters are doing/talking about. My feeling – and this could be completely wrong – is that workshops tend to focus more on individual exploration and technical skill rather than pedagogy. There’s obvious reasons for promoting technical skill since it’s important for teachers to develop their own applets and facilitate their students’ use of Geogebra. Doing this however should not be to the detriment of pedagogical use which I believe so often happens when technology is introduced.

Since I’ve recently put a workshop together – for the practical pedagogies event in Toulouse – I thought I’d share it online for two reasons which I explain in Part 1.  I’m sure it’s very much a common sense approach to GeoGebra so nothing especially ground breaking. I hope either you gain something from it, or I do with your comments. Apologies it’s broken up into lots of pieces – videos were too large for a standard vimeo account.

Click this link to go to the video

(Honourable mentions in the workshop: @dannytybrown, @Geogebrain, @mike_geogebra, @mathhombre, @SparksMaths, @tombutton, @a_mcsquared, @ATMMathematics)

Once your done with the main workshop, here’s the link to show how to make an animated Ferris Wheel from the Appendices.

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## Quick Post: Introducing Pascal’s Triangle with Set Theory

It occurred to me today that I could ‘meaningfully’ introduce Pascal’s triangle before students meet the binomial theorem.

When teaching basic set theory,  the most interesting part for me is to ask students to investigate the number of subsets of a set with n elements. For example, the set {a, b} has four distinct subsets – {}, {a}, {b}, {a, b}. See the table below for more data:

If you haven’t investigated this before, you’ll notice that a set with n elements has 2^n subsets (always a lovely link to exponential expressions/sequences/functions within the syllabus).

It’s common knowledge for those who love Pascal’s triangle that row n of Pascal’s triangle sums to 2^n.

Since position r, row n, of Pascal’s triangle represents n objects choose r of them, it is clear that this is equivalent to a set with n elements. E.g. 4 elements, choose 2 of them and there’d be 6 combinations. Indeed, this helps us show why the number of subsets of an n element set is 2^n by expanding (1 + 1)^n.

The illuminating part of this is that, if a student wants to know how many subsets with two elements exist from a set with 4 elements {a, b, c, d} , they simply need to generate the 4th row of Pascal’s triangle and take a look at the 2nd element (6 – {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}). Without even listing the number of subsets which have 4 elements in from a set with 7 elements, you can just generate the seventh row of Pascal’s triangle and see that there’d be 35 distinct subsets.

So not only is there a connection between set theory and exponential functions, there’s also a look into Pascal’s triangle without even needing to mention combinations if you didn’t think it appropriate. They just generate the triangle, observe that each row sums to 2^n just like the number of subsets of a set, and then convince themselves that each element in row n, position r, corresponds to the number of subsets of an n element set with r elements.

## Introducing Surds (Year 10/11)

This activity is inspired first by Dylan Wiliam’s chapter on “Eliciting Evidence of Learner’s Achievement” in Embedded Formative Assessment. My approach to the task is inspired by Danny Brown’s (@dannytybrown) posts on noticing and listening (http://www.squeaktime.com/).

Instead of setting a pre-test before going into our unit on surds – which can deter students and also be difficult for teachers to draw meaningful evidence from – I gave year 11 the group based task below (groups of four). Very little input from me with the exception of a few guiding questions here and there.

After 25 minutes I had a huge amount of rich, qualitative data from student comments and I now feel that I can teach the entirety of surds from what they currently know. I can use their comments as a starting point for most discussions we’ll have over the next week. (It was also a good chance to revise a few topics from last year as well – indices and standard form).

You’ll notice that I stuck a few questions in there that might reveal a few misconceptions (not just with surds). Can you identify where some of the students may have gone wrong due to a misconception?

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## Does an equation exist to describe a square? What about any polygon?

Yesterday I saw some awesome Mr Men Math posters created by @solvemymaths (www.solvemymaths.com) and have already stuck some of them up in the corridor at school. Mr Astroid particularly caught my eye as one of the interesting Mr Men shapes.

You can see, and understand intuitively, that the equation is a variation on a circle. In fact, the shape of the astroid looks quite close to a square. This got me thinking…does a function exist to describe any given polygon? So here goes…

Using Scratch: Year 7-9 Way of Creating a Polygon

So the standard way to create a polygon via programming is to get on a programme, such as hopscotch, logo or scratch – and create a mini-script to move and then rotate (by the exterior angle) and move and then rotate, etc. until you get back to where you started. Obviously you need to use values for the rotation angle (exterior angle) that are factors of 360 degrees in order to make a connected shape. Using 90 degrees as the rotation angle you get a square.

Using GeoGebra: Year 10-11 Way of Creating a Square

It occurred to me that, by increasing the power of the astroid equation to make it closer to one, it would approximate the linear function x + y = 1 and so the curve would approximate a square. The slider, n, in the picture below represents the numerator of the fraction and the denominator is n+1. The slider must increment by two in order to keep the numerator even. Can you figure out why this has to be the case? (Click the picture to go the GeoGebra File)

The next step in the journey was to realise that I was actually approximating the function below:

GeoGebra didn’t like this so I put it in Desmos:

Polar Coordinates in Desmos: Year 12-13 Way of Creating a Polygon Function

After I reached the modulus function above, I decided that polar coordinates might help with going further but couldn’t think of how this might work – decided to look it up. I found what I was looking for (see this post – thank you to the people who posted) but thought I’d describe the first part in a bit more detail so that students can understand the base of the function and then play around with second part which is fun and interesting. There’s no way I could’ve come up with this but by playing around on Desmos I’ve built up an understanding of why each cog in the function exists.

If you haven’t come across polar coordinates before, go and look up the standard mathematics and then observe the picture below. I decided to stick with a square for the example but could have chosen any polygon for this example.

The x co-ordinate of the right side of the square is x = cos(45). If it were any polygon with n sides, it would be x = cos(180/n), or in radians, x = cos(pi/n) (1).

Since in general with polar coordinates, x = rcos(theta) (2), with r in this case being the distance from the origin to an edge, combining equations (1) and (2) gives:

This is the equation for one side of a polygon. It’s the base function for any polygon with theta between -pi/n and pi/n. To get this base function for the first side to break off into the next side, you need the distance from the origin to stay the same as the equivalent point on the first side. This is very well explained in the link above so go ahead and play with/analyse the second part of the function to find out why it works. The entire function for any polygon is given by:

Click the picture to go to the Desmos File. The floor( ) function rounds a number down to the closest integer. It’s interesting to try the equation above without the floor( ) to see what happens and check this with different values. Indeed, without the floor we have                     r = cos(pi/n)/cos(-pi/n) .

Thanks for inspiring some new learning @solvemymaths!

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