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 (

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?

Posted in Number | Tagged | 2 Comments

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

Yesterday I saw some awesome Mr Men Math posters created by @solvemymaths ( 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.

mr astroid 3 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: astroideqn

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

desmos astroid

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!


Posted in Uncategorized | Tagged , , | 2 Comments

Examples of Linear Relationships: Proportion, Modelling, Correlation, Regression

Started a list for linear relationships. Not all linear across an infinite domain but that adds in discussion on piece-wise functions. Can anyone add more to the list?

Purely mathematical:

1) Diameter vs Circumference of a Circle

2) Length of Side vs Perimeter of a Square

3) Number of Sides of Polygon vs Sum of Interior Angles

Connected to Real Life:

1) Temperature of Coffee vs Time

2) Amount of Fuel in Tank vs Miles driven

3) Amount of Ink in Pen vs Pages Written

4) Profit made in first weekend of Film in Cinema vs Total Profit (

5) Altitude vs Temperature (data can be recorded on some airlines during descent)

6) Taxi Cost vs Time

7) Gym Membership Cost vs Time (Initial Payment to begin member)

8) Internet Cost vs Time (Initial Payment for Installation)

9) Distance-Time Graphs and Velocity-Time Graphs

10) Hooke’s Law and Newton’s Second Law  

– Physically Measure:

1) Length of Foot vs Height of Person

2) Pull back distance of Model Car vs Total Distance travelled of Model Car

3) Number of Bites of Liquorice vs Length of Liquorice (Discussion on the type of bites you would need to keep linear)

4) Queueing time vs Number of Items per Customer (Dan Meyer)

5) Number of Sips vs Amount of Water in Glass (Discussion of the types of sips you would need to keep linear)

6) Time Jogging on Treadmill vs Calories Burned  

– Service Related:

1) Number of Cigarettes Smoked per Day vs Life Expectancy

2) Number of Questions Practiced vs Grade in Test (in class data taken)

3) Percentage of Population in Extreme Poverty vs Time (ATM: MT245)

4) Income per Person vs CO2 Emissions (

5) Math Achievement 8th Grade vs Income per Person (

Things I’d like to know how strong the correlation would be:

1) Weight of Person vs 1 Rep Max on Benchpress in First Gym Session

2) Salary of Striker vs Number of Goals Scored

I did recently ask Grade 10 to look into Number of Tweets vs Number of Followers on Twitter (restricting the domain so as not to include anyone famous). Whilst I knew it wouldn’t be linear, I think it’s important to discuss examples of why relationships are not linear. Often there is a correlation between how many variables are involved and how linear the relationship is. For example, the start up capital of a business vs profit in first five years. There are simply too many variables at play in this, just as there are for number of followers on twitter. I’d love to add more to this list…

1. Charging an iPhone (Michael Fenton): Linear at first and then tails off.

2. UK shoe size vs Europe/US shoe size (mathematical beginnings)

Posted in Modelling | Tagged , , , , | 2 Comments

Starting with what Students know: Solving Quadratic Equations

I thought I’d share a first lesson on solving quadratic equations that I developed for year 10 recently. I think many would agree that starting with what the students already know about solving linear equations is an aspect of this learning journey which often gets side-stepped. I therefore tried to make this integral to the learning experience. Looking at the component parts of this lesson, each one is completely obvious to any experienced maths teacher. The reason therefore that I chose to blog about it is because I felt that the learning sequence and journey in this particular case was a quality way to promote real understanding of solving equations. More technical aspects, solving in different ways and real life application will all come later in the unit.


quad 1

Independent, quick and doable. All three aspects in this case were connected to the key parts of the lesson. Even though it pained me to do it, I taught double bracket factorisation before introducing solving equations this year. Usually I start by asking students to try to solve a quadratic and then introduce the factorisation as a means of necessity. This time however, I felt that to really hit home the concept behind why factorisation is used when solving quadratics, the students needed to be skilled at factorising before understanding why it is important (would be interested to know what other people do in this case).

Starter Q1 was important to remind them of balancing an equation and also to remind them of what the word solve actually means. Normally I would say that solving an equation gives you a value/s which, when substituted into the equation, make it work (e.g. satisfy it). In this lesson, I made the students discuss in more detailed terms what this means and we rested on, “The value/s that, when substituted into the equation, make the left hand side of the equation equal to the right hand side.” We plugged in a few numbers to make this clear, one of them being a decimal and the other being a negative whole number.

It’s obvious that starter Q2 helped to remind them of double bracket factorisation. Starter Q3 however is interesting in that I wanted this concept to be completely obvious and simple to the students before I introduced it as integral to the strategy of solving quadratics. Students said it was easy and I was glad that it was explicitly mentioned at the start.

Main (part 1):

quad 2

[The second question animated to appear after discussion of the first question]

The first question was easily solved by balancing although students forgot that there would also be a negative solution. It was very important to the lesson to show how balancing equations to solve them is a very general and wide-reaching tool for a mathematician.

The second question is then designed to show that balancing would not work; something the students were quickly able to identify. At that point I took it back to our definition of solving to ask how we might solve this particular quadratic. That is the reason behind the form of the quadratic; I wanted the student to be able to substitute a number into both sides. Students recommended trial and error to which I agreed but only if they use technology to maximise efficiency of the process. One student mentioned at this point that he’d already found the solution (x=2) to which I said that he could move straight onto the next task (I’ll come back to this). Luckily, Year 10 have TI-Nspire Graphic Display Calculators and so we utilised the spread sheet view. Excel would be just as good if you have quick and easy access to it.

I took the students through the use of the calculator with this example. Essentially we wanted to try lots of different input values for x and check whether the LHS of the equation would equal the RHS. (i.e. when is x^2+10 equal to 7x)

quad 4

Of course the interesting moment was when the student who had found the first solution (x=2) was amazed to see that this equation has another solution (x=5). I responded to his amazement with: “Just because you know a lot about linear equations it does not mean that you know a lot about quadratics. Alarms bells should have been raised when you saw that x^2+8=24 has two solutions. Mathematics requires both intuition and rigour.” His response was both perfectly timed and very ironic: “So all quadratic equations have two solutions?” I told him that he’d be able to answer that questions himself as time went on when he could apply more rigour to his conjecture. No one at this point picked up on the fact that we had only used positive integers as inputs but I was happy to leave unsaid for now.

Students had a go themselves with the following questions and then I wrote their solutions on the board:

quad 6

Of course some students were confused as to why Q2 didn’t fit the mould and only appeared to have one solution. Fortunately I got a “Ahhhh” moment when one student realised that the second solution may not be one of the input values we’d used in the calculator. That’s when I told them that there was a second solution, and in the interest of the pace of the lesson, I told them it (x=-1).

No one managed to connect the starter questions to the bottom question on this slide and so when we realised that the spread sheet method was unreliable, I went back to the starter questions to highlight Q2 and Q3 before stating that these integral techniques would provide us with a completely new way to look at solving equations. So I put up the next slide and facilitated the class through how factorising the left hand side allowed us to know with certainty that one of the brackets, or both of the brackets must be equal to zero.

[The second question on the slide was animated to show up after we’d discussed the first].

quad 7

It was completely obvious to them that the only way this method would work is if one side of the equation was equal to zero. I put this completely down to the fact that they had internalised both the skill of factorisation and the concept of multiplying two numbers to make zero, before I introduced the strategy. Therefore it was much easier for them to connect the dots and fully understand why factorisation works.

Here’s the objectives that I showed at the end of the lesson:

quad 8

Posted in Uncategorized | Leave a comment

Help Needed: Collaboration on a Rubric for Presentations about Pi

Quick one: Can anyone provide some advice on developing a rubric for student presentations on: “Understanding what pi actually is and why it is significant.”

It makes me cringe when people walk around reciting the number with no understanding of what pi is and so I’m determined this year to have my students educate the school community. I’ve got no problem at all with people memorising pi for fun and this can provide a motivation for certain students but I simply feel that we should aim to help students understand what pi is and why it is historically significant.

This is what I have so far but I feel like it could definitely be improved. Would hugely appreciate some help with this.

pi rubric

Posted in Assessment for Learning | Tagged , | Leave a comment

Exploring and Creating Mathematics Part 3: Investigating Powers using Pascal’s Triangle

Before reading this post, you might want to see the other two posts in the series:

Post 1 – Creating and Exploring Mathematics: How many different directions can we take a topic in 15 minutes?

Post 2 – Discovering and Creating Mathematics Part 2: Does investigating problems make it easier to pose interesting problems?

Whenever we get a chance as mathematics educators, I think it’s great if we can explore and discover our own mathematics. Of course it’s almost always been created before but I don’t see how can we ask students to investigate and explore if we’re not prepared to do so?

So today I spent a few hours investigating powers using a binomial theorem/Pascal’s triangle approach. Would love to see how other maths teachers would investigate this given the chance. Here’s some interesting stuff I discovered today:

1. The sum of row n of Pascal’s triangle is equal to 2^n.

2. Odd powers of 4 always have final digit ending in 4; Even powers of 4 always have final digit ending in 6.

3. The first four powers of 11 follow exactly the same digits in Pascal’s triangle. E.g.

11, 121, 1331, 14641, …

After that, the pattern doesn’t work. This is a great example to show why proof and reasoning are important in mathematics.

4. The sum of the digits of powers of 11 are always even.

Continue reading

Posted in Investigations | Tagged , , , , | 2 Comments

Exponential Functions: What’s been missing from my teaching?

To get straight into the meat of the post, I believe that one fundamental aspect has been missing from my teaching of exponential functions in the past.

To help students understand exponential functions, constant comparisons must be made to linear functions.

In the past, I’ve started discussions on exponential functions with a shocking statement (taken from Albert Barlett’s youtube lecture) – “The greatest shortcoming of the human race is our inability to understand the exponential function.” I’ve then tried to show, by getting students to do problems in which they struggle to predict the rapid, exponential growth, that this statement is on-the-whole correct. I still believe that this approach is valuable. However, when I reflect on how well students’ really understand the exponential function, there’s been something important missing from the learning equation. i.e. the connections and differences between exponential and linear growth.

The key difference between linear growth and exponential growth is that the rate of increase of linear growth is always the same, but the rate of increase of exponential growth always increases.  It sounds obvious, but I believe that in making this direct comparison, students’ are more likely to understand why exponential growth is so rapid.

I saw the WTF problem below  on Dan Meyer’s blog  (The basic principle behind “What the F” problems is that they display a concept in a surprising way), and so started the lesson with what the students’ already know about growth.

Continue reading

Posted in Algebra, Functions, Teaching Ideas | Tagged , , , | Leave a comment