Logarithms

In last week’s post I talked about the work that we had completed on indices and how we were using this to launch logarithms and exponentials this week. The benefits of this approach were shown up during one of the indices lessons when one of the students was tackling the following question (taken from Stuart Price’s Problem Book, @sxpmaths):

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He asked if there was an easier way to tackle problems like this, suggesting that if the numbers were much bigger it would make the problem more difficult. This was a perfect opening for introducing logarithms, which we did with a series of similar questions.

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When planning this lesson in discussion with Will, we felt that this approach really emphasised the link between indices and logarithms, something that we both felt had been missed by some students in previous years. We felt that the best way to do this was by talking about functions and their inverses. Yes, inverse functions is strictly a second year topic and they wouldn’t need it for their exams in the first year, but it is the connection that has been lost in the past and the reason students struggle with the topic.

In order to build these links we created a Geogebra file that allowed us to turn on an off a series of functions and their inverses. We had it set up to display a couple of linear graphs, a quadratic and then an exponential – the goal being to draw out that the inverses were a reflection in the line y=x. This had students already trying to tell us what shape the inverse of an exponential should be, before we even introduced what the function truly was. I’ve used a similar approach before, but not incorporated the graphs – so the connections were already stronger than they had been for students in the past.

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The next part of the lesson formalised the notation of logarithms, after which we went back to these questions and rewrote them as logarithms, solving the later ones using our new calculators as we went.

The next lesson began with a recap starter, but this time we took the students a bit further…

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They were all able to calculate the answers on their calculators, but actually explaining why was missing, and there was no real notation or workings out (yet). So when we went back through these questions we modeled taking logs of both sides of the equation, or using both sides as powers with an appropriate base. Then we could discuss that as the two functions that we’d composed were inverses that their effects cancel. All designed to reinforce the links between logarithms and exponentials as well as to lay the groundwork for exponential / logarithmic equations.

In the past we’d found that students can be quite unreliable at remembering the laws of logs, despite the connection to the rules of indices – perhaps down to the split in topics between C1 and C2. On our scheme of work this split is non-existent as we’ve run the topics together. We also decided that to further emphasise the link that we would start with the indices rules and from them actually derive the laws of logs – this might go over the heads of some of the students, but they’d have it to look back and reflect on, plus we knew that there would be a good proportion of our group that would embrace knowing why this rule exists. After doing the product law we let the students try to work out the quotient law, and even to have a go at creating the derivation on their own. This also built on our previous work on proof and on how to construct a solid mathematical argument.

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After we’d derived all the rules, the next step would be usually be to work through a series of examples, with students copying them down. As they’d already done a lot of writing I gave them the complete examples and we went through them with the students annotating why things were happening, and what rule was being used to do these things.

When we introduced ex we got the students to plot graphs in a template that we had created in Geogebra – the idea being that after they had plotted 2x and 3x and examined their gradients at each of the points we had given them that they would see that at each point,  ex has the same gradient as y-value. The group were fairly pleased with their discovery, and it allowed us to give them the reason why e is so special that it has been given its own letter. We did have one query though: “how did they calculate the value of e, so that it is the value that will them give it its own gradient?” – this came from a further mathematician, and that question became his homework.

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Much of the rest of the topic was fairly standard – log equations, hidden quadratics involving exponential equations, but with the addition of ex and ln( ) ≡ loge( ) for the new spec, however we did make one tweak to what will has been taught in the past. We felt that students could not pick up how to add logs to a term that does not have them in so that they can then combine them using the laws. Even just these 4 questions that we started together were enough to give them something to work from in the future.

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Overall we were very pleased with how the teaching of logarithms progressed over the week. In order to assess what we had covered we asked students to complete one of the Integral online assessments. Hopefully as this comes back we will see that students have a greater understanding than we have seen in previous years.

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Problem Solving and Technology

In our work on revamping the curriculum for the new specification we have been careful to make sure that we are considering the overarching themes of problem solving and use of technology. We are very keen to ensure that use of technology does not become ever more complicated and ‘interactive’ PowerPoint files that are demonstrated from the front of the class room with little chance for students to use and develop their own skills. We also want to introduce this aspect as early as possible to encourage students to think about technology as a vehicle for working on and solving problems when they get stuck. On Friday I met with Simon, Fiona Kitchen (from FMSP) and two colleagues from my department to discuss methods for this.

Our starting point was to use the worksheet “Problem Solving with Geogebra” from MEI’s scheme of work. We looked at solving the problems ourselves, trying to limit techniques to those that year 12 students were able to use. This proved rather difficult! After much wrestling (and a plea to twitter) we managed to create working models in Geogebra for the first three of the problems.

We had a lot of fun working on these problems but concluded that the level was a too high for students who are starting out in year 12. As such we will need to adapt to something closer to GCSE if we are going to introduce Geogebra in this way at the beginning of the course. One thing that struck me during the afternoon was that we persevered with the problems for a long period, around 2.5 hours. This was something that our students would have really struggled to achieve. The same morning one of my year 11 students, when confronted with a difficult question, said “It’s alright for you sir, you are good at maths and can do it easily.” I was unable to make her understand that I don’t find all maths easy and that I enjoy the struggle with harder problems. This perhaps sums up the major problem we have been fighting against with our A-level students over the last few years – the lack of resilience as soon as a problem gets complicated.

Hopefully this process of really concentrating on both problem solving and technology will help to improve this, which is certainly the focus of what we are looking at during this process. For now though here are the problems that we worked on.

Problem 1Problem 1

With this problem I found it very easy to create a polynomial fixed by the points A, B and C. This initially created a point D that moved as A, B and C moved. To do this I used the measuring tool in Geogebra to calculate how far away the points were from the origin.

My second attempt used a division at the start of the polynomial to allow me to control point D as well. It was at this point that I realised the shortcomings of my method of measuring distances – when I moved the points to negative values Geogebra continued to measure them as positive. To complete the problem Simon showed me how to use just the x-ordinate of A etc. in the calculation. My final solution is at: https://www.geogebra.org/m/bZD6fARj

Problem 2Problem 2

For this problem I started by drawing a circle centred on the origin with a point on the circumference fixed into the side AC. I then created another circle centred on C which connected to the first. I repeated this methodology to create a third circle centred on point B. The three circles could then be manipulated together until I had a solution that worked. This did not satisfy me – I wanted to be able to change the triangle and the circles to remain a solution to the puzzle.

My instincts for this puzzle were probably from spending time playing the mobile phone game Euclidea – maybe those hours were not completely wasted! I guessed that the points where the circles met on the edges of the triangle were those where the circle inscribed in the triangle also touched the sides of the triangle. My resulting solution can be found here: https://www.geogebra.org/m/jMSThZg8

Problem 3Problem 3

Problem 3 caused me the most problems. For a long time I was able to either create a line that was perpendicular to the tangent from A or a line that passed through the point B but not both. After a long time trying (and a plea to Twitter), Simon came up with a solution that can be found at: https://www.geogebra.org/m/MFgcJaAd

Problem 4

We ran out of time before tackling problem 4 and I have not yet returned to it. I have some ideas about using a quadratic function with roots that are the x-ordinates of A and B and integrating it but have not progressed any further yet. I leave that one with you…