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):
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.
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.
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…
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.
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.
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.
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.