I am in the process of moving house due to various job changes. Anyone who has ever even glanced at someone at any point in the house moving process will instantly feel a pang of empathy, and anyone who has gone through the process of selling their house and buying another house may need to go sit in a dark room while their PTSD-induced panic attacks and hallucinations of mortgage advisors roaming their halls subsides.

While I was driving to a viewing with my wife, we were discussing our estate agent’s fee, which is 1.25% of the sale value of our house. She asked “would it ever not be worth it to get more money for our house?” This struck me as odd, but after a bit of back and forth I realised that she was effectively wondering whether or not increasing the sale price by an amount would lead to a net loss because the rise in sale price is less than the marginal increase in the estate agent’s fee from the house being sold. How would you respond? Have a think and write something down before reading on – I’d be interested in what you think!

My actual response wasn’t particularly inspiring – I could use the excuse that I was driving at the time and I very much had my summer holiday brain wired in. I stumbled over an explanation talking about the fact that a percentage retains its value in proportion to the given amount. If we go from 100 to 101 then that 1 rise can’t be swamped by the rise in 1% of 100 to 1% of 101. My wife seemed placated, or sufficiently bored, and we carried on to the minefield of “how many toilets do we need in a house?” (At least 2 is the answer).

I wasn’t happy with my answer, however. I feel that it didn’t really deal with the issue in sufficient depth and it would be easy for a student to then make any number of misconceptions: that the amount due to the estate agents never changes or that it would always be a really small change (which my wallet actively disagrees with). It also didn’t tackle head on the potential to explore the maths around the idea – what if it was 1% of the extra amount as opposed to taking 1% of the new total? Would that be better – why or why not?

Colin Foster recently released a video on the NCETM around armchair discussions on the mathematics classroom – the basic idea being that there are questions or events that come up during the classroom that open up a myriad of different responses and that in the moment we may choose any given response without fully appreciating the power of the different paths. Colin suggests that it would be a good practise to take a bit of time after a lesson to think about how we may have answered questions differently – he calls this an armchair reflection. Given the present issue, it struck me that it was a good opportunity to indulge in such a reflection. So, how else could I have responded to the prompt: “If you an increase an amount will 1% of the new amount ever be larger than the actual increase in the total?”

It might be a good idea to use specific examples at different extremes. This is what I tend to do myself when getting a feel for the generality of a concept, or finding out what a function seems to be doing (e.g. sketching graphs). What happens to X when Y is really big? What if it’s really small? In this case this might be a prompt to the student of “make some calculations when the price increases by a little bit and also by a large amount. I would be really interested in what you find out.”

Alternatively I may decide to use algebra to help shortcut to a general result. To do this you would need to do a bit of coding: call the current price X and the new price Y – how can we phrase our question? Ideally we need to identify what we are comparing – the absolute difference between X and Y and the the absolute difference in 1% of X and 1% of Y. The next issue is what sort of relationship are we positing? A student may be tempted to bang an equality sign in there without really thinking about it. However it is important to understand the implications of doing so. If we write Y-X = 0.01(Y-X) we are trying to find the break even point, the value of the variables where this must be true. Dividing through and getting 1=0.01 is a curious result – what does it tell us? How can we guide a student to interpret that answer? “Could this ever be true? Have you done anything incorrectly in your working? So what must that tell us about our starting point?”.

We could also bust out a bar model, but I’m not sure I could use it effectively here. My initial idea is to frame the argument as “what would it look like if the fee rose by more than the increase?”. Using diagrams it would look ‘wrong’ in the sense that the portion of the bar ascribed to the fee would look over 1%. I feel like there is definitely a better way of going about this, but not sure how, definitely something to give deeper thought to at a later date. I am sure that the bar modelling gurus of twitter will show me something that will blow my mind and make me go “WHY DIDN’T I THINK OF THAT?!”

Finally, I could use graphs. Writing that down, it feels like the quintessential mathematician’s phrase, but plotting a graph of y=x and y=0.01x is a great way to get a feel for the quantities involved and beginning a conjecture, and one could quickly look at gradients and consider them as the rate of change. Linking back to the original question, how might this system of graphs show us that the rise in the fee will always be less the rise in the absolute value?

Thinking on it, I would probably plump for the idea of using concrete examples at a range of magnitudes for most of my classes. The algebraic approach has a lot of baggage to it – you feel like to embark on it you need to get students to sign some sort of disclaimer – and is probably more suitable in a typical A level classroom. Not to say it is a bad approach, but it could well take up the vast majority of lesson time and distract from the main point you want to focus on. Although this may shock you, I’m still not proficient enough with bar modelling to be comfortable winging it in the moment. The graphical approach would potentially be my other go to, but it depends a lot on how comfortable students are with straight line graphs.

I hope you’ve found this interesting, and I wonder how you would approach such a question!

-Hisham