Friday 5 July 2013

#MakeoverMonday - Modelling cost per minute

 This is my attempt at this week's #MakeoverMonday. This is the original problem:

I would love to know what you think, so give it a read and let me know. Ok, lets get started!


Tell a story (optional): "This is me travelling in South America.

I'm on a really tight budget (hence hitching a ride on top of a truck!).

Here are some pictures I'd love to post on facebook from my trek to Machu Picchu:

So I go to the nearest town and see these two internet cafés

Pose the problem: Which should I choose?

What factors might be important to consider (Cost, internet speed, how nice they are inside, etc.)? How could I find the information needed to make a smart decision?

At this point give them the image from the original problem, "This was all the information I could find:"

Subtext: the story introduction is something I've put in because one of my goals this year has been lesson personalisation. Its something that has paid off great dividends in terms of my relationship with many of the pupils.

Also, storytelling is supposedly hard-wired in to all humans, which means that the problem is framed in a relatable and memorable manor, even if the pupils are not capable of relating to the algebra on the horizon. Its optional because it may not suit your style or there may be classes who may just get confused/distracted with the extra detail.

The problem is posed simply and honestly. There are no 'write an equation' or other scary sentences, and it feels like this is information that you would readily be able to get from each internet café.

By the way: to make the problem more applicable to internet cafés I would change the <per hour> in to <per 10 minute slot>.

Building in the Maths

Specific: Some bright soul may say that it depends how long you're going to be on the internet for. Cement this idea by asking:
  • Which is cheaper if I'm online for 1 hour? 2 hours? 3 hours? 4 hours?
  • Will that café always be cheaper?
  • Try to find an amount of time that would make 'internet action' cheaper.

General: Scaffold the writing of equations (if needed):
  • How are you working out the cost each time (for each café)?
  • Could you write what you're doing in words?
  • What about in symbols?
Representing on a graph:
  • Could you plot these equations as lines?
  • What does the point where they meet represent?

Subtext: In the original problem, part d is a significant step down the ladder of abstraction and is much easier to solve than a-c. I've switched it around, going from specific cases to the general equations.

I've added my way of scaffolding writing equations (where you write down what calculation you are doing in words first). I've found it really helps pupils to connect algebra to what it actually means.

You could also have a 'take a guess when they will cost the same' section before moving between the specific and general case. This may improve motivation for the general case, but in this case pupils may feel its easier to use trial and improvement (thus demotivating the need for the general case).

Follow-up and Extension

Offer similar problems: e.g. Dueling Discounts or Stacking cups or just written questions.

Extend the problem: Put the question backwards by giving the costs for various amounts of time online and get the pupils to work out the price-scheme. You can also extend this by having a messier version of the same problem.

Subtext: You have to be careful here that the similar questions are not so similar that they become mindless. I like the duelling discounts and the toaster regression for this as it is not immediately obvious that the same technique would be applicable here. One of the 'big ideas' I would like my pupils to learn about maths is the portability of problem solving techniques in to different types of problem.

I used the classic extension method of reversing the question here. Offering a problem with messy numbers or without perfect correlation is another portable extension tool I often use.

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