A nice collection of Problem solving resources arranged by topics on this site by @LeanneShawAHS
The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.
What is the ratio of the height of the red triangle to the height of the pink triangle?
Can you solve using trigonometry or only using Pythagoras?
The border is now 2cm, whilst the side length of the red triangle remains 4cm. What is the ratio of heights now?
Explore what happens for other border widths. Can you generalise for any border width w?
Geogebra file here.
I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.
These problems are ones that are made much clearer by drawing a rectangle to represent the “whole” and then deciding how to divide it into equal parts. The numbers are not too tricky but interpreting the question might be:
These are not intended to be fraction of an amount questions. An approach could be to decide upon an amount, but the intention is to direct students to drawing a representation of each question.
UPDATED POST. I used this task at my workshop at #mixedattainmentmaths on Saturday. I asked all teachers to have a go at this task but to do it in what they thought was the most obvious / simplest way. An interesting experiment: what is obvious to some is not to others. Of the solutions that I managed to take in, these were the choices:
This looks like a very useful open-ended task which provides an opportunity for creative solutions and rich discussion.
In my view, the value in this activity is in representing each area as a fraction calculation.
According the Australian blog where I first read about this task, this is one of the most common first solutions
I’d be looking for some rationalising as to why the red area is a quarter. For example:
There are 100 solutions posted here!
And on a Prezi here enabling you to zoom into each one individually.
This is potentially very high ceiling. If students are struggling to come up with suitably challenging solutions of their own, you could always ask:
Have a go first yourself. I think this is a pretty mammoth task! This one caught my eye, but you might want to have a look at the 100 solutions to find something a bit easier!
My high attaining Year 9 class didn’t quite get this on their own yesterday but they enjoyed the challenge and were able to follow the explanation.
The crux of the problem is getting a right angle triangle with sides 1, (1+x) and (2-x). It is the (2-x) side which is hardest to spot. There were groans when I finally showed them.
Then it requires some algebra – namely expanding (1+x)² and (2-x)² which Year 9 hadn’t had much practice in, so it was good to show them why (1+x)²≠1²+x²
I gave them the problem printed out, here they are 2 to a page.
Here is link to the Geogebra file that I created this on.
I had a few Twitter responses to this including @ProfSmudge who kindly set us an extension question:
It’s an example of an Apollonian gasket, apparently (thanks to @mathforge for pointing that out!). That gets properly hard, involving Cosine Rule. Certainly not something I’d give to my students, but I’ve got a few teachers working on it!
My Year 10 class did these 3 this week:
They have their GCSE RS and Science exams next week, so I needed to find something a bit “fun” but still wanted it to be “mathsy”. They are a strong group with enough keen beans amongst them for me to feel confident that something like this would work. I was impressed with their teamwork and tenacity. At first it seems hard because you are faced with a blank sheet (their words). It took the quickest group 20 mins, 1 out of the five didn’t finish after 30 minutes.
Something I always bear in mind when doing any sort of team activity (see my earlier blog post) is that everyone should have a clear role to play and something to contribute. In this task everyone has their own set of clues, that they are not allowed to show to the others. I encouraged them to make sure they were the experts in their own clues so that when proposals were made they could say if this proposal broke any of their rules.
There is no need to cut up these sheets into “cards” as the resource suggests. Quicker and better if you just cut them into strips:
That way you can give a strip to each student. They are less likely to drop a card on the floor which kind of ruins it! If you cut some horizontally and some vertically as shown you can cater for groups of 4 or 5 students.
We finished with 10 minutes to spare and had a pretty good discussion about the task. Questions I asked them:
How did they start the task? (Talking or just reading)
Did it help to group the clues in any way?
Were they tempted to share the clues? (I told them I’d give them 30s time penalty if they were caught sharing.) Or were they happy having their own? (most were)
How did you work well as a team? What did you think went well?
Was it hard? What made it hard?
FMSP isn’t all about A level Further Maths. They have some great GCSE resources too.
This page contains some superb resources for problem solving and group work with some very helpful discussion prompts and worked solutions for each question.
These look like nice resources for introducing group work, as each member of the group has “their” cards but they work collaboratively to complete the task.
Also, this post from Number Loving has some good suggestions.