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.
I built this on Geogebra. It’s pretty simple but might be a good way in to 3D coordinates and more generally explaining the concept of dimensions.
You can download and open the original Geogebra file here which gives more control than just playing the gif.
- Start with all sliders at zero.
- As you increase slider a, talk about the first dimension. Any point in the 1D world can be described by a single number which shows how far along the line you are. Every object in a 1D world is just a line. Long or short. It can be described by a single number which we can call length.
- Once a has reached 1, talk about the second dimension. This is now like a floor, or the surface of the earth. We call this a plane.
- Increase the slider b.
- The world in 2D contains two dimensions, which we can call length and width. There are other words: e.g. breadth, depth.
- Every point in a 2D world can be described by 2 coordinates. These are the x and y coordinates. It’s important to notice that the x-direction (i.e. the x-axis) and the y-directions (the y-axis) are a right-angles to each other or orthogonal. Why is this?
- Once b has reached 1, what shape to we have? How many vertices does this shape have? How many edges? What are the coordinates of its vertices? Do we need 3 figure or 2 figure coordinates for a 2D shape?
- Now we can bring things into the real world in which we live, the 3D world where shapes also have height.
- Increase the slider c to grow the cube upwards.
- When c has reached 1, what shape do we have? How many edges, vertices, faces does it have? What are the coordinates of the vertices of this shape?
From there, you could always say:
Why stop at 3 Dimensions?
Origami is one of those things that I think I would love to spend more time exploring but rarely do. I have used the Origami Player with my students (it works really well as an App within Chrome), which gives excellent visual instructions on making things. The timings have been well thought out and it gives a little timer prompt so you know how long you have got to do that fold before looking up at the screen for the next one. It’s been an end of term, easy lesson. Nothing wrong with that.
The first session on the nRich Teacher Inspiration Day last week where we looked at some of the activities here got me thinking about how I could make it a slightly more meaningful learning experience.
Still not highly mathematical, but at least it gets students working together and struggling with something. To build resilience in our students they need see the struggle as a positive and not something to be avoided at all costs. It was a bit of a metaphor for all learning. A discussion that can be had with students when reflecting on this task might be along the lines of:
- Did you need help from someone at some point? (yes, good)
- Did you help someone else at some point? (yes, good)
- Did you struggle at some point? (yes, good)
- Did you give up? (hopefully no, good)
- Did you achieve something you didn’t think you could do before?
This type of discussion can be a powerful motivator and more useful than vague questions like “Did you enjoy it?” or “Did you have fun?”
There are lots of Origami ideas on this page of nRich’s new Wild Maths site. I really like the idea of modular Origami, i.e. each person makes a module and then they come together to create something beautiful. I have an end-of-term cross curricular session with Year 8 to plan. My Origami paper arrived this morning. You can use A4 paper and fold down the corner to make a square, but proper origami paper is really lovely and this pack was only £8 for 500 sheets. It looks great so time to start practising!
I will post an update on this as planning progresses and share some pictures of the final event. In the meantime, if anyone has an ideas for how to make Origami more mathematical (without spoiling the enjoyment of it!) then let me know.
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?
Some questioning as a way to get to Platonic Solids:
- What is special about a cube? (compared to a cuboid)
- What is special about squares? (the sides on a square)
- What do we call 2D shapes where all the sides are the same?
- So, can we generalise, what is special about a cube?
- What other regular 2D shapes do you know. Draw them now.
- What other 3D shapes can you picture that can be constructed only with regular 2D shapes? These are called Platonic Solids.
Have a go. See if you can find some dusty old 2 shapes in a cupboard somewhere.
These are quite cheap or you could make your own from card using this template.
So, here they are in all their glory. The wikipedia page has some nice animations.
The Greeks discovered these 5 and mathematicians since have proved that there are definitely no more, whichever polygons we use.
Next you could complete this table.
Once you have done that, can you spot any patterns in a relationship between F, E and V? You might even discover Euler’s formula, which holds true for any convex polyhedra, not just the platonic solids.
V – E + F = 2
Or use (a lot!) of 2d shapes to make nets of these shapes.
This page gives all the answers in a nice table!
I liked the look of this Nrich task thinking that my Year 7s, who have shown some appetite for investigative tasks, would enjoy it.
It’s a great task, but like so many investigations, you really need to have a go at it first yourself. I did it with my daughter (Yr7) and we took nearly an hour to find the 3×3. And I was trying pretty hard! Interestingly the 4×4 was much easier. Here are our combined efforts!
It’s going to be critical how I introduce / explain the task, so I will create a notebook file to help with this.
I have also created a worksheet as I feel that my class will need a bit more scaffolding on this task. I now know that I will encourage them to move onto the 4×4 if they get fed up with the 3×3.
I’ll update this post tomorrow once I have taught the lesson.
Based on a task from nrich but I felt that the context was a bit confusing. And I thought some colour would help!