My (not so little) monkeys in KS3 have been discovering and using Pythagoras’ Theorem. They usually deal with open questions quite well, however this one took a fair bit of discussion. This challenge requires no worksheets or fancy resources, just write it on the board. The context is modified, but the essential question remains the same.
Challenge
Zookeepers have attached eight bolts in a cuboid formation (sides 3m, 4m and 5m) to the trees in a chimpanzee enclosure. The keepers attach taut ropes between the bolts for the chimps to climb on. Each length of rope is individually cut. No length is lost in knots.
What is the maximum length of any one piece of rope?
What is the total amount used, if every corner is joined without duplication?
Solution
The first step to solving it is a good diagram of the problem. Students then need to break it down into triangles. The solution has several levels of difficulty:
Total of the edges of the cuboid
Total of the diagonals on the faces
Total of the diagonals across the inside of the cuboid
This diagram demonstrates the levels of the problem – have fun!
Joey? Why yes! Joey Tribbianni from ‘Friends’ (random 90s TV reference). When the friends came to London, from New York, Joey demonstrated his unique technique for navigation:
He jumps into the map! And that is what my students did for vectors.
In the first lesson we started looking at the basics of vectors. I found this PowerPoint from TES resources to be really good for starting discussions: Introduction to Vectors by rhemsley
In the second lesson, we started to solve problems and moved outside.
Objective
To develop problem solving skills with vectors.
To understand how vectors relate to more complex diagrams
Resources
Chalk (coloured is good)
Vector questions from a textbook, worksheet or exam database
Mini whiteboards – optional for working out/calculations on the move
Activity
I drew one side of a regular (ish) hexagon on the wall (I really would like an outside chalk board). We labelled the ends A & B and the vector joining them was a. I decided to use a different colour chalk for each vector to make them stand out.This was followed by vector b (joining B & C) and vector c (joining C & D). To be honest, I had to get a student to draw in D and the line – I’d accidentally* drawn the diagram higher than my arm reach.
(*I’m over 5ft 6in, but my lively lads are nearly 6ft – gallantry meant they felt they had to help and hence engaged with the task very quickly!)
Back to the problem:
This hexagon wasn’t going to draw itself, but which of the existing vectors to use to create point E? I explained that vectors a, b & c are our building blocks, which helped us move on.
The students were quick to identify vector a. They were very picky about the direction too. My gallant helpers added it to the diagram. We used the same logic to finish off the hexagon.
The students then went away with their worksheets and drew out their own versions of this diagram. As I walked around there were some heated debates as to which vector went where and in which direction. They addressed many misconceptions before they even looked at the question – it comes back to the old rule of ‘ Write down what you know’. They’d already answered questions on the diagram that they hadn’t yet been asked.
The questions were based on this diagram:
They started easy and quickly moved on to trickier concepts:
Why can’t you just invent a letter for BE?
Does it matter which route you take around the diagram?
Can vectors be simplified like algebra?
All of these questions can be solved by ‘doing a Joey’.
When I teach vectors I always use the analogy that vectors are like a one-way system of roads. For example, the vector from E to B is drawn in, but has no specific vector – this is like a closed road, you must detour down the other roads to get there.
As you walk down the other roads (vectors), are you going the correct way?
If the answer is ‘yes’, just make a note of the vector and carry on.
If the answer if ‘no’, then a minus sign gives you the permission to go the wrong way – write the the vector with a negative. (This then leads to the follow up discussion of why this works through use of column vector examples)
There is more than one route. What do you chose?
Clockwise or anticlockwise?
EDBC or EFAB?
Walk them both and see what happens – you can see that the vectors are the same, just arranged differently.
Finally, when you start to compare different routes, you can see that vectors cancel out, just like algebra. In fact, it quickly becomes clear that basic rules for simplification still apply.
Student Reaction
The sight of pupils walking around diagrams looked like something from a Monty Python funny walk sketch. As we walked back to class, after half an hour in the sun, I overheard two different groups of students. One group said: ‘That was a really good lesson’ and ‘Yeah, I enjoyed that!’ – that may have been the sunshine though. Another group however said: ‘I get it now. I didn’t get it last lesson the board, but now I get it!’ – and that wasn’t just about the sunshine!
We may only be a few weeks into the summer term, but I can safely say this is my book of the term. A gently inspiring, pick up a pencil and relax book.
‘This is not a Maths book’ by Anna Weltman (RRP £9.99) takes all the beautiful ideas we maths teachers wish we could use more often and collects them into a wonderful book.
The pages are full colour and the paper quality is excellent – almost tactile. And the best bit is that no-one can tell you off for doing students’ work or wasting your time making that wall display just right. It’s your book … just for you … you can be as possessive and OCD about the colouring pencils as you want!
It would make a good end of term prize too – a bit different to the usual geometry set or calculator. If you are a forward planner, you could even buy this book for your mathematical someone in a departmental ‘Secret Santa’.
I usually tick these worksheets until I find a mistake. I then tell the student to have a rethink. Obviously the correct answer is the other option, but the working out will need to be corrected. I also do not tell them how many of the remaining questions are actually correct – they then recheck these before I mark it again.
The only difference with this worksheet is that students have space for working out – no more guessing! The extension task asks students to try and figure out where the wrong (misconception) answers come from – that can be quite tricky and tests their understanding.
A quick reverse percentages resource for you. I explain reverse percentages by using both calculations and diagrams. These resources can be used as a starter activity or as a selective discussion point. The presentations are editable and the pdf is identical to them. I hope they are useful in addressing the reverse percentages misconceptions!
It’s beginning to look a lot like Easter … scrawny plastic chicks and over-priced chocolate eggs everywhere! This little ‘egg’ of an idea was totally inspired by some lovely Tweeters who mentioned ways to use empty plastic eggs.
Equipment
I bought these two-part plastic eggs from a local craft shop. They are available from lots of places on the high street and online. My pack has 30 eggs in six different colours. You may be able to see that I’ve numbered the top and bottom of each shell – just to avoid arguments.
Activity Now, I used these eggs for revision with my GCSE class. Each colour represents a different topic. There are 30 questions and the answers are the numbers 1 to 30. I hid the eggs in our main hall due to the unpredictable nature of the British weather. You could hide them inside or outside the classroom and give a prize to the person/group who correctly completes the most questions. Points could be deducted for trying to sabotage other groups. If you don’t feel that adventurous or it’s impossible to go outside, you could copy the questions and do this as a desktop activity.
Topics
Sometimes we get tunnel-vision on the focus for passing exams. We keep the ‘fun’ stuff for younger pupils. This revision activity is a treat for my hard-working students in KS4. They aren’t the easiest of topics, but they are perfect for students working at GCSE grade C and above.
Feedback
I was surprised to get feedback from this activity from a form teacher, who said their students had arrived at registration bouncing and saying how much they had enjoyed the lesson!
If you taught in England while mathematical coursework still existed, this post may not be new to you. However those who did not may be pleasantly surprised by the simple complexity of ‘Squashed Tomatoes’!
Aim
To investigate a growth pattern, which follows a simple rule.
Equipment
Squared paper
Coloured pens/pencils
Ruler & pencil
Rules
Imagine a warehouse full of crates of tomatoes. One crate in the middle goes rotten. After an hour it infects the neighbouring crates which share one whole crate side. This second generation of rot infects all boxes which share exactly one side. Once a box is rotten it can only infect for an hour, then ceases to affect others. This sounds complicated, but trust me … it’s simple!
Picture Rules
The first box goes rotten – colour in one square to represent the crate. The noughts represent the squares it will infect.
The second set of crates becomes rotten – use a different colour. The noughts represent what will become rotten next:
The third set of crates becomes rotten – change colour again. At this point it is useful to tell students to keep track of how many crates go rotten after each hour and how many are rotten in total:
The fourth set of crates forms a square:
The fifth hour returns the pattern to adding one to each corner:
The sixth hour adds three onto each corner:
Now you can continue this pattern on for as big as your paper is. Students can investigate the rate of growth of rot or the pattern of rot per hour. As the pattern grows, the counting can get tricky. This is when my students started spotting shortcuts. They counted how many new squares were added onto each ‘arm’ and multiplied by number of ‘arms’.
Here are some examples of my students work:
This is a lovely part-completed diagram:
This piece of work includes a table of calculations – you can see the pattern of 1s, 4s and multiples of 12.
This is just amazing – you can see that alternate squares are coloured (except for the centre arms).
On this large scale you can see the fractal nature of this investigation.
Extension: Does this work for other types of paper? Isometric? Hexagonal?
