If you’ve been a regular reader of this blog you may remember a post in 2014 advocating the use Duck Tape to help with practical investigations. The post was: http://mathssandpit.co.uk/blog/?p=1585

I’ve recently reused the mechanics activity with a new class of Year 12 students. They had only just started mechanics and were familiar with models and suvat equations. We timed four different objects being dropped, under gravity, down a stairwell. The items had a variety of masses: a paper helicopter, a light plastic ball, a small sponge and a dense juggling ball. We meticulously timed each drop and double checked the height.

I asked the students to work out the velocity of each object on impact with the ground. It was akin to lighting the blue touch paper and standing back …

They are a competitive bunch and raced ahead to use the correct suvat equations to calculate the velocity. Then the fireworks started!

Part of the class insisted that the juggling ball must have the highest velocity. Part of the class insisted the velocities were the same for all of the objects. The rest were catching up and wondering what all the discussion was about. I innocently gathered their ideas on the board and asked them what was going on. Those who had used initial velocity, time & distance to find the final velocity had differing answers for each object. Those who had used initial velocity, distance and acceleration had a consistent answer.

Suddenly a hand shot up and said “Because we model objects as particles, their mass doesn’t matter so we can’t use the times”. This was followed by assorted groans from the class – especially those who’d used the individual times of the objects.

The variation of the original activity was to emphasise prior learning on setting up mathematical models for solving mechanics problems. Objective achieved!

(Obviously later in the course we’ll look at the impact of mass on mechanical models, but this was early days)

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!

Here is a quick fun starter to get your class thinking about dimensions and volume.

Question: How many students can you fit into a metre cube?

The discussion will probably include:

‘How big is a metre?’ (find a metre stick)

‘Did you mean standing in a square?’ (no)

‘How big is the student?’ (average – that answer annoys students)

And finally ‘Huh?’ (ask the person next to you to explain)

Whilst this is going on make a metre square on the wall and the floor, using duct tape. The inner measurements of the cube are 1m, the rest is just tape border.

The Predictions
Draw up a quick tally chart of how many students they think will fit. A bright child will usually ask how are you going to find out. Easy …

Put students in the cube
Let them put themselves into the confines of the cube. Cue bouncy boys squashing up. Then remind them it can’t be higher than a metre. You might find it useful to have two spare students hold metre sticks vertically at the non-wall end to define the end of the cube.

We managed nine boys, plus gaps at the top for bits of a tenth boy – it wasn’t ethical to chop one up and sprinkle the bits. So we imagined the tenth person balanced on the gaps around their shoulders.

Ten? That is a new record for this activity!

The Point

Volume is the space inside a 3D shape.

One metre cube is bigger than you think.

It’s a memorable activity to refer back to.

Even better if …

I’d love to get sturdy board covered in birthday (or Christmas) paper to put under and around the cube to start a discussion about surface area. You could make a big show of unfolding the cube and laying the wrapping out on the floor to form a huge net.

Note
I used to do this by taping metre sticks into a cube, but they fell apart easily. In some schools three metre sticks is a challenge, twelve would be a miracle find. Duct tape works much better!

So I’m all ready to teach a lesson recapping number patterns from the basics for a lower ability group … then a visitor to the Department arrives and asks if it’s okay if they observe my lesson. They’ve been told that there is usually something ‘off the wall’ happening in my room. Thanks … I think!

Well, I’m not one to disappoint. A little fun with the starter perhaps? The sun is shining and I’ve got whiteboards and chalk …

We’ve all seen fence panel number patterns. Here is a fence:

What can you see?

We discussed the pattern linking number of posts and spacers. We then represented the fence in colour coded symbols (yes, we have chalk in more than one colour!) and annotated it.

The class were then sent off to find their own patterns. They found repeating patterns and made notes on their whiteboards. Once they were happy with their work they could chalk it out.

This group looked at number of slats on a bench with number of benches.

They represented each bench as an ‘L’ and each slat with an ‘o’.

They worked out:
No of benches x 6 = No of slats

Other groups looked at number of windows & number of classrooms and number of benches & number of picnic tables.

We then went back to our quiet number pattern work in the classroom.

This task is easily adaptable for many aspects of number, including ratio and proportion.

Most elements of Core Maths can be visualised with a good diagram, but volume of revolution can be tricky if your technical drawing skills leave something to be desired. My colleague JA came up with a visualisation which is simple and elegant, yet also fun and memorable.

Step 1
Start with a curve. Introduce the limits a and b. Discuss what shape a thin strip would make: a disc.

Step 2
What would several discs make?
Now this is the cool bit:

This innocent looking shape is a pop up gift tag:

You can demonstrate what happens if the curve rotates 180 degrees around the x-axis.

Step 3
Now the really fun bit: dig out those interesting honeycomb christmas decorations, a metre stick and some tape:

The metre stick represents the scale on the x axis. The decoration represents the full 360 degree revolution about the axis.

Hint
Since these decorations are made from paper and card. You can use a sturdy craft knife to cut them into other curves. They also make great wall displays.

Just a quick picture to share today. My colleague, D, went to the same TeachMeet as me and was equally impressed by the use of gaffer tape in the ‘Big Maths’ presentation.

Today his class were doing box plots and took the idea of averages even further. They made a vertical box-plot on a wall of the class heights. Brilliant!