Investigating Enlargement and Area

Recently I was looking with pupils at the link between enlargement and area. This was only in Year 8 so we weren't looking at ratio at this stage, rather sentences like "If the lengths are enlarged by scale factor 3, then the area is enlarged by scale factor ...". We did an investigation with a 3cm x 5cm rectangle, with pupils exploring different scale factors and the effect this had on the area.

Pupils were very quick to notice that the numbers at the end were the square numbers, allowing us to make the link and answer questions. It seems basic I know but I think this reinforced for me the idea that sometimes pupils discovering results and relationships for themselves is much more powerful than being shown a relationship and told what it is.

Visual approaches to Algebra

A rare day away from school today, at the Leicestershire Heads of Maths meeting run by the wonderful Jan Parry. She did a great session on concrete and visual approaches to algebra, which reminded me about the first time I saw the visual representation of the process of completing the square (it was actually on "The Story of Maths" starring Marcus Du Sautoy on Channel 4. I shared it recently when I presented at Maths Hubs and Spokes as part of my session on concrete approaches for conceptual understanding (technically not concrete I know, but still) and I thought I would share it here.

Du Sautoy showed that by chopping off half of the extra length at the end of the rectangle and moving it to the top, you created a near-square; just 3 short in each direction (i.e. 9 square units short). When I first saw this my mind just exploded - as a maths graduate I of course knew how to complete the square and what it told me about the graph/roots/minimum point etc of the quadratic but I hadn't even realised that I had no real understanding of what that process looked like in a non-abstract representation. Now I always start completing the square by showing this to pupils and getting pupils to work with similar images before going to the abstract. Hopefully, if you haven't seen this before yourself this will allow you to have that "Oh yeah!!" moment that I had when I first saw it - which will hopefully mean your pupils understand from the beginning what it means to physically "complete the square".

Concrete approaches to abstract mathematics

Did my session today for about 38 teachers entitled "Concrete approaches to abstract mathematics"; basically hands on Maths approaches. Informal feedback seemed positive and was a great opportunity  to catch up with some people that I hadn't seen in a long time. The link to the prezi I used is here http://goo.gl/UvUx3R and I thought I would share one of my favourite activities from the session - the positive and negative mood cards.

It works a bit like this - each pupil has two cards in front of them, one something they like or something they don't like (for the teachers I did pictures of a £10 note and a pile of marking on the cards, for kids I usually use a ferrari or something) and they work in pairs with a mini-whiteboard with an arrow drawn on it. The activity then goes as follows:

1) Get one person to give the other person their positive picture, and the person who receives hold the mini-whiteboard up to say if their happiness goes up or down (hopefully it will go up because they are getting something positive). This shows that when you give somebody something nice, their happiness goes up (++ = +).

2) Get one person to give the other person their negative picture and again the person who receives hold up the mini-whiteboard to say if their happiness goes up or down. This time it should go down because they are are getting something negative. This shows that when you give somebody something horrible, their happiness goes down (+- = -).

3) Get one person to take away the other person's positive picture and have the person who was taken from hold up the mini-whiteboard. The arrow should be down showing that when somebody takes away something nice your happiness goes down (-+ = -).

4) Finally get one person to take away the other person's negative picture and again the person who was taken from holds up the mini-whiteboard, which should point up showing that when somebody takes away something horrible your happiness goes up (-- = +).

I find it just a nice little gimmick to help people remember (not understand, but at least remember) how signs combine.

National Library Virtual Manipulatives

Teaching Year 8 pupils equation solving through balancing today; for me an important skill particularly for an average attaining group that will be going through the higher GCSE paper in the future. I am using my cups and counters tomorrow for a presentation to a CPD conference so decided not to use them today and so used an alternative I haven't used in a long time - Balancing Equations for the National Library of Virtual Manipulatives from Utah State University.

I really like these applets, and if you can navigate through the American grade system (UK teachers anyway, US teachers wouldn't have the problem!) you can find some real gems here.Some of my favourites are:

1) Multiplication of Fractions - A great applet showing fraction multiplication using overlapping shading of rectangles.

2) Sieve of Eratosthenes - Set how big you want a grid and then click numbers to remove multiples of that number.

3) Algebra Balance Scales - Great for physically showing what happens when solving equations through balancing; is a bit limited because of size of the scales and the need for whole number solutions but choosing careful equations can illustrate the point.

4) Function Transformations - Shows the effect of changing constants in the equations of well know graphs.

5) Stick or Switch - Great little applet for the Monty Hall Problem.

6) How High - Great volume of cuboids problems: how high will the liquid be in one container when it is poured in from another container?

7) Tessellations - Allows pupils to experiment with single shapes to find regular tessellations and different combinations of shapes to find semi-regular tessellations.

8) Spinners - Use different spinners to produce data for probability investigations or questions.

All of these are simple interactives that pupils can use at the board or in computer rooms to explore or demonstrate these ideas. Well worth a look.

First to 10...

The rich task advocates (to be fair, of which I am one) are going to slam me for this one; but resorted to a good old fashioned here are 10 (could be any number to fair) equations to solve today. And got the kids working furiously to solve them. Here is how:

As the title suggests the activity is first to solve all 10, but I don't stop when the first has finished. I tell the kids the first to finish will be given number 1, and they have to close their book at that point - no checking, no correcting, nothing. The second person to finish will be given the number 2, the third number 3 and so on until the first 10 people are done (or until a specified time limit). I then reveal the answers and they mark (checked by their peers to ensure they are not cheating if needs be). 

What happens next is what catches them - if number 1 gets them all correct they get the reward but if not it passes to number 2. If they haven't answered all correct it passes to number 3 and so on. If none of the first 10 to finish has them all correct then the reward goes to whomever has most correct. Generally it will actually be the 2nd or 3rd person rather than the first that gets the reward but today it turned out to be the eighth person! I like this task occasionally is it encourages kids to balance working hard with working accurately and rewards the person (or with a few adaptations people) that get it right.

We all know (I think at least) there are some times when we just feel kids need to sit down and practice a load of a similar problem and I think this at least provides motivation to approach the task sensibly.

Outstanding differentiation with one resource - or so the head tells me!

Had my observation with the head today - waiting for the official feedback but was told it was outstanding. Apparently part of the reason centred around my use of this resource:

The task came in three parts, with pupils able to choose which one they did.

1) Red Task -  Take the 7 'main' quadrilaterals (Square, Rectangle, Rhombus Parallelogram, Kite, Arrowhead and Trapezium - they were given pictures) and use the diagram to decide which box they go in.

2) Amber Task - Figure out which two of those shapes go in the same box. Can you add another question to separate them?

3) Green Task - This decision tree uses the side and angle properties of quadrilaterals. Can you design one that will sort all of the same quadrilaterals using their diagonal and symmetry properties?

The head loved the higher order thinking that was encouraged through the Amber and Green tasks. Just goes to show you don't need lots of resources for differentiation; just think about different things that your pupils can do with the resource.

Complete lesson (minus a few printed resources) here: https://prezi.com/xprmsjucot1e/quadrilateral-properties/

Balancing versus Inverse function machine

Had an interesting conversation today with my intern; she is teaching equation solving to a year 8 group that will be going on to the Higher tier at GCSE when they get there. She was going to teach them equation solving exclusively using function machines, with unknowns on both sides dealt with by moving them and then drawing the function machine when you have a single variable. I asked her to use a balancing method for at least part of the topic (basically use function machines in the first lesson and balancing in the second) because they will eventually need to be confident with balancing to do higher tier rearranging formulae. Got me thinking on two fronts:

1) Should we be worried about introducing balancing in Year 8 to support GCSE, or would it be OK to wait until next year, or even Year 10?

2) Can you approach rearranging formulae using a function machine? I have tried to visualise how it might work before but never been comfortable with it. Particularly can't see how you would rearrange for y in something like 3x - yx = 2y + 7 without balancing.