# 259. Squashed Tomatoes

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?

# 258. Indexed Learning

Back at the start of the academic year I set my Key Stage 4 classes an easy little homework: number the pages in your exercise books.

Why on earth would anyone set that kind of old fashioned pointless homework? Well…

1. It meant that any pages torn out of books would be more obvious
2. It gave me a ‘heads up’ on who may have issues with homework in the coming year
3. It allows me to reference specific work in feedback
4. It allows students to take responsibility for their learning

Not convinced? Here’s more proof:

1. Anorexic books

Over the course of two terms I’ve handed out fewer new exercise books, saving a little bit of money for the Department. Their books are full size and full of work – no super skinny books. The side-effect was students who forgot their books had to see me for paper, rather than tearing a page out of a friend’s book and hoping I wouldn’t notice. This means everyone has been more organised – an unexpected and pleasant surprise!

2. Sinners

It’s amazing how accurate that little task was at predicting work ethic. Those who numbered the whole book have been great at meeting deadlines, those who partially completed it have been a touch unreliable and those who didn’t do it have been regular homework sinners!

3. Focussed Feedback

I have been giving specific feedback more effectively. For example: ‘On page 34 you forgot to expand the second part of the bracket’ or ‘Now we’ve reviewed this work, go back and have another go at Q2 on page 18’ or ‘Excellent method on page 8, make sure you keep doing this technique!’.

4. Indexed Learning

At the start of the year, students were told to write their learning objectives and page number at the back of their books to create a personal index. I didn’t nag them to do this after the first month. Some carried on writing objectives, some abbreviated, some fell into disuse. Even then I didn’t nag them. Fast forward to the end of the second term (we start our new timetable in May, so February is the end of two terms): students are on their second/third books. Most of them numbered their subsequent books without being told to. The indexes continue to be added to in the majority of books too – even the least engaged student has a perfect index.

Then the ‘Eureka’ moment happened! We were using a Corbett Maths 5-a-day starter and a student asked whether a particular question was on Pythagoras. This student doesn’t immediately engage with work: there can be minor issues with socialising too much and also too little confidence. I agreed that it was and before I could offer to help, the student had flicked to the back of the book, looked up which page the Pythagoras notes were on and refreshed their understanding. As I moved on to the next person, this student was already half way through a correct method. I was so impressed by the maturity and organisation of this student – characteristics which had been missing at the start of the course. After this I looked more carefully at how my class were progressing with these starters – students were regularly looking back at their own notes and had become more willing to look up information in textbooks too.

A simple idea at the start of the course has snowballed into a tool for enabling independent learning and I can’t recommend it enough!

# 257. Making the absurd Rational

Here’s a nifty little resource for you, once again inspired by @MrReddyMaths

This worksheet takes you through the process of rationalising fractions where the denominator is a surd. All of the numerators are integers to make the focus the denominator.

Updated version of (pdf)

This new version is A4 sized to allow more space for working out.

If you like this, why not try out these:

232. Steps in Volume

241. Histogram Hysteria

# 256. Health Warning!

Test reviews/reflections … possibly one of the dullest type of lesson! I ask my students to write a list containing three topic specific things they are proud of, three topic specific things that could be improved upon and any ‘Maths Health Warnings’.

Image designed at: http://www.speedystamps.co.uk/

What is a ‘Maths Health Warning’?
They are silly mistakes, mild misconceptions, badly managed arithmetic and not reading the questions. They are bad for your mathematical health because a minor error at the start of a solution can cripple your chances of correctly solving a problem. These errors can infect any topic from grade G data to A* trigonometry.

I discuss common mistakes and it is up to pupils to decide which health warnings they need – this usually involves highlighter/feltpens and circling/starring the issue.

Examples
Having recently marked topic tests from grade E to A, here is a list of transferable mistakes:

• Mismanaging negatives
• Not using the correct equipment for diagrams
• Expanding brackets correctly and simplifying incorrectly
• Mishandling decimals in calculations
• Not reading the question (it happens a lot)!
• Rounding the calculation too soon
• Rounding incorrectly to decimal places/significant figures
• Typing a calculation into a calculator incorrectly
• Forgetting there are only 60mins in an hour
• Not using 2dp for money
• ….this list could get rather long!

Once students start identifying these types of errors themselves, they become more aware of them in their classwork, homework and assessments. This results in improved understanding and progress.

Next steps

You might want to generate your own ‘Maths Health Warning’ stampers or stickers. You could put warnings in your students books and ask them to identify the error.