# 335. The power of colour

As Mathematicians we appreciate the importance of getting the basics right and building a firm foundation. With this in mind I’ve been an absolute harridan with my Y8 students regarding presentation and technique for solving equations. If they can nail good algebraic presentation now, their future studies will be be much easier.

When we started there were students doing everything in their head, not always correctly. Some insisted on working backwards, which is great for basic cases but not for unknowns on both sides. Most frustratingly some students were breaking up the logic by putting extra working out between steps and losing track of what they were doing.

For example:

2x – 10 = 5x + 8

5x – 2x = 3x

3x – 10 = 8

So we had a really good discussion about logical presentation. We decided to write down what we were doing in the margin, try and keep the = sign lined up in the working and put any extra working out on the right.

This worked really well for most of the class, but I had a small group of students who just lost track of what they were doing and why. They knew things had to balance, but struggled to cope with equations with an unknown on both sides.

While I was talking things over with them using a mini whiteboard, I noticed they had a profusion of coloured pens and highlighters. Bring on the colour!

By highlighting the key point of each line of algebra and matching it with the balancing step they started to build the structure of good solutions. It was slow work to start with, but a couple of lessons later and these same struggling students are now hitting the extension work every time. And most of them no longer feel the need to highlight key information.

# 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

# 186. Fantastical algebra

Have you ever played the parlour game ‘Fantastical Creatures’? Click for a lovely description and example of it by Little Cotton Rabbits.

I’ve adapted this concept for teaching aspects of number and algebra.

Topics
Basic arithmetic
Inverse operations
Order of operations
Setting up simple equations
Using brackets with numbers/letters
Solving single sided equations

Equipment
Strips of paper – one sheet of A4 makes about 6 strips
Coloured pens (optional)

Basic instructions
1. Write an instruction on the top of the strip (portrait orientation). Label it (a).

2. Fold over the strip twice to hide the writing. Write (b).

3. Pass on the strip, do not unfold it.
4. By the ‘(letter label)’, write the next instruction. The letters help you keep track of how many times it has been passed on.
5. Fold over the strip twice and put a label for the next letter of the alphabet.
6. Repeat steps 3 – 5 as required.

The beauty of this activity is that each problem is constructed by a group of pupils and they are in control of the level of difficulty.

Activity 1: Setting up simple equations

Follow the basic activities with the following instructions:
(a) I think of a number and write an instruction
(b) & (c) Now I write an instruction
(d) The answer is write a number

Pupils fold the puzzle up tight and either pass it on one last time or hand them in (to be randomly distributed).

Pupils unfold their mystery puzzle and construct the equation, step by step. My pupils quickly realised the importance of simplifying, but many forgot the importance of using brackets. This was a useful misconception to identify.

Pupils then use inverse operations to calculate the unknown.

The algebraic operations and numerical operations can then be compared.

Activity 2: Problem solving

This follows the same structure as the equation activity, but pupils are describing a geometric problem. In the examples the blue sections are up to the pupils to choose.

Example 1

Example 2

In the second example pupils can visualise the problem as well as using algebraic terms.

Activity 3: Number Skills

This activity can also be used for setting up BIDMAS problems by omitting the algebra.

# 164. Plant a Learning Tree

Do you know that feeling when you are starting a topic which is building on existing knowledge and you are not sure how much to recap? Too much recap and they start the topic bored, too little recap and the new work is too difficult. What to do?

To quote an old UK TV ad: “I want to be a tree!” (Prudential, 1989).

I have a bright class of 13/14 year olds and needed to start some algebra work. We ended up making a tree.

Equipment

• Coloured paper
• Felt pens or markers
• Glue
• Scissors
• Roll of backing paper or wallpaper (I cut mine to fit on the back of a door)
• Optional: mini-whiteboards for mindmaps

Activity 1
In small groups, pupils draw mindmaps for the word ‘Algebra’. Encourage them to group or link topics.

Activity 2
Collect the answers on the main board. Any concepts which are not specifically algebra can be categorised as foundation skills eg understand calculating with negative numbers.

Activity 3
Split the diagram into parts:
Stones: foundation skills which are essential for algebraic success

Branches: subdivisions of algebra
Leaves: specific topics or objectives

Fruit: examples

Activity 4
Assign the different stones, branches, leaves and fruit to pupils to complete.

Activity 5
Assemble your tree. I added an owl and a disembodied voice asking ‘which careers need algebra?’. My branch labels were quickly covered by leaves, so I substituted extra leaves with these labels instead.

Variation
This could work for any topic in any subject. Imagine how good a tree lined corridor would look – a new tree for every area of study.

Review
I moved around the room chatting to pupils as they worked and got a good idea for where I need to start the next lesson. The pupils now have a visual representation of how algebraic concepts link and overlap. In hindsight, I’d probably make the leaves and fruit smaller so that links are clearer.

Show me your learning trees on twitter and I’ll share them on here. @Ms_KMP

# 141. Book(s) of the week 3

If you remember ‘The Wonder Years’ you are probably old enough to remember grunge the first time around and television programmes that didn’t involve so called ‘Reality TV’.

So what happens to female child stars?

Some have a rocky youth, work really hard and become hugely successful (Drew Barrymore). Some have a rocky youth and become hugely notorious (Lindsay Lohan). Some work really hard, do research, writing and acting, have a theorem named after them and become advocates for women and maths education!

Step forward Danica McKellar!

Apart from playing ‘Winnie Cooper’ in ‘The Wonder Years’, Danica is also a successful mathematician. She has written four books aimed at promoting maths to high school students, in particular girls. I strongly suggest you have a look at them or get your school library to purchase them as they are full of inspirational ideas and new ways to think about ‘dusty’ topics.

Her books to date are:

Girls Get Curves: Geometry Takes Shape (2013)

Hot X: Algebra Exposed! (2011)

Maths Doesn’t Suck: How to survive year 6 through year 9 maths without losing your mind or breaking a nail (2010)

Kiss my Math: Showing Pre-Algebra who’s boss (2009)

# 61. St George’s Day Investigation

Here is a quick St George’s day area investigation for the 23rd April.

What size cross must be drawn for the areas of red and white to be equal?

Assume the flag is a rectangle and the strips of the cross are the same width.

KS2/3: investigate by counting squares or working out areas.

KS3/4: extend to an algebraic solution if appropriate

If you like this post why not follow Maths Sandpit on twitter: @Ms_KMP