# 348. A-Level colouring (Updated)

Those of you who follow this blog will know I have a thing for explaining with colours. This isn’t just a gimmick for younger students, it also works for 16-18 year olds.

In the picture below we were looking at proving a statement involving reciprocal trigonometric functions and fractions. A common source of misconception with this kind of question is that students split the question into working with the numerator and denominator separately, then make mistakes when they put them back together. They can’t see the big picture.

Image credit: Mathssandpit

When I discussed this on the board I used separate colours for the expressions in the numerator and denominator. The class could follow the logic so easily. It’s probably my most successful introduction to this topic. I saw that some students used highlighter on their notes after I’d gone through it, so they could track the solution.

The second type of question we looked at was solving a trigonometric equation. The straight forward expansion was all in one colour, but the roots of the quadratic were highlighted in different colours. The reasoning behind this was that students often solve half the quadratic and neglect the other impossible solution. Our exam board likes to see students consider the other solution and formally reject it. It makes the solution complete. By using a colour, the impossible solution stands out and reminds students to provide a whole solution.

Image credit: Mathssandpit

So when you are planning for misconceptions at A-level, remember that coloured pens aren’t just for younger students.

Update: 22nd October

The brilliant Mr B has shared how he uses colour to identify the forces in perpendicular directions in Mechanics.

# 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

# 195. Marshmallow Maths

It’s our first birthday at the MathsSandpit and this post is party themed. Remember a few years ago, when chocolate fountains were the ‘in thing’ at celebrations and parties. The healthy guests stuck to strawberries drenched in chocolate. The unhealthy went for marshmallows on sticks and … well … all I’ll say is Geraldine Granger (Vicar of Dibley – Chocolate Fountain)

I’m trying to decompartmentalise the maths in my students heads. They struggle to see the links between different topics. So I introduced ‘Marshmalllow Maths’ – they were intrigued/hungry as soon as I mentioned it.

Equipment

• Cocktail sticks
• Pink and white marshmallows

Step 1

Step 2

What mathematical characteristics do the marshmallows have? I’ve summarised my classes’ responses below:

Two marshmallows lead to ratio, percentages, fractions, decimals and probability. The links between these topics start to emerge.

Step 3

How have the ratios, fractions, decimals, percentages changed?

Step 4

Make another 1:2 ratio marshmallow, identical to the previous one. How have the mathematical facts changed? In fact although the numbers have changed, the proportions have stayed the same which is proved when you simplify the numbers. Physically you can prove it by stacking the structures on top of each other – from above it looks like the original structure.

At this point I went cross-curricular and discussed the similarities between the marshmallow structure and water (H20). I was going to label the marshmallows with H and O, but my food-colouring pen wasn’t working. My logic was that water always has hydrogen and oxygen in the same ratio – this means we know we can drink it. If the ratio suddenly changed to H2O2, we would be in trouble! As far as I can remember H2O2 is hydrogen peroxide and is better for bleaching than drinking. This actually got the idea across quite well – no-one tried to fudge their ratios.

Step 5

I then allowed the class to make their own simple structures using their own piles of marshmallows. They had to make at least three identical structures, work out the related maths and prove that their numbers could be simplified to the basic form. In doing so they also looked at converting ratios to fractions and also found fractions of amounts.

Step 6

Eat marshmallows (whilst doing some related questions).

Optional: Step 7

Calculate the percentage increase in body mass on results day! It was marshmallows today, a chocolate prize for cracking a code earlier in the week and they say they learn better when they eat. I think it’s all a ploy to scrounge more food … but if it works … maybe fruit next time!

# 146. Sales Fractions

A quick idea for you today:

What is the cheapest each item could have been?

Which of these is the odd one out?

Why is it the odd one out? What discount does it represent? What is this as a decimal or percentage?

Keep your tags the next time you go to the sales – you never know what questions you’ll find.

# 115. Decimate

Here’s a nice (gory) problem to use as a plenary.

What does the word ‘Decimate’ or ‘Decimation’ mean?

The word derives from Latin meaning ‘to remove one tenth’.

The original context is the punishment of a Roman Legion. The unit was divided into groups of ten and lots were drawn. The unlucky soldier was executed. A Cohort consisted of  480-500 soldiers, a Legion consisted of 10000 soldiers. How many would survive in a Cohort? How many would be executed in a Legion?

Now if the class were an unruly unit, how many would draw an unlucky lot?

A legend suggests that the Theban Legion was decimated in the third century AD. The Legion had refused, to a man, to accede to an order of the Emperor, and the process was repeated until none were left. They became known as the Martyrs of Agaunum (there are several sources on this).

Rather than execute your class I use this legend as a logic problem. Which pupil in the class is safe? Either the class stands, sitting when they are ‘executed’ or stand and then line up to leave. You can stop every few counts and ask who thinks they are safe. Last soldier standing leaves the lesson first!

Don’t have nightmares!

# 43. Visualising percentages

Percentages are all to do with proportion, but this seems to escape the understanding of some. If you calculate 20% of £15, this is different to 20% of £25. The 20% is not a fixed quantity. How can you explain this to visual learners?

Visual Percentages/Proportion

Equipment
Pencil
Ruler
Paper – squared makes the task easier
Coloured pencils (optional)

Calculations
Find 20% of 15, 10 and 0.

Construction
Draw a 15cm line, mark 3cm along it.
Move down 5cm.
Draw a line, mark 2cm along it.

Join the ends of the lines with a ruler and indicate this with a cross.

This should be 10cm lower than the bottom line.
Repeat, joining the 3cm and 2cm points.

Label the lines.

The Maths bit
The width of the triangle indicates the whole amount (100%).

The whole diagram represents 20% of any number from 0 to 15.
This can be adapted for any number and percent. It visually shows that as a number gets bigger the percent increases proportionally.

You can also use this to investigate fractions.

Note: This is for comparing widths. You can challenge your students to prove whether it is also true for the areas of the triangles.

# 26. FDP Pyramid

This nifty little pyramid summarises how to convert between fractions, decimals and percentages.

Equipment
A5 paper or lightweight card
Scissors
Pens
Glue/tape
Compasses & pencil

Make a square
Fold the paper over to make a perfect 45 degree angle. Cut off the excess paper to make a square.

Fold & Cut
Unfold the square and fold the opposite diagonal. Cut from one corner to the middle along the fold.

Label
Draw an X on one of the quarters next to the cut. You will glue this piece later.

Either side of each fold label ‘Fraction’, ‘Percent’, ‘Decimal’.

Arrows
Using compasses and pencil, lightly draw two circles. Go over these lines with a pen to create one set of arrows going clockwise and one set anti-clockwise.

Facts
Label each arrow with the correct conversion fact and example.

Stick
Fold the X flap behind the next section and glue in place.

Summary
This is a tactile activity which could be used on a wall display. It can also be collapsed down flat where it can be taped on one side into a book and ‘pop up’ when required.