## 30 October 2018

### Map Scale

This question on map scale in AQA's 2018 Foundation tier GCSE paper was answered really badly:
Most students had no idea. In fact, they would have done better if they'd just guessed. The most common answer given was 1:1000.

The question requires students to understand two things:
1. how to convert from cm to km.
2. how to interpret a map scale given as a ratio with no units (eg knowing that 1:100000 means that 1cm on the map represents 100,000cm in real life).
I think that metric conversions are relatively easy to memorise (most students know that there are a hundred centimetres in a metre from their familiarity with a metre ruler and the prefix cent, and they should be aware that kilo means thousand). So this leads me to believe that the main difficulty with this question was a lack of familiarity with map scales given as a ratio.

I have a feeling that this topic is sometimes skipped over by maths teachers. Some students may only see one maths lesson on map scale in their entire time at secondary school. It gets buried in amongst other topics on schemes of work (it's normally in with either ratio or with bearings and scale drawings, though I have seen it in with similarity too). I think it sometimes goes unnoticed and doesn't get the time it deserves. Perhaps this is because it rarely comes up in GCSE exams.

Curriculum
You may be wondering if the ability to use a map scale of the form 1:50000 comes up in geography. In the geography GCSE syllabus it says that students must "use and interpret OS maps at a range of scales, including 1:50 000 and 1:25 000 and other maps appropriate to the topic". It doesn't specifically say that students have to measure and convert distances, though this is implied. I found this on a geography revision website:
"The scale number on an OS map indicates how many centimetres on the ground are represented by a centimetre on the map. On a 1:100,000 scale map, one centimetre on the map represents 100,000 cm on the ground, in other words, one centimetre on the map represents one kilometre in reality. A scale of 1:5,000 therefore means that a centimetre on the map represents a distance in real life of 5,000 centimetres (50 metres). This method of representing the scale of a map is called the fractional method, but you will also see graphical representations or written representations like 2 cm = 1 km."
I wasn't aware that the method of representing map scale in the form 1:50000 is called the fractional method. I like knowing proper names for things. Though I'm not sure whether this term is used consistently - I've seen other sites refer to it as a ratio scale or a fractional scale or a representative fraction.

Wikipedia lists different types of map scale including lexical (ie expressed in words - also known as verbal or stated scales), linear or graphical scales (represented as a bar), ratio scales, and fractional scales. It points out that a lexical scale in a language known to the user may be easier to visualise than a ratio, but lexical scales may cause problems if expressed in a language that the user does not understand or in obsolete or ill-defined units (eg one inch to a furlong or one pouce to one league). So ratio scales have pros and cons. When I read this I straight away thought that there could be some really nice opportunities for enrichment in this topic - I'd love to talk to my students about antiquated units of measurement!

Anyway, it looks like 'map skills' in GCSE geography focuses mainly on recognising symbols and using grid references. I haven't found many geography resources on ratio scale, other than a couple of PowerPoints on TES that run through it very quickly. So it seems that measuring lengths on maps and performing unit conversions using ratio scales isn't something they spend much time on in the GCSE geography course.
We definitely do need to spend some time on it in maths lessons, and it fits well at both Key Stage 3 and 4. The maths GCSE specification says (in both the ratio and geometry sections) that students should know how to use scale diagrams and maps. AQA's Teaching Guidance helpfully provides additional clarification: "Scale could be given as a ratio (for example 1:500 000) or as a key (for example 1cm represents 5 km)."

Approach and Resources
Without sound knowledge of both place value and metric unit conversions we can't even get started on this topic. So that's the first thing we need to check. I remember once giving my top set Year 10 a simple starter asking them to put these lengths in ascending order:
It took way longer than expected.

If metric conversions need teaching, there are loads of good resources for this, including:
Students who struggle with simple unit conversions might find it helpful to draw out a ratio table each time, writing their known fact at the top and using multiplicative reasoning to fill in the gap:
Once students are fluent in unit conversions, it would be sensible to remind them of how ratios work before moving onto map scale. There are lots of great resources available for this wide ranging topic (see my post on ratio), but the focus here is simplifying ratios with mixed units (ie converting the antecedent and consequent to the same units), and on expressing ratios in the format 1:n. Useful resources include:

Now we just need to combine these ideas to understand map scale.

If a map has scale 1:50000, how do we work out what 6cm on the map represents in km on the ground? The two steps involved (the unit conversion and the measurement conversion) can be done in either order. I'd suggest something like this:
Here I've start by writing the ratio scale with units - any units work but cm is usually preferable. Instead of writing 1cm = 50,000cm (which is a horrible use of the equals sign!) I've used a table.

I did my unit conversion in two steps, going via metres as the base unit.

Using a similar approach to answer the question: "If 4cm on a map represents 100km on the ground, what's the ratio scale?", we have the following process:
So the answer is 1:2500000.

Of course there are lots of different ways to set the workings out here - the table is optional.

Most resources I've seen online for ratio scales skip through it very quickly - it's often covered in a couple of slides at the end of a related lesson. Teaching it properly - in depth - should probably take two or three dedicated lessons. The CIMT material 'KS3 Scale Drawing' is very useful, as are the Boss Maths lessons 'Using scale diagrams and maps' and 'Scale drawings'. Corbett Maths has some exam style questions on this topic.

I am eagerly awaiting something on variationtheory.com on all this!

Once students are fluent in metric unit conversions and working with ratio scales, they might enjoy a bit of map work to consolidate their learning. MathsPad have a free online map scale tool which is helpful for demonstrating map skills on the board. The Mapzone website shows what different OS map scales look like - this is not on the maths curriculum but might be of interest to students. This Reading Map Scales Activity from emtay on TES gives students the chance to practise using a ratio scale on a map of Europe.

No doubt someone will tell me that when teaching this topic I *must* give my students full size OS maps and send them outside on some big orienteering project! Hmm. I'm not sure that's practical on a main road in Croydon... I also believe that although they may remember the activity, it probably won't help them either understand or remember the maths. So I'll probably skip that.
Since we've all now got sat nav on our phones this topic isn't as much of a 'life skill' as it once was. That's ok though. Thankfully we don't teach mathematics for its utility.

What's good about this topic is that as well as sewing together two key areas of school maths (ratio and metric units), we get the chance to come back to it when we teach bearings, and again when we do area and volume scale factor with questions like this:
A map has a scale of 1:50 000. A park is shown on the map as a rectangle measuring 6cm by 4.2cm . What is the actual area of the park?
The Boss Maths lesson 'Converting between metric units of measures of area and volume' covers this.

I'm involved in the London Thames Maths Hub workgroup on Challenging GCSE Topics in which I hope to look at unit conversions and ratio. Do get in touch if you want to get involved in developing some resources for this topic.

In the meantime - let's all make sure that map scale gets the time it deserves in maths!