# Revisiting the Vitruvian Man

As mentioned in my last post, Walking the walk, I recently completed a 2-week casual teaching block and agonised about walking MY walk. So, here’s another story…this time with Year 8s and on the topic of Rates and Ratios.

I had one lesson (yes ONE, the 3 with the year7s was a luxury in comparison) by the time I got my act together! Enter the Vitruvian Man again (yes, I’ve used him before, on Percentages and with more prep). Remixing my original idea was easier than starting from scratch! By the way, the previous time was with strugglers and this time with high achievers….note the differences in the activity, not just topic-wise but in also tasks…which also show how my teaching approach is evolving towards inquiry and reflective practice.

8mata1_vitruvian – have a squeeze (pdf) and remix if you like…the Vitruvian Man is a maths teacher’s friend!

In a nutshell, the students were meant to investigate the “accuracy” of the Vitruvian Man ratios, extrapolating their own, apply some ratio skills AND reflect on their leaning. That’s not too much to ask in one lesson, was it? And by the way, they were going to use 2 technologies they’ve never used before: Wallwisher (which turned out to be blocked. gah!) and Dropittome.  They had to submit TO ME and they won’t see me again. CRAZY!

So, I completely forgot about this until a week later …. mostly, I didn’t think any of them would submit work to a casual teacher … I mean, really.  No grades attached.

I finally remembered to check my Dropbox and – surprise! I got submissions. I wasn’t even going to blog about this activity but, how could I not?  Here are some excerpts:

### “New” ratios

• A man equals 96 fingers – 1:96
• A foot equals 16 fingers – 1:16
• Palm to foot: 4:1
• Length of outspread arms : 4 cubits

### Questions

• Is the theory 100% accurate?
• Is the growth of your arms in proportion to the growth of your height?
• Is the length of your foot, equivalent to the length of your forearm?
• Do these ratios work for everyone?
• What fraction of your body is one palm? (I love the use of the term fraction. yes!)
• What is the ratio of the length of a hand to a full arm span?
• When was the Vitruvian man first drawn/created?

### Reflections

• Ratios are good because they make it possible to work things out rather than have to measure them every single time.
• They may not always be 100% correct, as it was proven with my measurements.
• Although this task was reasonably difficult in the beginning, I found that it was reasonably accurate and correct, as we further investigated Da Vinci’s theory. My results displayed similar findings to the theory; I further developed my rates and ratio skills. Prior to this maths topic, I didn’t completely understand it, however now I find that I can use ratios in many other places to simplify, numerals I am given or to find and approve theory’s such as the Vitruvian theory.
and the icing on the cake…

I really enjoyed this lesson. I liked how we got a chance to put our maths into practice.

So yeah, I’m happy.

That kid saying ‘reasonably difficult’ was being truthful! I nearly abandoned the activity! I’m glad I didn’t. Unlike my strugglers before who were keen to get straight into measuring, this high-achiever class had to be prodded away from their calculations and musings. I’m sure there’s something there but I’m not going into it for this blogpost.

# Problems vs Exercises

## Problems vs. Exercises

Inspired by a blogpost by David Cox (@dcox21), I decided to use the same problem and added an extra fraction. Anyone following me for some time would see that I value mathematical thinking and, as David said, this is a good one to show problem-solving skills. I think it’s also a good one to revise, reinforce and connect mathematical skills (or tools as David calls them).

Simplify:

I asked the class what their first thoughts were upon seeing this problem on the board:

1. “It looks complicated”
2. “It’s hard”
3. “It looks like a big problem with lots of little problems”

They were all correct, of course, and I told them so (affirmation is good). I then told them that, in fact, they already have all the skills to solve this problem and they looked at me to as if I’ve gone mad. I suggested that, because it did look complicated, we look at the little problems that make up the big problem to make it easier (note the use of their responses; affirmation is good – oops, said that already).

I then asked what the problem looked like, that is, what’s familiar about the problem.

1. Dividing fractions
3. Algebra – use of pronumerals/letters

Upon revising the above, we added 2 more:

4. Multiplying fractions (division of fractions as multiply by inverse/reciprocal)
5. Order of Operations (fraction bar as a grouping symbol)

And so we set off to solve the problem and they asked for more to practice on, an Exercise as David puts it. This was the desired and expected effect. I used a similar approach when I introduced Decimals to my year 7 class last year (note to self: must share/blog this resource).

I should have also pointed out that this sort of thinking/questioning/problem-solving approach can be applied in real life. Sometimes problems we face in life can seem hard and complex yet often, with chunking (or breaking down into smaller bits), we find that we have the skills/tools to solve them.

# The Vitruvian Man – a context for learning

Finding a quantity given a percentage (or fraction) is a useful skill yet considered to be an extension topic for year 8. I thought I could give it a go but set the scene, so-to-speak, without the oft-used context of shopping and sales.

Enter the Vitruvian Man. Wikipedia has a good image and brief explanation of this drawing by Da Vinci interpreting ideal (hu)man proportions according to Vitruvius. Wikipedia also provides a list of these proportions.

### Lesson Activity

My ‘hook’ question to the class was “How do forensic scientists figure out the height of victims given minimal data?”

I showed and explained the Vitruvian Man. We even managed to do a quick revision of properties of squares and how this was used in the drawing. Anyway, here’s how I used this “tool”.

1. Divide the class into pairs (or small groups).
2. For each pair, give a card which showed one of the proportions (e.g. 1/4 of height = shoulder width) as well as a measuring tape
3. They take the fractional measurement of their partner, i.e. the item to the right of the equation, e.g. shoulder width
4. Demonstrate how to calculate the height given a known percentage (or fraction); in my example, multiply each side of the equation by 4
5. Finally, measure the actual height and compare to the calculated height

### Discussion points

• How do the actual and calculated height compare?
• What are the reasons for differences?
• Why multiply both sides (make a point to mention working with Algebra and equations)?

## Reflection

The class really enjoyed the activity and didn’t really mind the ‘maths’ at all. Given that this class is deemed below-average, the level of engagement was good. The students – all girls – already associate percentages with sales but, for most if not all of them, this is the first time they’ve associated it with the human body. I know that I will use the Vitruvian Man again.

This was a lively lesson with talking and standing up and discussing. This isn’t every teacher’s cup-of-tea but it suits me just fine.

# Percentage Composition

I wanted to make this lesson a bit more interesting as well as incorporate opportunities for connections with other maths topics.

I knew my year 8 class students loved music so I devised a very simple poll on Rob Thomas (one of the students was going to his concert): like him, don’t like him, don’t know him. This was a good opportunity to revise Frequency Distribution.

Just by fluke, our 3 fractions turned out percentages of recurring decimals (another connection). In that sense, it’s really not a very good first example! Anyway, someone asked if the percentages will total a hundred (I love it when they think!). So, we added and of course it didn’t. We got a result of 99.8%.  This opened up a discussion on the consequences of rounding/truncating decimals.

I made a statement that the most accurate way to represent not whole (fractional) values is via fractions. Nods all around until I asked, “‘Do you believe me?” The consensus was yes because I was the teacher. I was quick to point out that they should not always believe everything teachers say – they have to think if it’s reasonable; teachers are humans, after all, and can make mistakes. Besides, I really wanted to challenge my students to develop their thinking and reasoning.

So, I went back to the Frequency Distribution Table and 3 fractions. Adding the fractions naturally gave a total of 1 and nearly everyone said that’s 100%, as expected. Joy!

Were it a more able class (we stream our Maths classes), I could have pursued more connectionist opportunities but I did sense I’ve pushed enough today. I know I’ll refer back to this lesson when we actually do Data later in the school year. The class was engaged because they knew the meaning behind the numbers.

By the way, the statement is a fact I had truly learned in my previous career in IT. Particularly when I worked for a bank calculating interests, calculations in programs (software rather than curricular) were designed to stick to fractions as much as possible to minimise rounding errors.

In summary (more for my future reference),

Learning strategies: connectionism, motivation and engagement via meaningful and relevant examples, question/reason validity, compare/contrast

Fractions, Decimals and Percentages – provide different ways to represent fractional values, of which fractions are the most accurate