### The Vitruvian Man, or Thanks Bruce

I am always quite inspired by the posts made at Authentic Inquiry Maths (even though he puts an 's' at the end of math when there is clearly no need for one).  I read them whenever they popup in my google reader (Feedly for the iPhone is awesome) and bookmark many to do later.  It never happens that we are studying the same thing at the same time.

Until...

The other day a lovely post about Da Vinci and the Vitruvian Man.  Bruce made a wonderful list of Da Vinci's observations about the human body:

- a palm is four fingers
- a foot is four palms
- a cubit is six palms
- four cubits make a man
- a pace is four cubits
- a man is 24 palms
- the length of the outspread arms is equal to the height of a man
- from the hairline to the bottom of the chin is one-tenth of the height of a man
- from below the chin to the top of the head is one-eighth of the height of a man
- from above the chest to the top of the head is one-sixth of the height of a man
- from above the chest to the hairline is one-seventh of the height of a man.
- the maximum width of the shoulders is a quarter of the height of a man.
- from the breasts to the top of the head is a quarter of the height of a man.
- the distance from the elbow to the tip of the hand is a quarter of the height of a man.
- the distance from the elbow to the armpit is one-eighth of the height of a man.
- the length of the hand is one-tenth of the height of a man.
- the root of the penis is at half the height of a man.
- the foot is one-seventh of the height of a man.
- from below the foot to below the knee is a quarter of the height of a man.
- from below the knee to the root of the penis is a quarter of the height of a man.
- the distances from the below the chin to the nose and the eyebrows and the hairline are equal to the ears and to one-third of the face.
I noticed that fractions were all over this.  I also noticed that one of the difficult aspects my kids are having with fractions is that they get confused about what the whole is.  If I take 1/2 of N's pizza, and I take 1/2 of M's pizza, are they equal?  Well, that depends on what the whole was.  Who pizza was bigger?  1/2 of a square cm is not the same as 1/2 of a square meter, but both are half.

Bruce's post inspired me to use the thing that we can most relate to as our whole (1).

Ourselves.

Setting the Stage

I showed them a picture of the sketch and tried to evoke some discussion about its purpose.  A couple of the kids had seen it before, and knew it was drawn by Da Vinci, but nobody knew the purpose of it.  I asked them to brainstorm what they thought was the point of this diagram.  There were some silly answers and theories (this is what people will look like in the future, with four arms and legs) but eventually somebody asked what the square was for.

S1: What is there a square around the body?
S2: It must be to measure.
S1: Measure height.
S2: Yeah, and measuring how long the arms for.
S3: Then what is the circle?
S4: To measure the stretched body (I think he meant spread eagle).
T: So, what is the purpose of this diagram?
S1: Measurement.

I explained that Da Vinci used this diagram to make a whole list of proportions of the human body.  I told them that we were going to check his work and see if his work was accurate.

But, we are going to take them one at a time.

Conjecture

The first one I presented was the observation above.  Before we started, we needed to know what we needed to know.  And why we needed to know it.  How could we find this out?

In groups they came up with a list of ideas and methods.  All the groups knew that we need the following measurements; from the chin to the top of the head, and height.  There was also more than one way to analyze this data.

1) Height ÷ 8 = the chin to the top of the head
If we take our height, and divide it by 8, we should get a number that is very close to the number we got for the ch

2) chin to the top of the head x 8 = Height
If the chin to the top of the head is 1/8, then it would make sense that 8 of those pieces would equal the overall height.  Or thereabouts.

3) Height ÷ chin to the top of the head = 8
If we divided the chin to the top of the head measurement into the height, we should get 8, or near 8.

After a short discussion, we decided that this is the method we would use to figure it out.  We need consistency if we wanted to compare all the answers.

Measure

I stressed the importance of accuracy.  I wanted them to be as precise as possible.  While they were measuring, I walked around and criticized their methods, pointing out flaws and asking them to try something else.  The height was easy.  Stand back straight against a flat surface, put something flat on the head at a 90 degree angle and mark the wall.  Now, measure from the floor to the line.  They have been doing this for years, at home and at school.

Measuring from the chin to the top of the head though, this was a new one, even for me.  The kids came up with some great strategies:

- Measure from the chin to the floor and then minus it off the overall height
- Lay your head flat down on a table and trace it and then measure from chin to top of head
- Put a book under your chin, and a book on your head (both at 90 degree angles), and measure the distance between books.

Analysis

We collected all of our data on the table (I also started a google spreadsheet with class measurements in it, it could come in handy later and we should add a lot to it) and took a look at our findings.  It seems the average ratio was not 8, but 6.87.  It seems that

height ÷ chin to the top of the head = 8

was not accurate.  I posed the question, was Da Vinci wrong?  They let that sink in and thought for a long moment.

S: Maybe we did something wrong?
S: Our measurements weren't accurate.
S:  No, it is because we are kids are still growing!
S: Yeah!  We did this on adults right Mr D?
T:  Yes he did.
S: So because our bodies are still growing, we are not the same.
S: Yeah, look at Mr Dwyer's ratio.  He is an adult man and it is 7.8!  Almost 8.
S: Was Da Vinci just talking about men, or does this include women as well?

That last question is a great inquiry waiting to happen....

Thanks Bruce.

1. Hi Craig - looks like you had some fun with this one! Great stuff!

2. We did! Thanks again.

3. Anonymous10.3.13

Thanks so much for sharing this. I think this engagement is just brilliant. I like the whole idea of integrating Math and a bit of history into this inquiry. Inspiring. Makes learning not at all boring.