## Monday, June 30, 2014

### Heads and Feet (and multi-variable equations)

NRICH has a fun little problem on its top page for lower primary: heads and feet.

The challenge: A farmer has chickens and sheep. If the animals all together have 8 heads and 22 feet, how many sheep does he have and how many chickens?

My question: Is this low threshold and high ceiling?

Threshold
The basic task seems to be one unit, but there are a couple of progressions to make it more accessible. My guess is that most children who get stuck will need to be guided toward alternative versions that will let them build strategies for understanding the challenge. Perhaps starting with a version that has fewer animals will be accessible. If necessary, drawing pictures of animals, counting them, counting body parts, etc can help get the exploration started.

Note that drawing realistic animals doesn't necessarily help with the mathematical content (and it can actually get in the way of one solution strategy). However, if you want to practice, here are some instructions that will lead you to a much better animals than I draw!

Plan: have Jane counting objects, Jate drawing pictures of animals and counting body parts, and Jin to get the challenge as originally stated.

Ceiling
The obvious route to a (super) high ceiling is linear equations of multiple variables and the whole world of linear algebra. My idea to open that route is to ask if Jin can write an equation to describe the number of animal heads and another equation to describe the number of feet.

The solution strategy I've seen him use in the past is to draw a picture, starting with the assumption that all the animals are chickens, like this, with heads and feet helpfully labelled:

and then adding feet to make sheep until you get the right total number of feet (the sheep bodies are shaded to make the distinction between sheep and chickens more clear):

I've been wondering about whether this strategy is generalizable, since even three variables seems unclear. My questions, if we get this far:
- can we write an equation or equations that describe the relationships between sheep, chicken and heads? What about sheep, chicken and feet?
- Is always a solution, no matter how many feet are given? Do we notice anything special about the number of feet in this type of problem?
- With the right number of chickens and sheep, can we get all combinations of (heads, feet) = (n, 2m)? Are there any restrictions we can identify?
- Can you think of any problems like this? Maybe chairs (4 legs) and stools (3 legs), bicycles (2 wheels) and tricycles (3 wheels) etc.

Post-Mortem
This activity wasn't loved and didn't capture Jin's attention during the evening rush.  Maybe something to save for a quieter time or perhaps the lesson is to have various investigations available? Mommy's post mortem:  there wasn't time to do this anyway, so nothing lost.

Well, we ended up with another nice chicken picture, at least: