Solving equations physically

I enter the classroom, with a bag of cash, a small digital kitchen scale and a glint in my eyes.

I don't need to even need to ask the question, the students ask for me.

"How much cash is that?"

Without skipping a beat, I put the coins into a cup and put it on the scales...

3320 grams

Well there is a tricky problem. How many coins does it take to weigh that much?

The students split into 4 groups, each with 2 cups with difference selection of coins in each. We weigh each cup and then after subtracting the weight of the cup (5 grams), the students developed equations to represent the weights.

Let "a" be the weight of 5 cent coin, "b" for 10 cents coin and "c" for 20 cent coin.
4a + 5b + 2c = 310 grams
3a + 2b + c = 210 grams

The students attempted to discover the weight of each coin, through application of a range of algebraic techniques.
Given the 3 variables and only 2 equations, they find it impossible to solve, until eventually one group shares with another group, and the class as a whole understand that sometimes, the answer requires asking someone else.

Finally some answers start to emerge... however there are still differences from groups. Once all groups have had a chance, I get each group to submit their answers and we create a simple table on the white board.

A students calls out, "We should average them"

(I hadn't planned on that but what a brilliant idea)

"Great... how do we do that?" I innocently inquire.

"Sum them, then divide by four" The chorus replies.

So I add another row, for Class Average, as the students call out the numbers.

How could we check?
Well who makes these coins?
Lets look up the Royal Australian Mint
(http://www.ramint.gov.au/designs/ram-designs/5c.cfm).

Despite the questionable quality of the scales, minimal quantity of coins and potential for algebraic mistakes the class average arrive within 0.2grams for each coin. I think to myself, well there goes my discussion on sources of errors, but one student notices that a group has consistently predicted nearly a gram too high for all of the coins. What could that be? The students happily provide suggestions, and the list creates itself.

With time vanishing, and students generally feeling positive, I don't feel like I even need to answer the original question. Instead it is time for some reflection...

1) What did we just do?
2) What skills did we use?
3) What new did I learn?

I circulate to get a feel for how students went. The response was clear. The more capable students had been working through the problem and developing their ability, however the lower ability students required considerable scaffolding. The activity simply led to frustration for the lower ability students.

While having a problem solving activity, for groups running over a whole lesson allowed the high performers to shine through persistence and application of logic, it had let some lower abilities coast. Perhaps they had seen something new, or perhaps they will recall the activity in future discussion, but they didn't feel ownership over the solution.

This lesson was great, but next time I need to be more aware of when students are letting the stars of the team carry the rest.

As a final thought- I was glad that I weighed the coins initially. It gave me an easy way to check that all of the coins had been returned (which they were).