We did an Estimation 180 involving the value of coins today: Here are their estimates and some of the reasoning they gave me: We had looked at the previous day's Estimation 180 to see that there were 450 pennies in the glass. The first student took the value of the pennies and multiplied it by 5 and rounded down as nickels take up more room. The second student has originally estimated $54 so I wanted to figure out where he had made a mistake. He estimated the number of nickels to be about half the number of pennies as they are about twice as thick. He then divided by 4, thinking there are 4 quarters in a dollar, instead of thinking that there are 20 nickels in a dollar. We fixed that up and his new estimate was much more reasonable. Next, we solved systems of linear equations. We did this in the first cycle with Smarties and jujubes and pennies. Today, we used elimination. Here was the first problem: I do my best to keep this as concrete as possible, so we drew pictures to represent the situation: How can this help us find the prices? At least one student noticed that the difference between the orders was 2 coffees so they had to account for the difference in price. They could now use the price of a coffee to work out the price of a doughnut: On to the second example: In this case, we could not compare the two orders as they did not have the same quantity of either item. However, if we double the first order, both will have the same number of cookies. This was more direct instruction than I would have liked and their behaviour echoed my sentiments. But we carried on with the third example: I also showed them a more algebraic solution as some seemed like they would prefer to do it this way: I really emphasize that my students need to understand why they are doing what they are doing. The idea of doubling or tripling an order which will in turn double or triple the cost is an important step in solving these systems. Understanding that they need to be able to compare the orders by making them such that there are the same number of one item in both orders is fundamental to understanding how to solve these systems. I wonder how many of our academic students actually understand and how many are just working through an algorithm.