The group approached the problem by comparing the T-tables to the graphs and identifying common coordinates between the two (x,y). This led to some confusion to begin with as graphs and T-tables were grouped if they shared a single point. However, relationships y=3x and y=2x+1 both have the point (1,3), and they cards could not be assigned based on that one point alone. Realizing this error, the group used the graph to compare the change in y with each unit of x to the change in each row of the T-table. Overall this was a reasonable method of comparing relationships, and the group drew from their past knowledge in order to represent each as an algebraic expression.
The group did not really use the counting block diagrams, and they were also an unfamiliar way of representing the equations to me. I did like how the blocks seemed to be laid out so that there was a multiplication component and separate addition component for each (they were not labelled but seemed to be physically segregated to indicate they were different), as this allowed us to clearly see which portion was growing with the change in x. I liked the exercise where we constructed the relationships with blocks ourselves, and could physically represent what was happening, and even colour-code the addition and multiplication components.
We spent some time in this class discussing how to ask effective questions. At the time it seemed a little overwhelming, as it was so different from my experiences learning math and science. The Province of Ontario document Asking Effective Questions gives many examples of how to develop student understanding and critical thinking, and the ideas were reflected in a chart that we were given in our cohort group.
While the chart is not as thorough, it offered a clear starting point in working towards promoting student critical thinking. The factual section of the grid is what we are generally used to asking and answering in math, but these are what we might consider low-order content-based questions. They examine a student's knowledge of details and are easy to assess, but do not consider higher-level thought processes such as analysis, prediction, and application. The chart is a nice way for me to visualize these deeper questions and the phrasing (how could, why might, why would...?) makes for a much softer introduction than provincial documents! That said, once this mindset is activated, there are many other factors that Asking Effective Questions goes into more detail about, such as wait time and making questions meaningful.
The first activity of the day dealt with building a bridge framework out of Q-tips. Each extension of the framework resulted in two new Q-tips being added to the original three. We wrote an algebraic expression for this pattern, using given points (1,3), (2,5), (3,7), (4,9). This corresponded to an increase of two Q-tips for every step, so the slope of the expression would be 2. For the first step there are three Q-tips despite the slope indicating two Q-tips for every step. To make up this extra Q-tip we can add it as what would be the y-intercept, giving an expression y=2x+1 for the bridge supports. This expression was then used to solve for step 45, and then we were to find the number of supports for both sides of the bridge at step 45. Something I noticed for this activity was the choice of step 45, an odd number, for the total bridge length. When adding steps to the bridge, each odd step is the shape of a trapezoid for x>1, and a parallelogram for all even numbers of steps. A parallelogram would not sit well over a span, but a trapezoid would be appropriate for building a bridge support.
The second activity dealt with a series of operations on a number of your choice. It was a fun activity because it did have the essence of a magic trick. You did a series of seemingly random operations to get a very different number than you began with, but what was hidden in the calculations was that you were more or less adding (1+9)/2 to any number, then subtracting the number you picked, giving you again (1+9)/2=5 as an answer. The third question was very similar to the first, and the second was supposed to equate to the same value forward and backward, but I wasn't able to get it to work.
The last activity was one that I found really relevant. It dealt with two workers banking overtime hours in order to go on vacation. While this may not speak to children, I liked it because it can represent most ways we budget or save our money (I worked a lot of overtime at Canada Post to save tuition). This activity provided a background story with a T-table to illustrate the relationship between the data. We then used this to find different inputs and outputs, eventually arriving at the expression y=3x+1 for Colin's overtime worked. The second question reversed the scenario and Steve worked much less overtime than Colin. It is stated that Steve worked 1 hour of overtime in the first week. Comparing this to the T-table we find an input/output of (1,4). This was a problem however because Steve had only worked one hour, not four. I thought that perhaps the input and output columns had been reversed. In this case, it implies that (4,1) is the first value in the table, and Steve has worked only 1 hour of overtime even after four weeks. With this information, it was then possible to write the expression y=(1/2)x-1.
