I will start off by asking the essential question. In the last lesson we wrote equations to represent the constant of proportionality as m = y/x. I will then display the graph from this section's resource section. I will write m = y / x on the graph. I will point out that this graph represents m, the constant of proportionality. We will find m. Then I'll ask: Could we rewrite the equation to solve for y, the total number of square feet? We will test students thinking using the graph. This should lead us finally to seeing the equation as y = mx.
We will the do the example problem. The point is to examine the table to find the constant of proportionality. I'll ask students what the total cost is based on n cookies. There is a blank spot on the table for this. Watch out here as many students will give the cost for 5 cookies as opposed to writing n * 0.75 or 0.75 *n.
As we work through parts of the example, students should be lead to the equation in the form y = mx or in this case t = 0.75n.
Finally students will be asked to describe in words the meaning of the equation. They should say something like, the total cost is equal to the constant of proportionality multiplied by the number of cookies. This shows students are able to reason abstractly and quantitatively (MP2) about the equation.
There is only one problem here. It is in the same structure as the example problem. The only difference is the data is represented in a graph. It would be a good idea to briefly ask if the graph represents a proportional relationship. Students will have already learned that this graph has the characteristics of a proportional relationship. It will be a nice review of the key characteristics.
Students will work through this problem 1 part at a time, so that we can stop to discuss solutions.
There are two problems with 5 parts each. The main thing to be on the look out for in both problems is whether students are first finding the correct unit rate. On the first problem we are specifically asked to find the cost per pizza, not how much pizza per cost of 1 dollar. The second problem is presented as a graph. Students may be tempted to say the cost per apple is $2. Here I could ask the students to tell me the meaning of cost per apple. When they are able to say that it means how much does 1 apple cost, I can then ask where this can be found on the graph. The should find the value of 1 on the x-axis and then its corresponding price.
Before we begin the exit ticket we will summarize. We can look to part v of each problem. In each the constant of proportionality is being multiplied by a quantity of items.
Students then take a 4 part exit ticket that is identical in structure to all of the problems explored today.
I view question i-iii as a scaffold to get students to answer iv correctly. That being said, a successful student should be able to answer all 4 questions.