I will begin with the essential question. This question will be immediately answered as we review the vocabulary for the lesson. My students should now be very familiar with unit rates but the term constant of proportionality will be brand new. When defining the term, I will provide a simple example: 15 dollars for 3 sandwiches. I'll will write it on the board as m = 15 / 3. I will ask the class what the unit rate is. They should all confidently say $5 per sandwich!
Next we will go through a model problem for the lesson. I will pick 4 ordered pairs and actually plot a point and label the ordered pairs on the graph for clarity. This will help make sure students are not confusing the x and y coordinates. In part ii, I will explicitly write the equation in the form of m equaling the y-coordinate over the x-coordinate, then I will simplify.
Part iii will be a discussion question where students must explain their thinking (MP3).
Part iv is meant to assess that students understand the meaning of the points on the graph. So, x = 0.5 has a y value of 2 which means half of a dollar equals 2 quarters or 0.5 / 2 = 4.
The guided problem solving mirrors the example problem from the previous section. The scale of the y-axis may cause some problems for several students. Students will see labeled values of 50, 100, 150, etc but may have difficulty interpreting the unlabeled values. I will ask: how many intervals (or spaces) are there between 0 and 50? When they answer 5, I'll ask what must be the value of each interval. They should more readily say 10 now. If not I will have to show an easier example by drawing a number line by two's from 0 to 10 but only labeling the 0 and the 10.
The first question of this section follows the same structure as the previous problems.
The second question put the onus of graphing on the students. I will pass out rulers here so that students can connect the points to make a neatly drawn straight line. It will be fun to explore values (2, 6) or (1, 3) as they were not actually graphed yet they have the same values as the graphed points.
Before beginning the exit ticket, we will recall that the constant of proportionality can be written as the ratio y/x.
The first two problems are the heart of the lesson. Students should be able to successfully complete both of these.
The question in part iii, will be covered in another lesson, so this problem is not as critical for now. This question will be a bit harder than what students saw in the lesson because it is more difficult to locate the x-coordinate 1 due to the scale of the graph. Students who answer this question correctly will have a very good understanding of the constant of proportionality as it relates to graphs.