I ask students to hypothesize how a grid is useful in today’s world. Students report out, and I make an anchor chart. This can then be used later, at the end of the mini-unit, for students to create their own relationships over time problem/graph. (Students add to anchor chart started yesterday.)
As a class, we use a graph to show how much money Henry earned working. We analyze the relationship between the number of hours Henry works and the amount of money he earns. Using MP1 - Make sense of problems and persevere in solving them, I ask students what problems have we solved like this before. Students respond that it is similar to the puppy dog problem from the other day.
In pairs, students discuss:
a) how much money earned after working 2 hours,
b) how much money earned after working 6 hours,
c) how much money earned after working 5 hours,
d) how much money earned after each hour,
e) how much money earned after working 10 hours.
We specifically discuss how the creator of the graph chose to use increments of 2 on the "x" axis and 10 on the "y" axis. We then create a table, and work "backwards".
To find (c), students are challenged to determine the pay after 5 hours, because 5 hours was not an increment of the graph. Knowing that his money increases by $20 every two hours, students have to analyze the graph to determine a mid-point between the 4th hour and 6th hour.
To find (d), students are challenged to determine the pay after 1 hour, because 1 hour is not an increment on the graph. Students have to envision where a plot point would be to indicate 1 hour. I expect students to get this pretty easily, and some students need some assistance. I encourage those students to look at the earnings for 2 hours ($20) and then determine what the rate would be for one hour. This is an easy problem here; my students are then eager to tell me the rate for 30 minutes too (which isn't relevant to the question, but my students are excited to use their skills here.) Since students are already determined (c), this is relatively easy to determine; students already knew this answer (but may not have verbalized it) if they solved (c) correctly.
To find (e), students had to apply what was known from figuring out (c) and (d), to apply to to extend the graph.
For Independent Practice, students worked to first use a table to determine which increments to label their self created graph. This was really challenging for some of the students, even though we've done this multiple times before. My students very much want things to be simple and easy, and are sometimes really not willing to put in very much effort to solve problems. Some attempt to simply just "wait it out". Knowing this, I know who I need to focus on to get them started. To get students more interested in this problem, I incorporated science into math; this problem dealt with Liquid Temperature Change. Students had to graph the change in temperature over time. After labeling the "x" and "y" axis, students then plotted the points onto the graph from the chart. Students find the temperature of the liquid after 5 minutes. Students then analyzed the graph to tell a story that would have gone with it (to explain the relationship shown in the graph.)
Using MP3 - Construct viable arguments and critique the reasoning of others, students now have the opportunity to explain their thinking to others, and respond to others' thinking about the relationship shown in the graph. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get that?” and “Why is that true?” Many students realize that the temperature decreased very rapidly within an 8 minute time frame, and predicted that a (snow) storm could potentially occur, or cold front was coming.