Today's Opener jumps off from the assumption that students buy what they saw in the Everyday Situations handout and we've accurately matched the four algebraic rules to the descriptions of four situations. From there, we can really think about how an algebraic rule can help reveal what a graph should look like.
As I've described in previous lessons, I really like the arc of the "Everyday Situations" assignment.
With all of that in mind, the opener references one of the situations on the "Everyday Situations" assignment:
A group of people rent a bus for a day. The total cost of the bus is shared equally among all the passengers. x = the number of passengers; y = the cost for each passenger in dollars
Students had trouble coming up with what the graph of this situation should look like. On today's opener, I add what we learned from the algebraic rule that matches this situation: that the cost of the bus rental is $300. Then, I instruct students to "make a table of values showing how much everyone would have to pay if there were 1, 2, 3, 4, 5, 6, 10, 15 or 30 passengers.
Making the table is an accessible task. I can count on almost all of my students jumping right into this one. I circulate to make sure they've got it, and then as they work, I prepare some axes on the front board. After a few minutes, I elicit help from the class to fill in the table, and then I say that we're going to use these points to learn what the graph of this situation should look like. As soon as we plot the points, the shape of this function becomes clear.
Now we continue with the "Car Value" situation:
My car loses about half of its value each year. x = the time that has elapsed in years; y = the value of my car in dollars.
As an example of exponential decay, this function actually gets at the focus of this unit. We follow the same process, making a table of values and seeing the shape of the curve.
At a few points during this mini-lesson, I summarize for students by saying, "There are two things I want you to think about as you consider each situation. First, as x increases, does y increase or decrease? Second, should the graph be a straight line or a curve?"
To conclude the week, students have a go at another version of the "Four Situations" assignment. This comes from a Formative Assessment Lesson on the Shell Center's Mathematics Assessment Project site, and you can find the specific documents I'm using here: the original "Four Situations" assignment is on pages 15-16, and the similar "Another Four Situations" assignment that I'm using today is on pages 23-24.
I make sure to give students at least 25 minutes to work in groups and complete the assignment. As they start, I reiterate my advice to determine which algebraic rule makes the most sense for each situation, and that tables of values can be used to figure what a graph looks like. I also remind students once again that the should consider whether each graph is increasing or decreasing, and straight or curved.
It's great to watch kids work, because so many conversations can happen! I really try to stay out of it, and I can assess my success this week by the level of student discourse I hear as they're working. I do my best to stay out of things.
To end the week, I collect this. Still the work isn't perfect, particularly on Situation "D: Saving Up", which will require a closer reading on Monday. On the other hand, solid foundations for next week have been laid. Next week, we'll go much deeper into the parameters, interpretation, and focusing specifically on exponential and linear functions.