Finishing Up the Spending & Saving Project, With Extensions

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SWBAT construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, a table, or two input-output pairs.

Big Idea

Students get back into the swing of things after a break by finishing up a project, and exploring the distinguishing characteristics of linear and exponential functions.

Opener: Matching Equations and Graphs

8 minutes

Today's opener gets the class started toward those goals.  There are four function rules to be matched to four colored graphs.  Each function is represented by a different letter (f(x), g(x), etc.), which gives me a low-pressure to review the idea that functions can be denoted by letters other than f.  

This is a pretty informal opener, and I don't insist that kids write anything down.  I give kids a few minutes to think about it, and then we discuss it.  The most straightforward distinctions to be made are those between linear and exponential functions, and increasing or decreasing functions.  If students can make those distinctions, then the solutions take care of themselves.

What I really hope is that this opener gets a conversation going.  Students will remember the differences between these kinds of functions, and in addition to the distinctions I've already outlined, we might be able to talk about the y-intercepts of linear and exponential functions, both of which play a role in the work that students are finishing on the Spending & Saving Project.  

After I use this opener to frame our work for the day and for the week, I leave it up during work time, because it's helpful to be able to reference it during the rest of the lesson.

Project Work Time: Digging Back In, and Finishing Up

25 minutes

The previous six lessons contain my descriptions of each part of the Spending and Saving Project, and students will have 20-25 minutes to work on that right now.  I will collect the project two days from now, on Wednesday.  

Even though I've already laid out my plans and expectations for the project on this site, the fact is that every time I give kids more time to work on it, I think of another half-dozen points to write about here on BetterLesson.  Rather than going on and on, here's a list of a few things that my students and I discussed on this first day back at school:

  • When a quantity is increasing or decreasing by a certain percent, we can use two steps (calculate the percentage, then add it to the original value) or one (multiply by 1+r) to get from one value to the next.  Kids are excited to understand why this shortcut works, so we often spend some time on that.  
  • Revising graphs (see here, for example), rescaling axes, and deciding on parameters that make it "easier" to see what's happening on a graph (this is for Part 2 of the project) always leads to rich conversations about how math involves choices and that revision shouldn't only be reserved for research papers and other writing assignments.
  • What is the minimum number of points needed to make a graph?  This project makes it clear that we can save time by understanding that only two points are required to define a line.  Additionally, when we're skip-counting on the the x- and y-axes, there are some input-output pairs that are easier to plot on a graph.  It's prudent to try to find those "easier" points.  For this and the previous bullet-point, the idea of making things "easy" certainly entails knowing enough mathematics to accomplish the trick.
  • A complement to that previous point, of course, is that exponential functions will require us to plot more than two points to create an accurate graph by hand.
  • Some students want to know about how to write a function rule for a savings or debt scenario that involves both interest being added and a monthly deposit (to a savings account) or payment (of debt).  I explain that it's hard to write an explicit function rule for such a scenario, but that we can easily model it with a recursive rule, which is precisely what we use in Excel to do the same thing.
  • I frame the extension to Part 3 of the project.  Here are some examples of the work I receive from students.  Please see the narrative video narrative video I've posted here.  The video includes some of my reflections on the project as a whole, and some ideas for how how to get kids started on the brochure.

What is most important is the general structure of this work time.  When we give kids tasks and give them time to try things, I believe that the best kind of conversations happen.  When we as teachers know our content inside and out, and we're prepared enough to talk about any number of things, we no longer need to plan every moment of a lesson, because we can let learning happen.  Of course, we have to make sure that the kids know we love them and want them to achieve the best.  Really make sure of this.  As they work, help out, then see what conversations transpire.  You'll be thrilled.  

Check In Quiz: Linear vs Exponential Functions

10 minutes

My final move for getting back into the swing of things after a long break is to give this quiz on SLT 4.1.  The learning target states that students should be able to differentiate between linear and exponential functions, expressed in various forms.  This quiz is made to get directly at that point.  If students get it, then they'll be starting by earning a great mark right away, which I hope will set the tone for the rest of the week.

If any students are not successful on this quiz, I can quickly target who needs help.  I can see if students are confident with one representation, like graphs, but not another.  Even more importantly, students can identify that for themselves as well.

If a student does know what's going on, this quiz should not take more than three minutes.  I provide ten minutes.  I don't want students to feel to much pressure for time.  If they finish early, I tell them the extension: to write an equation that models each situation.  This, of course, is much more challenging.  It gives everyone plenty to grapple with, and it will provide a starting point for tomorrow's lesson.