What are the key features of a linear function?

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SWBAT identify and interpret the key features of a linear function.

Big Idea

Does the y-intercept of a linear cost function imply that someone is paying for nothing?

Opener: Netflix DVD Plans

15 minutes

Today's opener is about the pricing of Netflix DVD plans.  It's funny how quickly the world changes: I have to be prepared to spend a moment of two explaining that there's this other, non-streaming, service that Netflix offers, where they mail DVD's to your home.  "The DVD selection is better than the streaming selection," I say to one class, to which a student responds, "You should just get your DVD's from a Redbox."  I reply with a smile, "That sounds like a perfect math project!  We can compare the cost of renting DVD's on Netflix and Redbox."  

But I'm getting ahead of myself.  To start today's class, I post the opener, and I ask students to see what they can figure out.  If necessary, I describe how the DVD service works, and set students to it.  Students will notice that there are actually two problems here: the cost of the regular DVDs and the cost of a plan that includes Blu-Ray.  For both, I want students to notice that the costs of borrowing one, two, or three discs at a time form an arithmetic sequence, and can therefore be described with a linear model.  (Actually, the cost of borrowing more than three DVD's departs from this model, but I'm saving that for next week.)

The first slide asks students to figure out what a 10-disc Netflix plan should cost.  After a few minutes, many students are satisfied with their answer.  For today's lesson, I'm much more interested in creating and interpreting an equation to represent this data, so rather than confirming whether or not student solutions are correct, I move to the second slide, which prompts everyone to write that equation in slope-intercept form.  I mix this with the language of arithmetic sequences by asking for the common difference and revisiting the algorithm we used last week.

Soon, we've got the equation:

y = 4x + 3.99

I stand at the board and point to the number 4.  "What is this number?"  I ask.  I want my students to recognize that this is the slope, and if some students call it the common difference, then that's a bonus.  If everyone is with me, I'll ask about the domain of this function - can we use negative values for x? decimals? - and we might decide that it's better to think of this as a sequence than as a boundless linear function.  I'll take as much as I can get with this line of discussion, but I don't push it.  Tomorrow's opener gives us another opportunity to differentiate between discrete and continuous data.  Continuing to point to the 4, I ask, "What does this number mean?"  I want students to recognize that this is how much the cost of the plan increases for each additional disc.

I repeat my two questions - what is this number, and what does it mean? - pointing to the 3.99.  The best thing that can happen here is for a student to note aloud that the 3.99 doesn't make sense.  Often, at least one student will say something like, "Why would you pay $3.99 for nothing?"  If someone does, I'll say, "That's a great point!  And now you're interpreting this function."  We can discuss how well this interpretation holds up.  "Does anyone actually pay Netflix $3.99 per month for nothing?"  And if not, then what does that money represent.  Now kids will throw out other interpretations: it's the tax, the "background costs", or the "starting costs."  

Written on the side board is the learning target, and now I reference it:

I can identify and interpret the key features of a linear function, from an equation, a table, or a graph.

I say, "When I ask 'What is this number?' I'm asking you to identify it as the slope.  When I ask, 'What does it mean?' I'm asking you to interpret it.  That's what this learning target says you have to be able to do."

Work Return and Framing

5 minutes

As students initially worked on the opener, I returned three pieces of work.  There are two quizzes, and the individual/group assignment from our previous lesson.  I'm listing them here, with links to the lessons:

After finishing the opener, I say that there is a unit exam coming up on Friday.  I tell students that one way to begin to study is by taking a look at these three assignments - their work and my feedback - and that I'll be happy to answer questions any of these.

What are the key features of a linear function?

23 minutes

For the remainder of class, it's time to practice some algebra skills.  Once again, we review the learning target:


Today's class started with a lesson about identification and interpretation, so now I ask the obvious question: "What are the key features of a linear function?"  I write the question on the board.  If kids can't tell me, I add "y=mx+b", and this is enough of a hint for students to say "slope and y-intercept", and these are the two key features on which we'll focus today.  We'll get to the third key feature, the x-intercept, tomorrow.

I distribute a Kuta worksheet full of three kinds of practice problems:

  • finding slope, given two points
  • find the missing value in one of two points, given slope
  • writing the equation for a line, given the slope and one point

The first two types of problems are about slope.  In this video, I describe how I teach students to complete these exercises.  The third type of problem is an opportunity to find the y-intercept.  For this, I ask students to follow the traditional method of substituting numbers and solving for b.

Students have the rest of class to work on this assignment.  I circulate to answer questions, and I identify experts on each type of problem, so I can ask them to provide a little tutoring.  Along with the work I've just returned, this assignment is another study-tool for Friday's exam.