James Bialasik SWEET HOME SENIOR HIGH SCHOOL, AMHERST, NY
Algebra I : Unit #2 - Linear Functions : Lesson #9

Rate of Change

Objective: SWBAT determine the rate of change between two variables by examining tables of values.
Standards: HSF-IF.B.6 MP1 MP2 MP3 MP7
Subject(s): Math
60 minutes
1 Open - 3 minutes

In this opening activity we are asking students to reason quantitatively (MP2) about the meaning of a car traveling at 60 miles per hour.  Have students work on writing down an explanation by themselves at first and then share with a partner (Think-Pair-Share).  When students share out, guide the conversation towards an understanding that "60 miles per hour" is a comparison of miles and hours (rise and run).  This in turn is also connected to the concept of slope.

2 Investigation - 20 minutes

In this investigation, students will be working in pairs to determine the rate of change between pairs of variables set into a context.  Students will need to examine the structure of each table to determine the rate of change (MP7).  The context of the problem will help students understand and visualize rate of change in a way that may be more familiar to them (not so abstract).  For example, students will be investigating scenarios involving "dollars per hour", or "meters per second." 

I will structure this investigation as follows.  First I will print off 2 copies of the slides in the accompanying powerpoint rate_of_change_day1_launch.pptx (1 on each page).  I hang the 12 slides around the room (there will be two of each slide).  I then number each slide 1-1, 1-2, 1-3, 1-4, 1-5 and 1-6.  For the other identical slides I label them 2-1, 2-2, 2-3, 2-4, 2-5, and 2-6.  I make index cards with the same numbering system on them.  For this activity, I will allow students to choose their own partner.  Once they choose a partner they come to see me and get an index card.  This tells the pair of students which slide to start at.  I give students about 3-4 minutes per slide to answer the questions in the rate_of_change_day 1_graphic organizer. Students move to the next slide when given the signal.

Instructional Note: If you want students to travel in groups of 3-4 then you would only have to print one copy of each slide and they could be more spaced out around the room.

Investigation Reflection
Transitions

This was a great investigation for the students because they were investigating and discussing rate of change in contexts that they could understand.  Some students (about 6) were still having trouble coming up with the generalized formula for calculating rate of change and needed some scaffolding around this skill.  I continued to use the concept of "the constant value" as a way to adjust the rate to get a value in the table.  For example, in the "studying vs. grades" table, students could come up with a rate of change of 8.  From here, students would would see that if they studied for 1 hour they would have a grade of 60.  They knew that a correct equation would look like 60=8(1)+____.  Looking for this constant value that made the equation true helped them to write the final equation: y=8x+52.

I also liked this investigation because students naturally found unit rates as part of their work.  With real world application, more meaning is derived from a unit rate.  For example, in the "sprinting" table students could see that the sprinter traveled 50 meters in 10 seconds.  However, they naturally wanted to reduce this fraction to 5 meters every second.  This idea of reducing to a unit rate when possible will also make more abstract rate of change questions (such as those on the coordinate plane between random points) easier to understand.

3 Debrief - 12 minutes

For this portion of the lesson, I project each slide on the screen at the front of the room.  Two pairs of students will come together to share their solution for the table that they started with.  For example, the pair of students that had 1-1 and the pair that had 2-1 will come together (a group of 4) to share their solutions for the "sprinting" table.  At this point the rest of the class can critique their reasoning and ask any clarifying questions (MP3).  Each groups is given about 2 minutes to present their findings to the class and all students in the group need to share at least one of the solutions from their graphic organizer.

4 Closure - 5 minutes

This Ticket out the Door (rate_of_change_day1_closure.doc) will give students an opportunity to show that they can analyze a table independently.  Students will find the rate of change for the table given and find a formula to generalize the values in the table.  Students will also have an opportunity to explain (MP3) how they know their formula is correct based on the values in the table.  This explanation will give you some insight into how the students are thinking about the connection between the rate of change, the table and the general formula.