Graphs of Linear Functions

Logic of the Lesson

Shannon Coombs and Pat Thompson

 

The following is a lesson logic for teaching how to reason about situations involving initial conditions, constant rate of change, and different ways to represent those situations.

 

A lesson logic is the outline of how you will develop the lesson's main ideas. It does not pay attention to time, meaning that the "lesson" may transcend several class periods. It does not give the level of detail that a lesson plan gives, meaning it might not say how you will organize the classroom, how you will transition from one activity to another, etc. Instead, it focuses on the ideas you will develop, the way you develop them, and why you take the approach you take.

 

The following lesson logic provides a structure in which the surrounding conversation unfolds these ideas:

 

1.     Constant rate of change of quantity A with respect to quantity B means that all changes in A are the same multiple of a corresponding change in B.

2.     A graphical representation of an initial condition is a point.

3.     Rate of change is about changes, so rate of change with an initial condition is about changes from that state.

4.     Ideas 1-3 together allow one to think about the graphical representation of a function that has an initial condition and changes at a constant rate.

Meanings or schemes students must have before the lesson:

 

1)    Covariation of one quantity with another

2)    Finger tool thinking

3)    Coordinates of a point tell the states of two variables simultaneously.

Steps in the Lesson Logic

 

Step

Action

Reason

1)              

What does it mean that someone travels at a constant speed?

-      Anticipate "speed never changes"

-      What does that mean about distance and time?

-      (Anticipating difficulty saying it) Suppose we are told how far this person traveled in some 16 second time period. What do we now know?

-   Just this 16 second period or other 16 second periods, too?

-   Does this tell us about how far he will travel in 8 seconds? 4 seconds? 1 second? 1 minute? 1 hour?

-      End with this summary: To say that a person travels at Z mi/hr means that however many hours he travels, he will travel Z times as many miles as that number of hours.

 

2)              

Set up scenario of traveling from home to store. Ask how to represent graphically that I am 3.8 miles from home and my watch says it is 7 minutes since I first looked at it.

 

3)              

I am traveling at the constant speed of ½ mile per minute (what does that mean?). I am at 3.8 miles and I first looked at my watch 7 minutes ago. How much farther will I travel in the next 1/10 minute? How much farther will I travel in the next 3/10 minute? How much farther will I travel in the next 10 minutes? (Remind them of the constant multiplier meaning of constant rate.)

 

4)              

Let's show these changes graphically (do the calculus triangle thing). Be sure to talk about changes in number of minutes and changes in number of miles.

 

5)              

What is this graph going to look like?

Draw attention to the graph will not deviate up or down from the pattern already established.

6)              (Maybe)

(New phase)

How far will I be from home after the next 1/10 minute? The next 3/10 minute? Does this knowledge change my graph? (No. It just makes explicit something we hadn't noticed yet.)

The point is that knowing an initial condition and knowing the changes is the same thing as knowing the total states.

7)              

Was I at home when I started my watch?

 

8)              

How far was I from home when I started my watch?

 

9)              

Recap: What did we do?

-       Represented given information (where I was after some number of minutes)

-       Decided what the graph would look like knowing that my speed was constant

-       Figured out where I was when I started my clock by using the initial information and by knowing what constant speed means.

 

10)           

(New scenario)

After 7.2 minutes