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.
1) Covariation of one quantity with another
2) Finger tool thinking
3) Coordinates of a point tell the states of two variables simultaneously.
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. |
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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? |
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8)
|
How far was I from home when I started my watch? |
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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 |
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