Pointslope
Formula and Pointpoint Formula:
Logic of the
Lesson
Pat
Thompson
The following example is of a lesson logic for teaching the
pointslope formula (it really should be called the pointrate formula) and the
pointpoint formula in Algebra I.
A lesson logic is an 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. All lesson logics assume that the lesson unfolds by way of the teacher leading a conceptual conversation around the lesson's central goals.
The following lesson logic unfolds these ideas:
1.
You can determine the coordinates of all the points on a
linear function's graph by knowing just its rate of change and one point
through which the graph passes. Thus, you can determine the coordinates of the
yintercept just by knowing the function's rate of change and one point its
graph passes through. Therefore, you can put the function in standard form y = mx +
b, where m is the function's rate of change and b is its yintercept.
2.
If you know the coordinates of two points that a linear
function's graph passes through, then you can calculate the function's rate of
change from those coordinates. Then you are back to the situation in (1), where
you know the function's rate of change and the coordinates of one point that
its graph passes through.
1) The rate of change of a function describes how y changes with respect to changes in x (ex: if the rate of change is 3.5, when x changes by 1, y changes by 3.5 * 1; when x increases by 1/100, y increases by 3.5 × 1/100);
2)
Lines are collections of an infinite number of points.
Step 
Action 
Reason 
1. 
Give the students a point in the xyplane and a rate of
change. [Note: Here it may be helpful to give them this rate of change in
a context they are familiar with. If they have encountered rates of change
when talking about distance a car travels over some amount of time, refer
back to such an example so students can relate this to their prior knowledge
about the subject of rate of change.]
Now, ask the students to find a second point where the rate of change between
these two points is the given rate of change. 
This forces students to think about the relationship
between the x and y coordinates for points on a linear functionÕs
graph. They need to think ÒIf I am given a rate of change, this means that
when x increases by some
amount, y increases by the rate
of change times the amount by which x changes.Ó 
2. 
Ask different students in the class the point that they chose and have them explain how they found it. Most likely, different students will find different points that satisfy the above conditions. 
This emphasizes the fact that many different pairs of points can share the same rate of change between them. 
3. 
Assume that at least one of the students found his or her
point by finding the value of y when x
increased by 1.Ask students what would
happen if x changed by 1/100,
2/100, .0375, or by 0.5281? By 7, 90.3, or by 204.8? By ^{}2, ^{}3.4,
or by ^{}7.3? Does their reasoning change even though the numbers
are different? 
This helps students to see once again that y changes by the rates times the amount by which x changes. 
4. 
Ask the students if they have found all the points that satisfy the above scenario that the rate of change between their new point and the initial point is the given rate of change. If not, ask them where the other points are that do satisfy the scenario. Point out to students that they have established that all the points on a linear functionÕs graph are
determined by knowing just the functionÕs rate of change and one point on the
graph. 
Students should see that any point on the line determined
by their two points would also satisfy this relationship: However much the
xcoordinate of the second point changes from the xcoordinate of the first
point, the ycoordinate of the second point changes by the rate times the
change in the xcoordinates. Symbolize this as Òif x changes by Æx, then y
changes by rÆx.Ó 
5. 
Tell them that you are thinking of a function that has a line as its graph. The graph passes through the point (7.5, 2.4), and the function has a rate of change of 3. Ask them, ÒWhat will be the value of y when x is 0? Do this again with a different point and a different slope. Focus on the similarity in reasoning between the two cases. 
This is really to find the yintercept. Students should eventually reason that moving from
(7.5,2.4) is like changing x by
7.5. If the functionÕs rate of change is 3, then y will change by 3«(7.5),
or 22.5. So y will change from 2.4 to (2.422.5), hence y will be 20.1 when
x is 0. Asking students to do it twice will highlight the steps in reasoning
over the steps in arithmetic. 
6. 
Ask students, ÒA function has a line as its graph. The
graph passes through the point (a,b) with slope 2.5. What is the value of y when x
is 0? 
This asks students to generalize to arbitrary points._{} 
7. 
Tell students, ÒThe students in another school do not know
how to find the y intercept for a
linear function when all theyÕre given is a point on the graph and the
functionÕs rate of change. You do not know the precise points they will work
with. Write them a letter to tell them how to do this so that your plan will
work for whatever points they pick.Ó 
This step forces students to formalize their method for finding a linear functionÕs yintercept. They will need to decide how to represent the given information and how to help the other students to apply this method. Students should end up with something like: . If we move from x_{1}
to 0, then y_{1} will
change by r(^{}x_{1}). 
8. 
Say, ÒThe graph of a linear function passes through the
point (^{}3,^{}4.1) with slope r. Write a formula that represents this function in
standard form (i.e., in the form y
= mx + b).Ó 
This is the pointslope formula. Students should be able
to answer this question now that they know how, in general, to calculate the
functionÕs yintercept. 
9. 
Say, ÒThe graph of a linear function passes through the point (x_{1},y_{1}) with slope r. Write a formula that represents this function in standard form (i.e., in the form y = mx + b).Ó 
Ditto the above, except this is the most general case. 
10. 
Tell students, ÒIÕm thinking of a linear function whose
graph passes through the points (7.2, 5.5) and (12.7, 2.1). Write a formula
that represents this function in standard form (i.e., in the form y = mx
+ b).Ó 
They have only the added step of determining the functionÕs rate of change. Once they have this, they also have one point the graph passes through, so the problem is reduced to the prior case. 
11. 
Tell students, ÒIÕm thinking of a linear function whose
graph passes through the points (x_{1},y_{1}) and (x_{2},y_{2}). Write a formula that represents this function in
standard form (i.e., in the form y
= mx + b).Ó 
This is an occasion to generalize from the concrete
example in 10. The function's rate of change r is 
12. 
Recap what they have done:
