Checkpoint
- Course 2, Unit 4
Power Models
In each unit, there is a final lesson and Checkpoint that helps students
summarize the key ideas in the unit. The final Checkpoint will generally
be discussed in class, with the teacher facilitating the summarizing,
and students making notes in their Math Toolkits (teachers
may just refer to this as "notes") of any points they need to remember,
adding illustrative examples as needed. If your student is having difficulty
with any investigation in this unit, this Checkpoint and the accompanying
answers may help you recall the concepts involved, and give you the big
picture of what the entire unit is about. If your student has completed
the unit, then a version of this should be in his or her notes or toolkit.
Students should also have Technology Tips in their toolkits, which may
be useful for this unit.
Possible
Responses to Unit Summary Checkpoint
| In this unit, students investigated many situations
where power and quadratic models are useful. The
bolded words are vocabulary and concepts your student should be familiar
with. |
| a. |
What situations would you
choose as good illustrations of the following?
- A direct variation power model. The relationship between
volume of a cube and edge length is a direct variation model.
We say that the volume varies directly with the cube of
the edge length, because volume is given by the equation
v = s3. Thus, as s3 increases,
v increases with it. Another example would be that the distance
fallen by a dropped object varies directly with the square
of the time elapsed. The distance is given by an equation in
the form d = at2. Thus, as t2 increases,
the distance fallen increases. In this example, the value of
a is 0.5g, where g is the acceleration caused by gravity. The
first example is a cubic, or power 3, model. The second is
a power 2 model. There were other examples investigated in
class.
- An inverse variation power model. The relationship between
time and average speed for a trip of fixed distance is an inverse
variation function. We say that time varies inversely with
speed, because the time is given by the equation t = d/s,
where d is the fixed distance. Thus, as s increases, t decreases.
Another example would that the intensity of light varies
inversely with the square of the distance from the light
source. The intensity is given by the formula I = a/d2,
where a is a constant in the situation, fixed by the strength
of the light source. Thus, as d2 increases, the
intensity of the light weakens. The first example is degree
1, the second degree 2. There were other examples investigated
in class.
- A quadratic model. The relationship between height of a
kicked ball and time in flight is modeled reasonably well by
a quadratic function. The height depends on the initial height
(a constant in a specific situation), and the initial velocity
of the ball (again constant in a given situation), and on gravity.
The initial height will supply the constant term in the quadratic
equation, the initial velocity will supply the linear term,
since distance = (velocity)(time), and gravity will
supply the coefficient of the degree-2 term (since distance
fallen under the influence of gravity varies with the square
of time - see the direct power model). Another example would
be that income for an entertainment event might be a quadratic
function of ticket price (if number of tickets sold is a linear
function of ticket price). This is because Income = (price per
ticket)(number of tickets). There were other examples investigated
in class.
- A radical or fractional power model. The relationship between
the length of a side of a square and the area of the square
is given by S =
 .
The relationship between the length of the side of a cube and
its volume is given by S =  .
These can also be written in fractional notation as
S =  or
S =  .
|
| b. |
What patterns in graphs would
you sketch in each case?
- Direct variation: A direct variation graph should pass through
the origin. If the power is odd, then the end behavior is in
opposite directions, and the curve has rotational symmetry around
the origin (the point where the x- and y-axes intersect). This
makes sense because when you cube, for example, a positive
number the answer is positive, but when you cube a negative
number the answer is negative. So we expect to have points
in the first and third quadrants.
If the power is even, then the end behavior is in the same direction,
and the curve has reflection symmetry across the y-axis.
This makes sense because when you square, for example, a positive
number you get a positive answer, AND you also get a positive
answer when you square a negative number. So we expect to have
points in the first and second quadrants.
- Inverse variation: An inverse variation graph does not
cross the axes, the axes are asymptotes. This makes
sense because the equations involve division by a variable,
so if that variable is zero then there is no related value
for the other variable. The graph has two branches. As with
direct power models, the odd powers are symmetric around the
origin. The even powers are symmetric across the y-axis.
- Quadratic: The graph of a quadratic function will have a maximum or
a minimum point and be symmetric about a vertical axis.
The rate of change is not constant.
|
| c. |
What kinds of questions would
you ask in each situation?
- Direct variation: For a cube with specific edge length,
what is the volume? For a particular lapse of time, how far
has an object fallen?
- Inverse variation: For a given speed, how long will the
journey take? At a particular distance, how intense will the
light be?
- Quadratic: How long will it take for a ball to reach a specific
height? What is the maximum height of the ball? When is the
income a maximum?
- Radical: For a given volume of cube what is the length of
an edge?
|
| d. |
For questions that call for
solving quadratic equations,
- How would you find the solution? By using the table or graph,
students can find values of one variable to correspond to specified
values of the other variable. Typically, students will graph
the relation then focus on a particular point by drawing a
horizontal line at the specified value of y, and look for intersection
points.
Student solutions at this stage are calculator
oriented. By changing the increment in the table, or by using
pre-programmed algorithms in the calculator, students can find
coordinates to any desired degree of accuracy. In Course 3, students
work with the quadratic formula to find exact answers to quadratic
equations.
- How would you check the solution? By substituting the value
of the found solution back into the equation.
- How many solutions would you expect, and how is that shown
by the graphs of the quadratic relations? For a quadratic
equation, there might be 2 distinct solutions, or 1 solution,
or no solutions. If the equation to be solved is d = ax2 + bx + c,
then we would graph the relation y = ax2 + bx + c,
and the equation y = d. This horizontal line will either
cross the curve at two points (a, d) and (b, d) or
will just touch at 1 point, or will not intersect with the curve
at all, as shown below.
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If you would like to see specific problems from Course 2, Unit 4, a
link is provided to Examples of Tasks from
Course 2, Unit 4. If you are interested in following up on the Algebra
strand in general, then the Scope and
Sequence will help you see where different concepts are introduced.
On the Algebra page, you can read an explanation
of the main algebra concepts as they are developed in all four courses.
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