Checkpoint
- Course 1, Unit 3
Linear 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
| a. |
Describe how you can tell whether a situation can be
(or is) represented by a linear model by looking at:
- a scatterplot: Most of the data points should be close to
a possible line. The points may look like an elliptical cloud.
The existence of a very few outliers does not spoil the overall
trend.
- a table of values: There should be a constant (or nearly
constant, in the case of experimental or observational data) difference
between y values as the x values change uniformly. The ratio of
the change in y compared to the change in x is the rate at which
y is changing relative to x. This rate appears as the slope of
the line on the graph, or as the coefficient of the x in the equation.
It does not matter which two points we take to measure this rate,
nor how far apart the points are, since the rate is constant throughout
the situation.
- the form of the modeling equation: The equation will have
an x term with a positive or negative number multiplying the x,
and there may be a constant term added or subtracted. The general
form of the equation is y = a + bx. Sometimes
the number multiplying the x is zero, in which case the line will
be horizontal, because the y values are not changing. The numbers
in the equation might be fractions, but the x is not in the denominator
of the fraction. There are no other powers of x, and x itself
is not a power. (Of course, variables other than x and y are frequently
used, particularly for problems coming from contexts. See student book page 195.)
- a description of the problem: The problem has to have one
variable changing as another changes. There must be some information
that makes us think that a constant rate of change makes sense
in the situation.
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| b. |
Linear models often describe relationships between an
input variable x and an output variable y.
- Write a general form for the rule of a linear model. What do
the parts of the equation tell you about the relation being
modeled?
y = a + bx. The number a is the y-intercept
value. When x = 0,
y = a. The pair (0, a) appears in the table. Students
often think of this as the "start" though, of course, it will
not be the "first" point in many situations. The b is the rate
at which y is changing compared to x. Between any two points,
the ratio of the difference of y values to the difference of
x
values gives the value b. This also gives the slope of the line.
If b is positive, say 2/3, then y increases 2 units for every
3 units that x increases, or 2/3 of a unit for every 1 unit increase
in x. If b is negative, then y decreases as x increases. (At
this time, students have not met other forms of linear equations
such as ax + by = c.
This form will be introduced in Course 2, Unit 1.)
- Explain how to find a value of y corresponding to a given
value of x, using:
- a graph. Find the given numerical value of x along the
x-axis. Then move to the linear graph and identify the
y-value for the point on the line that has the given x-value.
- a table. Scan the table for the given value of x (resetting
the increment in the calculator table if necessary) and read
the corresponding y-value.
- a rule. Replace the x in the equation with the given
numerical value of x, and solve for y.
- Explain how you can solve a linear equation using:
- a graph. On a graph, trace to the given x or y value
and read the other value. For example, 3x – 1.3 = 4.7.
In this case, we graph y = 3x – 1.3 and
find the particular value of x that goes with the given value
of y = 4.7. (It is more efficient to graph y = 4.7
and y = 3x – 1.3, and find the intersection.)
Or if x is on both sides of the equation, then we graph a
line for each side, and find the intersection. For example,
3x – 1.3 = 0.7x – 5.1. In this
case, we graph y = 3x – 1.3 and y = 0.7x – 5.1,
and find the particular value of x that makes the two y values
equal, that is, the x value of the intersection point.
- a table. In a table, scan to the given x or y value and
read off the other value. If the x occurs on both sides of
the equation then we make tables for both equations as above,
and scan for an x value that makes the two y values equal.
- symbolic reasoning. Use algebraic rules to get the variable
isolated. Students may have a "balance" strategy, of doing
the same to each side to rewrite the equation as "x =."
Or they may have an "undoing" strategy. For example, if 3x – 5 = 11,
then 3x must have had a value of 16 before the 5 was subtracted.
If 3x has a value of 16, then x must have had a value of 16/3
before it was multiplied by 3. The "balance" strategy is more
effective in more general linear cases, and is probably more
familiar to you. (Note: In the past, these symbolic strategies
were most common because graphing calculators were not available.
However, this "balance" strategy is not effective with nonlinear
equations, such as 2x2 – x = 2,
or 2x = x + 3. The graphing
strategy is more generally useful.)
- Explain how you can solve a system of linear equations using:
- a graph. Trace to where the lines intersect and read
the coordinates of the point of intersection.
- a table. Scan to where the same x value produces equal
y values. (Assuming "y =" format.)
- symbolic reasoning. Assuming both equations are in "y ="
format, set the right-hand sides of the equations equal,
and solve by "balancing." Then find the y value corresponding
to the x value.
See the Quick Summary for this unit for an example.
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If you would like to see specific problems from Course 1, Unit 3, a link
is provided to Examples of Tasks from Course
1, Unit 3. If you are interested in following up on the Statistics
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 statistics concepts as they are developed in all four courses.
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