Point-Slope Form—Life Expectancy
Life expectancies are the average (mean) lengths of life for persons in various
groups. For example, the life expectancy of a person born in 1940 is 62.9 years.
This means that the anticipated average age of death of all people born that year is
62.9 years (though many of them haven’t died yet). Life expectancy is predicted on
the basis of many factors. Considering past data on life expectancies, what would
you predict the life expectancy of someone born in 2010 to be?
Q1
Make a conjecture about the life expectancy of someone born in 2010.
Q2
Do you think the life expectancy of females born that year will be more or less
than that of males born that year?
INVESTIGATE
1. Open the Fathom document Life.ftm and drag a graph from the shelf. Drag
the Birth_Year attribute to the horizontal axis and one of the other attributes
(Female, Male, or Combined) to the vertical axis to make a scatter plot.
To make a prediction about 2010, you want a line that seems to fit the data set.
It needs to be plotted, rather than movable, so you can slide a red dot along it.
However, because the data points aren’t in a straight line, you’ll want to be able to
manipulate the line easily. For that, you’ll use a slider.
To find the slope,
divide the difference
between the life
expectancies
( y1 ⫺ y2) by the
difference between the
birth years (x1 ⫺ x2).
This answer is an
expression using both
Birth_Year and Slope in
place of the numbers
you used in Q7.
Q3
Write down the coordinates of one of the data points. Select a second point and
use it to calculate the slope of the line between the points.
Q4
How many birth years are between the points? How many birth years are between
your first point and another point with horizontal coordinate Birth_Year? (This
answer will be an expression that includes Birth_Year in place of a number.)
Q5
How many life expectancy years are between your two selected points? How can
you use the slope (Q3) and the number of birth years between the points (Q4)
to calculate this number?
Q6
Find an expression for the rise or fall (change in life expectancy) between your
start point and a point with horizontal coordinate Birth_Year; use the variable
Slope in place of the number for slope. (This answer will be an expression using
both Birth_Year and Slope in place of the numbers you used in Q5.)
Q7
How can you find the vertical coordinate of your second point using the vertical
coordinate of your first point and the answer to Q5?
Q8
How can you find the vertical coordinate of any point on the line using the
vertical coordinate of your first point and the answer to Q6?
Point-Slope Form—Life Expectancy
continued
Double-click the
slider’s thumb to open
the inspector.
2. Drag a slider from the shelf. The slider name, V1, will be selected; rename the
slider by entering Slope. Adjust the scale on the slider or set Upper_ and Lower_
limits in the slider’s inspector so that it runs from about 0 to about 0.4.
3. Choose Plot Function from the Graph menu.
Enter the expression you wrote for Q8, using
the variable Slope. The slider will now control
the slope.
You might need to
adjust the scale on the
graph.
4. Adjust the slider so that the line appears to fit the data points as closely as
possible.
5. If you think a line through a different data point will represent the data better,
double-click on the line’s equation and enter coordinates of that other data
point. Adjust the line this way until you think you can make a good prediction.
Q9
Record the equation. This graph
shows one possible equation.
6. Click on the line and move the
red dot, watching the coordinates
in the status bar.
You might need to
rescale the vertical
axis.
Q10
What prediction does your line
make for life expectancy in 2010?
Q11
Why might the form you’ve been
using, y ⫽ y1 ⫹ b(x ⫺ x1), be
called the point-slope form of a
linear equation?
For any given year, the residual is the difference between the actual life expectancy
and the life expectancy predicted by the line. You can use squares of the residuals to
improve the fit of the line.
7. Select the graph and choose Show Squares from the Graph menu. Change the
slope to minimize the sum of the residual squares.
8. Choose Least Squares Line from the Graph menu to plot the line that
minimizes the sum of residual squares.
Q12
Was your first line a good approximation of the least squares line? What is your
evidence?
Point-Slope Form—Life Expectancy
continued
EXPLORE MORE
1. To adjust your line even more flexibly, you can use sliders for the point to pass
through, rather than being limited to a data point. You might use sliders called
x1 and y1 for the two coordinates of the point. Change your formula accordingly.
What equation do you think gives the best prediction, and what is that prediction?
2. Adapt your graph, or make a new one, with both Male and Female on the vertical
axis. Do so by having one of the two attributes already on this axis and then
dragging one toward the axis. A plus sign appears; drop the second attribute on
the plus sign. Both data sets will appear on the scatter plot. Find an equation to
predict whether the life expectancy for females will still be greater than that for
males and how much the life expectancies will differ. Are the life expectancies
getting farther apart or closer together? What measure(s) indicate(s) that?
Point-Slope Form—Life Expectancy
Objective: Students develop and use the point-slope form
of the equation of the line.
Activity Notes
Q1 Answers will vary considerably.
Q2 Women will probably have a longer life expectancy.
Student Audience: Algebra 1
Activity Time: 35–50 minutes
Setting: Paired/Individual Activity, Exploration, or
Whole-Class Presentation (use Life.ftm for any setting)
Mathematics Prerequisites: Students can calculate slope
from two points.
Fathom Prerequisites: Students can open an existing
Fathom file, make a scatter plot, plot a function, and edit
the equation of a plotted function.
Fathom Skills: Students will use a slider and trace a
function graph with a red dot.
Notes: This activity introduces and explains the point-slope
form as y ⫽ y1 ⫹ b(x ⫺ x1), where y1 represents some
starting value for the dependent variable, b is the rate of
change, and x ⫺ x1 is the distance moved from the starting
point. The location of the points on the line are given in
terms of some starting point (x1, y1) and the movement
from that point.
If students are stuck on how to write an expression, have
them think about what they did with specific values before
they try to generalize. Suggest they repeat Q3 and Q5 for
more than one choice of second point before introducing
a variable in the expression in Q4 and Q6. As you listen in
on groups that are struggling with Q7, you might suggest
that they substitute the points they chose in the question.
As groups share their predictions, ask them to describe the
points and equations that lead to those predictions and
explain why they chose those points.
For a Presentation: As your class discusses the answers,
use several points for Q3 and Q5 before writing the
expressions with the variables. Once you have a line of fit
drawn, ask students to suggest different points the line
could go through to give a better line of fit. For each of
those points, write an equation and predict the 2010 life
expectancy.
INVESTIGATE
Q3 Students might choose any data point. The next few
answers are based on the choice of (1970, 70.8) and
(1990, 75.4) for (Birth_Year, Combined). The line
75.4 ⫺ 70.8
between the points has slope ________
1990 ⫺ 1970 , or 0.23.
Q4 1990 ⫺ 1970, or 20, years between the points;
Birth_Year ⫺ 1970 or 1970 ⫺ Birth_Year (The form
Birth_Year ⫺ 1970, which gives positive values
after 1970 and negative values before 1970, is more
conventional and will be used below.)
Q5 The difference in life expectancy rose 75.4 ⫺ 70.8, or
4.6, but also 0.23(20) ⫽ 4.6; multiply the slope times
the number of years between the two points.
Q6 Slope (Birth_Year ⫺ 1970)
Q7 70.8 ⫹ 4.6 ⫽ 75.4; add the vertical coordinate to the
answer to Q5.
Q8 70.8 + Slope (Birth_Year ⫺ 1970)
Q9 One good line has the equation Combined ⫽ 72.6 ⫹
0.17(Birth_Year ⫺ 1975).
Q10 The line in Q9 predicts a life expectancy of about 78.6
in 2010. Men’s life expectancy will be about 76 years,
and women’s will be about 83 years.
Q11 This form incorporates the coordinates of a point
(x1, y1) as well as the slope (b).
Q12 Answers will vary.
EXPLORE MORE
1. Answers will vary a good deal. Some will be close to
the answer to Q10.
2. Most models will show the male life expectancy
to have a smaller slope than the higher female life
expectancy, so they would be moving apart.