Exploring Data and Linear Equations
Objectives
In this activity you will:
·
Develop and use correct
mathematical vocabulary.
·
Enter the equation of a line in
slope-intercept form.
o
y = mx + b
·
Calculate the slope, m, and y-intercept, b, of a line.
·
Observe and investigate the
effects of m and b on the graph of a line.
o
y = mx
o
y = x + b
·
Enter data into lists.
·
Be able to name lists.
·
Create a scatter plot of the
data.
·
Find the Line of Best Fit by
doing a Linear Regression on data.
·
Look at the graphs and write
your conclusion as to the effect m
has on the lines.
·
Use the Transformation Graphing Application to enhance your skills with
slopes and y-intercepts.
Materials need:
·
TI-84+SE
·
Transformation Graphing Application
Algebra 6.0: Students will graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = -4). They are also able to sketch the region defined by a linear inequality (e.g., sketch the graph defined by 2x + 6y < -4).
Overview:
In this
lesson, students will explore linear equations, input data into lists, create
scatter plots, use an application to further explore linear equations, and use
correct mathematical vocabulary in explanations and discussions.
Vocabulary:
Cartesian coordinate system Axes
Ordered Pair Plot
Points
X-Coordinate Y-Coordinate
Domain Range
Independent Variable Dependent
Variable
Abscissa Ordinate
Quadrants Graphing
Slope Intercept
Linear Equation Non-Linear
Equation
Standard Form Slope-Intercept
Form
X-Intercept Y-Intercept
X-intercept in Ordered Pair Form Y-Intercept in
Ordered Pair Form
Prediction Estimation
We will begin by
going over the graphs of liner equations.
In order to graph equations on the TI-84+SE, the equations must be in
slope-intercept form, y = mx + b.
|
Begin in Function (Func) mode and turn off any
plots or equations previously turned on.
To turn off a
plot, press o, use cursor keys (|~}†)
to move to the highlighted plot, and press Í.
To clear an
equation, position the cursor anywhere in the right side of the equation, and
then press ‘
|
All Stat Plots are off and all
equations are cleared.
|
|
Enter the
equations into the o screen. Look at the graph
of Y1 = -7X; Y2
= -3X; and Y3 = -1X.
Use the r
to check values.
|
The equations are entered Y1 = -7x
Separately.
|
|
What effect does
m have on the lines? What would the lines look like if the
slopes were positive?
|
Y2
= -3x Y3
= -1x
|
|
Next, look at
the graphs of
Y1 = X - 5; Y2 = X - 1; Y3
= X + 5.
|
Y1 = x – 5
|
|
What effect does
b have on the lines?
|
Y2
= x - 1 Y3 = x + 5
|
We will now look
at data that you will enter into lists.
|
Birth Year
|
Female
|
Male
|
Combined
|
|
1940
|
65.2
|
60.8
|
62.9
|
|
1950
|
71.1
|
65.6
|
68.2
|
|
1960
|
73.1
|
66.6
|
69.7
|
|
1970
|
74.7
|
67.1
|
70.8
|
|
1975
|
76.6
|
68.6
|
72.6
|
|
1980
|
77.5
|
70.0
|
73.7
|
|
1985
|
78.2
|
71.2
|
74.7
|
|
1990
|
78.8
|
71.8
|
75.4
|
|
1995
|
78.9
|
72.5
|
75.8
|
|
1998
|
79.4
|
73.9
|
76.7
|
(2000 World Almanac, p. 891)
Questions:
- Which set of data will be your independent
variable, or x-values?
2.Which set of data will be your
dependent variable, or y-values?
3.Find the Minimum and Maximum
values of the Domain and Range for each set of data.
|
Clear all lists
by going to … 4:ClrList d, e, f, g, h, i then press Í.
When entering
the data into lists, you can use the generic L1, L2, L3, and L4.
|
Standard
Headings for Lists
|
|
Or you can enter
the name as a title of the list. Go to
the blank lists and enter the title of the list using only five letters. The lists have been changed to BYEAR for
Birth Year in L1, WOMEN
for Female in L2, MEN
for Male in L3, and
COMB for Combined in L4.
|
Changed
Headings for Lists
|
|
Press y, 1 to select 1:Plot1 and press [. Turn the plot on and
enter the settings as shown in the diagram.
For Xlist and Ylist, be sure to use the correct
lists in which your data were entered.
Press q 9 to select 9:ZoomStat and display the plot using
the data in the lists.
|
BYEAR vs WOMEN StatPlot using Zoom 9.
Remember to
choose different marks for each Stat Plot.
|
|
Create Plot1 for BYEAR vs WOMEN and enter as
in the diagram. Do as above. Create Plot2
for BYEAR vs MEN.
|
BYEAR vs
MEN StatPlot
using Zoom 9.
|
|
Create Plot3 for BYEAR vs COMB and enter as
in the diagram. Do as before.
|
BYEAR vs
COMB StatPlot
using Zoom 9.
|
|
Display the three
Plots by turning on all three ,’s. What can you say about
the three plots? Do you see any trends? Is there a consistent trend in the
relationship between the birth year and women, men, and the combined data?
|
Turn on all Stat Plots and use Zoom 9.
Use ▫ for Women, ⁺ for Men, and
. for Comb.
|
|
To find the
Linear Regression Equation, go to … CALC 4:LinReg(ax+b) . Then Í…include
LBYEAR, LWOMEN,
Y1 Í for the linear regression equation
for BYEAR vs WOMEN. In
the o screen, the Linear Regression Equation is entered as Y1.
|
By putting Y1 in the LinReg, the
equation will be entered into Y1
|
|
When you go to s,
press q 9:ZoomStat.
Why would you
use ZoomStat? Could you have used another method to plot
BYEAR vs WOMEN?
|
|
By using an application called Transformation Graphing, we will explore the slope and y-intercept of a linear equation in more
depth.
|
Press Œ
key. Select Transfrm. (This may have a
number, letter, or nothing in front of it depending on the number of
applications you have loaded).
Press any key
(except y or ƒ) to install the application.
|
This is a built-in App for the TI-84+SE but can be
loaded from the TI Website.
|
|
Begin in Function (Func) mode and turn off any
plots or equations previously turned on.
To turn off a plot, press o, use the cursor keys (|}~†)
to move to the highlighted plat, and press Í. To clear an equation, position the cursor
anywhere in the right side of the equation, and then press ‘.
|
Plots turned off, equations cleared. Notice that the left-hand markings are
different and note that this App will only graph one equation at a time.
|
|
Move the cursor
to Y1 = , and enter A „ Ã B
|
Initial starting set up. The only active variables in this App are
A, B, C, and D.
|
|
Press s
to display the plot with the app enabled.
The line Y=AX+B is
displayed with the most recent values of A and B. These values will probably have no relationship
to your plot. Enter your estimated
values for the slope and y-intercept
for A and B. To enter these values,
use the up and down cursor keys (} and †) to highlight either A= or
B=. Adjust the values for A and B till you have
a reasonable visual model.
|
Initial set up
After
trial and error or using a strategy
to find the equation of the line.
|
Questions:
- As the value of A increases, what happens to the graph?
2.As the value of A decreases, what happens to the graph?
3.Would the value of A be negative? Why or why not?
4.As the value of B increases, what happens to the graph?
5.As the value of B decreases, what happens to the graph?
6.Could the value of B be negative? Why or why not?
7.Looking at y = mx + b, why do you think it is called the
slope-intercept form of a line?
8. What other strategy could you have used to find the equation of
the line of best fit?
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