Jeffrey Nielsen
Standards:
EALR: 1.2.6: Understand and apply estimation strategies to
obtain reasonable measurements at an appropriate level of precision.
EALR: 1.4.5: Use bivariate data in tables and displays to
predict mathematical relationships.
EALR: 1.5.2:
Determine an equation or rule for a linear function represented in a pattern,
table, graph, or model.
NETS- basically EALRÕs for technology ideaÕs for lesson
plan.
1.)
Creativity and Innovation
c.) Use
models and simulations to explore complex systems and issues
d.)
Identify trends and forecast possibilities.
3.) Research and Information Fluency
d.)
Process data and report results
In
the Barbie Bungee exercise the students are going to be constructing best fit
lines for linear equations using the data they gather from their experiments
with the equation they create using TI-83 calculators they must interpret the
data so that they understand what each part of the equation means, and how they
will use it to predict the out come of how many rubber bands they will need to
drop the doll from a certain height. By performing these tasks the students
should satisfy the NETS listed above in that they have constructed a model of
the data they have gathered during their experiment when they get the height the
doll falls when they change the variable in the experiment in this case the
amount of rubber bands attached to the doll. After processing the data they have
gathered and observing the equation the students have obtained from their model
they will report and test the conclusion they come to. The main point of the
experiment in the first place was for the students to create their own best fit
line so the technology part of the experiment will be for the students to put
their data into a TI-83 calculator and observe if they got an accurate result
from their data. So the technology part of the lesson plan is going to start at
step 6.
Objectives: Students will be able to gather data and build a
scatter plot using the data that they found to construct a best fit line, and
use the information they have gathered to predict how many rubber bands it will
take for the doll to jump a safe distance.
Recall: For this lesson plan the students will need to know
how to solve simple arithmetic problems as well as solving for a variable by
balancing the equation. So to warm up the student should answer some basic
questions.
Procedure:
For this lesson the students will be presented with the task
to give Barbie the
greatest thrill while still ensuring that she is safe. In other words the
students will be using a Barbie doll, rubber bands, and a piece of paper, yard
stick or measuring tap. To gather data so that they can predict how many rubber
bands the students will need to have the doll jump from a certain height and
come as close as possible without hitting the floor.
With the data the students gather
they should be able to predict the maximum number of rubber bands that should
be used to give Barbie a safe jump from a height of 400cm. At the end of the
lesson plan the students will be given the opportunity to test their
predictions.
Step 1: Break the students into
small groups of about 3-4 students each. Distribute the Barbie Bungee activity
packet to each student, along with a Barbie doll, 15-20 rubber bands, and a
large piece of paper, some tape and a measuring tool to each group. Make sure
that all rubber bands are the same size and thickness. Differences in rubber
band elasticity will affect the results.
Step 2: Before the students begin, demonstrate how to create
the double-loop that attaches to BarbieÕs feet. Also show how to attach more
rubber bands to one another to form a rope or bungee.
Step 3: Allow the students to do the experiment and record
their results for the data table in question 2 of their packet. Suggest to the
students that they run the experiment several time for each length they have
and to use the average of those points to get a better representation of their
information at that point.
Step 4: Using the data that the students have collected it
is time for them to draw a best fit line for their graph what the students have
to do here is to draw a line that will represent their data in the best
possible way, how the students do this is by drawing a line that will cut their
data in half. To demonstrate this construct a graph on the board and use the
data [(2, 6), (4, 9), (6, 12.5), (8, 14), (10, 16.7), (12, 19)] after plotting
the data show the students how to make a best fit line.
Step 5: To find the equation for the best fit line the
students will need to come up with an equation y = m*x + b, since we know that
Barbie cannot jump safely with no rubber band, we know that the y-intercept or
b = 0 so to find the equation for y = m*x we just need to find the slope of the
best fit line to do this I would suggest to the students to find the slope
between all of the data points they had and then average the slope that they
had to come close to the slope of their best fit line.
Step 6: Now that the students have found what they believe
to be the best fit line for the equation it is time to introduce them to
another way they can check their data to see if the best fit line they made is
accurate to do this the students will need a TI-83 calculator.
Following are the direction the students will need to follow
to obtain a best fit line from their data.
Press STAT then ENTER.
You should see three columns, headed by the names L1, L2,
L3. If your columns do not display these lists, press STAT and then 5:SetUpEditor and ENTER, to load
the default lists L1, L2, etc. into the statistical
editor, and finally press STAT then ENTER to access the statistical editor. If the columns have numbers in them, you want to clear
them. To clear column 1, use the up cursor to highlight L1,
press CLEAR, then ENTER. Similarly clear L2 and L3.
To enter data into L1 (these numbers
should be the x-values of the points you wish to plot), place the cursor on the
first position in the column (see the left screen below), type in the first
number, press ENTER, which moves the
cursor to the next position in that column. Enter the next number, press ENTER, and continue the process until you have all the values
entered in column L1. Then use the right cursor to go to
column L2 and enter the y-values in this column. The x and y
values of each pair of data points should be side by side in these two
columns. Also, you must have the same number of entries in each column. Once the data points have been entered, you are
ready to plot them with a scatter plot.
Press 2nd then Y=
(this is STAT PLOT). Press ENTER.
On the screen that appears, you navigate with the arrow or cursor keys (left,
right, up and down) and make selections by pressing ENTER. Select PLOT 1 (if
it is not already highlighted), then highlight On and Press ENTER.
The next line, Type:, should have the
picture on the upper left highlighted (which represents a scatterplot, see the
right screen above), the Xlist should
be L1 and the Ylist should
be L2. The Mark can be the
leftmost (small box) or middle (+) selections. If these are already at
these settings, do nothing; otherwise, highlight what needs to be changed and
press ENTER for each change.
Before plotting the data points, do two things:
To graph the scatterplot, press GRAPH.
(2) Observe the scatterplot. Do the
points appear to lie almost on a line?
If so, then we want to find a line that is a good "fit." There are
many ways to do this.
If your points
are on graph paper, you can just use a clear plastic straightedge and
manipulate it until it seems to best fit the points, then draw the line.
Once the line is drawn, you can calculate the slope of the line, and find a
point on the line, and then give the equation of the line using the point-slope formula (slope = m, (a, b) on line, equation is y = m(x - a) + b).
You can also use the calculator to find the "least-squares"
regression line.
(3) Regression line (least squares fit).
Statisticians often use a standard method to find
a "fit" line, called a least squares fit (the "fit" line is called a regression
line). This method finds the line that minimizes the sum of the areas of
the squares that have their vertical edges drawn from a data point to the line.
The TI-83 can calculate the equation of this regression line with a few
keystrokes.
Press STAT,
and use the cursor to highlight CALC,
then press 8:LinReg(a+bx) then tell the
calculator to store the resulting regression equation in Y1 by
pressing VARS, right-arrow (to select the Y-VARS menu), ENTER (for FUNCTION), then [ENTER] (for Y1) and finally press ENTER
The screen displays the slope b and the y-intercept a of the regression line, and this equation is automatically entered as Y1
on the Y= screen. You can press GRAPH to see how well it fits the data (assuming STATPLOT 1 is
still displayed and you have a good window chosen). You can access the value of
the correlation, r, by pressing [VARS] [5] (to access the statistical variables) then scroll to the [EQ] menu (press the right arrow key twice) and finally press [7:r] and [ENTER]. If
you'd like your calculator to always display the correlation r (and also r2)
whenever it does a linear regression (like the right screen above), press [2nd][CATALOG], then [D] to
jump to the commands which begin with the letter D, scroll down a few lines to the command DiagnosticsOn and press [ENTER]
(twice). Now r will automatically be
displayed whenever you execute a linear regression command. The residuals of
the least squares regression line (the values found by subtracting the true
y-values from the regression calculated values) will be stored in a list named RESID (or perhaps LRESID), which can be accessed by pressing [2nd][LIST] and then scrolling down to the line with RESID on it and pressing [ENTER]. You can see a residual plot by setting stat-plot 2 as a
scatterplot, with XList = L1
and YList = RESID.
Step 7: Have the students finish the rest of the worksheet
in which the make their predictions and give their opinion what the data they
gathered means.
Step 8: After the students predictions are done supply them
with the rubber bands they think they will need and allow them to try their
experiment out.
Closure: This activity should get the students interested in
the work as well as showing them how the things they learn now can be used in
real life situations.
Assessment: To asses the students on what this lesson plan
has covered they are to complete the reflection part of the worksheet as well
as complete another worksheet in which they will have to build a scatter plot,
and draw a best fit line from the data that is given to them.
Materials: Paper, pencil, grid paper, Barbie doll, rubber
bands, yard sticks, calculator.
Post Lesson: Did the students understand the lesson plan?
Are there any areaÕs in which they require more work or further explanation?
Did the students respond well to this activity setting?
Giving credit: This lesson plan was found at the website: http://illuminations.nctm.org/LessonDetail.aspx?id=L646
The how to plot a best fit line instructions for a TI-83 calculator can be found at:
http://pages.central.edu/emp/LintonT/ti83/html/linreg/linreg.html
After hearing about it in class I couldnÕt pass up the opportunity to use it for one of my lesson plans it should get the students interested in the lesson plan and will get them talking between themselves about mathematical concepts.