SUBJECT: Algebra
GRADE: High School
TECHNOLOGY: Spreadsheet
BY: Sophie Billekens
Target Grade/Subject:
This is usually taught
around 10th grade. Students are
generally not too excited by Algebra II (or any school subject for that
matter), and dislike the idea of functions and graphs even more. This lesson hopes to help this aversion
by letting the students use the computer to compute function values and graph
them as well, to supplement their own tries at what the functions ought to look
like.
Objectives:
Students will find the
domain, range, and zeros of basic functions (y = x^2, y = x^3, y = x^1/2, y =
x^1/3, y = |x|, y = e^x, y = ln(x)) using a spreadsheet (i.e. Excel) to compute
values of functions given by the teacher on an interval (x = -10 to 10
incremented by half), having first sketched what they think the function ought
to look like based on trying a few values.
Students will find the
inverse of several basic functions by exchanging y and x and solving for x, and
see that some of the basic functions are each other's inverse.
Using the spreadsheet,
students will create their own multiple functions and see what they look like
on the interval.
Students will find examples
of where these functions come up in real life.
Materials:
A computer lab with
Microsoft Excel or ClarisWorks or a similar spreadsheet program. I will assume Excel is used, since that
is what I used, and I finally now know some of those tricks.
Description:
The students will be given a
list with the functions the teacher wants them to look at. The list of functions ought to include
at least the following items: y =
x^2, y = x^3, y = x^1/2, y = x^1/3, y = |x|, y = e^x, y = ln(x).
Using scratch paper, or
their notebooks, students will plug in some values between -10 and 10 and
sketch these. For e^x and ln(x),
it is probably a good idea to have them use the spreadsheet to calculate the values
of the functions at those points, since these are a bit tricky to calculate by
hand. If the students have not
been introduced to e^x or lnx, it is probably best to leave these two off the
list for now, since the aim of this lesson is to work with basic functions that
students have at least heard of.
The students will now use
the spreadsheet to create a table of numbers. They should put x-values in one column, ranging from -10 to
10, incremented by 0.5. The next
columns should contain the y-values for the equations listed above. Use the "paste function"
button on the toolbar to find such functions as "power,"
"sqrt" (square root) "exp" (exponent) and "ln"
(natural log) if you are not familiar with the commands on how to set up
equations with a spreadsheet. Do
not forget that you want to enter the cell number of the value that you want,
not the value itself! (Also, do
not forget that the student can paste the function down the column by pointing
the arrow at the right bottom corner of the cell, where it becomes a little
black cross, clicking down when it is this cross, and finally scrolling down
the desired number of cells.
Once the students have at
least two columns (x-values, and corresponding y-values for the equation y =
x^2, for example), they will use the "Chart Wizard" tool from the
tool bar to graph their findings.
They will check these findings with what they sketched themselves. By looking at the graph, students will
find the range and domain of the functions, as well as the zeros, and x- and
y-intercepts (if the teacher is feeling demanding/inspired). The students should find the y-values
for all the functions listed (using the spreadsheet) and the corresponding
graphs. Ask the students what is
happening when the value returned is "#NUM!" - they should realize
that the function does not exist at that x-value then.
Once again turning to scrap
paper or notebook, the students should find the inverse of each function by
swapping x and y in the equation and then solving for x. For example:
Y
= x^2 becomes
X
= y^2
(x)^1/2
= (y^2)^1/2
x^1/2
= y
y
= x^1/2
If the students do not
notice themselves, the teacher should be sure to tell them that y = x^2 is the
inverse of y = x^1/2, y = x^3 is the inverse of y = x^1/3, and y = e^x is the
inverse of y = lnx. If the graphs
can be printed out, ask the students to do so, and have them compare the graphs
of the inverses. Point out, if
they do not see it for themselves, that the graphs reflect over the line y =x.
As an introduction to
multiple functions, students should now create their own equations and see how
these behave in the interval from -10 to 10.
Finally, students will look
at their graphs and try to think of where these curves occur in the real
world. For example, the graph of y
= x^4 (or any high order even power) looks quite like a skateboarding
ramp/pipeline, and the graph of y = x^3 looks quite like what someone's neck-
and shoulderline form. This should
be done as a discussion at the end of the lesson.
Evaluations:
Students will show or hand
in a printout of their chart and the graphs that accompany the data in it,
including the multiple functions created by them (see the attached documents
for examples). The teacher should
look during the lesson if they are plugging in values, sketching the graphs,
and solving for inverse functions, since this is done either on scrap paper or
in their notebooks. Students
should participate in the discussion of where these shapes can be found in the
real world.
Note: This lesson plan can be easily adopted
to use with trigonometric functions, too!
Excel has those equations available also, and even converts to radians
for you!