BACK
From arithmetic operations with real numbers to
composition of functions using dynamic number lines.
Workshop
John Olive - The University of Georgia/Mathematics and Science
Education; Athens, GA; USA
Description
This workshop will engage participants in the construction of
dynamic number lines and "dynagraphs" (parallel number lines for
representing functions) using The Geometer's Sketchpad 4.06.
Participants will explore relations between pairs of real numbers
and their sums and products qualitatively (without computation),
and more complex arithmetic combinations of the two numbers that
lead to investigations of functions of two variables.
The Dynagraph representation (a pair of dynamic number lines, one
for the input variable and the second for the output) will be
constructed and used to investigate different categories of
functions (linear, polynomial, step, trigonometric, exponential
and logarithmic), eventually leading to modeling compositions of
two or more functions using a series of linked dynagraphs. A
geometric transformation from the pair of parallel numberlines to
rectangular coordinates will be constructed and used to relate
the dynamic motion of the dynagraph to the coordinate graph of a
function.
Plan of Activities
Part I: Arithmetic operations on a dynamic number line
1. Introduction to GSP coordinate system.
2. Construction of a number line with two free points.
3. Construction of the product of the two free points by
dilation.
4. Construction of the sum of the two free points by
translation.
5. Exploration of “ m ystery machines.gsp ” to locate origin and
scale point on an unmarked number line.
6. Exploration of all four arithmetic operations using the
sketch “ A rithmetic Machines.gsp ”
7. Construction of a plotted point on the number line using the
GSP calculator to define an expression relating the measures
of the two free points.
8. Exploration of “ m ystery relations ” using the “ n umberline
tool.gsp” sketch.
Part II: Representing functions on parallel number lines
(Dynagraphs)
1. Introduction to dynagraphs using the “ d ynagraphs.gsp ”
sketch (guess-my-function).
2. Construction of a dynagraph using two parallel number lines.
3. Construction of a “ dependent ” output point using a “ f ree ”
input point and the GSP “ N ew Function ” capability.
4. Exploration of different categories of functions using
dynagraphs.
5. Exploration of domain and range of particular categories of
functions using the trace function with dynagraphs (see
sketch Domain Range.gsp).
6. Construction of the composition of two functions by
splitting the input point from one dynagraph number line and
merging it with the output point of the other dynagraph’s
output point. Thus, the output of one function becomes the
input for the second function.
7. Exploration of the composition of two or more functions
(from different classes of functions) using dynagraphs (see
sketch Composition.gsp).
Part III: From dynagraphs to rectangular coordinate systems
1. Construction of an oblique coordinate system through
rotation of the output number line.
2. Coordination of the motions of input and output points using
lines through each point that are parallel to the other
point’s number line (coordinate axis).
3. Construction of the “ coordinate point ” (intersection of
the two lines parallel to the oblique number lines).
4. Construction of the graph of the function as the locus of
the coordinate point.
5. Exploration of the graphs of different functions using
oblique axes.
6. Rotation of output axis to go from oblique to rectangular
coordinates.
7. Dynamic comparison of both dynagraph and rectangular
coordinate representations of the same function (see sketch
DynaToCart.gsp).
8. Construction and exploration of arithmetic operations on
functions using several linked coordinate systems (see
sketch function_operations.gsp).
Following: from the book Transforming Mathematics with the Geometer´s Sketchpad:
Activity 6.2: Constructing a product of two numbers on a number line using
dilation
We can use dynamic dilation to locate the product of any two numbers on a single number line.
The following steps will create a number line in Sketchpad:
• Open a new sketch.
• Choose Define Coordinate System from the Graph menu.
• Click on the vertical axis to select it and choose Hide Axis from the Display menu.
• Choose Hide Grid from the Graph menu.
• Label the origin and unit points as 0 and 1 on your number line.
• Place two free points on your number line and label them A and B.
Your sketch should now look like figure 6.3:
A
-10
B
0
-5
1
5
10
Figure 6.3: A number line in Sketchpad
•
•
•
Select points A and B and choose Abscissa (x) from the Measure menu.
Using the Label tool re-label the two measures (XA and XB) A and B respectively.
Change the precision of these two measures to tenths by selecting a measure and
choosing Properties from the Edit menu and the Values tab from the properties panel.
Move your two points around on your number line and check that their measures change
accordingly. The following steps will create a new point on the number line that will
represent the product of your two measures, A and B:
• Select the points 0, 1 and B IN THAT ORDER and choose Mark Ratio from the
Transform menu.
• Double click on point 0 to mark it as the center of dilation.
• Select point A and choose Dilate from the Transform menu.
[At this point, a new point should appear on your number line. If you do not see a new
point then move either A or B closer to zero.]
• Label this new point C and measure its abscissa (x-coordinate).
• Change the label of this measure from XC to C.
Your sketch should now look something like figure 6.4.
A = -2.0
B = 4.0
C = -8.08
A
-10
C
-5
B
0
1
5
Figure 6.4: Product of A and B using dilation on a number line
10
Move points A and B around on your number line and test to see if the measure of point C gives
the product of the measures of points A and B.
In figure 6.4, my sketch shows the product of –2.0 and 4.0 as –8.08. Why might this be so? Does
it make mathematical sense to have the product’s precision set to hundredths when the precision
of the factors is set to tenths? What would the measure of C in figure 6.4 indicate if its precision
was set to tenths? What could you do with these measures if you wanted your students to work
with integer arithmetic rather than with rational numbers? Modify your sketch to show integer
calculations only.
Activity 6.3: Representing arithmetic operations dynamically on a number
line
In the above example, the product of two numbers was constructed using dynamic dilation.
Sums and differences can be constructed using translations on the number line rather than
dilations. Each number is represented as a horizontal vector from the origin (zero), and the sum
(or difference) is constructed by translating the end-point representing one number by the vector
(or its reverse) indicated by the zero point and the point representing the other number. For
example, in figure 6.4 above, the sum of A and B could be constructed by marking the points 0
and B as a vector using the Transform menu and then translating point A by this marked vector.
Construct the sum of A and B on your number line using the “marked vector” method described
above. Call the new point D and investigate relations between the product point, C and the sumpoint, D.
• When is the sum greater than the product (i.e. when is D to the right of C)?
• When is C to the right of D?
• When is D between A and B?
Activity 6.4: Investigating arithmetic relations dynamically
In the previous two activities you constructed the product and sum of your two independent
points on the number line using geometric transformations (dilation and translation). In this
section we shall use the GSP calculator to compute arithmetic relations based on the positions of
the two independent points. Start with a new number line as in figure 6.3 above and measure the
abscissa (x-coordinate) of each point. Re-label these measures A and B as in figure 6.4. The
following steps create a new point, C, on the number line based on a defined mathematical
relation using the measures of A and B:
• From the Measure menu choose Calculate. The GSP pop-up calculator should appear
(you may need to move the calculator in order to see the measures of A and B).
• Create an arithmetic expression in the calculator by typing values, arithmetic operators
(+, -, *, ÷) and clicking on the measures for A and B. For example, to create the
expression for 2A-B you would click on the 2, the * (for multiplication), the measure of
A (the label A should appear in the calculator window), the – key and finally the measure
B. You would then see the expression 2*A-B in the calculator window.
• Click on the OK button on the calculator. The measure of 2A-B should appear in your
sketch.
• With the label tool, double click on this new measure and change its label to C.
•
•
•
•
With the select arrow, select just the origin point of the number line and measure its
ordinate (y-coordinate) from the Measure menu. The measure y0=0 should appear in
your sketch.
Deselect everything by clicking in open space and then select IN THIS ORDER the
measure C and the y0=0.
With these two measures selected (and nothing else) choose Plot as (x, y) from the Graph
menu. A new point should appear on your number line at the position corresponding to
the value of measure C. (Note: you may have to move point A or B to create a value for
C that is within the visible portion of your number line.)
With the label tool, label this new point C.
Investigate the behavior of your plotted point C as you vary your independent points A and B.
Does it behave as you would predict? What positional relations can you create with these three
points? Will point C ever be between points A and B? To the left of A and B? To the right of A
and B?
You can also use the Numberline Tool.gsp sketch for this activity. This tool provides colored
tags attached to your independent points and a free tagged point labeled C that you can merge
with your plotted point C after following all of the above steps (simply select both the plotted
point C on your number line and the free tagged point C and choose Merge Points from the Edit
menu). Figure 6.5 illustrates the situation with both points C selected just before merging.
Figure 6.5: About to merge the free tagged point C with the plotted point C on the number line
using the Numberline Tool.gsp sketch.
Figure 6.6 shows the number line after the points have been merged.
Figure 6.6: Number line with tagged points.
Creating Dynagraphs Algebraically in GSP 4
The secret to constructing Dynagraphs algebraically using GSP 4 is to create two
horizontal number lines (an input axis and an output axis) and to use the function calculator to
calculate the value of your function for some variable point on the input axis. You then use this
calculated value to plot a point on the output axis. As you move your variable point on the input
axis the plotted point on the output axis moves appropriately. The major concern in using this
method is to make sure the appropriate coordinate system is marked when you are calculating or
plotting coordinates. Use the following steps as a guide:
1. Open a new sketch and choose Define Coordinate System from the Graph menu.
2. Place a point on the y-axis about an inch below the x-axis.
3. Hide the y-axis and hide the grid.
4. Select your point below the visible x-axis and choose Define Origin from the Graph
menu. A warning dialog will ask you if you really want to define a new coordinate
system. Click on the Yes button.
5. Hide the grid (choose Hide Grid from the Graph menu) and hide the new y-axis.
6. Label the top axis input and the bottom axis output (click on each axis with the label tool
and edit each label).
At this point you should have two horizontal axes as in figure 7.8 below.
input
-10
-5
5
10
5
10
output
-10
-5
Figure 7.8: Two horizontal axes
7. Place a free point on the input axis and label it “x”.
8. VERY IMPORTANT STEP: Select the origin point of your input axis and choose Mark
Coordinate System from the Graph menu (this step makes the input axis the active
coordinate system for coordinate measurements).
9. Select your free point x on the input axis and measure its x-abscissa (from the Measure
Menu).
10. Select New Function from the Graph menu and create your own function (e.g. f(x)=ex)
11. Select Calculate from the Measure menu. Click on your f(x) then click on your inputx
value. Click on the OK button. The value of f(inputx) should be displayed.
At this point your sketch should look something like figure 7.9 below:
f ( x ) = ex
input
x
= 3.63
input
-10
f (input
x
-5
x
)=
5
10
5
10
37.85
output
-10
-5
Figure 7.9: Input and function values
At this point there are two different ways to plot the output point on the output axis. You can
select the origin of the output axis and mark it as the active coordinate system (see step 8 above)
and then plot the point (f(inputx), 0) on this output axis. You will need a value 0 calculated or
measured in order to do this. I prefer the following method, however, as it doesn’t require
changing coordinate systems or creating a zero value. The following method uses the function
value as a scale factor for dilation and then dilates the unit point of the output axis about the
origin of the output axis using this scale factor:
12. Select the function value that you calculated in step 11 and then choose Mark Scale
Factor from the Transform menu.
13. Double click on the origin point of your output axis. This selects it as a center of dilation.
14. Select the unit point on your output axis and then choose Dilate from the Transform
menu. A dialog box should appear that indicates that you are to dilate by the scale factor
of f(inputx). Click on the Dilate button. A new point should appear on your output axis.
If you cannot see a point then move your input point, x, close to one or zero (depending
on the function you used) until the point appears on your output axis.
15. Label this new point f(x). Construct a segment between points x and f(x).
At this point your sketch should look something like figure 7.10 below. This completes your
construction of a Dynagraph. You can make a custom tool of this construction for creating more
Dynagraphs. The givens for your dynagraph tool should be the origin point of a new input axis
and a new function. (For tips on creating custom tools see your GSP manual.)
f ( x ) = ex
input
x
= 2.01
input
-10
f (input
x
-5
x
)=
5
10
7.47
output
-10
-5
5
f(x)
10
Figure 7.10: Completed Dynagraph in GSP 4
Exloring Range and Domain with Dynagraphs
Open the file Dynagraphs.gsp inside the Algebra folder. This sketch is from the CD
accompanying Exploring Algebra with the Geometer’s Sketchpad (2003). You will find two
pages of mystery dynagraphs. Explore each of them in turn and try and deduce the functions for
each dynagraph. There are buttons to provide more information (such as the scale point – i.e. the
unit point for each axis, or the actual numbers on the axes). There is also a button to show the
functions but resist using this until you have thoroughly explored all of the dynagraphs on that
page.
Comparing the range and domain of these different functions may help in your
exploration. The domain of a function is the set of input values for which the function produces
an output; the range of a function is the set of output values that the function can produce. In
order to visually record the range of one of the dynagraphs in the Dynagraphs.gsp sketch, first
select the triangular region attached to the output point and choose Trace Locus from the
Display menu. Move the input point slowly to see the trace of the output triangle along the
bottom of the output axis. Figure 7.11 shows the trace for functions h and i from page one of the
sketch.
Figure 7.11: Two Dynagraphs with Ranges Traced
Note that the range for function h appears to be discrete rather than continuous, taking
only certain values along the number line, whereas the range for function i appears to be only
positive real numbers, but does appear to be continuous. What possibilities do these traces of the
ranges of these functions suggest for the type of function in each case?
Explore the range and domain of the functions on page 2 of this sketch. Are there
functions that have a limited domain (not all real numbers)? Are there functions that have a
bounded range? Are there functions that appear to have “holes” in their range (a value or values
that the function can not achieve)?
Activity 7.3: Composition of Functions using Dynagraphs
When we form the composition of two functions, such as g(f(x)), the output of the inner
function (f(x)) becomes the input for the outer function. We can make this connection explicit
using two dynagraphs. For instance, with the two dynagraphs in figure 7.11, in order to form the
composed function i(h(x) we would want the output of h (the point h(c)) to be the input point for
function i. The input point for function i is D. We need to make point D become point h(C). We
can do this by splitting point D from its axis and then merging it with point h(C). The following
steps achieve this process:
1. Deselect everything by clicking on a blank part of the sketch.
2. Select point D and then choose Split point from axis under the Edit menu. Point D (with
its attached pentagon) will move away from the axis.
3. Leave point D selected and also select point h(C).
4. Choose Merge Points from the Edit menu.
Your two dynagraphs should now look like figure 7.12
input
h
C
h(C)
output
input
i
i(D)
output
Figure 7.12: A Composed Dynagraph
Now explore the range of this composed function. How is it related to the ranges of the two
original functions? Describe the set of numbers that comprise this composed range.
Note: It is very important to make sure that the unit scales on both dynagraphs are the
same and that the origins are lined up vertically, otherwise the output values of h(x) and the input
values of i(x) will not be the same!
The process you used to compose i(h(x)) can be used to compose more than two
functions. Working from the input point of the outer function, split it from its axis and merge it
with the output point of the next function, then split the input point of this next function from its
axis and merge that point with the output point of the next inner function, and so on. Experiment
by composing several of the functions on page 2 of Dynagraphs.gsp. Investigate the ranges of
the composed functions and compare them to the ranges of the individual component functions.
Assignment 7.3
Create three functions of your own, each of which belongs to a different family (e.g, step,
quadratic, and trigonometric) and investigate the composition of your three functions. Write-up
your investigations, highlighting any interesting or surprising characteristics you discovered for
your particular composition.
Asymptotic Behavior with Dynagraphs
Recall that in the Cartesian (2-D) representation of the graph of a function, some
functions have what are called asymptotes. These are imaginary lines that the function tends
towards as the input approaches a discontinuity that produces an undefined output (e.g. the
function 1/x as x approaches zero). The following problems (designed by Dr. Tanya Cofer for
students in a Mathematics Education course at the University of Georgia) ask you to explore how
asymptotic behavior appears within a dynagraph:
1. How would you characterize vertical asymptotes using the dynagraph representation?
(see function w on page. 2 of the Dynagraph.gsp sketch). Come up with other rational
functions that behave differently about the vertical asymptote.
2. How would you characterize horizontal asymptotes using this representation? Start by
analyzing the dynagraph of: y=(2x–3)/(x+1) (look at its “behavior at infinity”) then come
up with other examples of functions with horizontal asymptotes.
3. How would you characterize slant asymptotes using this representation? Start by
analyzing the dynagraph of: y=(x2+3x-5)/(x+1) (look at its “behavior at infinity”) then
come up with other examples of functions with slant asymptotes.
4. How would you characterize the asymptotes of: y=(x2+3x-5)/(x2+1)? What do you notice
about the range of this function?
From Dynagraphs to Cartesian Coordinates
The transition from the parallel number lines representation of functions to the more
traditional Cartesian Coordinate representation can be made dynamically using your GSP
dynagraph. Even though both GSP 3 and GSP 4 have built-in coordinate systems, it can be
enlightening and interesting to create your own two-dimensional coordinate system from your
one-dimensional number line or dynagraph. What you will be doing, in fact, is transforming a
mathematical mapping from R1 to R1 (the real numbers) into a mapping from R1 to R2 (2space).
A pair of parallel number lines can be transformed into non-parallel, intersecting number
lines to form coordinate axes in 2-space. The axes can be oblique as well as rectangular, and do
not have to share a common origin. One simple way to construct such a flexible coordinate
system is to construct a rotated image of the second (output) number line about its origin. The
following steps are provided as a guide:
1. Move the output axis of your dynagraph above the input axis.
2. Mark the origin of the output axis as a center of rotation.
3. Place a free point somewhere above the unit point of the output axis and label this
point tilt.
4. Select (in this order) the unit point and origin point of your output axis AND the point
tilt and choose Mark Angle from the Transform menu.
5. Select the output axis and the output point (f(x)) and choose Rotate from the
Transform menu. The output axis and point should rotate about its origin by the
marked angle. Your dynagraph will look something like figure 7.13.
6. Label the rotated f(x) point and hide the original output axis and point.
f(x)
tilt
output
0
-5
0
-5
input
f(x)
0' 1'
0 1
inputx = 4.80
x
5
10
15
5
10
15
f (x) = 2⋅ (x-3 )2
(
)
f inputx = 6.46
Figure 7.13: A Rotated Dynagraph Output Axis
By moving the two origin points together you will have what is called an Oblique
Cartesian Coordinate System, named after the French philosopher and mathematician, René
Descartes (1596-1650) who is considered one of the fathers of analytic geometry. When the two
axes are perpendicular to one another, the system is called a Rectangular Cartesian Coordinate
System. Research the web to find out about René Descartes and his coordinate system. Did
Descartes ever use oblique coordinates? If so, for what purpose?
A point is plotted in these systems by constructing lines through the input and output
points that are parallel to the other axis. That is, a line through x parallel to the rotated output
axis and a line through f(x) parallel to the horizontal input axis. Where these two lines intersect
will be the point in the coordinate plane corresponding to the ordered pair (x, f(x)) or (x, y) using
the conventional label of y for the output axis. It is important to realize that the intersection of
these two lines is what coordinates the motion of the input and output points on their respective
axes. Thus, the coordinate graph of a function is the locus of this coordinate point; i.e. the path
of the coordinate point as the input point x varies.
Hide the horizontal output axis and move the two origin points together. [NOTE: You
can use a Movement button to move 0’ to 0 automatically and accurately. Select 0’and 0 IN
THAT ORDER and choose Edit/Action Buttons/Movement from the menu bar. Click OK to
create your movement button. Click on the movement button to move 0’ to 0.] Construct the
parallel lines on your oblique coordinate system and the coordinate point (x,y) at their
intersection. Trace this point as you move your input point x. You can also construct the locus
of this coordinate point by selecting point x and point (x,y) and then choosing Locus under the
Construct menu. You should have something like the picture in figure 7.14.
Move 0' -> 0
(x,y)
f(x)
tilt
input
0
-5
x
0 1
0'
inputx = 4.80
5
f (x) =
10
2⋅ (x-3 )2
(
15
)
f inputx = 6.46
Figure 7.14: Oblique Coordinate Graph of a Quadratic Function
Investigate the properties of this oblique graph of a quadratic function.
• Where does it meet or intersect the x-axis?
• Where does the graph intersect the y-axis?
• Do these points change when you change the tilt angle of the y-axis?
• Does this graph pass the “vertical line test” for a function?
• What would be an appropriate “test” for a function graphed on oblique axes?
Edit your function by double clicking on the f(x) expression and test your conjectures with
different kinds of functions.
Sketchpad has a built-in rectangular coordinate graphing system that automatically
constructs the locus of the ordered pair (x, f(x)) when a function is plotted using the Plot
Function option under the Graph menu. Select the function expression in your sketch and plot its
graph using this feature of Sketchpad. You should have a figure that looks something like figure
7.15.
Move 0' -> 0
(x,y)
f(x)
tilt
input
0
-5
x
0 1
0'
inputx = 4.76
5
f (x) =
10
2⋅(x-3 )2
(
15
)
f inputx = 6.21
Figure 7.15: Oblique and Rectangular Graphs of a Quadratic Function
Rotate the tilt point until both graphs coincide. Move your input point x and observe how
the plotted point (x,y) moves along the plotted function graph.