MAE 5868 E-FOLIO
FLORIDA STATE UNIVERSITY
SPRING 2006
MARSHA COPELAND
This e-folio is a compilation of solutions to various
problems. These solutions were discovered via
three mediums: Geo-Sketchpad, Excel and
Graphing Calculator. Discussions of the relevance
of these problems and the tools used to solve them
to educational standards in math and technology
are also included.
:
Click on the link to access information:
:

The International Society for Technology in Education
(ISTE) is a nonprofit organization whose mission is to
“provide leadership and service to improve teaching and
learning by advancing the effective use of technology in
education.” Great diversity is utilized to accomplish this
mission. The organization is involved in professional
development workshops, presenting innovative books and
programs, research, and conducting forums and symposia
to promote the use of effective technology in education.
:
 ISTE
provides its members (>85,000 teachers,
worldwide) with information, networking
opportunities, and guidance in the use of
technology in their programs. Forums, such as
NECC, offers the opportunity to learn, exchange
and survey the field of educational technology.
:
 ISTE
supported Special Interest Groups (SIGs)
allow members to “share best practices worldwide,
sponsor workshops, review articles and programs
and advance knowledge in their particular areas of
interest.”
:
 ISTE
is the home of NETS ( National Educational
Technology Standards) which have been adopted
by many districts, states and colleges. These can be
accessed at
http://cnets.iste.org/teachers/t_book.html
BACK TO CONTENTS
:
National Educational Technology Standard 3:
Students use technology tools to enhance learning,
increase productivity and promote creativity.
Students use productivity tools to collaborate in
constructing technology-enhanced models, prepare
publications and produce other creative works.
LINK TO ACTIVITIES LINK TO REFLECTIONS
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ACTIVITY 1
One of the Pythagorean Theorem typically uses squares constructed on each side of a
right triangle. The area of the square constructed on the hypotenuse is equal to the sum
of the squares constructed on each of the legs of the triangle. (50 Points).
Instead of using squares, can we use equilateral triangles?
How about regular hexagons?
Would these or any other figures give the same results?
Plan: I will construct a right triangle, then using the rotate and dilate tools, will attempt
to construct a tessellation of equilateral triangles along each side to form a larger
equilateral triangle on each side.
Solution: See diagrams with measurements below. These are constructed in Geo
Sketchpad and show that the areas of the shapes along each side of the right triangle’s
legs sum to the area of the shape that shares a side with the hypotenuse.
m A'''A = 8.1 cm
m BA''' = 4.9 cm
m AB = 6.5 cm
NINE EQUALATERAL TRIANGLES
Area BA''''A''' = 10.2 cm 2
Area BB'A = 18.2 cm 2
Area AA'''A' = 28.4 cm 2
m ABA''' = 90°
B'
A''''
B
SIXTEEN
EQUALATERAL
TRIANGLES
A'''
A
TWENTY-FIVE EQUALATERAL
TRIANGLES
PYTHAGOREAN THEOREM STATES THAT
THE SUM OF THE SQUARES OF THE
LEGS EQUALS THE SQUARE OF THE
HYPOTENUSE. LOOKING AT THE
DIAGRAM, THIS HOLDS TRUE FOR
EQUALATERAL TRIANGLES, NOT ONLY
IN NUMBER BUT IN AREAS.
A'
m AB = 3.5 cm
m BC = 2.1 cm
m CA = 4.0 cm
Area p1 = 31.0 cm 2
+
Area p2 = 11.1 cm 2
=
Area p3 = 42.1 cm 2
(Area p1 ) + (Area p2 ) = 42.10 cm 2
Area = 11.1
Area = 31.0
B
m ABC = 90°
p2
p1
C
A
Area =
42.1
p3
Area CBC'B ' =31.6 cm 2
C
This also holds true for parallelograms and pentagons.
m
l
Area CA'C'A =6.3 cm2
A
m CAB =90°
j
B
Area ABA'B ' =25.3 cm 2
Area p5 = 14.3 cm 2
F
Area p6 = 33.0 cm 2
m FDE = 90°
E
D
Area p4 = 18.7 cm 2
Analysis: Teaching view: Great activity to reinforce that there are many
ways to solve a problem. Students can see that the Pythagorean theorem
works for any shape
Learning view: Students will have fun manipulating the shapes and sizes
of the triangles, and verifying that the sum of the squares of the legs of a
right triangle equal the square of the hypotenuse. As they manipulate the
figures, they will learn the theorem.
Reflection: Interesting problem…easy to construct, easy to see the
outcome.
ACTIVITY 2
A prime number is a positive integer, which is greater than one and evenly divisible only by itself and one. For example,
2,3,5,7,11,13,17 are all prime numbers. For centuries, prime numbers have fascinated both professional and amateur
mathematicians.
In 1742 C. Goldbach wrote a letter to the eminent Euler, stating therein the celebrated "Goldbach conjecture," essentially that
every even integer exceeding 2 is a sum of two primes.
For example:
2+2=4
3+5=8
3+7 =10
5+5 =10
Euler replied that Goldbach's conjecture did indeed seem plausible. However, neither Goldbach nor Euler was able to prove
it.
According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like
'Goldbach's Theorem,' which have never been proved and which any fool could have guessed."
Faber and Faber offered a $1,000,000 prize to anyone who proved Goldbach's conjecture between March 20, 2000 and March
20, 2002, but the prize went unclaimed and the conjecture remains open.
Finding prime numbers with EXCEL
1-Your assignment is to find ALL of the prime numbers between 1 & 50 using the Excel functions. You can use any function or
create your own formula. We want to see your process. If students were to follow your completed steps then they should be
able to generalize to find the prime numbers between 1 & n, where n is a positive Integer. (15 points)
2-As a follow up after you have found the primes, try to confirm or disconfirm the Goldbach conjecture for the numbers 130. Basically find the primes (p1, q1) such that p1 + q1 = an even numbers between 1 & 30 (find for ALL even numbers).
Sometimes you will find more than one pairs of primes whose sum makes up an even number (ex: 5+5=10; 3+7=10), find all
possible alternatives. (30 points)
PLAN: I will use a form of sieve to establish the primes, 2-30. then use excel to add each prime in turn
to all of the primes established with the sieve. Once this is done, I will find sums of primes that result
in all even numbers 4-30, making sure to notice all combinations and pairs.
SOLUTION:
First of all, list all integers from 2-50 in the first column. The number 1 is not listed because the smallest prime is 2. Then
divide each integer in the list by 2. Those numbers not evenly divisible by 2 (except 2) are incorporated in a new list. Now
each element in the new list is divided by 3, and all numbers except 3 not evenly divisible by 3 are used to make another new
list. One need not divide by 4, since any number divisible by 4 is also divisible by 2 and would have been eliminated as a
prime then. Now divide each element in the newest list by 5, follow this procedure for 7 also. Six, eight and nine need not
be “tested” for the same reason there was no need to test 4. After checking for divisibility by 7, the final list includes all
primes from 2-50.
ANALYSIS: TEACHING VIEW: This is a good activity to allow students to get an
understanding of prime numbers and the apparent “sporadity” of their
occurrence. This activity uses technology to mimic the Sieve of Eratosthenes to
find primes between 1 and n. Once the primes are found, then students would
have to solve the problem of determining all the pairs of primes that add to even
numbers…an activity that promotes problem solving. LEARNING VIEW:
Students will find this activity engaging and interesting, providing them with a
historical activity that can be related to technology and with a situation that makes
it necessary for them to plan a strategy and problem solve to find a way to
efficiently find all the pairs that add to an even number.
REFLECTION: I enjoyed this activity. I am not particularly proficient with
EXCEL, but was able to use this activity to become more adept at using it. If I
used this with my students, I would be sure to help them with some of the function
tools in the program and allow them to work in groups to brain-storm the problem
and how best to set up the program in order to solve the problem.
LINKACTIVITY 2 EFOLIO.doc TO DATA
ACTIVITY 3
Using TI-83 Find the sum sequence from k=15 to k=19 (45 points)
 (k2 +3k2/3 –5)
PLAN: A sum sequence is as follows 1,2,3,6,12,24…so each element is the
cumulative sum of those before it. Finding the sum sequence of (k^2 + 3k^(2/3) –5),
for k= 15 to 19 is done in the graphing calculator by first forming a sequence of values
generated by substituting the integers 15-19 into the expression in order. Once this is
done, the cumulative sum function can be applied to the sequence to obtain the sum
sequence
SOLUTION:
THE ENTRIES TO THE LEFT SHOW
THE TERMS OF THE SEQUENCE
FOTHE ENTRIES TO THE LEFT
SHOW THE TERMS OF THE
SEQUENCE FORMED BY
SUBSTITUTING THE VALUES 15-19
IN FOR K IN THE EQUATION.
RMED BY SUBSTITUTING THE
VALUES 15-19 IN FOR K IN THE
EQUATION.
Now, having created the above sequence {238.246606, 270.0488126, 303.8344671, 339.6048564,
377.3611021}, this will be used to generate the sum sequence by adding the terms prior to get the next term
in the new sequence:
As can be seen from the screens above, the sum sequence generated is; {238.246606,508.2954186,
812.1298857, 1151.734742, 1529.095844}
ANALYSIS: TEACHING VIEW: From a teaching standpoint, it is important to teach
students to use the graphing calculator. Even more important is teaching them to use
the manual that comes with it to accomplish the calculations they wish to do. I think
that is the main focus of this activity and that is the way I would approach it with my
students. I would give them the manual and the problem and ask them to attempt to
learn to use the calculator by reading the manual. LEARNING VIEW: Students who
become self-sufficient at learning how to use technology via the manuals and help
guides in on the web will be better able to deal with anything they encounter in the
careers they choose. This activity would be challenging, not so much in the
calculations, but in finding the procedures and manipulating the calculator to
accomplish what it is you wish to do.
REFLECTION: It took me a while to figure out how to do this in the calculator, using
the list menu and operations for lists. I could have simply used the home screen,
accomplished the calculations myself and then summed to get the sum sequence.
However, I felt it was important to learn to use the manual and determine the most
effective way to accomplish the task.
BACK TO STANDARD 1
.
Each of the activities chosen to model possible ways to approach
this standard represent each of the three tools used in solving the
problems in this class. The Pythagorean Theorem problem was
solved via the geo sketchpad program and demonstrates creativity,
and model construction that could be used by a student to solve the
problem presented. Prime numbers and the Goldbach conjecture
was solved using Excel. Students using this tool can, as modeled by
the solution, increase productivity by organizing and manipulating
the data and numbers via the EXCEL tools. The third problem, Sum
Sequence, was accomplished with a graphing calculator. The use of
the program in the computer to quickly solve the problem helps
students realize the importance of technology to enhance their
learning and increase their productivity.
BACK TO CONTENTS
.
National Educational Technology Standard 6:
Students use technology resources for solving
problems and making informed decisions.
Students employ technology in the development of
strategies for solving problems in the real world.
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ACTIVITY 1:
8. X-Intercept Transformation: Identify the change in the x-intercepts of a function
when you reverse its leading coefficient and constant term.(25 pts)
a.
What happens to the x-intercept of y = ax + b when you reverse the leading coefficient, a,
and the constant, b?
b. What happens to the x-intercept(s) of y = ax2 + bx +c when you reverse the leading
coefficient, a, and the constant, c?
What happens to the x-intercepts of a polynomial in the form y = axn + bxn-1 +cxn-2+...+z when
you reverse the leading coefficient, a, and the constant, z?
PLAN: Working with the linear equation, I plan to graph pairs of equations with the leading
coefficients switched, then trace the lines to determine the x-intercept of each, and compare the xintercepts to determine their relationship to each other. I will also try this with the quadratic
equations, then with higher order polynomials. If the trace will not “stop” at exactly y=0, I will
use the programming function of the calculator to program in the quadratic formula and find the
roots or x-intercepts that way. On higher level polynomials I can use the “solver” mode or the
graphing mode and zoom in on the x-intercept until I can get close enough coordinates to make a
determination about the x-intercepts.
As can be seen from the graphs of the linear equations above, if the leading coefficients
are exchanged, i.e. the values for the slope and the y-intercept are exchanged, the xintercepts are reciprocals of each other. Remember, the x=intercept results when y = 0.
Using the programming capabilities of the graphing calculator, I programmed in the
quadratic formula to help find the x-intercepts or roots of various quadratic equations
and proceeded to enter in values for a, b, and c using the form of quadratic ax^2 + bx +
c = 0, exchanging the leading coefficients a and c and finding the different values for
the x intercepts.
FOR A=1, C=-2, THE ROOTS ARE {1,-2}
FOR A=2, C=1, THE ROOTS ARE {1, -.5}
1 IS THE RECIPROCAL OF 1 AND –2 IS THE RECIPROCAL OF -.5
FOR A=3, AND C=2, THE ROOTS ARE {2, -.33…}
FOR A=2, AND A=3, THE ROOTS ARE {.5, -3}
2 IS THE RECIPROCAL OF .5 AND -.33 IS THE RECIPROCAL OF –3.
FOR A=2 AND C=-4, THE ROOTS OR X-INTERCEPTS ARE {4, -.5}
FOR A = -4, AND C = 2, THE ROOTS OR X-INTERCEPTS ARE {.25, -2}
THE RECIPROCAL OF 4 IS .25 AND THE RECIPROCAL OF -.5 IS -2
Judging from the above examples, the resulting x-intercepts are again reciprocals of
each other.
c. By graphing and tracing higher degree polynomials, I found that the x-intercepts are
always reciprocals of each other when the leading coefficients are exchanged.
ANALYSIS: TEACHING VIEW: Teaching relationships and patterns in math is
important. Using this activity helps students realize there are patterns that exist even if
they are not at first evident or taught as part of the concept.
LEARNING VIEW: Students will be challenged trying to figure out how they can find
the x-intercepts for each type of equation. Suggestions of what they might use will be
helpful, and allowing them to program the quadratic equation will also be interesting to
them..(and they will have to completely understand the arithmetic in the formula).
REFLECTION: This was an interesting activity. When I first thought about how to
solve the problem, it seemed the graphing function of the calculator would be the most
effective. However, the trace does not always give the coordinates you need to
determine the exact x-intercept. It was necessary to know the exact values in order to
determine exactly what the relationship between the x-intercepts are when you switch
the leading coefficients. Therefore, I had to find another way to find the x-intercepts for
quadratic and higher level polynomials. Programming in the quadratic formula gave
me exact values for the quadratic equations, and this allowed generalization to higherlevel equations.
ACTIVITY 2:
1. Now use Excel to find the sum sequence from k=1 to k=10 (20 points)
 (k2 –7k+5)
2. Now use Excel to find the sum sequence from k=15 to k=19 (20 points)
 (k2 +3k2/3 –5)
3. Now use Excel to find the sum sequence from k=25 to k=33 (20 points)

ln x
Reflect on the differences, from a teaching window, between using TI and
Excel when solving these. Which is better, IF ANY, to use? Why?
LINK TO SPREADSHEET
ACTIVITY 3
Suppose Sammy the swimmer at the tip of pier H wants to swim to the tip of pier I.
Pier H is 2km long and pier 1 is 1km long. Since the swim is very long from H to I,
Sammy thinks he will need to stip off at the beach to take a break (at point J). Sammy
can stip at any point on the beach between the two piers. If Sammy takes the break,
where should he sip on the beach if he wants to swim the least distance for the entire
trip? What is the shortest distance Sammy can swim for the entire trip (30 points).
Plan: I will construct the problem, then manipulate the distance between the piers and
for each of these manipulations, drag the point on the beach to determine the shortest
distance of his swim.
A'
m B'A' = 2.0 cm
x
x = 2.0 cm
y = 1.8 cm
B'
m AA' = 2.0 cm
y
uH
A
j
y + x = 3.8 cm
w = 1.0 cm
w
B
The distance from the end of Pier H to the end of
Pier I is 2.1. The smallest combined distance of x
and y is 3.5, however, Sammy would not be at an
advantage to do this because the distance x is
larger than the distance between the ends of the
piers.
m B'A' = 2.1 cm
A'
x
j = 1.8 cm
x = 2.4 cm
y = 1.1 cm
m AA' = 2.0 cm
u
m AH = 0.3 cm
A
y + x = 3.5 cm
= 2.6 cm
= 1.2 cm
+ x = 3.9 cm
m B'A' = 2.6 cm
A'
x
m AA' = 2.0 cm
y
H
u
A
j
j = 2.4 cm
B'
w
B
w = 1.0 cm
j
B'
w
B
j = 1.8 cm
m AH = 1.2 cm
The distance from the end of Pier H to the end of
Pier I is 2.6. The smallest combined distance of x
and y is 3.9, however, Sammy would not be at an
advantage to do this because the distance x is equal
to the distance between the ends of the piers.
y
H
w = 1.0 cm
The distance from the end of Pier H to the end of
Pier I is 2.8. The smallest combined distance of x
and y is 4.0, however, Sammy would accomplish
what he set out to do because the distance x is
smaller than the distance between the ends of the
piers.
m B'A' = 2.8 cm
A'
x
x = 2.6 cm
y = 1.4 cm
m AA' = 2.0 cm
y
u
A
y + x = 4.0 cm
H
j
B'
w
B
w = 1.0 cm
j = 2.6 cm
m AH = 1.6 cm
This is the smalles distance, travelling along the red
path that would accomplish what Sammy wants to
do, i.e. travel less than the distance from the ends of
the piers to the beach, rest, then continue on.
y + x = 20.2 cm
m B'A' = 20.0 cm
A'
x
x = 14.0 cm
H
u
j
A
m AH = 13.9 cm
B'
y = 6.2 cm y
w = 1.0 cm
w
B
j = 20.0 cm
As the distance between the piers increases, the smallest combined
distance of x and y gets closer and closer to the distance between the
ends of the piers.
Analysis: Teaching view: The depth of this problem is not evident at first. Students will have a problem
because with the initial reading, there does not seem to be enough information. This is a good problem to
help develop higher thinking skills.
Learning view: Students will learn to delve deeper into seemingly simple problems. They will learn try the
whole spectrum of a problem and avoid jumping to conclusions.
Reflection: I enjoyed this problem. It was subtle and required higher level thinking skills.
BACK TO STANDARD 2
.
The three activities chosen to model this standard are 1. X-Intercept
Transformation, 2. Sum Sequence using EXCEL, and 3. the
Swimmer problem. The first was done using a graphing calculator.
Students will use the calculator to transform the coefficients in an
equation and observe the changes in the x- intercept(s), and use
these observations to make an informed decision of the affects of
the transformations. The Sum Sequence carried out via EXCEL
requires a student to develop an organized method for obtaining the
values necessary using this program and is similar to situations in
the real world. The swimmer problem solved using Geo-Sketchpad
allows a student to manipulate the diagram and use the measurement
tools to determine the solution(s) of the problem while realizing the
subtlety of the problem.
BACK TO CONTENTS
.
Curriculum and Content Area Standard (Mathematics
Standards) 3:
Mathematics instructional programs should include attention to geometry and spatial sense
so that all students
analyze characteristics and properties of two-and three-dimensional geometric objects

select and use different representational systems, including coordinate geometry and
graph theory

recognize the usefulness of transformations and symmetry in analyzing mathematical
situations

use visualization and spatial reasoning to solve problems both within and outside of
mathematics
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ACTIVITY 1
Tangential Circles: compare the area and circumference of a circle inscribed in a region
between two intersecting circles:
1.
Two larger circles with equal radii pass through each other’s centers. A smaller circle can
be created inside the overlapping region so that it is tangent to the other circles. Compare
the area and circumference of the smaller circle to the area and circumference of the larger
circle.
Using geosketchpad, construct a circle with center A. Now shift-select a point on the circle
and point A and construct a circle with the point on circle A as the center of this new circle.
This will create a region or area of intersection. Choose a point on the perimeter of this
region on each circle. Construct a segment, and the midpoint of that segment. Use the
midpoint as the center of another circle construction with the end of the radius on one of the
large circles. Use the measure tool to have the program indicate the length of the radius, and
the circumference and the area of one of the large circles and the small, inscribed circle.
Shift-select the areas of each circle and use the plot as (x,y) to set up a graph which compares
the areas of the circles. Trace this point on the graph as you move the endpoint of one of the
radii to change the size of the two circles.
650
640
630
620
610
600
590
580
570
560
550
540
530
520
510
500
490
480
80
470
460
450
1. Tangential
Circles: 15
70
pts.
440
430
420
410
Circumference AB = 12.3 cm
Circumference EB = 6.1 cm
Area AB = 12.0 cm 2
Area EB = 3.0 cm 2
Radius AB = 2.0 cm
Radius EB = 1.0 cm
400
390
380
60
H: (3.0, 12.0)
370
Area of
large
circle
360
350
340
50
Circumference BA = 12.3 cm
330
320
Area BA = 12.0 cm 2
Radius BA = 2.0 cm
310
300
290
40
280
270
30
260
A
250
E
240
230
20
220
210
200
190
10
180
170
160
150
140
130
-150
120
-140
-130
-120
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
10
20
30
40
110
100
-10
90
80
Circumference
Circumference
AB = 14.2 cm
Area EB = 4.0 cm 2
Radius
Radius
AB = 2.3 cm
1. Tangential
Circles: 15
70
pts.
EB = 7.1 cm
Area AB = 16.1 cm 2
EB = 1.1 cm
60
H: (4.0, 16.1)
Circumference
A rea of
large
c irc le
50
BA = 14.2 cm
Area BA = 16.1 cm 2
Radius BA = 2.3 cm
40
30
E
A
20
10
-140
-130
-120
-110
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
10
-10
T he area of the s mall c irc le is always one-fourth the area of the larger circ le. B ec aus e the diameter
of the small c ircle is equal to the radius of the large c irc le, the radius of the s mall c irc le is half that of
-20
the large c irc le. A rea is a s quared meas ure therefore the area of the larger circ le is 2 x 2 or 4 times
that of the smaller circ le.
20
30
40
A rea of s mall circ le
50
60
70
80
90
100
110
120
130
140
150
160
The area of the small circle is always one-fourth the area of the larger circle. Because the diameter
of the small circle is equal to the radius of the large circle, the radius of the small circle is half that of
the large circle. Area is a squared measure therefore the area of the larger circle is 2 x 2 or 4 times
that of the smaller circle. A = pi x r x r for the smaller circle, therefore A = pi x 2r x 2r or 4 x pi x r x
r for the larger circle
The circumference of the smaller circle is half that of the larger circles. Circumference is a
magnitude 1 measurement, therefore, when the diameter of the smaller circle is doubled, the
circumference is doubled.
-20
-30
-40
Area of small circle
50
IV. Closing Analysis:
Learning window: Working with the geosketchpad to construct and manipulate the
circles necessitates higher level thinking about the criteria of the problem and how the
spaces relate to each other. It is interesting to “play” with the size of the circles and see
how the data traces on the graph. It also becomes very clear not only how the areas
relate to each other but how graphs can be used to make these relationships easier to
see. Once I collected the data and looked at the graph, it was much clearer why these
areas are related the way they are.
Teaching window:
As a teacher, I am always looking for ways to help students make connections between
math concepts. The ability to tie the graph with the measures of the areas, then watch
the graph form as the circles are manipulated in a strong way of accomplishing this.
Not only will students be made to realize there is a connection, they will be able to
visualize just why the relationship works as it does.
V. Reflection: Given this type of technology, not only in a geometry classroom, but in
an algebra classroom, I would definitely use it to promote the connections between
different concepts and areas of mathematics. I enjoyed working with GSP to
accomplish this and realized more about the relationship of the areas than I expected to.
Once I was familiar with the program, it was relatively easy to construct and use the
figures to determine the solution to the problem.
ACTIVITY 2:
Triangles in a Trapezoid: Determine the relationship of the areas of the triangles
formed by the diagonals of a trapezoid. (15 points)
How do the areas of the red triangles relate to each other ( triangles share a side with the
legs of the trapezoid). How do the areas of the yellow triangles relate to each other?
I. Plan: I will construct the trapezoid with diagonals and measure the areas of the
triangles along with the lengths of the bases and the legs and also the area of the
trapezoid itself. Comparisons will be done with these measurements in order to
determine the relationships between the areas.
Area ABG = 1.4 cm 2
Area CGE = 25.0 cm 2
Area AGC = 5.9 cm 2
Area BGE = 5.9 cm 2
A
Area ABEC = 38.1 cm 2
B
m AB = 4.4 cm
G
m EC = 18.6 cm
m BE = 11.3 cm
m CA = 4.8 cm
m BC = 8.5 cm
m AE = 15.5 cm
(Area ABG)
(Area CGE) = 0.06
(Area ABG)
(Area ABEC )
= 0.04
(Area CGE)
(Area ABEC )
= 0.66
E
(Area AGC)
= 0.15
(Area ABEC )
m BAC = 136°
m ACE = 44°
m CEB = 17°
m EBA = 163°
m BAC
= 7.95
m CEB
m ACE
= 0.27
m EBA
Area(Polygon AGC)/Area(Polyg...
Area(Polygon CGE)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
0.19
0.54
0.07
Area(Polygon ABG)/Area(Polyg...
0.13
Area(Polygon ABG) 1.35
Area(Polygon CGE) 10.33
Area(Polygon AGC) 3.73
Area(Polygon BGE) 3.73
Area(Polygon ABEC) 19.15
Length(Segment EC) 15.58
Length(Segment BE)
3.37
Length(Segment AE) 8.67
Length(Segment AB) 5.63
Length(Segment CA)
7.33
Length(Segment BC) 12.86
Area(Polygon AGC)/Area(Polyg...
Area(Polygon CGE)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
0.19
0.54
0.07
Area(Polygon ABG)/Area(Polyg...
Area(Polygon ABG)
0.13
5.39
Area(Polygon CGE)
Area(Polygon AGC)
Area(Polygon BGE)
Area(Polygon ABEC)
Length(Segment EC)
41.33
14.93
14.93
76.59
15.58
Length(Segment BE) 11.24
Length(Segment AE) 15.97
Length(Segment AB) 5.63
Length(Segment CA)
7.35
Length(Segment BC) 10.04
Area(Polygon ABG)/Area(Polyg...
0.06
Area(Polygon ABG)/Area(Polyg...
0.04
Area(Polygon CGE)/Area(Polyg...
0.66
Area(Polygon AGC)/Area(Polyg...
0.15
Area(Polygon BGE) 2.43
Area(Polygon AGC) 2.43
Area(Polygon CGE) 10.34
Area(Polygon ABG) 0.57
Length(Segment AE) 12.98
Length(Segment BE)
8.18
Area(Polygon ABG)/Area(Polyg...
0.06
Area(Polygon ABG)/Area(Polyg...
0.04
Area(Polygon CGE)/Area(Polyg...
0.66
Area(Polygon AGC)/Area(Polyg...
0.15
Area(Polygon BGE) 6.26
Area(Polygon AGC) 6.26
Area(Polygon CGE) 26.62
Area(Polygon ABG) 1.47
Length(Segment EC) 20.58
Area(Polygon ABEC) 15.78
Length(Segment EC) 20.58
Area(Polygon ABEC) 40.62
Length(Segment BC) 11.15
Length(Segment BC)
Length(Segment CA)
12.56
7.75
Length(Segment AB)
4.84
Length(Segment AE) 15.08
Length(Segment BE) 10.40
Length(Segment CA)
Length(Segment AB)
6.66
4.84
Area(Polygon AGC)/Area(Polyg...
Area(Polygon CGE)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
0.19
0.54
0.07
0.13
Area(Polygon ABG)
Area(Polygon CGE)
Area(Polygon AGC)
Area(Polygon BGE)
Area(Polygon ABEC)
Length(Segment EC)
Length(Segment BE)
4.96
37.97
13.72
13.72
70.36
20.12
9.18
Length(Segment AE) 15.73
Length(Segment AB) 7.27
Length(Segment CA)
7.34
Length(Segment BC) 13.53
Area(Polygon AGC)/Area(Polyg...
0.19
Area(Polygon CGE)/Area(Polyg...
0.54
Area(Polygon ABG)/Area(Polyg...
0.07
Area(Polygon ABG)/Area(Polyg...
0.13
Area(Polygon ABG) 6.52
Area(Polygon CGE) 49.99
Area(Polygon AGC) 18.06
Area(Polygon BGE)
Area(Polygon ABEC)
Length(Segment EC)
Length(Segment BE)
Length(Segment AE)
18.06
92.63
20.12
14.17
20.85
Length(Segment AB)
Length(Segment CA)
7.27
6.78
Length(Segment BC)
10.22
The triangles that share the legs are of equal area. (
Polygons AGC and BGE, shown in red). The ratio of
the areas of the other two triangles CGE and ABG is
.54 in the four upper trapezoids, and remains constant
when pts A, B or C are dragged, as shown in the four
upper charts. Whe Pt E is dragged, the ratio changes
but remains constant once again when pts A, B, or C
are dragged in this new trapezoid.
Area ABG = 10.9 cm 2
B
Area CGE = 2.3 cm 2
Area AGC = 5.0 cm 2
Area BGE = 5.0 cm 2
Area ABEC = 23.2 cm 2
m AB = 17.6 cm
m EC = 8.0 cm
m BE = 5.7 cm
m CA = 4.6 cm
E
m BC = 13.5 cm
m AE = 12.3 cm
(Area ABG)
(Area CGE)
= 4.81
(Area ABG)
(Area ABEC )
= 0.47
(Area CGE)
(Area ABEC )
= 0.10
(Area AGC )
(Area ABEC )
= 0.22
(m AB)
(m CA)
(m BE)
(m EC)
G
C
= 3.86
A
= 0.71
(m AB) + (m EC) + (m BE) + (m CA) = 35.9 cm
(Area ABG)
((m AB) + (m EC) + (m BE) + (m CA))
= 0.3 cm
(Area CGE)
((m AB) + (m EC) + (m BE) + (m CA))
= 0.1 cm
Area(Polygon ABG)
10.94
Area(Polygon ABG)
18.38
Area(Polygon ABG)
21.90
Area(Polygon CGE)
2.27
Area(Polygon CGE)
3.82
Area(Polygon CGE)
4.56
Area(Polygon AGC)
4.99
Area(Polygon AGC)
8.38
Area(Polygon AGC)
9.99
Area(Polygon BGE)
4.99
Area(Polygon BGE)
8.38
Area(Polygon BGE)
9.99
Area(Polygon ABEC)
23.19
Area(Polygon ABEC)
38.98
Area(Polygon ABEC)
46.43
Length(Segment AB)
17.60
Length(Segment AB)
12.02
Length(Segment AB)
12.02
Length(Segment EC)
8.02
Length(Segment EC)
5.48
Length(Segment EC)
5.48
Length(Segment BE)
5.68
Length(Segment BE)
5.80
Length(Segment BE)
7.48
Length(Segment CA)
4.56
Length(Segment CA)
5.28
Length(Segment CA)
5.46
Length(Segment BC)
13.53
Length(Segment BC)
10.21
Length(Segment BC)
11.99
Length(Segment AE)
12.35
Length(Segment AE)
9.43
Length(Segment AE)
8.58
Area(Polygon ABG)/Area(Polyg...
4.81
Area(Polygon ABG)/Area(Polyg...
4.81
Area(Polygon ABG)/Area(Polyg...
4.81
Area(Polygon ABG)/Area(Polyg...
0.47
Area(Polygon ABG)/Area(Polyg...
0.47
Area(Polygon ABG)/Area(Polyg...
0.47
Area(Polygon CGE)/Area(Polyg...
0.10 Area(Polygon CGE)/Area(Polyg...
0.22 Area(Polygon AGC)/Area(Polyg...
0.10
Area(Polygon CGE)/Area(Polyg...
0.10
0.22
Area(Polygon AGC)/Area(Polyg...
0.22
Area(Polygon AGC)/Area(Polyg...
Area(Polygon ABG)
32.89
Area(Polygon ABG)
21.97
Area(Polygon ABG)
11.00
Area(Polygon CGE)
6.84
Area(Polygon CGE)
4.57
Area(Polygon CGE)
2.29
Area(Polygon AGC)
15.00
Area(Polygon AGC)
10.02
Area(Polygon AGC)
5.02
Area(Polygon BGE)
15.00
Area(Polygon BGE)
10.02
Area(Polygon BGE)
5.02
Area(Polygon ABEC)
69.73
Area(Polygon ABEC)
46.57
Area(Polygon ABEC)
23.32
Length(Segment AB)
12.02
Length(Segment AB)
11.12
Length(Segment AB)
6.61
Length(Segment EC)
5.48
Length(Segment EC)
5.07
Length(Segment EC)
3.02
Length(Segment BE)
11.42
Length(Segment BE)
11.25
Length(Segment BE)
9.67
Length(Segment CA)
8.14
Length(Segment CA)
6.80
Length(Segment CA)
6.80
Length(Segment BC)
15.82
Length(Segment BC)
15.82
Length(Segment BC)
12.37
Length(Segment AE)
8.84
Length(Segment AE)
5.93
Length(Segment AE)
5.15
Area(Polygon ABG)/Area(Polyg...
4.81
Area(Polygon ABG)/Area(Polyg...
4.81
Area(Polygon ABG)/Area(Polyg...
4.81
Area(Polygon ABG)/Area(Polyg...
0.47
Area(Polygon ABG)/Area(Polyg...
0.47
Area(Polygon ABG)/Area(Polyg...
0.47
Area(Polygon CGE)/Area(Polyg...
0.10
Area(Polygon CGE)/Area(Polyg...
0.10
Area(Polygon CGE)/Area(Polyg...
0.10
Area(Polygon AGC)/Area(Polyg...
0.22
Area(Polygon AGC)/Area(Polyg...
0.22
Area(Polygon AGC)/Area(Polyg...
0.22
B
A
G
E
C
Area(Polygon AGC)/Area(Polyg...
0.19
Area(Polygon AGC)/Area(Polyg...
0.19
Area(Polygon AGC)/Area(Polyg...
Area(Polygon CGE)/Area(Polyg...
0.19
0.54
Area(Polygon CGE)/Area(Polyg...
0.54
0.54
0.07
0.07
0.07
Area(Polygon CGE)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
Area(Polygon ABG)/Area(Polyg...
0.13
Area(Polygon ABG)/Area(Polyg...
0.13
Area(Polygon ABG)/Area(Polyg...
0.13
Area(Polygon ABG)
4.96
Area(Polygon ABG)
1.35
Area(Polygon ABG)
5.39
Area(Polygon CGE) 37.97
Area(Polygon CGE) 10.33
Area(Polygon CGE) 41.33
Area(Polygon AGC) 13.72
Area(Polygon AGC)
3.73
Area(Polygon AGC) 14.93
Area(Polygon BGE) 13.72
Area(Polygon BGE)
3.73
Area(Polygon BGE) 14.93
Area(Polygon ABEC) 70.36
Area(Polygon ABEC) 19.15
Area(Polygon ABEC) 76.59
Length(Segment EC)
15.58
Length(Segment EC)
20.12
Length(Segment BE)
9.18
Length(Segment EC)
15.58
Length(Segment BE)
3.37
Length(Segment BE) 11.24
Length(Segment AE) 15.73
Length(Segment AE)
Length(Segment AB)
8.67
5.63
Length(Segment AE) 15.97
Length(Segment AB)
5.63
Length(Segment AB)
Length(Segment CA)
7.27
7.34
Length(Segment CA)
7.33
7.35
10.04
13.53
12.86
Length(Segment CA)
Length(Segment BC)
Length(Segment BC)
Length(Segment BC)
Area(Polygon AGC)/Area(Polyg...
Area(Polygon CGE)/Area(Polyg...
0.19
0.54
Area(Polygon ABG)/Area(Polyg...
0.07
Area(Polygon ABG)/Area(Polyg...
0.13
Area(Polygon ABG)
6.52
Area(Polygon CGE) 49.99
Area(Polygon AGC) 18.06
Area(Polygon BGE) 18.06
Area(Polygon ABEC) 92.63
Length(Segment EC)
20.12
Length(Segment BE) 14.17
Length(Segment AE) 20.85
Length(Segment AB)
Length(Segment CA)
7.27
6.78
Length(Segment BC)
10.22
Analysis: Teaching view: This would be difficult to use in a classroom
unless you restrict it to dragging only the vertex that does not change the
ratio of the unequal area triangles. Students will see that the
relationships of the triangles remain constant.
Learning View: Students will find the construction and manipulation of
the shape fun and interesting. It would be difficult for them to see why
the ratios of the areas change as they do when certain of the vertices are
dragged and do not change when others are dragged.
ACTIVITY 3
1. Varied Slopes and Intercepts: Discover what happens to the graph when you vary
the
values for m and b in the equation y = mx + b. (25 pts)
.
How does changing m affect the graph of the line?
Now plug in different values for b, keeping m fixed.
What does changing b do to the graph of the line?
The equation of a line can be written in the form y = mx + b, where m and b are real
numbers.
Plug in different values for m, keeping b fixed.
Make sure to use positive and negative values as well as fractional values, such as
PLAN: Using the graphing calculator I will enter various equations on the “y=” screen,
first keeping the b fixed and varying the values for “m”, then keeping the m fixed and
varying the values for “b”. Observing the changes in the lines when graphed and using
the trace button to ascertain the values of “b” or the y-intercept will give me the
information to answer the questions posed.
Holding “b” the same, changing “m” or slope of the line…Positive slopes for this group
of graphs:
For negative “m” or slope of the line:
Now, keeping “m” or the slope the same and varying “b” or the y-intercept:
How does changing m affect the graph of the line?
Since “m” represents the slope of the line, as can be seen in the first two sets of graphs, when
“m” varies, the direction and “steepness” of the line varies. The larger the absolute value of
“m”, the steeper the slope of the line. If the value of “m” is positive, the line slopes up from left
to right, if it is negative, the line slopes down from left to right.
What does changing b do to the graph of the line?
The variable “b” in the equation of a line y= mx + b, represents the y-intercept of the line or
where the graph of the line intersects the y-axis. Varying the values for “b” results in changing
the point where the line crosses the y-axis, but does not change the slope of the line
BACK TO STANDARD 3
.
Use of the Geo-Sketchpad lends itself well to incorporating standard 6 of the
curriculum and content area standards for mathematics into the classroom. Two
of the activities presented used this program. Students working to solve the
Tangential Circles problem will be able to use visualization to solve the problem
via the sketchpad. It is also possible to graph the data collected to a coordinate
plane and represent the relationship between the areas of the circles using the
graph that results. The Triangles in a Trapezoid problem allows students working
with the sketchpad to visualize the shapes and how transforming the outer
trapezoid affects the inner triangles. The third activity was the Varied Slopes and
Intercept problem solved via the graphing calculator. Use of this tool for this
problem enables students to represent the linear equations in a system other than
pen and pencil and helps them recognize the usefulness of transformations in
analyzing mathematical situations.
BACK TO CONTENTS
.
Curriculum and Content Area Standard (Mathematics
Standard) 5: Mathematics instructional programs should



include attention to data analysis, statistics and probability so that
all studentspose questions and collect, organize and represent data to answer
those questions
develop and evaluate inferences, predictions, and arguments that
are based on data
interpret data using methods of exploratory data analysis
LINK TO ACTIVITIES
LINK TO REFLECTION
BACK TO CONTENTS
ACTIVITY 1
2. Transformations of Data: Explore what changes in the mean, median and mode as
you add, subtract, multiply, and divide elements in a data set. (25 points)
Use spreadsheets to compute the mean, median, and mode for any set of data.
•What happens to the mean, median and mode if you add 0.7 to each
element of the data? (5 pts)
•What happens to these values if you subtract 0.5 from each element of
the data? (5 pts)
•If the mean, median, and mode of the data set a, b, c, d are 16, 11 and 4,
what are the mean, median, and mode of the data set a + x, b + x, c + x, d
+ x, where x is any real number? (5 pts)
•Describe what happens to the mean, median and mode of a data set
when you multiply each data point by a number k. (10 pts)
LINK TO SPREADSHEET
THE FIRST MEAN, MEDIAN AND MODE ARE FOR THE ORIGINAL SETS OF DATA.
THE SECOND MEAN, MEDIAN AND MODE ARE FOR THE SETS AFTER THEY HAVE HAD .7
ADDED TO EACH ELEMENT. ALL THREE VALUES ARE ALSO .7 MORE THAN THE ORIGINAL.
THE THIRD SET OF VALUES ARE CALCULATED ON THE SETS OF DATA AFTER .5 WAS
SUBTRACTED FROM EACH ELEMENT, AND THESE VALUES ARE ALSO .5 LESS THAN THE
ORIGINAL MEAN, MEDIAN AND MODE.
C. If the mean, median, and mode of the data set a,b,c,d are 16,11, and 4, what are the mean,
median, mode of the data set a + x, b +x, c + x, and d + x, where x is any real number? The
mean, median, and mode would be 16 + x, 11 + x and 4 + x.
D. The final sets of means, medians and modes reflect those values after data set 1 has been
multiplied by 2,3,4 and 10, respectively. Notice that these values are consistently, 2,3,4 and 10
times the mean, median, and mode of those calculated on the original data set 1. Therefore, when
each element of a data set is multiplied by a number, k, the mean, median, and mode of the
resulting transformed data set will also be multiplied by k.
ANALYSIS: TEACHING VIEW: Students rarely realize how the mean, median, and mode
values are affected by increasing, decreasing, multiplying or dividing each data element by a
constant value. This activity will help students understand the relationship between the values
and the effect these transformations have. LEARNING VIEW: Students will learn how
transforming data changes the measures of central tendency and will do so using a program the
students will need to know how to manipulate in their careers at a later date. This is a good
activity to help students learn something that they will use “in the real world”.
REFLECTIONS: I found this to be a straight-forward, simple problem, but will definitely use it
with my students. The activity helped me continue to learn to manipulate the EXCEL program
and the functions set up in the program. This is an effective activity to use not only with
statistics, but to help students develop number sense and see how transforming values can affect
other calculations.
ACTIVITY 2
5. Creating Integers: Explore the relationship between a fraction and its decimal
equivalent. (40 points – 10 pts each)
a.What is the smallest positive integer that will "work"?
a.What are the smallest positive integers that will work for other
decimals such as 0.05, 1.25, 0.375, 0.444...,
a.Compare your answers with the fraction equivalents of 0.20, 0.05, 1.25,
0.375, 0.444.... and
a.. Describe how a fraction equivalent can be determined from a decimal
number.
a.Will this method work for all decimal numbers? Why? Why Not?
PLAN: I will generate a list of integers, 1-50, as a list in the first column. Once this is
done, the next six columns will be used to multiply the original integers by .20, .05,
1.25, .3755, .444… and .142857…., respectively. Once this is done, I will make
comparisons in order to answer the above questions.
LINK TO SPREADSHEET
THE SMALLEST INTEGER FOR WHICH MULTIPLYING BY .20 WILL
RESULT IN AN INTEGER IS 5. 5 X .20 = 1.
B.) FOR .05, THE SMALLEST INTEGER THAT WILL WORK IS 20.
FOR 1.25, THE SMALLEST INTEGER THAT WILL WORK IS 4. FOR .375, THE
SMALLEST NUMBER IS 8
FOR .444 REPEATING, THE SMALLEST INTEGER THAT WILL WORK IS 9.
FOR .142857, THE SMALLEST INTEGER THAT WIL WORK IS 7.
A.)
C. Noticing that the smallest integer that works for multiplying the decimal to get an
integer is also the denominator of the fraction equivalent of the decimal, it becomes
apparent that there is a relationship that could help determine a fraction equivalent from
the decimal. Using a calculator or excel, one can quickly determine the smallest integer
that must be multiplied by the decimal to obtain a fraction. This is the denominator of
the fraction equivalent. The numerator is the integer obtained when the multiplication
is done.
This will work for all rational decimals, since any rational number can be written
as a fraction, so: decimal = integer a/ integer b. This yields, by multiplying both sides
by the denominator: integer a * decimal = integer a. Of course irrational numbers
cannot be written as a fraction, therefore this procedure will not work on decimals of
that type.
ANALYSIS: TEACHING VIEW: This is a great activity for helping students
understand the relationship between fractions, and decimals. I would give a few more
“guiding” questions or hints to help the students see the procedure, but it would be
very effective for a better understanding of how fractions and decimals relate to each
other. LEARNING VIEW: Students working on this activity will develop a deeper
understanding of the concept of terminating and repeating decimals and how they
compare to their equivalent fractions. This also gives students another way of finding
equivalent fractions for rational numbers.
REFLECTIONS: This is an activity that could be adapted to help students find
connections between fractions and decimals, especially repeating decimals. The
algorithm for finding decimal equivalents for repeating fractions can be confusing for
students, this activity allows a student to develop an alternative procedure for finding
decimal equivalents for any rational equation.
D.
ACTIVITY 3
Problem: Using value sets provided for x and y, create a scatter plot and find
an equation that fits the graph. Then using data for the life expectation of women
for various years, create a scatterplot, find the equation that best fits the data, and
predict the life expectancy for women in the year 2000. (The tables in the solution
show the data provided.)
I.
II.
Plan: Using the TI-83 graphing calculator, a connectivity link to the
computer, and the instruction book for the calculator (extensively  ), I will save
the data into a table under the STAT menu, then create a scatter plot using the
STAT PLOT, and find the equation using the LINREG ( AX +B) function.
Table of data L1 contains the “x” values, L2 contains the “y”
values:
Scatter plot showing the above points graphed on a coordinate
plane:
Equation for line of best fit: y = .0230877122 x + .429626236
Exercise: Table Showing Life Expectancies (L4) of Women for Various Years (L3 where 0 = 1950, 10 = 1960, etc.)
Scatter Plot of the above data ( L3 is on the “x” axis, L4 is on the “y” axis)
Equation which best models the above data and graph
Y = .197 x + 71.06
Prediction of life expectancy for year 2000
( x = 50) is 80.91 years.
Closing Analysis:
Learning Window: For students using the graphing calculator for the first
time, the logistics of it can be quite daunting. While GSP is fairly selfexplanatory and uses commands that are “logical”, the TI-83 has many menus
and sub-menus to navigate. It is imperative that the student be taught to refer
to the instruction book. This activity helps students learn how to store data in a
table and use that data to construct a visual (scatter plot) of the relationship of
the data.
Teaching Window: It is important that students know how to construct a
scatter plot and what it can be used to model. Once they know how to do this
manually, it is extremely effective for them to learn technology that can make
the job easier for them and take the monotony from the job. The use of the
“best-fit” equation helps them to realize that algebra and statistics are related
and work together to describe this type of situation.
I.
Reflection: I had to work hard and use the instruction book to remember how
to store data in the lists and then use the data to create the scatter plot and bestfit equation. I think this is a good activity for students. It is effective in helping
BACK TO STANDARD 4
.
The three activities chosen to represent the fifth curriculum and content area standard
for mathematics involve, once again, reflect more than one technical tool used in
the solutions. The first activity, Transformation of Data was accomplished in
EXCEL. Students utilizing such a program can collect, organize and manipulate
large amounts of data in order to observe relationships and determine affects of
transformations in the data. The second activity, Creating Integers was also
accomplished using EXCEL. The solution of this problem using EXCEL uses
interpretation of data to develop a relationship between fractions and decimals.
The third activity, Linear Regression, was solved using the graphing calculator.
This activity uses a set of data to set up a scatter plot, and the capabilities of the
graphing calculator to interpret the data and analyze it by finding an equation of
best fit. Students working on this problem will develop their abilities to evaluate
the data, interpret and analyze the date, then make predictions using the best fit
equation and scatter plot.
BACK TO :CONTENTS
.
 http://www.iste.org/
 http://cnets.iste.org/teachers/t_book.html
 http://cnets.iste.org/
 Geo-sketchpad