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TRANSCRIPT
Kelly Mosier
Activity #1: Coins and Dice
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The activity Coins and Dice was very interesting for me (Stohl, 2002). As I was going
through the steps in the lesson and as I played with the spreadsheet, I learned a little bit more
about probability. I learned that if I flip a coin 100 times, the probability is that 50 of those tosses
will turn out to be heads and 50 will be tails but the likelihood is a completely different story.
Sure, the frequency of heads and tails might be pretty similar but it’s not always split down the
middle (Stohl, 2002). A student could end up generating a completely random event of 30 heads
and 70 tails. That’s a pretty big change in the frequency! But what’s even more interesting for
the students to discover is that they are both equally likely to occur. I never really considered this
before. Since we had already conducted an activity about rolling dice and probability in class,
this part of the activity wasn’t really enlightening for me but, for my students, it will be. Most
students have an idea that 7 is the most frequent total rolled from two dice but now with this
activity, they will learn why. Relating previous knowledge to new material is what learning is all
about. I really think they’re going to enjoy this activity; I know I did.
I chose this activity because probability has always been my weakness. I was never
required to take a statistics course in middle or high school so, when I took statistics in college, I
was very far behind. I know how difficult it can be to separate the probability from reality. For
instance, a student might assume that flipping a coin 8 times will result in 4 tails and 4 heads but
the probability turns out to be the same for any combination of results. That might seem counter-
intuitive to some students. I chose this lesson so I can help any students that are still stuck on this
subject. In addition, by seeing an activity created with Excel, I am now tempted to find more
lessons that incorporate Excel into the classroom. Excel is practically on every computer and it’s
a required program to know for most jobs. By exposing my students to this technology (and this
lesson), they will be better prepared for the future.
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This activity is aligned with NCTM standards. The students will be required to
understand and apply the basic concepts of probability: randomization, sample space, outcomes,
and frequencies (“NCTM”, 2004). They will be able to “make and investigate mathematical
conjectures, organize and consolidate their mathematical thinking through communication, use
the language of mathematics to express mathematical ideas precisely, and recognize and apply
mathematics in contexts outside of mathematics” (“NCTM”, 2004). This activity satisfies my
goals because I have learned how to make a lesson using technology that wasn’t very familiar to
me. Now I can incorporate Excel into my future lessons since I have seen a glimpse of the power
it holds. Not only has this lesson benefitted me, it will benefit my students. They will be excited
to have a new form of learning in the classroom. In addition, this activity itself will expand their
reasoning skills and educate them on a topic that might be unfamiliar to them. Overall, this
lesson was well chosen.
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Activity #2: Proof Without Words: Completing the Square
I really enjoyed the activity Proof Without Words: Completing the Square (“Proof”). It
concerned the reason and method for completing the square: x2 + ax = (x + a/2)2 – (a/2)2. I never
quite understood why, when you are completing the square, you must divide the coefficient in
front of x by two and then square that value. I never understood where that algorithm came from.
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This applet taught me the reason why. Basically, you have a yellow square and a blue square
(“Proof”). The yellow square’s area is your x2 and the blue square’s area is your ax (“Proof”).
Therefore, to make a perfect square, you need to split the blue square in half (thereby making
two rectangles of length a/2 and width x) and align each piece next to the yellow square but you
also need to add the missing piece: (a/2)2 (“Proof”). So when you add your square pieces, x2 + ax
+ (a/2)2, you must then subtract your (a/2)2 from both sides (“Proof”). This is how we get the
right hand side of our equation. This is why we must divide the length of the blue square in half,
a/2, form a square, and subtract the missing piece (“Proof”). The applet was the best part of this
whole activity. Of course my students can just multiply out the right hand side and see that it
equals the left hand side but that wouldn’t have taught them anything. This activity teaches them
the reason behind the algorithm to completing a square. In addition, the visualization aids in their
conceptual understanding of what’s going on. This is why I really enjoyed this activity.
I chose this activity because I never understood the reason for the method of completing
the square. I always had a curiosity for how it was developed but I never truly understood why
until I found this activity. The applet helped me immensely. I was able to visualize what happens
as you complete a square and how the values come about. Since I knew how to calculate the area
of a rectangle and square, I was able to bridge my knowledge on completing the square to a
visualization of the act of completing the square. Before I chose this activity, I also noticed that
my students would appreciate the applet as well. Students love to use technology, especially in
math class when they are so used to lectures and worksheets. I figured that this applet would
show the students why math is fun and exciting.
This lesson is aligned with NCTM standards. It “allows the students to understand the
meaning of equivalent forms of equations and it gives them the opportunity to write equivalent
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forms of equations and solve them with fluency using technology” (“NCTM”, 2004). Students
will be required to “recognize and use connections among mathematical ideas, understand how
mathematical ideas interconnect and build on one another to produce a coherent whole, analyze
and evaluate the mathematical thinking and strategies of others, and finally, use representations
to model and interpret physical, social, and mathematical phenomena” (“NCTM”, 2004). This
lesson is also aligned with my goals because it showed me another interesting way to use
technology in the classroom. Applets seem to be very beneficial for students since they can see
what is happening and not have to take your word for something. They get to explore and make
conjectures on their own. They seem to have more creativity and enthusiasm for math. Also,
since this lesson concerned algebra and I want to teach that in the future, another one of my goals
was satisfied. I always wondered how I would approach this topic in the future and now I have a
great solution.
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Activity #3: The Slope Game
As I was exploring the activity, The Slope Game, I realized that it is a very simple but
educational activity for middle school students who are just being introduced to algebra (“The
Slope Game”). The students will be paired off and one student will construct 5 lines and use GSP
to calculate the slopes of each of those lines. Then that student will hide the points formed and
his/her partner will need to use the slopes on the left side to match each slope to its respective
line. I found it to be a fun and interesting game. Surprisingly, I learned something as I was doing
the activity. I created an almost vertical line and the slope of that line was about -32. I actually
wasn’t sure what the slope of that line was until I looked at the answer. I just figured that the
almost vertical line had an undefined slope but, since that wasn’t listed, I had to reevaluate my
answer. I had to stop and consider what the slope of an almost vertical line should be. This
activity made me realize that if you pick two points on an almost vertical line, the change in y
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would be so much greater than the change in x and that is why the number was so huge for the
slope. I never really thought of a line having that big of a slope before. Therefore, this lesson was
enlightening.
The Slope Game was pretty short but it will educate the students. It uses GSP to create
multiple lines on a coordinate plane and the students will test each other to see if they can match
a given slope to a line (“The Slope Game”). It’s interesting for the students because it’s similar to
a game for them. They get to compete against each other in a positive manner. I chose this
activity because I feel that games are a great way to challenge students. They become more
invested in the activity and try to excel more than if you just gave them a boring worksheet.
Also, since it involved GSP, I figured the students would be excited to play around with the
software and create their own graphs. Finally, since the lesson concerned algebra, I figured that I
could use this activity when I teach.
This lesson is aligned with the NCTM standards. The students will need to “represent,
analyze, and generalize a variety of patterns with graphs, explore relationships between symbolic
expressions and graphs of lines, and use graphs to analyze the nature of changes in quantities in
linear relationships” (“NCTM”, 2004). In addition, the students will be required to
“communicate their mathematical thinking coherently and clearly to peers, teachers, and others,
analyze and evaluate the mathematical thinking and strategies of others, and finally, select,
apply, and translate among mathematical representations to solve problems” (“NCTM”, 2004).
This lesson is aligned with my goals because, as already stated, I can use this is my algebra
classes. The students will benefit from creating and analyzing lines and their slopes. Through this
lesson, I also learned how to create a lesson through Geometer’s Sketchpad. I have been curious
as to how I would create a lesson with this program. Luckily, this lesson gave me the opportunity
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to learn just how amazing the program is. It will foster student’s conceptual understanding of
mathematics while still being entertaining.
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Activity #4: Unit Circle and Right Triangle Functions
This activity was probably my most favorite of them all. It concerns trigonometry and
relating the unit circle to triangles and even to the graphs of the sine, cosine, and tangent
functions (“Unit Circle”). Students can see the unit circle and, when they animate point C, they
see point C traveling around the unit circle counterclockwise (“Unit Circle”). The students will
then use the action button “Measure Arc Angle” to calculate the angle of point C at each point as
it travels around the unit circle (“Unit Circle”). Students can also measure the x and y-
coordinates as well as the slope for the line connecting the origin to point C (“Unit Circle”). As
for the triangle, students will be able to learn about how the trigonometric functions are formed
and what their respective ratios are (“Unit Circle”). They can then merge the triangle onto the
circle to see the connection between the two topics (“Unit Circle”). As the point C moves around
the circle, the value of sine is the y-coordinate of point C, the value of cosine is the x-coordinate
of point C, and the value of tangent is the slope of the line connecting the origin and point C
(“Unit Circle”). I never realized how the graph of a sine function evolved from the unit circle.
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Obviously, I knew how to find the sine of each angle on the unit circle but I never equated that
with the graph of sin(x). It was very enlightening to see it graphed out. I honestly gasped when I
saw what happened. Everything made more sense at that moment. I was able to connect what I
already knew to another representation of the same thing. That moment when everything clicked
was the most powerful feeling. This is why I chose this activity; I want my students to have that
same rush when they are in math class or even in the real world. I want them to appreciate
mathematics and how it relates to everything around them.
As already stated, I chose this activity once I had that ah-ha moment. It’s an amazing
feeling and I hope to give my students that experience someday. I want them to see math like I
see it; fascinating and always evolving. I also chose this activity because I like trigonometry and
I haven’t seen very many applications of it yet in our technology class. I decided to venture out
on my own and see what I could find. I am glad that I discovered this activity.
This activity is aligned with the NCTM standards. The students will be following the
Next Generation Sunshine State Standards which state that students must “define and determine
sine, cosine, and tangent using the unit circle, find approximate values for trigonometric
functions using appropriate technology, make connections between right triangle ratios, and
decide whether a solution is reasonable in the context of the original situation” (“Standards”,
2012). Also, the students will “recognize and use connections among mathematical ideas,
understand how mathematical ideas interconnect and build on one another to produce a coherent
whole, and finally, create and use representations to organize, record, and communicate
mathematical ideas” (“NCTM”, 2004). This activity is aligned with my goals because I have
seen yet another example of how Geometer’s Sketchpad can be used in the mathematics
classroom. I have learned how to create a lesson designed for students taking trigonometry. This
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lesson was very eye-opening for me so I have no doubt that it will be for them as well. This
lesson should excite and inspire my students to learn more about the topic and branch off their
knowledge to other areas as well. After showing them this lesson, I could have them make
connections among what they learned and the real world. For example, I could give them a
problem relating to a Ferris Wheel at a fair and how the motion of the individuals on the Ferris
Wheel changes as the wheel goes around and around. They could then graph this movement
using the knowledge they just gained.
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References
NCTM standards & focal points (2004). NCTM. Retrieved from NCTM Online Website:
http://www.nctm.org/standards/default.aspx?id=58
Proof without words: Completing the square. Retrieved from Illuminations Online Website:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
The slope game [PDF document]. Retrieved from Key Curriculum Online Website:
http://www.keycurriculum.com/resources/sketchpad-resources/free-activities/sketchpad-
algebra-activities
Standards (2012). CPALMS. Retrieved from CPALMS Online Website:
http://www.cpalms.org/Standards/FLStandardSearch.aspx
Stohl, Hollylynne Drier (2002). Coins and dice [Excel worksheet]. Retrieved from Center for
Technology and Teacher Education: Mathematics Activities Online Website:
http://www.teacherlink.org/content/math/activities/ex-randomevents/guide.html
Stohl, Hollylynne Drier (2002). Simulating Random Events. Retrieved from Center for
Technology and Teacher Education: Mathematics Activities Online Website:
http://www.teacherlink.org/content/math/activities/ex-randomevents/guide.html
Unit circle and right triangle functions [PDF document]. Retrieved from Key Curriculum Online
Website: http://www.keycurriculum.com/resources/sketchpad-resources/free-activities/
sketchpad-trigonometry-conics-and-precalculus-activities
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Unit circle and right triangle functions [Sketchpad document]. The Geometer’s Sketchpad
(Version 5.05) [Software]. Available from
http://www.keycurriculum.com/resources/sketchpad-resources/free-activities/sketchpad-
trigonometry-conics-and-precalculus-activities