kellymosier.files.wordpress.com€¦ · Web viewThis applet taught me the reason why. Basically, you have a yellow square and a blue square (“Proof”). The yellow square’s

Kelly Mosier

Activity #1: Coins and Dice

2 | Reflections of Each Activity

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.


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.


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.


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


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.


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


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


to learn just how amazing the program is. It will foster student’s conceptual understanding of

mathematics while still being entertaining.


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.


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


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.


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


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

Documents

kellymosier.files.wordpress.com€¦ · Web viewThis applet taught me the reason why. Basically, you have a yellow square and a blue square (“Proof”). The yellow square’s