Communication in Mathematics

Communication is an essential part of Mathematics. Mathematicians not only need to know how to add and subtract numbers, but also how to communicate to the world the new discoveries that are found each day. This communication starts at an early age with students in a classroom explaining the way the student discover the answer to a problem and learning the unique language of mathematics. Communication will be used in a variety of mode and settings. Students will also know how to communicate effectively using mathematical language and symbols because the ideas will be generated and shared.

Communication involves a variety of modes: pictures, written symbols, spoken language, relevant situations, and manipulative. Each way the students has to find the link between each one and how the variety of modes represent the same problem, but in multiple ways. Students use pictures, whether told to draw a picture by the teacher or the student creates a picture on their own, to create a better understanding of the problem and use this picture to help think through the process needed to take in order to find the solution. Manipulatives are objects that can be touched, moved, and stacked. This allows the students to physically see the problem being manipulated and how a small piece is related to the whole. Students are able to communicate in number and symbols, but they are also able to communicate in the spoken language of mathematics. The students can explain their reasoning and process of the result they came up with to help better understand problems related to the one the student solved. A relevant situation can be any context that involves appropriate mathematical ideas and holds interest for children, and connected to a real-life situation. This simple act will create interest in the problem for the students and allow them to invest time into the problem to come up with the solution. Using relevant situation can also show the student how the material can be used in everyday life.

Students are able to use these variety of modes in a variety of setting: small group work, whole class work, partner work, and individual work. This will allow the students to think together and discover the solution to a problem that they can all understand using the different modes of communication. With the discovery  of the solution, the students will be able to write their process of thinking using the language of mathematics and symbols. This written report or oral report of their process can help better understand the problem as a whole, and the student are then able to help others in sharing their ideas for the problem.

When students understand the problem and how they discovered the solution, this will create confidence in mathematics. When they gain the confidence, the students will use communication to share their ideas and understanding of many real world problems.  

References

Clement, Lisa. Pictures Written Symbols Manipulatives Relevant Situations Spoken Language (n.d.): n. pag. Web.

"THE FIRST FOUR STANDARDS STANDARD 2 - COMMUNICATION." New Jersey Mathematics Curriculum Framework. New Jersey Mathematics Coalition, n.d. Web. 26 Nov. 2014.

Trigonometric Identities

As math students and future teachers, we all have to memorize all the different Trigonometric Identities. With this simple diagram you are able to get 21 different Trigonometric Identities with four different tricks. If I was presented this diagram in High School (this was presented to me in my college education class), I would have been able to memorize some of the Trigonometric Identities a lot faster. Below is the Trig Identity Trick:

Now I will break down each one of the four tricks in this one diagram. The first trick is finding the Reciprocal Identities. 

We will follow the red lines in the diagram. From the start of one identity on the line equals 1 divided by the end identity of line. 

 The second trick is finding the Pythagorean Identities. 

We will now look at the triangles. We will look at the left corner of the triangle and travel around the triangle clockwise. The point of the triangle is what the Pythagorean Identity is equal to and the top two points are added together. Remember that all the identities are squared!

The last two tricks go hand in hand. The third and fourth trick is finding the Quotient Identities and relations of the other identities.

Quotient Identities

Other Relations 

going around clockwise

going around counter clockwise

This is a great tool in which students and teachers can use all through out Trigonometry. This will relieve stress in memorizing all the trigonometric identities and focus more on applying the identity to problems. This graph is a great tool to use int he classroom because it uses only the six basic Trig Identities and creates many different formulas that they students will use multiple times through out the life time of math. Instead of the students memorizing all the different formulas they just have to remember how to construct this graph and how to use it. All of the formals are within this graph! 

Math Superheros

When teachers start a lesson for new material, they need to create a "hook lesson" for students to become interested in the new material. The following activity is a GREAT game to start an introduction to the topic of exponents. 

POWER UP! 

Create your Hero: 

Step one: Choose your abilities

Super Speed        Super Strength        Invulnerability       Telepathy           Telekinesis 

Lightning              Fire                         Ice                       Earthquake        (Creature) power 

High teach Gadgets   Blaster                Magic                  Power ring/gem   Stretchy 

Greater Ability 1: (base: 4)    Lesser Ability 2: (base: 3)    Random Ability 3: (base: die roll)

In the battle, you choose which ability you're using each round before you roll the dice. For Greater and Lesser Ability, you will use one die and whatever the outcome of the roll that number will become the exponent. For Random Ability, you will roll two dice and choose which is number is the base and which number is the exponent. 

Step Two: Origin Story 

Name: 

How did you get your powers?   

Mutant                Story: 

Alien 

Magic 

Science

Other 

Step Three: Are you a             Hero   or     Villain 

Step Four: Picture of your character. 

BATTLE: 

Each battle you will use each power once. *Note: it is ok for heroes to fight heroes and villains to fight villains.

Battle last for three rounds. Roll the dice to see who goes first. 

ROUND 1: first player chooses an ability. Second player chooses an ability. Roll the dice at the same time to discover the exponent for the base of the ability. Highest exponent total wins round one. 

ROUND 2: Switch! Second player and first player chooses an ability they haven't used in round 1. Roll the dice at the same time to discover the exponent for the base of the ability. Highest exponent total wins round two. 

ROUND 3: Both players will use the remaining ability. Roll the dice at the same time to discover the exponent for the base of the ability. Highest exponent total wins round three. 

TIEBREAKER: The players will use their random ability one more time. 

VARIATIONS: 

• Team Play: a team of super heroes fight another team of villains or heroes. Teammates multiply their scores each round.  
• Character design: players assign their power base. The student choose the base (1 or higher) for each power so that the three abilities add up to 12.

This game can be used in a lesson plan to teach exponents and to make the connection that we sometimes refer to the exponents as a power. This will also get the imagination of the students and get them really into the math lesson that will follow. The teacher could also have the student describe the battles in a paper and explain how they either won or lost the battle. They would have to explain what the exponents mean and the total of at least six different exponents. They would also have to explain for their random ability, why they choose the base number and the exponent number. This is a great connection to refer back to through the chapter about exponents. 

As the students learn more about the exponent rules, the teacher could bring back this game and create new rules about using the exponent rules.

City of Angles

City of Angles

This project was presented to Mrs. Meyer's 9th and 10th grade geometry class after the introduction of interior, exterior, and corresponding angles. This project brought the students into using these three angles into a real world problem. After the student complete the activity, they will be able to demonstrate their knowledge of parallel lines with a transversal and will be able to show when angles are congruent or supplementary given parallel lines and a transversal. 

Overview of Project:  

For this project, each student will make his or her own street map for a fictional city. This city will consist of: 

1. Six streets that are parallel to each other (each will be named for reference). 

2. Two transversal streets (each will be named for reference). 

3. Traffic lights or stop signs at four different intersections. 

4. The following building, represented in your city: 

     - Post Office 

     - Bank 

     - Fire Department 

     - Police Station 

     - Gas Station 

     - School 

     - Restaurant 

     - Grocery Store 

     - Courthouse 

     - Your House 

Instructions: 

Please place the buildings in the following locations: 

1. Your house and the school at congruent alternate interior angles. 

2. The post office and the bank at same side interior angles. 

3. The fire department and police station at congruent alternate exterior angles.
4. The restaurant and courthouse at non-congruent alternate interior angles.
5. the gas station and grocery store at congruent corresponding angles.

BE CREATIVE!!

Above is my creation of: City of Math! (sorry about not being artistic, I cant draw for the life of me)

My thinking process as I created my City of Angle
I created six parallel lines. I had to think what a parallel line was and I know that they could not intersect each other. Then I created two transversals that intersected the six parallel lines. I created mine so that they did not intersect each other on my graph, but a student could create that type of transversals. I then ha to place stop signs at the intersections of a few lines. I know that intersections are where two lines cross each other creating a t-shape in the image. I next had to place my building in the city. I first read all of the placements of the city to see if any of the building corresponds to more than just one building. I noticed that each building only corresponds to another building within the city. I started with the buildings that had congruent angles (your house, the school, the fire department, the police station, the gas station and the grocery store). Once I placed these buildings, I had four more buildings to place. I first did they non-congruent angles ( the restaurant and the courthouse) because I knew that it was the opposite of congruent angles. Lastly, I placed the final two buildings (The post office and the bank) because these building did not relate to corresponding or non-corresponding, but had to deal with interior angles.

This is one way of thinking to create this city. Each students will take different steps in creating their own city. A great follow up question would be asking the students explain why they place their building in the spot and relate it back to corresponding angles.


Below are some student's cities in the making!

FINAL City (Millbrooks Town) by a student:

When the project is completed the students will have a better understand of interior, exterior, and corresponding angles and it also allows the students get artistic and have a little bite of fun with math. 

Problem of the Week

Problem of the Week
I was sitting in a 10 grade Geometry class at Union High School that I observe Tuesdays and Thursdays, and the teacher, Mrs. G, passed out a worksheet that was titled: Problem of the week. I looked at the worksheet and this is what I saw:

Four Step Problem Solving Sequence 

Problem of the week that the students and teacher are going to work on. 

Understanding the Problem

Analyze the problem and establish what is asking you to do or find. Underline any important words or facts. Define any key vocabulary words. 

Create a Plan 

Devise a plan for solving the problem and/or make an illustration to demonstrate your understanding of the problem. 

Carry of the Plan 

Write a complete sentence explaining the answer and remember to attend to precision. 

Reflecting 

Evaluate your solution, is it reasonable? Can you find another valid approach to solve the problem?

Each part of the problem is done in class with the students and the teacher. There are four steps for one problem, one step for each day. On the final day, the students will do a problem to the one similar to the one they worked on all week all on their own. 

Mrs. G finds the problems that are either review from the previous week or a problem that will be coming up in the new week. This problem is challenging to the students, but when the week is done, they be able to do a similar problem.

Below is one of the problems Mrs G presented the class, and I completed the worksheet.

This worksheet can help focus the students attention to what the question is asking and following through with the answer but also another way of looking at the same problem. This worksheet breaks down a problem into steps of thinking, so the students will receive the full understanding of a problem. 

Counting Circle

Counting Circle
"What in the world are you talking about? What in the world is a counting circle?"; you might be asking yourself as you read the title. I will tell you it is exactly as the title says...you are in a circle counting.You may be thinking: "O you are counting 1,2,3,4...1010. that seems way easy." In all reality, the counting circle can be that easy or it can be created into a harder game.

The Game
The set up of the game is very simple. You gather your group/students into a large circle. The teacher (or student) will pick a number for the counting circle to start with. Then the teacher (or student) will decide on the number that will be added or subtracted to the starting number. The first student in the circle will take the original number and add or subtract the teacher's number in his/her head. The next student in the circle will take the previous students answer and add or subtract the teacher's number in his.her head. This will continue around the circle until the teacher decides the number line is done. Then the teacher will pick a student and ask the group what their number would be once it was their turn.

Example
Starting number: 432
Teacher's number: +32
1st student: 464
2nd student: 496
3rd student: 528
continue...

Experience from Game
What can we learn from counting in a circle? When the game is over ask the students to think about the way they got their number in their heads. Then the teacher will randomly pick a student and ask to explain how they came up with their answer. This way will be written on the board. This will continue until all possible ways to come up with the answer is up on the board. This shows students that there are many ways to get the correct answer. Some ways will be easier than others, while others might make more sense to the student might not make much sense to the student standing next to them.

Side note: Continue the counting circle even if a student gets the number wrong-just write the number on the board. This will create a challenge for the following students.