Friday, December 10, 2010
Thursday, November 18, 2010
Math Project: Codes and Ciphers
EDCP 342A “Math Projects” Assignment
Names: Hong Jiang, Howard Hu, Esther Yang
Grade level: Any level in high school (grade 8-12)
Purpose: The purpose is to encourage students to be creative, arouse interests, and promote analytical skills.
Original project tasks:
1. Read description of your chosen code and work to make sense of it.
2. Research more information about your code from at least 2 other resources.
3. Make a poster and teach us about your code.
4. Prepare a 5 minute presentation, include a puzzle for the class to work on as homework.
5. Take notes on other people's presentations, there will be a question on the next test about codes.
Sources: Students can use these suggest and/or other books and Internet sources on codes:
Martin Gardner (1972). Codes, ciphers and secret writing. NY: Simon & Schuster, (J652.8 G22c)
Fred Wrixon (1992). codes and ciphers. NY: Prentice Hall. (BUS 652.8 W95c)
Handouts, graphics, etc.
Example:
Handout #1: The Keyword cipher: substituting part of the alphabet with a keyword
To create a substitution alphabet from a keyword, first write down the alphabet.
Then write down the keyword below the alphabet, followed by the remaining
unused letters of the alphabet.
To create a secret message, convert all letters from the top row to their corresponding letter on the bottom row.
These types of simple substitution ciphers can be easily cracked by using frequency analysis and some educated guessing.
Keyword: orange
A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
o | r | a | n | g | e | B | C | D | F | H | I | J | K | L | M | P | Q | S | T | U | V | W | X | Y | Z |
(might not be a good one since the last part of the alphabet has not been changed)
Keyword: zhujiang
A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
z | h | u | j | i | a | n | g | B | C | D | E | F | K | L | M | N | P | Q | R | S | T | V | W | X | Y |
Try this:
Encipher this massage: vizpiuzsngrgiem
Benefits of this project:
We have found that the original project would be a great opportunity for students to explore new and interesting mathematical things, and it would be a fun project for students to try and hopefully spark some curiosity about mathematics. Through research students would realize that mathematics is not all about numbers and calculations. We think that this project would be a good way to introduce different branches in mathematics. Also, this group project will help students to develop skills in research, presentation, communication, and teamwork.
We have found that the original project would be a great opportunity for students to explore new and interesting mathematical things, and it would be a fun project for students to try and hopefully spark some curiosity about mathematics. Through research students would realize that mathematics is not all about numbers and calculations. We think that this project would be a good way to introduce different branches in mathematics. Also, this group project will help students to develop skills in research, presentation, communication, and teamwork.
Weaknesses of this project:
The topic is a bit advanced for some grades, and some students may find it too complicated to understand some coding methods. Since every group is doing a poster presentation, it may become too routine and boring. Also, it is often times difficult for students to take notes while listening to presentations and retain information for the test question about the presentations.
Modified Project:
Grade level: Any level in high school (grade 8-12)
Purpose: The purpose is to encourage students to be creative, arouse interests, and promote analytical skills.
Sources: Students can use these suggest and/or other books and Internet sources on codes:
Martin Gardner (1972). Codes, ciphers and secret writing. NY: Simon & Schuster, (J652.8 G22c)
Fred Wrixon (1992). codes and ciphers. NY: Prentice Hall. (BUS 652.8 W95c)
Modified Project tasks:
Project Format (5-7 minutes long): Video or Skit
Theme: History of the code and how to use the code OR Creative presentation incorporating the code somehow
Project Requirements: Use of props, everyone must have equal act/talk time in the presentation (thus, everyone MUST participate), explanation of how the code works
Benefits: Practice public speaking, boost of confidence, memorable experience to refer to for the test question, incorporating something different into a math classroom
Marking criteria:
Rubric for the presentation (skit or video)
5 | 4 | 3 | 2 | 1 | |
Organization and preparation | |||||
Creative approach of presentation | |||||
Use of props effectively | |||||
Clarity and Relativity of Presentation | |||||
Ability to answer questions from audience |
Saturday, November 13, 2010
Response to creativity, flexibility, adaptivity, and strategy used in mathematics
I found the story about Ferit's strategy to be very interesting, yet the ending result of Ferit's habit of continuously using his method for every question was also expected. Students often tend to follow this patterning when they have realized that they have come up with something useful and try to apply it to as many cases as possible, regardless of how inefficient as it may seem in certain cases. Although such habits are hard to break, it is up to the teacher to help correct such habits by helping the students expand their knowledge of mathematical methods of approaching and solving questions/problems.
The emphasis of the article are mainly focused on creativity, flexibility, and adaptivity. These three aspects are interlinked with each other when it comes to mathematics because for students who are capable of achieving all of these skills, they will be capable of dealing with math problems/questions easier due to their arsenal of mathematical knowledge.
Of course being capable of attaining and mastering all three of these aspects at the secondary school level is indeed a complicated task to achieve. Teachers should try to motivate their students to create their own opinions and thoughts on how to solve a problem and then guide them to formulate and eventually construct the actual formulas and equations. During my short practicum, I was glad to see many mathematics teachers avoiding the lecturing method of teaching and more of retrieving information out of students that led to the grand formula needed to solve all the questions in that section.
It is to be noted that it is human nature that people choose the path that works for them best, regardless of how inefficient it may seem to others. These habits of using certain methods for certain situations are also a way of how people have become accustomed to dealing with these problems. What works well for someone, may not be as useful for someone else, but there is no harm in suggesting other methods because people may gain some insight on either adapting to the newly learned method or gain some sort of self realization on how to improve their own methods.
The emphasis of the article are mainly focused on creativity, flexibility, and adaptivity. These three aspects are interlinked with each other when it comes to mathematics because for students who are capable of achieving all of these skills, they will be capable of dealing with math problems/questions easier due to their arsenal of mathematical knowledge.
Of course being capable of attaining and mastering all three of these aspects at the secondary school level is indeed a complicated task to achieve. Teachers should try to motivate their students to create their own opinions and thoughts on how to solve a problem and then guide them to formulate and eventually construct the actual formulas and equations. During my short practicum, I was glad to see many mathematics teachers avoiding the lecturing method of teaching and more of retrieving information out of students that led to the grand formula needed to solve all the questions in that section.
It is to be noted that it is human nature that people choose the path that works for them best, regardless of how inefficient it may seem to others. These habits of using certain methods for certain situations are also a way of how people have become accustomed to dealing with these problems. What works well for someone, may not be as useful for someone else, but there is no harm in suggesting other methods because people may gain some insight on either adapting to the newly learned method or gain some sort of self realization on how to improve their own methods.
Wednesday, November 10, 2010
Hundred Squares - Math Problem
How few straight lines are required on a page in order to have drawn exactly 100 squares?
Online Word Problem Analysis
A Hat of a Different Color
The wise teacher offered the three noisiest students a deal. He showed them that he had two red hats and three blue hats.
The deal worked like this:
The three students would close their eyes, and while their eyes were closed, the teacher would put a hat on each of their heads (and hide the other two hats). Then, one at a time, the students would open their eyes, look at the other two students' heads, and try to guess which color hat was on their own head.
Any students that guessed correctly would have no homework to do the rest of the semester. But any students that guessed wrong would not only have to do their own homework, but they would have to help grade everyone else's work also. The students drew numbers to see who would guess first. Then they closed their eyes and the wise teacher put a hat on each one's head. Arturo, who was to go first, opened his eyes, looked at the others' heads, and said he didn't really want to play. He couldn't tell for sure and he didn't want to guess in case he was wrong. Next, Belicia opened her eyes and looked at the others' heads. She also thought about the fact that Arturo had said he couldn't tell. Then she said she didn't want to risk it either. She couldn't tell for sure.
Carletta was third. She just stood there with her eyes still closed tightly and a big grin on her face. "I know what color hat I have on," she said. And she gave the right answer.
Your problem is to figure out what color hat Carletta had on and how she knew for sure. Remember: Carletta didn't even look!
1. is it practical?
2. is the imagery memorable?
3. can it be interpreted in more than one way?
4. can it be solved with the given information?
5. would kids be able to interpret it as intended or not?
6. is there anything strange about it?
7. how would you rewrite it, expand it, use it?
1. Yes, this question seems practical to me because it is easily applicable to any students in a classroom.
2. Since this situation is easy to set up, yet the answer to this question is tricky, students would remember this question's imagery easily (especially since the reward and penalty are at such high stakes).
3. I do not think that it is possible to find this word problem to be ambiguous in any other way... it is quite straight forward.
4. Yes, this question can be solved with the given information! There is a simple solution and a more complex solution as well, if you are interested, you can check this question out at the following link: http://mathforum.org/library/drmath/view/55638.html
5. Students should be able to interpret this question easily. If not, then the teacher can always do a quick and easy demo in class to demonstrate.
6. There is nothing strange about this question at all, except that I can see students wondering why the first two students could not guess the color of their own hats, but if you take this information as a hint, then you can solve this question easily!
7. I would not rewrite this question in anyway as it is already a well structured and logical word problem (at least for me), I would expand this question to adding more hats or more students or more hats of different colors to make this question more interesting. This is a great question to lead into the topic of probability or logical reasoning.
The wise teacher offered the three noisiest students a deal. He showed them that he had two red hats and three blue hats.
The deal worked like this:
The three students would close their eyes, and while their eyes were closed, the teacher would put a hat on each of their heads (and hide the other two hats). Then, one at a time, the students would open their eyes, look at the other two students' heads, and try to guess which color hat was on their own head.
Any students that guessed correctly would have no homework to do the rest of the semester. But any students that guessed wrong would not only have to do their own homework, but they would have to help grade everyone else's work also. The students drew numbers to see who would guess first. Then they closed their eyes and the wise teacher put a hat on each one's head. Arturo, who was to go first, opened his eyes, looked at the others' heads, and said he didn't really want to play. He couldn't tell for sure and he didn't want to guess in case he was wrong. Next, Belicia opened her eyes and looked at the others' heads. She also thought about the fact that Arturo had said he couldn't tell. Then she said she didn't want to risk it either. She couldn't tell for sure.
Carletta was third. She just stood there with her eyes still closed tightly and a big grin on her face. "I know what color hat I have on," she said. And she gave the right answer.
Your problem is to figure out what color hat Carletta had on and how she knew for sure. Remember: Carletta didn't even look!
1. is it practical?
2. is the imagery memorable?
3. can it be interpreted in more than one way?
4. can it be solved with the given information?
5. would kids be able to interpret it as intended or not?
6. is there anything strange about it?
7. how would you rewrite it, expand it, use it?
1. Yes, this question seems practical to me because it is easily applicable to any students in a classroom.
2. Since this situation is easy to set up, yet the answer to this question is tricky, students would remember this question's imagery easily (especially since the reward and penalty are at such high stakes).
3. I do not think that it is possible to find this word problem to be ambiguous in any other way... it is quite straight forward.
4. Yes, this question can be solved with the given information! There is a simple solution and a more complex solution as well, if you are interested, you can check this question out at the following link: http://mathforum.org/library/drmath/view/55638.html
5. Students should be able to interpret this question easily. If not, then the teacher can always do a quick and easy demo in class to demonstrate.
6. There is nothing strange about this question at all, except that I can see students wondering why the first two students could not guess the color of their own hats, but if you take this information as a hint, then you can solve this question easily!
7. I would not rewrite this question in anyway as it is already a well structured and logical word problem (at least for me), I would expand this question to adding more hats or more students or more hats of different colors to make this question more interesting. This is a great question to lead into the topic of probability or logical reasoning.
Monday, November 1, 2010
Practicum story
I think the most interesting thing that occurred during my practicum was the first day that I met and taught a grade 9 class. I was nervous since it was my first actual lesson in a highschool classroom, but the lesson went the way I had planned with minimal distractions/disruptions. I noticed from last day of observing this grade 9 class, that there were a few students who were the noisy trouble makers who were being constantly reminded to pay attention. To my surprise they seemed quite relaxed and calm when I taught and during my class I assigned a few questions for the class to do, I began circulating the class as the students worked on the problems. As I walked by the supposed noisy trouble makers, one of them started a conversation with me, asking me about "how I got so big". I explained that I started working out since grade 10 and never used any supplements or drugs to enhance my physique and they payed attention to everything I said. Once the students seemed like they had finished working out the questions, I went back to the board and asked students to come up and write their solutions. Near the end of my lesson, I gave the class a tricky question, which was confusing to solve. However, the biggest surprise was about to come... the student who raised their hand to give the solution to this tricky question was actually one of the trouble making students, he approached the whiteboard to write his answer, while the class sat in their seats shocked that of all people, this student was the one to come up to the board. I heard comments as he approached the board like "oh my god, he's going to write something!", "how did he get it and we didn't?", "wow..." and to everyone's amazement, this student had shown the proper solution. Once the student finished writing his solution down, he asked me if his solution was correct, but he ended up nicknaming me, rather than calling me as Mr. Hu.
It was from that day and on that that student and his group of friends started calling me by the nickname they had given me... Mr. Muscles.
I don't mind that name honestly, but I laugh everytime I hear it going down the hallway and everyone stares, although it may sound funny, but given my physique, it helped create a connection to these students and for the remaining classes, when I needed the class's attention, these trouble makers actually help me do so at times too.
It was from that day and on that that student and his group of friends started calling me by the nickname they had given me... Mr. Muscles.
I don't mind that name honestly, but I laugh everytime I hear it going down the hallway and everyone stares, although it may sound funny, but given my physique, it helped create a connection to these students and for the remaining classes, when I needed the class's attention, these trouble makers actually help me do so at times too.
Thursday, October 14, 2010
Team Microteaching Reflection:
As usual, time management is always an issue when it comes to teaching new lessons. Although my group had an excellent lesson plan set up, there was simply too much to cover within the 15minute time span. We were able to complete the main objectives in our lesson plan, but it was sort of a rushed job. Majority of the group felt as if the introduction was way too fast for any student to follow and understand fully. The group also said that they needed more time to understand and use the algebra tiles, which is understandable since it is a new concept to learn.
Overall, everyone enjoyed our lesson and was able to learn multiple methods of factoring quadratic trinomials (algebraically, using algebra tiles, and the virtual manipulatives) and was able to see that factoring can be related to the concept of the area of a rectangle. I find these microteaching lessons to be highly useful in showing everyone what teachers have to deal with when teaching a lesson, that not everything will go as planned and adjustments may be needed due to timing or the lesson being interrupted by a student's curious question. I think that within a lesson plan, there should be spaces for teachers to mention when to create a 'check point(s)' to make sure everyone is paying attention and caught up with the lesson and just to confirm that the teacher isn't going too fast or slow for the student's understanding of the lesson.
Overall, everyone enjoyed our lesson and was able to learn multiple methods of factoring quadratic trinomials (algebraically, using algebra tiles, and the virtual manipulatives) and was able to see that factoring can be related to the concept of the area of a rectangle. I find these microteaching lessons to be highly useful in showing everyone what teachers have to deal with when teaching a lesson, that not everything will go as planned and adjustments may be needed due to timing or the lesson being interrupted by a student's curious question. I think that within a lesson plan, there should be spaces for teachers to mention when to create a 'check point(s)' to make sure everyone is paying attention and caught up with the lesson and just to confirm that the teacher isn't going too fast or slow for the student's understanding of the lesson.
Wednesday, October 13, 2010
Lesson Plan: Factoring Quadratic Trinomials Using Algebra Tiles
Lesson Plan: Factoring Quadratic Trinomials Using Algebra Tiles
EDCP 342: Lesson Plan
Topic: Factoring Quadratic Trinomials Using Algebra Tiles
Group: Howard, Maria, Raman
Intended Students: Grade 10 Fundamentals and Pre-calculus
Suggested student worksheet format:
Name: __________________ Topic: Factoring Quadratic Trinomials
1. Factor: x^2 + 5x + 6
2. 2x^2 + 5x + 2 = ( )( )
3. x^2 + 6x + 9 = ( )( ) = ( )
4. x^2 + 7x + 2 = ( )( )
5. x^2 + kx + 6 = ( )( ) or ( )( )
6. One thing I learned today is _____________________________________.
7. Algebra tiles do/do not help in understanding factoring trinomials because_______________________________________________________.
EDCP 342: Lesson Plan
Topic: Factoring Quadratic Trinomials Using Algebra Tiles
Group: Howard, Maria, Raman
Intended Students: Grade 10 Fundamentals and Pre-calculus
WHAT | HOW LONG | MATERIALS | |
BRIDGE | Give everyone a small sheet of paper. In 5 seconds, write as many factors of 60. | 1 minute | |
LEARNING OBJECTIVES | Using the algebra tiles, students will be able to: 1. Factor quadratic trinomials, including perfect square trinomials 2. Relate the dimensions of a rectangular area with finding the factors of a trinomial 3. Experience three modes of factoring trinomials: algebraic method, concrete algebra tiles, and virtual manipulatives | ||
TEACHING OBJECTIVES | 1. Maximum engagement of all students 2. Individual hands-on-learning using math manipulatives (algebra tiles) 3. Demonstration of using virtual manipulatives in factoring trinomials | ||
PRETEST | Each student will be given a worksheet sheet 1. Factor the trinomial: x^2 + 5x + 6. Write answer in worksheet. Ask for answer. Show of hands who got the correct answer. Ask a student to briefly explain his/her answer. | 2 minutes | |
PARTICIPATORY LEARNING | 1. State the learning objectives. Tie-up bridge and pre-test to objectives. 2. What are the factors of 6? (3 and 2) How can we illustrate this geometrically? (Draw a 3 by 2 rectangle, divided into 6 squares). How are factors related to dimensions (of length and width), and product related to area? (Finding the factors of a number is the same as finding the dimensions of a rectangle whose area is the number). Will this geometric representation work for finding factors of a trinomial? 3. Distribute/introduce the algebra tiles, as a geometric method of finding factors of trinomials. Each student will be given a complete set of tiles, with a transparent tile board. Walk the students through the 3 different tile sizes representing x^2 (green), x (white) and 1 (red). Explain that x is a variable that can represent any positive number. 4. Assemble 2-green x^2, 5-white x tiles and 2-red 1-tiles. If all the 9 pieces represent the area of a rectangle, what algebraic expression represents this area? (2x^2 + 5x + 2) How can we get the dimensions of this rectangle? * In your worksheet, complete equation #2: 2x^2 + 5x + 2 = (2x + 1)(x + 2) 5. Empty your tile board. For our second rectangle, assemble 1- green x^2, 6-white x and 9-red 1-tiles into a rectangle. What expression represents the area of this rectangle? (x^2 + 6x + 9). What are the factors? (x + 3) and (x + 3). What do you notice with our rectangle? (It is a square). Introduce the perfect square trinomial (PST). * In your worksheet, complete equation #3: x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2 6. Virtual Manipulatives: Reiterate that finding the linear factors of a quadratic trinomial is very much related to finding the dimensions of a rectangle that contain the trinomial. The internet is full of virtual manipulatives that offer fun, creative, and interactive ways of factoring trinomial, which may appeal to today’s technology-savvy students. Factor x^2 + 7x + 12. (x + 4)(x+3). | 9 minutes | * Algebra tiles * Virtual manipulatives |
POST-TEST | Using your algebra tiles, find values of k, where x^2 + kx + 6 factors into 2 binomials. (k = 5, 7). Write answer in #5 of your worksheet. | * Algebra tiles | |
SUMMARY & WRAP-UP | Ask students what they have learned today, which should touch the following points: 1. That to the concept of factoring is very much related to finding the dimensions of a rectangle of a given area. 2. That a quadratic trinomial factors only if one can arrange it into a rectangle. 3. That we know that a trinomial is a perfect square if the tiles neatly arranges into a square, with 2 equal dimensions. 4. Ask students to complete # 6 & 7 of their worksheet. Collect worksheets. | 3 minutes |
Suggested student worksheet format:
Name: __________________ Topic: Factoring Quadratic Trinomials
1. Factor: x^2 + 5x + 6
2. 2x^2 + 5x + 2 = ( )( )
3. x^2 + 6x + 9 = ( )( ) = ( )
4. x^2 + 7x + 2 = ( )( )
5. x^2 + kx + 6 = ( )( ) or ( )( )
6. One thing I learned today is _____________________________________.
7. Algebra tiles do/do not help in understanding factoring trinomials because_______________________________________________________.
Subscribe to:
Comments (Atom)