Kamis, 31 Desember 2009
Rabu, 30 Desember 2009
To Do + Math on Vacation
The Mathniños at the Modern Wing
Math Teachers at Play 22
I'm hosting this months blog-carnival, and I'd love your submissions from your own blog or nominations from blogs you read. Submit nominations at the blogcarnival site, or using the widget at the bottom of the right column on this page.
I never had a chance to write about Carnival 21 at Math Mama Writes... but Jonathan (JD2718) had a nice post about a subtle shift that produced a good effect.
An aside: I got to meet Math Mama over break and Sue was as nice and interesting as you'd think from her blog. We were going to meet up with Maria from Teaching College Math, but the weather prevented it (sorta).
Biggest Math Story of the Year
Time Magazine has submitted that the proof of Langlands' Fundamental Lemma was the 7th Biggest Scientific Discovery of 2009. A quite technical exposition of the Lemma (not the proof) is here. (Hat tip Peter Woit of Not Even Wrong). I do not have an understanding of the math, but it concerns a connection between Galois group theory (about solutions to algebraic equations) and general linear algebra representation theory (think matrices on steroids). One of the most famous theorems that uses the Lemma has Femat's Last Theorem as a corollary. Powerful stuff.
It's nice for students to know that math is ongoing, and also to hear terms like lemma, theorem and corollary used genuinely. I also like how this emphasizes the power of connections in mathematics, and how even (or especially) the world's most powerful mathematician's start problem solving by solving simpler problems or finding another way to put the problem. It also should be noted that Ngo's proof was submitted in 2008, but took until this year to verify.
Math (Art) on Vacation
My wife and I recently won tickets to the Cirque de Soleil show Banana Shpeel (which I would recommend.) On the quick trip to Chicago this week sans kids (hat tip: grandparents!) we had more time in the Art Institute than we would have had otherwise.
In the geometry class I've been writing about, we do one constructive project in which students build a polyhedron for the following context:
But at the Art Institute I had two ideas for new variations. The Modern Wing in itself is an inspiration as it is filled with beautiful proportions and rectangles with more connections and relationships than you could ever hope to count.
The Chess Set
This wonderfully sculpted set filled me with visions of two people working together to make a set of polyhedra vs non polyhedra. They could do found or constructed objects. It will require connections (what makes the difference between the king and queen, how will players know this is a rook, what's the difference in the bishops between the sides...) and analysis of the solids and their volume.
The Building
I so wanted to see the isometric drawings for this. I wouldn't insist that the building be built, which gives me pause, but I would want two different representations that would help someone build it. Students should design their own advanced building and think about it in scale
I'm curious to see where these ideas go, and what students make of them. I would like to hear more from other teachers about from where their ideas come, and the process they follow to implement them and refine them in the classroom. I think we teachers do a lot of our sharing as a product exchange.
Calder Mobile
Math Teachers at Play 22
I'm hosting this months blog-carnival, and I'd love your submissions from your own blog or nominations from blogs you read. Submit nominations at the blogcarnival site, or using the widget at the bottom of the right column on this page.
I never had a chance to write about Carnival 21 at Math Mama Writes... but Jonathan (JD2718) had a nice post about a subtle shift that produced a good effect.
An aside: I got to meet Math Mama over break and Sue was as nice and interesting as you'd think from her blog. We were going to meet up with Maria from Teaching College Math, but the weather prevented it (sorta).
Biggest Math Story of the Year
Time Magazine has submitted that the proof of Langlands' Fundamental Lemma was the 7th Biggest Scientific Discovery of 2009. A quite technical exposition of the Lemma (not the proof) is here. (Hat tip Peter Woit of Not Even Wrong). I do not have an understanding of the math, but it concerns a connection between Galois group theory (about solutions to algebraic equations) and general linear algebra representation theory (think matrices on steroids). One of the most famous theorems that uses the Lemma has Femat's Last Theorem as a corollary. Powerful stuff.
It's nice for students to know that math is ongoing, and also to hear terms like lemma, theorem and corollary used genuinely. I also like how this emphasizes the power of connections in mathematics, and how even (or especially) the world's most powerful mathematician's start problem solving by solving simpler problems or finding another way to put the problem. It also should be noted that Ngo's proof was submitted in 2008, but took until this year to verify.
Math (Art) on Vacation
My wife and I recently won tickets to the Cirque de Soleil show Banana Shpeel (which I would recommend.) On the quick trip to Chicago this week sans kids (hat tip: grandparents!) we had more time in the Art Institute than we would have had otherwise.
In the geometry class I've been writing about, we do one constructive project in which students build a polyhedron for the following context:
Hoity Toity, the upscale chain of Haute Couture for the masses, is having a competition to design new and original knick-knack boxes. Being an accomplished mathematician, you have an unfair advantage, which you intend to exploit to the maximum. Boxes must have a volume of at least 1 liter. (1000 cm3).Students do a terrific job in general. One student this semester made a dog-shaped polyhedron. Cool. Some classic Archimedean solids, as well as prisms and antiprisms and such.
But at the Art Institute I had two ideas for new variations. The Modern Wing in itself is an inspiration as it is filled with beautiful proportions and rectangles with more connections and relationships than you could ever hope to count.
The Chess Set
The Building
I'm curious to see where these ideas go, and what students make of them. I would like to hear more from other teachers about from where their ideas come, and the process they follow to implement them and refine them in the classroom. I think we teachers do a lot of our sharing as a product exchange.
Photograph puzzle maker
Create a Jigsaw puzzle using this online facility. It turns any uploaded picture to a jigsaw puzzle. You may use a Mathematical picture here.
Try it out!
Try it out!
Number triangle 1.1
Number triangle 1.1 is a freely downloadable Math puzzle.
The purpose of this task is to get the largest total at the top of the triangle by putting in numbers in the bottom line. The rules are that the numbers on the bottom line must add up to a specified number, for example 10.
To get the number on the line above we have to follow simple rules
Odd next to an even number = Multiply them and put answer in box above.
Odd next to Odd = Subtract and the difference goes in the box above.
Even next to Even = Add them and put the answer in the box above.
General Features include:
Simple Type in the base numbers and the totals will be automatically worked out.
4 different levels with different Totals. (only 3 available in Freeware version).
To get the number on the line above we have to follow simple rules
Odd next to an even number = Multiply them and put answer in box above.
Odd next to Odd = Subtract and the difference goes in the box above.
Even next to Even = Add them and put the answer in the box above.
General Features include:
Simple Type in the base numbers and the totals will be automatically worked out.
4 different levels with different Totals. (only 3 available in Freeware version).
Selasa, 29 Desember 2009
Senin, 28 Desember 2009
Math challenge 2
Its a free downloadable digital flash card system that can be customised to the level of learner.
Try it!
Download link
Try it!
Download link
Minggu, 20 Desember 2009
Rabu, 16 Desember 2009
Selasa, 15 Desember 2009
Macro Plan: Formative assessment in Mathematics
Formative tests: These tests are assessments/tools designed for learning. They are designed for the improvement in the knowledge and understanding of concepts of learners. Mathematics teaching learning requires focus on development of various skills viz. arithmetic, logical, analytical, drawing, plotting etc. A variety of formative testing tools are available which may test these desired skills.
In Mathematics students may have concept understanding difficulties resulting in conceptual errors ,may do calculation errors ,may wrongly interpret word problems etc.
All these kinds of problems are to be resolved at the right time for improving further learning. Based on the feedback of learning of students via various tools, problems of students are diagnosed .Continuing further, remedies are designed.
In Mathematics students may have concept understanding difficulties resulting in conceptual errors ,may do calculation errors ,may wrongly interpret word problems etc.
All these kinds of problems are to be resolved at the right time for improving further learning. Based on the feedback of learning of students via various tools, problems of students are diagnosed .Continuing further, remedies are designed.
ASSESSMENT TOOL #1
C.W/H.W/ASSIGNMENTS:
C.W. /H.W. notebooks of students are to be assessed by the teacher in specified duration of each Formative assessment.
Following four parameters are to be taken care of:
1. Regularity: Submission on time (1 mark)
2. Maintenance/neatness (1 mark)
3. Completion of work (2 marks)
4. Correction work (if any) (1 mark)
Assignments:
Assignments are to be given in each chapter. It may include all types of questions according to the design of question paper (Multiple choice questions, Short answer type and long answer type).
Following four parameters are to be taken care of:
1. Regularity: Submission on time (1 mark)
2. Maintenance/neatness (1 mark)
3. Completion of work (2 marks)
4. Correction work (if any) (1 mark)
Note: Questions in assignments should not be a replica of questions given in the prescribed text book. The number of questions should not exceed 15.
Please Note:
Talk to students that C.W./H.W./Assignments will be assessed.
Observe at least 4-5 class work / home work records in each Formative assessment duration
If a child is unable to complete the work due to some reason, then give him/her time for the same.
ASSESSMENT TOOL #2
Oral test: (Class response)
It is quite a reliable formative testing tool. It focuses on testing understanding of concept, Logical thinking and speaking skills. In Mathematics children may be asked basic concepts, formulae, important results, theorems etc. The class response of students is observed while teaching learning of each unit and on the basis of observations during a period of each formative assessment students are given marks accordingly.
Some suggestive ways of conducting an oral test:
1. While teaching teacher may ask questions to students Roll No. wise and keep the record in the diary. The whole task can be distributed in the duration in which the unit is taught.
2. One day may be assigned for taking oral test. Students need to be informed before the test.
3. Oral test may be taken in the Math lab period also.
4. Teacher may initiate a discussion in the classroom on some general topics like use of Mathematics in daily life, brainstorming on the utility of contents of chapter taught, recreational Mathematics etc and observe the participation of students.
Please Note:
Although it is quite a good tool for assessment, but at the same time it is to be kept in mind that it creates maximum anxiety in students.
When oral assessment process is on, a teacher needs to show a high degree of patience.
Our purpose of oral assessment is to help a child to learn and not to create any fear for learning.
ASSESSMENT TOOL#3
Pen and Paper test:
Total marks allotted for pen and paper test in each Formative assessment duration is 15 marks.
It may include classroom worksheets and class tests taken during teaching learning in each formative assessment duration. The number of classroom worksheets/class tests may vary according to the need of the chapter.
Classroom Worksheets (5 marks)
Teacher may prepare worksheets of 10 minutes duration which can be given to students either before a lesson (for testing previous knowledge) or after teaching a topic (as a recapitulation exercise).
Suggestive format for Worksheet Class Worksheet Type 1
Total No. of questions 10
4 MCQ’s
4 Fill in the blanks
2 True/False
Note: Design MCQ’s for testing understanding of concept taught, Fill in the blanks for testing arithmetical skills and true/false questions type for testing analytical and logical reasoning.
Class Worksheet Type 2
Crossword
Class test (10 marks)
Duration of each test would be 30 min to 40 min. Short answer type and long answer type questions may be asked.
Please Note:
At least one class test should be taken during one Formative assessment period.
If a good number of students do not perform well in a class test, then following remedies are suggested to
revise the chapter and discuss problem areas
Give some support material in the form of list of basic formulae and concepts, practice worksheets, practice tests etc.
If some selected students do not perform well, then they can be suggested to prepare well for the next test.
ASSESSMENT TOOL #4
Math activity /Math projects:
Math activities :( To be performed individually)
There are three parameters for assessing Math activities
1. File Record (4 marks)
2. Performance of activity (4 marks)
3. Viva (2 marks)
Please note:
Each child is required to perform hands on activities as suggested by CBSE.
Record of the same has to be maintained in a separate record file.
Teacher is suggested to provide guidelines of performing the activities through activity sheets / a demonstration.
Students are to be observed in Math lab while doing hands on .
Teacher may ask questions in between related to Math concept learnt.
Math Projects: (Group work) (To be performed in a group of 3 to 4 students)
Suggestive Math projects:
1. Make a dictionary of terms in Geometry with figures. (Making a Pictorial dictionary)
2. Survey in Statistics
Minggu, 13 Desember 2009
Triangle Detective
This game is a variation of a variation of a game. There's a terrific game called Polygon Capture which originally came from a Mathematics Teaching in the Middle School article (Oct 98, William M. Carroll). The idea is to turn over a sides property card and an angle property card and capture the polygons that fit the description. For my classes I'm on about version 3 of it, and it's a solid middle school game. (Contact me if you want my copy.) At one point a class was interested in adapting it to triangles, so we did a version of the game that focused on them.
Michigan has moved a lot of the triangle identification objectives down to 4th grade, though, so I thought I should adapt that to be a 4th grade game, and that's what I'm posting today. Mrs. Bruckbauer's students decided it should be Triangle Detective, because you inspect the triangles to see if they fit the card. They are so right. As usual. They also suggested the three points per triangle scoring, and preferred it to the 'most triangles wins' rule.
Triangle Detective
(click here for a PDF version, with cards and triangles to cut out)
Rules
• Put the triangles into the middle and shuffle or mix up the cards.
• On a player's turn they flip over a card. They catch a triangle if they can find a triangle that matches the card's description. All players have to agree it fits the description.
• Master Cards are special challenges. Are you a triangle master?
• STEAL cards are the only way to take a triangle someone else has captured.
• Play until the deck is empty. Players get three points for each triangle, and the most points wins.
Remember:
Two things are congruent in geometry if they are exactly the same size and shape.
An angle is right if its sides are perpendicular, like the corner of a square.
An angle is obtuse if it is bigger than a right angle.
An angle is acute if it is smaller than a right angle.
A triangle is acute if it has ALL acute angles.
A triangle is right if it has ONE right angle.
A triangle is obtuse if it has ONE obtuse angle.
A triangle is equilateral if it has THREE congruent sides.
A triangle is isosceles if it has TWO congruent sides.
A triangle is scalene if it has NO congruent sides.
Cards:
Triangles: (Click for full size image) (3 of each type... including some borderline cases.)
Michigan has moved a lot of the triangle identification objectives down to 4th grade, though, so I thought I should adapt that to be a 4th grade game, and that's what I'm posting today. Mrs. Bruckbauer's students decided it should be Triangle Detective, because you inspect the triangles to see if they fit the card. They are so right. As usual. They also suggested the three points per triangle scoring, and preferred it to the 'most triangles wins' rule.
Triangle Detective
(click here for a PDF version, with cards and triangles to cut out)
Rules
• Put the triangles into the middle and shuffle or mix up the cards.
• On a player's turn they flip over a card. They catch a triangle if they can find a triangle that matches the card's description. All players have to agree it fits the description.
• Master Cards are special challenges. Are you a triangle master?
• STEAL cards are the only way to take a triangle someone else has captured.
• Play until the deck is empty. Players get three points for each triangle, and the most points wins.
Remember:
Two things are congruent in geometry if they are exactly the same size and shape.
An angle is right if its sides are perpendicular, like the corner of a square.
An angle is obtuse if it is bigger than a right angle.
An angle is acute if it is smaller than a right angle.
A triangle is acute if it has ALL acute angles.
A triangle is right if it has ONE right angle.
A triangle is obtuse if it has ONE obtuse angle.
A triangle is equilateral if it has THREE congruent sides.
A triangle is isosceles if it has TWO congruent sides.
A triangle is scalene if it has NO congruent sides.
Cards:
| Obtuse Triangle | Has an obtuse angle | Has NO congruent sides | Scalene Triangle |
| Right Triangle | Has two or more acute angles | Has two or more congruent sides | Isosceles Triangle |
| Acute Triangle | Has a right angle | Has three congruent sides | Equilateral Triangle |
| Has at least two congruent angles | Has three acute angles | Has three sides | Has a line of symmetry |
| Master Card Take a triangle IF you can name its side type AND its angle type | Master Card Take a triangle IF you can explain the type of each angle in the triangle | STEAL! If you can find someone with an acute scalene triangle. | STEAL! If you can find someone with a right isosceles triangle. |
Triangles: (Click for full size image) (3 of each type... including some borderline cases.)
Teaching Notes: The game worked really well for review. The students were engaged and asking good questions. Students were motivated to ask about vocabulary they didn't know, got to see other people apply vocabulary, and see triangles in many different orientations. The statements on the cards led to them thinking about alternative ways to say the same thing, and to think about properties in combination.
Acute triangles are an issue, because it seems to them that one acute angle should be enough. Which is actually nice parallel reasoning from how right and obtuse triangles are explained. We talked about how each angle has a type and each triangle has one angle type. So if it's not obtuse nor right...
We used square and triangle pattern blocks to help check the triangles angles, and talked about how equilateral triangles have all the same angle as well, and that's a way to check them.
The students got quite expert at checking side lengths, and quickly weren't satisfied with 'they look the same.'
They were generally quite helpful to each other, to the point where one student asked them to stop helping because she wanted to find her own.
Good luck to if you try it. And, of course, I'd love to hear how it goes!
Acute triangles are an issue, because it seems to them that one acute angle should be enough. Which is actually nice parallel reasoning from how right and obtuse triangles are explained. We talked about how each angle has a type and each triangle has one angle type. So if it's not obtuse nor right...
We used square and triangle pattern blocks to help check the triangles angles, and talked about how equilateral triangles have all the same angle as well, and that's a way to check them.
The students got quite expert at checking side lengths, and quickly weren't satisfied with 'they look the same.'
They were generally quite helpful to each other, to the point where one student asked them to stop helping because she wanted to find her own.
Good luck to if you try it. And, of course, I'd love to hear how it goes!
Rabu, 09 Desember 2009
Mathematics Recordings is 'Label of the Month' on Resident Advisor
"Jamal Moss' Chicago-based label has had a prolific 24 month period, seeing the imprint's catalogue nearly double. RA's Lee Smith calls up the man called Hieroglyphic Being to see why there's so much noise to be heard."
Free Hieroglyphic Being mix! (right click + save as)
Tracklisting:
01. Rotating Assembly - Illumination - Sound Signature
02. Abacus - Collectors Edition - Prescription03. BeBe Winans - Thank You - Atlantic
04. DJ Genesis - Back In The Middle - Dynamite Soul
05. Idema + Co - Cascade - INFRACom!
06. Madonna - Erotic - Maverick
07. Noni - Be My (Friend) - Prescription
08. Rose Royce - Love Don't Live Here Anymore - Hispavox
09. Slam Mode - Eternal - Ibadan
10. William S - I'll Never Let You Go - Trax Records
11. Usutu Juma - Bongolistics - i! Records
12. Frontline Orchestra - Don't Turn Your Back - ICE
13. Mariah Carey - Fly Away (Butterfly Reprise)
14. Inner City - Share My Life - (Kenny Larkin Detroit Remix) - 6x6 Records
15. Peven Everett - I Can't Believe I Loved Her - Nite Grooves
16. U get 4 - Edit - White
17. Hieroglyphic Being - Machines for Lovers - Spectral Sound
Senin, 07 Desember 2009
Other People's Geogebra
Transformations on a Graph
I've been looking for Geogebra applications for function transformations, and wanted to share a couple of the neat sketches I've found.
Michael Higdon, a math teacher at Kincaid, a college prep school in Texas, has a quadratic function in vertex form, y=a(x-h)^2+k, with a, h and k as sliders to study transformations.
Geogebra webpage: Transformation of Functions
Mike May, a Jesuit math teacher at St. Louis University, has a beautiful applet where you can input the function, and control vertical and horizontal shifts and scaling with sliders.
Geogebra webpage: Translation Compression
An overall great collection of interactive webpages appears at The Interactive Mathematics Classroom. It has a nice search feature and a good breakdown by area of mathematics.
My first attempt at a transformations sketch is with a cubic as the starting function. Although you can change the function.
As a webpage, and as the geogebra file.
Slope in Linear Equations
A nice collection of middle school or Algebra I activities for linear equations from mathcasts.org: Slope Explorations
Mathcasts are screencasts of writing with voice-overs. They have mathcasts for K-12, and a nice collection of interactive math activities.
Here's my first slope sketch. It tries to get at the idea of the slope being constant on a line regardless of what points are selected.
As a webpage or a geogebra file.
I've been looking for Geogebra applications for function transformations, and wanted to share a couple of the neat sketches I've found.
Michael Higdon, a math teacher at Kincaid, a college prep school in Texas, has a quadratic function in vertex form, y=a(x-h)^2+k, with a, h and k as sliders to study transformations.
Geogebra webpage: Transformation of Functions
Mike May, a Jesuit math teacher at St. Louis University, has a beautiful applet where you can input the function, and control vertical and horizontal shifts and scaling with sliders.
Geogebra webpage: Translation Compression
An overall great collection of interactive webpages appears at The Interactive Mathematics Classroom. It has a nice search feature and a good breakdown by area of mathematics.
My first attempt at a transformations sketch is with a cubic as the starting function. Although you can change the function.
As a webpage, and as the geogebra file.Slope in Linear Equations
A nice collection of middle school or Algebra I activities for linear equations from mathcasts.org: Slope Explorations
Mathcasts are screencasts of writing with voice-overs. They have mathcasts for K-12, and a nice collection of interactive math activities.
Here's my first slope sketch. It tries to get at the idea of the slope being constant on a line regardless of what points are selected.
As a webpage or a geogebra file.
Minggu, 06 Desember 2009
A Bigger Hex
As a webpage, and as a geogebra file.Made a simple hexagon dilation sketch in geogebra for my geometry class. Let's you vary the objects, measures area and perimeter, and control scale factor. Algebra view let's you see individual side lengths also.
It's part of a set of four similarity problems for stations. Here's the pdf.
Family Mathematics Videos
A former student at GVSU, Jamie Trost, is showing a lot of initiative, and posting videos to youtube providing instruction on elementary topics. They could be used as support for students who are struggling, extension for students succeeding in one method by exposure to another, or as support for parents trying to make sense of their children's math class.
She's just getting started, and is very open to suggestions, for topics and for improving the videos. If you get a chance, please check out Jamie's Family Mathematics, and give her some feedback. Her initial series is on multiplication.
Sabtu, 05 Desember 2009
Rumus Matematika
Tidak bisa dipungkiri salah satu hal yang membuat orang tidak tertarik atau bahkan tertarik pada matematika adalah rumus matematika. Menjadi tidak menarik karena begitu banyak rumus yang harus dihapal menjadi menarik ketika beberapa permasalahan dalam matematika dapat diselesaikan menggunakan rumus praktis atau sering juga disebut rumus cepatnya.Sebelumnya saya pernah menuliskan kegusaran saya
Jumat, 04 Desember 2009
Why Pi?
Why Pi? is a DK picture book/encylcopedia by Johnny Ball. The author sounds as if he was the Mr. Wizard of Great Britain in the 70s, and is the author of many books, several of which are math and science related. But this is my first of his.
Xavier (9) says: it's really good. I bow to whoever suggested it. It was fun and funny.
Ysabela (10) says: it's kind of funny, but in an interesting way. It's got a lot of information, but is definitely enjoyable reading. And it has Egyptians, awesome.
John (45) says: the book was unexpectedly thorough. It has a lot of excellent math and science history, from several ancient cultures, and deeply explores the concept of measurement as it pertains to many topics. It is a great read, and a solid reference. Despite this exposure coming from the library, I will be looking to purchase it. As the kids have pointed out, it is long on humor, and deep in breadth. (Both difficult quantities to measure.)
Kamis, 03 Desember 2009
15 Places to Find Math Games and Activities Online
Math isn't everybody's favorite subject, but it can be made more interesting with fun games that entertain and educate at the same time. There are dozens of board games made specifically for this purpose. There are also a number of math games that can be played for free online. Here are 15 sites that offer interactive math games guaranteed to engage even the most reluctant math students:
Tutpup - Tutpup is a fun site that allows kids to play educational games and compete with other children all over the world. Math games include addition, subtraction, multiplication, division, and algebra.
iknowthat.com - This free children's educational site provides standards based math games for PreK-6 students. Kids play in a social environment and are able to save scores, documents, and other accomplishments to show friends and family members.
Funbrain Math Arcade - The Funbrain Math Arcade features 25 arcade-style math games for elementary and middle school children. Visitors choose their learning level to play their first game. Every time a game is beat a new game is unlocked.
Math Playground - Created for elementary and middle school students, this comprehensive math site features math games, puzzles, videos, and other math activities that can be enjoyed online.
Illuminations - In addition to providing lesson plans and information about math standards, this NCTM (National Council of Teachers of Mathematics) site offers more than 100 different online activities preK-12 students. Activities are sorted by grade level and can be easily located using the site's search feature.
NumberNut - NumberNut is a high quality site devoted entirely to math. Visitors can choose from basic math activities and advanced math activities.
Cool Math 4 Kids - Cool Math 4 Kids is a Net-Mom approved site where elementary and middle school students can play math games and solve puzzles online. The site also offers virtual math flashcards for extra math practice.
National Library of Virtual Manipulatives - This Utah State University site provides interactive, web-based virtual manipulatives for preK-12 students. Covered subjects include measurement, numbers and operations, algebra, geometry, and data analysis.
Arcademic Skill Builders - Arcademic Skill Builders features arcade-style math games that can be played alone or with a friend. The games can be played online and also work with the Nintendo Wii gaming system.
Sheppard Software - This site offers a new math quiz for every day of the week. Other site features include games that teach basic math, fractions, place value, money counting, telling time, and pre-algebra.
Math and Reading Help for Kids - In addition to offering hundreds of helpful articles and links to math resources, this site also provides several different fun and educational math games for kids. One of the best is Math Search, which combines addition, subtraction, multiplication, and division with the traditional word search.
CyberBee - CyberBee provides several standards-based online math games for grades K-5. Visitors who are struggling with math can also use the site's visual math dictionary to gain a better understanding of various subjects.
Multiplication.com - Multiplication.com provides dozens of free interactive games designed to teach students multiplication tables. A new game is added each month.
Wired Math - Wired Math offers fun online math games, challenges, and exercises for grades 7-9. Activities are sorted by grade level and subject.
Math TV - This site features an extensive collection of math videos that can be played online. Videos cover a wide range of math topics, including basic math, algebra, geometry, trigonometry, and calculus.
Guest post from education writer Karen Schweitzer. Karen is the About.com Guide to Business School. She also writes about online school for OnlineSchool.net.
Tutpup - Tutpup is a fun site that allows kids to play educational games and compete with other children all over the world. Math games include addition, subtraction, multiplication, division, and algebra.
iknowthat.com - This free children's educational site provides standards based math games for PreK-6 students. Kids play in a social environment and are able to save scores, documents, and other accomplishments to show friends and family members.
Funbrain Math Arcade - The Funbrain Math Arcade features 25 arcade-style math games for elementary and middle school children. Visitors choose their learning level to play their first game. Every time a game is beat a new game is unlocked.
Math Playground - Created for elementary and middle school students, this comprehensive math site features math games, puzzles, videos, and other math activities that can be enjoyed online.
Illuminations - In addition to providing lesson plans and information about math standards, this NCTM (National Council of Teachers of Mathematics) site offers more than 100 different online activities preK-12 students. Activities are sorted by grade level and can be easily located using the site's search feature.
NumberNut - NumberNut is a high quality site devoted entirely to math. Visitors can choose from basic math activities and advanced math activities.
Cool Math 4 Kids - Cool Math 4 Kids is a Net-Mom approved site where elementary and middle school students can play math games and solve puzzles online. The site also offers virtual math flashcards for extra math practice.
National Library of Virtual Manipulatives - This Utah State University site provides interactive, web-based virtual manipulatives for preK-12 students. Covered subjects include measurement, numbers and operations, algebra, geometry, and data analysis.
Arcademic Skill Builders - Arcademic Skill Builders features arcade-style math games that can be played alone or with a friend. The games can be played online and also work with the Nintendo Wii gaming system.
Sheppard Software - This site offers a new math quiz for every day of the week. Other site features include games that teach basic math, fractions, place value, money counting, telling time, and pre-algebra.
Math and Reading Help for Kids - In addition to offering hundreds of helpful articles and links to math resources, this site also provides several different fun and educational math games for kids. One of the best is Math Search, which combines addition, subtraction, multiplication, and division with the traditional word search.
CyberBee - CyberBee provides several standards-based online math games for grades K-5. Visitors who are struggling with math can also use the site's visual math dictionary to gain a better understanding of various subjects.
Multiplication.com - Multiplication.com provides dozens of free interactive games designed to teach students multiplication tables. A new game is added each month.
Wired Math - Wired Math offers fun online math games, challenges, and exercises for grades 7-9. Activities are sorted by grade level and subject.
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Guest post from education writer Karen Schweitzer. Karen is the About.com Guide to Business School. She also writes about online school for OnlineSchool.net.
Selasa, 01 Desember 2009
Jamal Moss Presents Marcello Napoletano - The Space Voodoo
MATHEMATICS 033
A1. The Space Voodoo - Celestral Intro
A2. The Calm B-4 The Storm
B1. Ottantotto
B2. LMBDA Blues - 2 My Heroes Rmx
B3. Expressions Of A Miles Man - Slowest Part 1
C1. Mood Jane - Black Keys & White Dust Rmx
C2. My Brothers Soul
D1. Music If For The People
D2. I Hope That U Can Feel It
D3. Somewhere In The Galaxy - Outro
Paradoks dalam Matematika
Banyak orang menganggap kalau matematika itu ilmu eksak, yang selalu pasti. Seolah tidak pernah terdapat keraguan sama sekali, hanya ada benar atau salah. Untunglah matematika terus berkembang seturut perkembangan peradaban manusia. Sehingga sekarang tidak hanya benar atau salah seperti dalam logika matematika dwi nilai, tapi kebenaran itu relatif. Bisa benar 90%, 50% atau hanya 1 % saja seiring dikembangkannya logika kabur dalam matematika.
Namun sebenarnya sejak awal, para ahli matematika yang biasanya juga ahli filsafat sekaligus ahli-ahli di bidang lainnya sudah menemukan bahwa antara benar dan salah kadang tidak bisa dipisahkan begitu saja, bahwa yang ini benar yang itu salah. Ternyata mereka kadang mendapati bahwa ada sesuatu yang benar sekaligus sesuatu itu salah. Inilah yang dikenal sebagai paradoks.
Paradoks adalah suatu situasi yang timbul dari sejumlah premis yang diakui kebenarannya yang bertolak dari suatu pernyataan dan akan tiba pada suatu kontradiksi. Paradoks juga dikenal dengan nama antinomi karena melanggar hukum kontradiksi principium contradictionis (law of contradiction). Sama seperti dilema, paradoks biasa digunakan untuk mematahkan argumentasi lawan dengan menempatkannya ke dalam situasi yang sulit dan serba salah.
Contoh dari paradoks adalah “Semua laki-laki adalah pembohong!” Para cewek pasti akan mengamini pernyataan di atas, sementara bagi para cowok membaca kalimat di atas mungkin hanya akan tersenyum simpul dengan sorot mata tidak bersalah sambil melenggang pergi.
Ketika saya mengatakan, “Semua laki-laki adalah pembohong!” maka ada sesuatu yang aneh di sana. Mengapa? Sebab saya seorang laki-laki, sehingga apa yang saya katakan bahwa “semua laki-laki adalah pembohong” juga merupakan suatu kebohongan. Artinya semua laki-laki bukan pembohong. Jadi mana yang benar, semua laki-laki adalah pembohong atau semua laki-laki bukan pembohong? Bingung kan?!
Paradoks di atas dikembangkan dari paradoks tertua dan sangat terkenal di dunia yaitu paradoks pembohong (liar paradox) atau Epimenides Paradox yang diungkapkan oleh Epimenides yang hidup diabad 6 sebelum masehi. Paradoks itu aslinya adalah sebagai berikut: ”Epimenides si orang Kreta mengatakan bahwa semua orang Kreta adalah pembohong.”
Rangkaian premis berikut ini akan membawa kita pada dua kesimpulan yang bertentangan:
• Jika apa yang dikatan Epimenides benar, ia bukan pembohong.
• Jika Epimenides bukan pembohong, apa yang dikatakannya tidak benar.
• Jika apa yang dikatakannya tidak benar, ia pembohong.
Kesimpulan pertama: Jadi, ia adalah pembohong dan bukan orang jujur.
• Jika yang dikatakan Epimenides tidak benar, ia adalah pembohong.
• Jika ia pembohong, apa yang dikatakannya tidak benar.
• Jika apa yang dikatakannya tidak benar, itu berarti bahwa ia adalah orang jujur.
Kesimpulan kedua: Jadi, ia adalah orang jujur dan bukan pembohong.
Apa yang dikatakan Epimenides sebenarnya secara bersama-sama sekaligus mengandung kebohongan dan kebenaran. Jika kebohongan, berarti ia benar-benar pembohong, dan jika kebenaran, ia adalah seorang yang jujur.
Paradoks diatas jadi masalah besar, terutama bagi para matematikawan, yang memandang dimana dunia itu adalah salah atau benar dan sebuah pernyatan harus punya nilai jelas antara 0 dan 1 atau True (T) dan False (F).
Paradoks terjadi karena kita mengambil referensi dari diri kita sendiri. Kurt Gödel, di tahun 1931, menjelaskan problema self-reference diatas dalam sebuah teorema yang dikenal dengan nama Godel’s Theorem, yang mengatakan: “To every ω-consistent recursive class χ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(&chi) (where v is the free variable of r.”)
Teorema Godel sendiri terlihat persis seperti sebuah paradoks juga. Intinya niscaya kita akan bertemu dengan kontradiksi kalau kita melakukan self-reference atau kalaupun kita melakukan self-reference pastikan kalau kita tahu bahwa itu adalah self-reference.
Berikut ini beberapa contoh paradoks sederhana dalam matematika.
1 = 2
Bukti:
Misalkan a = b
maka a2 = ab (kalikan kedua ruas dengan a)
a2 − b2 = ab − b2 (kedua ruas kurangi dengan b2)
(a − b)(a + b) = b(a − b) (kedua ruas difaktorkan)
a + b = b [bagi kedua ruas dengan (a − b)]
2b = b (substitusikan a = b)
2 = 1 (bagi kedua ruas dengan b)
Jadi terbukti 1 = 2!
Pada langkah dimana kita membagi dengan (a−b), sebenarnya kita melakukan pembagian dengan 0, karena a = b, sehingga a − b = 0. Dan dalam matematika pembagian dengan 0 tidak didefinisikan, sehingga bukti di atas yang tampaknya benar dan logis, sesungguhnya salah.
Dalam filosofi moral, paradoks memainkan peranan sentral dalam debat tentang etik. Misalnya, peringatan etis untuk "mencintai tetangga kita" adalah tidak hanya kontras, tetapi juga sangat kontradiktif jika tetangga kita itu bersenjata dan selalu mencoba membunuh kita: bila dia berhasil, kita tidak akan berhasil untuk mencintainya. Tetapi untuk menyerang mereka terlebih dahulu atau menahan mereka biasanya tidak dimengerti sebagai tindakan cinta. Ini dapat disebut sebagai dilema etik. Contoh lainnya, adalah konflik antara perintah untuk tidak mencuri dan untuk memberi perhatian kepada keluarga, yang kita tidak mampu memberi mereka makan tanpa kita mencuri uang.
Lalu buat kita yang bukan matematikawan dan hanya orang awam ini, apa artinya paradoks tersebut? Berhati-hatilah ketika ada orang atau pihak yang mengklaim memiliki kebenaran dan benar 100% sehingga semua yang lain salah. Karena sudah sama-sama kita ketahui dari paradoks bahwa benar dan salah adalah sekaligus sebuah kontradiksi.
Namun sebenarnya sejak awal, para ahli matematika yang biasanya juga ahli filsafat sekaligus ahli-ahli di bidang lainnya sudah menemukan bahwa antara benar dan salah kadang tidak bisa dipisahkan begitu saja, bahwa yang ini benar yang itu salah. Ternyata mereka kadang mendapati bahwa ada sesuatu yang benar sekaligus sesuatu itu salah. Inilah yang dikenal sebagai paradoks.
Paradoks adalah suatu situasi yang timbul dari sejumlah premis yang diakui kebenarannya yang bertolak dari suatu pernyataan dan akan tiba pada suatu kontradiksi. Paradoks juga dikenal dengan nama antinomi karena melanggar hukum kontradiksi principium contradictionis (law of contradiction). Sama seperti dilema, paradoks biasa digunakan untuk mematahkan argumentasi lawan dengan menempatkannya ke dalam situasi yang sulit dan serba salah.
Contoh dari paradoks adalah “Semua laki-laki adalah pembohong!” Para cewek pasti akan mengamini pernyataan di atas, sementara bagi para cowok membaca kalimat di atas mungkin hanya akan tersenyum simpul dengan sorot mata tidak bersalah sambil melenggang pergi.
Ketika saya mengatakan, “Semua laki-laki adalah pembohong!” maka ada sesuatu yang aneh di sana. Mengapa? Sebab saya seorang laki-laki, sehingga apa yang saya katakan bahwa “semua laki-laki adalah pembohong” juga merupakan suatu kebohongan. Artinya semua laki-laki bukan pembohong. Jadi mana yang benar, semua laki-laki adalah pembohong atau semua laki-laki bukan pembohong? Bingung kan?!
Paradoks di atas dikembangkan dari paradoks tertua dan sangat terkenal di dunia yaitu paradoks pembohong (liar paradox) atau Epimenides Paradox yang diungkapkan oleh Epimenides yang hidup diabad 6 sebelum masehi. Paradoks itu aslinya adalah sebagai berikut: ”Epimenides si orang Kreta mengatakan bahwa semua orang Kreta adalah pembohong.”
Rangkaian premis berikut ini akan membawa kita pada dua kesimpulan yang bertentangan:
• Jika apa yang dikatan Epimenides benar, ia bukan pembohong.
• Jika Epimenides bukan pembohong, apa yang dikatakannya tidak benar.
• Jika apa yang dikatakannya tidak benar, ia pembohong.
Kesimpulan pertama: Jadi, ia adalah pembohong dan bukan orang jujur.
• Jika yang dikatakan Epimenides tidak benar, ia adalah pembohong.
• Jika ia pembohong, apa yang dikatakannya tidak benar.
• Jika apa yang dikatakannya tidak benar, itu berarti bahwa ia adalah orang jujur.
Kesimpulan kedua: Jadi, ia adalah orang jujur dan bukan pembohong.
Apa yang dikatakan Epimenides sebenarnya secara bersama-sama sekaligus mengandung kebohongan dan kebenaran. Jika kebohongan, berarti ia benar-benar pembohong, dan jika kebenaran, ia adalah seorang yang jujur.
Paradoks diatas jadi masalah besar, terutama bagi para matematikawan, yang memandang dimana dunia itu adalah salah atau benar dan sebuah pernyatan harus punya nilai jelas antara 0 dan 1 atau True (T) dan False (F).
Paradoks terjadi karena kita mengambil referensi dari diri kita sendiri. Kurt Gödel, di tahun 1931, menjelaskan problema self-reference diatas dalam sebuah teorema yang dikenal dengan nama Godel’s Theorem, yang mengatakan: “To every ω-consistent recursive class χ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg(v Gen r) belongs to Flg(&chi) (where v is the free variable of r.”)
Teorema Godel sendiri terlihat persis seperti sebuah paradoks juga. Intinya niscaya kita akan bertemu dengan kontradiksi kalau kita melakukan self-reference atau kalaupun kita melakukan self-reference pastikan kalau kita tahu bahwa itu adalah self-reference.
Berikut ini beberapa contoh paradoks sederhana dalam matematika.
1 = 2
Bukti:
Misalkan a = b
maka a2 = ab (kalikan kedua ruas dengan a)
a2 − b2 = ab − b2 (kedua ruas kurangi dengan b2)
(a − b)(a + b) = b(a − b) (kedua ruas difaktorkan)
a + b = b [bagi kedua ruas dengan (a − b)]
2b = b (substitusikan a = b)
2 = 1 (bagi kedua ruas dengan b)
Jadi terbukti 1 = 2!
Pada langkah dimana kita membagi dengan (a−b), sebenarnya kita melakukan pembagian dengan 0, karena a = b, sehingga a − b = 0. Dan dalam matematika pembagian dengan 0 tidak didefinisikan, sehingga bukti di atas yang tampaknya benar dan logis, sesungguhnya salah.
Dalam filosofi moral, paradoks memainkan peranan sentral dalam debat tentang etik. Misalnya, peringatan etis untuk "mencintai tetangga kita" adalah tidak hanya kontras, tetapi juga sangat kontradiktif jika tetangga kita itu bersenjata dan selalu mencoba membunuh kita: bila dia berhasil, kita tidak akan berhasil untuk mencintainya. Tetapi untuk menyerang mereka terlebih dahulu atau menahan mereka biasanya tidak dimengerti sebagai tindakan cinta. Ini dapat disebut sebagai dilema etik. Contoh lainnya, adalah konflik antara perintah untuk tidak mencuri dan untuk memberi perhatian kepada keluarga, yang kita tidak mampu memberi mereka makan tanpa kita mencuri uang.
Lalu buat kita yang bukan matematikawan dan hanya orang awam ini, apa artinya paradoks tersebut? Berhati-hatilah ketika ada orang atau pihak yang mengklaim memiliki kebenaran dan benar 100% sehingga semua yang lain salah. Karena sudah sama-sama kita ketahui dari paradoks bahwa benar dan salah adalah sekaligus sebuah kontradiksi.
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