Tuesday, August 31, 2010
M2G1 - Geometry in Primary Grades (2)
Well, it has been a long time since I wrote an entry here. I apologize for those who posted their comments for not responding. I will try to keep updating this blog a bit more frequently.
Last time, I started writing on the geometry standards, and I will continue the discussion of geometry in this entry. In M2G1, students are now expected to classify plane figures "according to the number of edges and vertices and the sizes of angles (right angle, obtuse, acute)." The GPS includes the following figures: triangles, square, rectangle, trapezoid, quadrilateral, pentagon, hexagon, and irregular polygonal shapes. Last time, I questioned the appropriatenes! s of the expectation that kindergarteners to distinguish squares and rectangles. In Grade 2, students are learning about different types of angles, right, acute and obtuse. Therefore, it is in Grade 2 when it is appropriate for students to learn that rectangles are quadrilaterals with 4 right angles, and squares are rectangles with all sides equal.
However, let's think about how students can understand right angles. When I ask my adults, including some teachers and teacher candidates, what right angles are, they almost always respond by saying "90-degree angles." Although it is true that right angles measure 90 degrees, measuring angles using "degree" as the unit is a Grade 4 standard. Thus, how are second grade students to understand what right angles are? A Japanese elementary math textbook by Hironaka and Sugiyama has an interesting approach to this topic. They define a right angle to be the angle you obtain when you fold a piece of paper as shown in th! e figures below:
Note that the piece of paper can be any shape to start with. The second fold is made in such a way that the first fold line will be folded onto itself. Although it might also be helpful to point out to children that the corners of note papers, index cards, etc. are right angles, we cannot always be sure that corners of any piece of paper are right angles.
Interestingly, this definition of a right angle is very much comparable to Euclid's definition of right angles in his book The Elements. He defines that the angles you obtain by equally dividing a straight angle are right angles. When the second fold is made so that the first fold line will be folded onto itself, we are indeed dividing the straight line (the first fold line) into two equal angles.
I want to end this entry by raising another issue with! the standard, however. This standard expects students to des! cribe an d classify trapezoid. However, in order to describe trapezoids, children need to concept of parallelism. The Grade 2 Geometry unit of Math Frameworks define trapezoids as "quadrilaterals with two parallel sides." Unfortunately, parallelism is a Grade 4 topic. Therefore, it is very strange that we should expect students in Grade 2 to describe and classify trapezoids.
To make the matter even worse, Grade 4 Geometry unit of Math Frameworks defines trapezoids as quadrilaterals "with only one pair of parallel sides." This definition is different from the Grade 2 definition, which does not say anything about the other two sides of quadrilaterals. In a recent publication, Zalman Usiskin and his collaborators document how these two definitions of trapezoids has existed in US mathematics textbooks. Therefore, the fact that the definitions are different isn't too surprising. However, it is rather unfortunate that a document that emphasizes coherence will not try to ! be consistent in their definition of a geometric figure.
quadrilaterals definition
The economical tutoring service!!!
...the best tutoring services, I have seen... better than skype or anything.. just like being in front of the teacher.
! Students can schedule when they want to study, and in www.knowledgeonlineservices.com teachers are available at round the clock. The student can manage his time best, even late at night the teachers are available, without complaint.They offer free introductory session .
And best of all, www.knowledgeonlineservices.com is much cheaper than getting your child tutored personally by a teacher. www..knowledgeonlineservices.com. tutors only teach one child at a time, in one session. so there is complete focus on that students needs, speed, grasping power and everything. Tutoring for all the subjects are awe some and Great…!!!!!
online tutoring free
Creative Mathematics- Holidays Homework(Class 9 and 10)
Each student is required to work on any one of creative Maths activity work according to the following distribution:
Roll No. 1 to 10
- Self composed poem on Mathematics : Do you like writing poems? If yes then try your hands on writing a Math Poem. You may choose any topic and start writing.
OR
- Making a Fractal Card (You can make any one out of 4 suggested at http://classes.yale.edu/fractals/Labs/PaperFoldingLab/PFLProcedure.html
(Bring your cards after school re opens)
(Click pictures of every step while making and send it through e mail to your respective Maths activity teacher. You can make a presentation or write on a word file along with pictures.)
Roll No. 11 to 20
- ! Role play (Script writing explaining any Maths concept from chapters learnt till now)
Please note: there is no specific word limit. Be sure, it should not be vague and cater to the topic's need.
OR
- Making a 3D Snowflake (For instruction click on http://www.wikihow.com/Make-a-3D-Paper-Snowflake )
(Click pictures of every step while making and send it through e mail to your respective Maths activity teacher. You can make a presentation or write on a word file along with pictures.)
Roll No. 21 to 30
- Life sketch of a Mathematician (atleast 500 words)
OR
- Making designs and patterns using geometrical shapes
(Click pictures of your designs and send it through e mail to your respective Maths activity teacher. Write a small brief on your explorations.)
Roll No. 31 to 40
OR
- Making mathematical patterns using line segments .(Visit this link WIKI HOW? )
(Click pictures of your designs and send it through e mail to your respective Maths activity teacher. Write a small brief on your explorations.)
Roll No. 41 onwards
- Utility of a Maths topic (e.g. application of Linear equations in two variables etc. )
OR
- Making a paper snowflake (For instructions click on http://www.papersnowflakes.com/)
See Sample work by students http://mathematicsprojects.blogspot.com/2008/01/making-paper-snowflakes.html
(Click pictures of ever! y step while making and send it through e mail to your respective Maths activity teacher. You can make a presentation or write on a word file along with pictures.)
Please note:
- The work should be submitted online through e mail to your respective Maths activity teacher latest by 15. 06.2010 .
- Do mention your Name, Class/Section, Roll Number
- Original and innovative work will be highly appreciated.
- Click pictures if you are working on making a Fractal Card, a 3D Snowflake, designs and patterns using geometrical shapes, mathematical patterns using line segments or a paper snowflake.
- In case you choose the first one out of the two, then write it in your own words. If you find some relevant pictures, you may attach it with description. Please do not forget to mention the source of your information. ! ol>
For any query: E mail mathclass_khms@yahoo.co.in
Class 9A students will e mail their work to Mrs Mamta Katyal mamtakatyal1970@yahoo.co.in
Classes 9B and 9 G students will e mail their work to Mrs Roman Dhawan dhawanroman@gmail.com
Classes 9C ,9D, 9E and 9H students will e mail their work to Mrs Sujata Manchanda samolina6@yahoo.com
Class 9F students will e mail their work to Mrs Kaveri Ramachandran kaverikulachi@gmail.com
Classes 10A, 10B, 10K and 10I students will e mail their work to Mrs Ankita mathematics_khms@yahoo.co.in
Classess 10C, 10D, 10G and 10H students will e mail their work to Mrs Rashmi Kathuria mathclass_khms@yahoo.co.in
Classes 10E, 10F and ! 10J students will e mail their work to Mrs Mamta Katyal mamtakatyal1970@yahoo.co.in
Classes 9I, 9J and 9K students will e mail their work to Mr. Mohit Khanna mkmaths0706@yahoo.co.in
DO YOUR ASSIGNMENTS IN YOUR MATHS NOTE BOOK. (DOWNLOAD THEM FROM SCHOOL'S WEBSITE)
online polynomial factorizer
Holy Frijole Mole!
I just made it and oooooohhhhh baby! I have one great little lunch to take to work tomorrow -- scooping up this stuff with some FoodShouldTasteGood chips. O Happy Day!
Recipes for Green Beans: Frijole Mole
This recipe is from an inspirational book: Animal, Vegetable, Miracle by Barbara Kingsolver. You can eat this dip with crackers or in wraps. Make sure you steam the beans thoroughly.
- 1/2 lb fresh green beans, trimmed
- 1 coarsely chopped onions
- 1 tablespoon olive oil
- 3 hard-boiled eggs
- 2 cups fresh basil leaves
- 1 tablespoon ! lemo! n juice< /a>
- 1 to taste mayonnaise or yogurt
- Steam green beans until tender.
- Saute onions over medium heat in oil until transparent.
- Combine beans, cooked onions, eggs, basil, and lemon juice in food processor and blend into coarse paste.
- Remove puree to a bowl and combine with enough mayo or yogurt to hold mixture together. (Note from Kelly: I used Trader Joe's White Bean Hummus. Very good if I do say so myself)
- Add salt and pepper to taste.
mole frijole
Tutorial - Revit and Calculus - Part 1
"Formulas support the following arithmetic operations: addition, subtraction, multiplication, division, exponentiation, logarithms, and square roots. Formulas also support the following trigonometric functions: sine, cosine, tangent, arcsine, arccosine, and arctangent.
The valid formula abbreviations for arithmetic operations and trigonometric functions are
• Addition— +
• Subtraction— -
• Multiplication—*
• Division—/
• Exponentiation—^: x^y, x raised to the power of y
• Logarithm—log
• Square root—sqrt: sqrt(16)
• Sine—sin
• Cosine—cos• Tangent—tan
• Arcsine—asin 226; Arccosine—acos
• Arctangent—atan
• e raised to an x power—exp
• Absolute Value—abs
You can enter integers, decimals, and fractional values in formulas, using normal mathematical syntax, as shown in the examples below:
• Length = Height + Width + sqrt(Height*Width)
• Length = Wall 1 (11000mm)+ Wall 2 (15000mm)
• Area = Length (500mm) * Width (300mm)
• Volume = Length (500mm) * Width (300mm) * Height (800 mm)
• Width = 100m * cos(angle)
• x = 2*abs(a) + abs(b/2)
• ArrayNum = Length/Spacing.
Parameter names in formulas are case sensitive. For example, if a parameter name begins with a capital letter, such as Width, you must enter it in the formula with an initial capital letter. If you enter it in a formula using lower-case letters instead, for example, width * 2, the software will not recognize ! the formula."
- Creating a Conceptual Mass.
- Create From with Spline and Profile.
- Edit and Create formulas within Families.
- Drive formulas with parameters.
- Add Profile.
- Use CALCULUS IN REVIT!
modular arithmetic tutorial
Christian Gladiators? Athletics as a metaphor for Christian life
Image : http://www.flickr.com
When Paul went to Corinth, in the middle of his second missionary journey, Acts 18:2-3 reports that he joined Aquila and Priscilla, manufacture of tents. This fits well with what Paul writes to himself, at the same time. For example, in 1 Thessalonians. 1:9 says, "But remember, brothers, our toil and hardship, we have worked day and night to avoid becoming a burden to anyone while he was preaching the gospel of God for you." Even in 2 Thess. 3:7-8, Paul: "Wenot idle when we were with you, even eat without paying. Instead, we worked day and night, with difficulty, so it would not be a burden to be one of you. "
Paul in these two States still at Corinth. Shortly after Corinth, left, towards the end of the third missionary journey, Paul in his apostolic right, which supported intellectually benefit (1 Co! r. 9). The question is rhetorical irony, "OrThere are only I and Barnabas who must work for a living? "(1 Cor. 9: 6).
By Jerome Murphy-O'Connor:
View of the trade the first time in a tent-maker seems particularly appropriate for a department that focuses on the city, but there was a tendency among the craftsmen of that title, though not actually covered a much broader and more ... the same purpose as a craftsman of today, sometimes described as a carpenter. Paul was in allChance, leather work, hand in producing the large number of items contained in this material: "Can [sandal] has, in turn, would gourds for water and wine, crockery saddles, shields, etc. shops and footpaths are were also made of leather, and a ready market in Corinth (St. Paul, we Corinth: Texts and Archeology, 168).
One reason for this is true is that the Games of the Isthmus took place in nearby Isthmia. After the Olympics among the four major Panhellen! icGames, Isthmian Games were two times more likely than others! , is hel d every two years. Time Paul went to Corinth, the Isthmian Games were 500 years. They were not even in the century of its tradition of welcoming Corinth was abandoned almost completely disrupted (146-44 BC).
At the time of Paul to Corinth, 50 CE, are the games have been brought from Corinth to Isthmia. Another cesarean games simultaneously with another event in every location in the isthmusGames. The Commission presented its own site of action. Groups of people from all over the Roman Empire, met Isthmia, either to participate or attend events. According to Casio, a contemporary of Paul (in his speeches, 8.12), containing the basic sporting breeds Isthmian Games, wrestling, diving, boxing, javelin throw and threw it on the disc.
One time or another in the history of the Games, including the new phenomena of horse racing, racing, poetryReading, theater, singing, announcing, playing the lyre and the flute, and a painting competition. Yacht Race, near the Saroni! c Gulf, has a function, not the Games of Olympia, Delphi, Nemea, O. Demonstrations were planned for women, like men, and even for children. Large amounts of money in hand, not only for players who have won and lost, but that gives the winner.
Isthmia excavations began in 1883 with Paul Monceaux. RenewedSB 1930 by Jenkins and H. Megawati. These initial efforts yielded only meager results. Oscar Broner, however, that the site of excavation 1959-1967 discovered the temple of Poseidon, porticoes, the sanctuary of Palaemon, two stages, much earlier than others, and a Hellenistic settlement near "Rashi". Assistant Broner, Elizabeth Gebhard outdoor theater. From 1967 to 1976, Clemente excavated Roman baths and other buildings. Ms. Gebhard againExcavated in 1980 and 1989 in the central chapel and a prehistoric settlement "Rashi".
Archaeologists were unable to find permanent housing for the crowd of participants in the Games as a track in the firs! t century AD They were built only in the second century. With ! the opti on of dealing with several miles of walking per day to reflect events or shopping and to enter a tent, hundreds if not thousands, witnessed what you prefer. In other words, this city was theBest places in the Mediterranean world of Paul to open the tent-making shop. Small shops like yours (with lines of 10 feet, 10 meters), the squares in cities throughout the Hellenistic empire.
Games of Paul, while living in Corinth? We have no idea, for sure. The games began with a sacrifice to Poseidon, the patron of the local deities. In addition, many sporting events were conducted in the most simple men and women were likelyonly poor clothing. One might expect that this nerve to offend the scruples / Judeo-Christian. However, Murphy-O'Connor said:
It is difficult to decide whether Paul himself took part in the games. Jewish Palestinian resistance against these glasses is well documented ... but we think that the same attitude that has prevailed in the di! aspora. The wire was free, I had an all-in wrestling competition (see Probis Omnis, 26), we can be sure that many Hellenized Jews did not hesitateParticipation in the Games. Jews had special reserved seats in the theater of Miletus in western Asia Minor .... (17).
We do not know for sure is that Paul uses the familiarity with the games as a source of images in their teaching. A review of his speeches and letters, in roughly chronological order, presents a series of allusions to sports competition. (I want the fat and suggestions for a more literal translation.)
Delivered before arrival in Corinth, in a sermon13h25 to Antioch in Pisidia (Acts), Paul "(race)" (Drôme greek, after "rollerdrome" and "Racetrack") as a metaphor for God's purpose for the life of John the Baptist: "And while John ended his career, he continued: "Who do you think I am? I'm not him. "
Years later, Paul us! ed the same pictures again, their objectives in life. Older Ep! hesians (Acts 20:24) was rejected, said Paul"But I think my life is worthless for me, if I can finish the race and only fill the job that I received the Lord Jesus, to testify the gospel of the grace of God."
In Galatians 2:2, Paul describes as a next visit to Jerusalem, to write together, "I. .. first to preach the Gospel, are among the nations. But I did it in private, that seemed to be transported by a fear that walking or running, he had run in vain. "Later inthe same book (5.7), commented: "You did a good race. Who cut for you and obeying the truth?"
These metaphors Paul employs, before arriving in Corinth. The longer wheelbase, however, occurs in the first Corinthians 9:24-27. Shortly after the founding of the church at Corinth, Paul urges the Corinthians:
Do not you know that running in a race all the runners, but only one receives the prize? Run, so that price. AnyoneParticipate in games goe! s into strict training. They do it to get a crown that will not last, but we do it for a crown, the eternally preserved. I can not, like a man running aimlessly, not knowing who fights like a man in the air. No, I beat my body and my slave so that after preaching to others I am not disqualified.
In previous centuries, of course, (was in greek: Stefano), the crown as a prize at Isthmia, fromThe branches of pines, like the back of the coins and find contemporary sculpture Isthmia in the sample. The crown of pine branches is a symbol of the Games of the isthmus, there was no evidence that the plant than others, Selinon (a plant similar to celery or parsley was) in the first century BC and a votive size of gains Crowns shows isthmic Selinon used in a variety of plants, including pine y. Goes particularly well with the phrase "crownno less important, or more literally, "crown perish." At the moment the isthmus that athlete! s have received their crown of grass, as it had dried.
! In seve ral letters of Paul, he uses the word "fight" or "competence" (the agon greek, that "torture" and torment. ") In Rome. 15:30, for example," Please, my brothers through our Lord Jesus Christ, the love of the mind, in my struggle with prayer to God for me to come with me. "Similar steps are presentedPaul wrote in his letters from prison (see Eph. 6:12, Col. 1:29, 2:1, 4:12, Phil 1:30) in which he wrote after (edition 1 Tim was 4.: 10, 6.12).
In one of these letters in the same prison (Phil 3:13-14), Paul Racing photos on his experience: "One thing I do: forgetting what lies behind and strain forward to what you end the sentence to which I go God in Jesus Christ in heaven. "
The word translated "reward"(In brabeion greek) is used by a greek at least three words to describe the prize to the winner of the competition. The context indicates that Paul refers to forget his su! ccesses, not the mistakes of the past. Contrary to what many preachers use this verse does not say: "Do not let your past failures discouraged by their earnings." Says the opposite: "Do not trust your" bottom line "benefits." Every race is a completely new business, and all runnersincluding the captain, must rise again. "
When Paul is back in prison shortly before his execution, he wrote in his latest letter (2 Tim. 4:7-8):
I have fought the good fight (dying), which is the rate (Drôme), I concluded kept the faith. What I have to get the crown (Stefano), the righteousness which the Lord has given me that day when the just judge, Not only has given, but all that the love of her appearance.
As in modern times, has the honor to submit the transfer higher premiums, said Paul expected Jesus, his crown will be awarded by his victory in the Christian race.
Although! this may reduce some principles, some "rules":
!
D espite the victory of Christ on the cross and his desire for forgiveness, having continued to live the Christian life to a struggle between good andMal. This is a fight with the devil, where the stakes are high and the risk of loss is real (Eph. 6:12).
God wants us to win this competition, and all terms of our success. But we must be prepared to submit to the discipline and establishing a rigorous training, if we expect (with a gain of 1 Cor. 9:24-25).
God is the price you want to be with us (Acts 20:24). The boundaries of behavior that has staked its moral is, we movebeyond the limits, if you commit sins or "sins." Within this moral line, but we must ask ourselves the same freedom of our judgments themselves, the best way to "run our race" (2 Tim. 4:7).
Death is the goal, and life with Christ in eternal prize (Philippians 3:13-14, 2 Tim. 4:8).
In the sense of the word, who are already winners, as soon a! s you start the race. In the words of Paul, who are champions of the Super, "Him who loved us" (Romans 8:37). But we still have to runthe race and to avoid being disqualified (1 Cor too. 9:26-27).
When you are in your career, Christian? Are behind the scenes? Perhaps you have doubled the last corner and stared at the finish line ahead of me. Or maybe you're still in the stands watching people in the race.
Wherever you are, the challenge of God to enter the race, all I have to run and cross the finish line victorious. Allows you to equip and train you. But it is necessaryYour role: to take seriously their Christian life, to learn all about the "rules" to listen and obey their coaches. Then, run!
Want to go further?
Here are some useful sources:
1962 Oscar Bronner. "The Crown Victoria del Istmo. American Journal of Archeology. 66:259 et seq. (See also Broner writings of many others in Isthmia.) !
1967 Pfitzner, Victor C. And the reason Paul Agon: The! images of the traditional sport Pauline literature. Leiden: EJ Brill. (Ph.D.Thesis, Faculty of Evangelical Theology in Münster, Westphalia, Germany)
1973 Elizabeth Gebhard. The theater at Isthmia. Chicago: University of Chicago.
1983 Jerome Murphy-O'Connor. St. Paul's Corinth: Texts and Archeology. Wilmington, DE: Michael Glaser.
marapets word search answers
More Damn Vegan Pecan Pie
! div>
gwam test
Saved by the Elephant (again)
I couldn't do this problem because an elephant was blocking it!
Elephants have tails.
Is that -3 for not doing the question, or not adding a tail to the elephant?
Very similar to this one. I wonder if you can get in trouble for plagiarizing a technique for avoiding a question?
geometry test 1 answers
Knowledge of Math
The term algebra states the constants and variables in the equation. Algebra is one of the familiar topics in mathematics. Algebraic fraction is similar to the usual fraction but a difference is that the numerator and the denominator are algebraic expressions. Algebraic fraction simplifying is similar to that of algebraic fractions. Simplifying algebraic expressions is very easy and simple. In online, students can learn about algebraic expressions. In online, algebraic calculator is helpful to solve the given problem. In online calculator, when the denominator and numerator value is entered, calculator automatically simplifies the problem.
Example : Simplifying Algebraic Fractions
Simplify 3b/9b2.
Solution:
The given problem is an algebraic expression.,
3b = 3*b
9b2 9*b*b
Like terms gets cancelled,
On simplifying this,
Answer = 1/3b
Need More help with Algebra 1
expression simplifying calculator
Note on simplifying expressions calculator
In this article let me help you on simplifying expressions calculator. Algebra is the subdivision of science which deals with the learning of rules of operations and relations, and the concepts and constructions arising from them, including polynomials. Scheming simplifying exponential expressions is also a air of algebra.Exponents are represented by a force on a size or varied. Expressions are chainlike with exponential damage.
Index is a maths computation in algebra represented by two drawing 'a' & 'n', where a is the stand and n is the power. This could also help us on factors of 39. The index denotes the limit of nowadays the form is closed into, it is endeavour of algebra an! d they person positive properties, which enables us in simplifying the affine to them are surrendered in the following sections.
Let us simplify exponential expressions provision. Keep reading may be in the next session let me help you on tutoring math
expression simplifying calculator
Free algebra worksheets
Usually algebra textbooks provide lots of problems to practice the algebraic concepts and techniques, but some of you may still benefit from resources for free (or mostly so) printable algebra worksheets. Please see the list below, which I've originally compiled for my HomeschoolMath.net site.
Algebra worksheets
Worksheet Builder
Great and free worksheet maker software with nearly 7,000 built-in algebra and geometry questions.
www.jmap.org/JMAP_WORKSHEET_BUILDER_INSTALLATION_FILES.htm
Free Algebra Worksheets from KUTA Software
Free worksheets (PDF) for equations, exponents, inequalities, polynomials, radical & rational expressions and more.
www.kutasoftware.com/free.html
AlgebraHelp.com worksheets
Interactive worksheets that are checked online for most algebra 1 topics.
www.algebrahelp.com/worksheets/
Math.Com algebra worksheets generator
Generate worksheets for: linear equations, systems of equations, and quadratic equations.
www.math.com/students/worksheet/algebra_sp.htm
LessonCorner worksheets
These free worksheets include a few topics such as calculations with polynomials, factoring, and graphing lines.
www.lessoncorner.com/worksheets/
Algebra Fun Sheets
Worksheets that integrate algebra skills with fun activities including sudoku, word finds, riddles, color patterns, crosswords, games, matching cards, etc. A subscription is required.
www.algebrafunsheets.com
About.com Algebra Worksheets
An assorted collection of free algebra worksheets and answers. These pages are not very well organized, but they have lots of worksheets.
math.about.com/od/algebraworksheets/Algebra_Worksheets.htm
Algebra Worksheets from MathWorksheetCenter
Lots of worksheets for over 100 algebra topics. A few are free; most are accessible only by one-year a subscription.
www.mathworksheetscenter.com/mathskills/algebra/
These are for graphing linear equations and linear inequalities.
Online Math Work
Free multiple-choice worksheets for pre-algebra and algebra 1 topics. You can do them online, or copy to a word processor to print.
www.mathonlinework.com
Lastly... my own algebra worksheet collections, which aren't free but there are many free samples:
Math Mammoth Algebra 1 Worksheets Collection
A two-part collection (A and B) of 137 quality algebra worksheets covering all the topics in a typical algebra 1 curriculum. These worksheets are hand-crafted and contain lots of word problems and other variable problems. Free samples available. $11.50.
www.mathmammoth.com/worksheets/algebra_1.php
answers for algebra
Mr. Giraffe Solves Your Linear Algebra Problems
Please Note: If I made any mistakes in this test, perhaps this picture of a giraffe will convince otherwise.
algebra problems and answers
Learning Targets: Graph multiple inequalities using x and y intercepts, test a point to determine the feasible and non feasible region, and shade and label the feasible and non feasible region
Goals:
• Go Over L2I1 MIQ
• Go over the rest of the game problem: critical points, objective function, solving linear programming problems
• Work on L2I2 calendar work
HW: Finish your groups calendar work not finished in class.
Reminder: L2I2 MIQ tomorrow. Be able to complete a problem like what’s in the investigation. In order to retake the MIQ, make sure you have completed everything from the ENTIRE row labeled Investigation 2 including all the investigation work and the MIQ practice problem.
Reflection: If you need help with any of these things, just ask.
To Find Critical Points: Look at your shading. One side of the critical point should have all your colors shaded and the other side should have only one, two, or no colors shaded.
Prove x and y intercepts are Critical Points: If you put your inequalities in y= form to graph the lines, show me a table snapshot of the x and y-intercepts. If you used x and y-intercepts to graph the lines, then you have already shown all the work you need.
Prove Intersection Points are Critical Points: If you put your inequalities in y= form to graph the lines, show me a table snapshot of the intersection points. If you used x and y-intercepts to graph the lines, use matrices to solve the system of equations.
If a Critical Point Lies on a Horizontal or Vertical Line You Graphed: Use the x or y value from the horizontal or vertical line you graphed and plug it into the other intersecting equation to solve for the missing variable. For example, if you graphed x=3 and 2x+5y=8 you would plug 3 in for x and solve for y in the 2x+5y=8 equations.
Writing the Objective Equation: This information is usually the last bullet, last sentence in a paragraph, or a separate part of the problem. This is what you are maximizing or minimizing.
Maximizing or Minimizing the Objective Equation: Plug in every critical point to the objective equation and check to see which point maximizes or minimizes the objective. That point is the solution for the whole problem. Celebrate! You are finished!
x and y intercepts
BLAS, LAPACK, and Parrot-Linear-Algebra
I hadn't touched it in a while, for a variety of reasons. Austing Hastings has been very busy with other projects and hasn't cut an "official" release of Kakapo that works with the last supported Parrot release, 2.3. I've applied some hotfixes that make Kakapo build and pass tests on 2.3, but that's not quite good enough. When I make a PLA release I don't want "clone the Kakapo repository from Gitorious" in the instructions, because then the instructions get out of date when Kakapo is updated to be incompatible. What I want is to say "Download Kakapo version X from this URL", which will be invariant.
Last night I added some prototype new features to the NumMatrix2D PMC, the basic numeric matrix type. I'm going to mirror the majority of those changes to the two other general-purpose types; ComplexMatrix2D and PMCMatrix2D. I added methods to do basic elementary row operations (swap two rows, add a multiple of one row to another, and multiply the contents of a row by a non-zero constant), and used those operations to put together an example program to calculate both the Row Echelon and Reduced Row Echelon forms of a matrix. Those methods, combined with block manipulation methods I added previously and the new GEMM method I added last night as well, create a basis for us to create a comprehensive library of linear algebra routines.
But that brings me to my next concern: How to implement the remainder of the fundamental algorithms a linear algebra pack is going to want to provide? Obviously I would love to wrap the LAPACK library up and call the routines it provides directly. LAPACK provides a huge number of routines for doing things like singular value decomposition, QR, LU, and other types of decomposition, solving the eigen probem, solving systems of equations in two variables, etc. In fact, LAPACK provides almost all of the functionality I would want PLA to have in the next few months.
The problem, however, is that LAPACK is not nearly as accessible from C code as I would like. In fact, there is no "standard" for C-bindings to the library, and several lame attempts are available that tend to be incompatible with each other. The reference version, and only reliably-available version, of LAPACK is written in FORTRAN. The standard CLAPACK library is just a machine-lead translation of the FORTRAN sourcecode, with a few points in the code needing to be manually tweaked after conversion. It has a few problems, including the fact that every single function parameter (even basic ints and floats) must be passed by reference. The ATLAS library, which PLA currently prefers to provide BLAS bindings, provides some LAPACK C bindings of it's own, but only a very very small subset of all LAPACK functions are provided, and the ones it does have hardly support all the operations PLA is going to want to provide.
CLAPACK, being more or less the standard could be made to work, except that it doesn't come as any sort of user-friendly package. There are no CLAPACK packages for Debian (Ubuntu) or RedHat that I have seen, and that raises the barrier to entry significantly, something that I want to avoid.
I could use the LAPACK library directly, and twist my code to match the FORTRAN calling conventions and data alignments. That's not unthinkable, though it would require some more work on the part of PLA developers than any other solution.
I could skip LAPACK entirely, and instead rely on something like GSL with proper C bindings built-in. GSL does provide several important matrix operations and decompositions, though I would need to do more research into the capabilities of that library.. What I don't want is to lose focus and have this project grow to try and become a general wrapper for all of GSL. I want PLA to stay focused on Linear Algebra only. We could maybe create sister projects to encapsulate more of the GSL functionality and use PLA under the hood to implement the basic data structures, of course.
Maybe we will get lucky, and the new NCI system will have support for calling FORTRAN routines from shared libraries. I don't think we can just anticipate this kind of functionality, at least not during the GSoC program.
A "nuclear" option, perhaps, would be to not rely on any external library for these things and instead brew up all the basics myself. I'm not against such work per se, but it would be a huge investment in time and the case cannot be made that it's the best use of my limited development time. I did put together a Gauss-Jordan elimination routine last night, it wouldn't be too too much effort to put together a generalized QR decomposition algorithm and a singular-value decomposition algorithm, followed by routines to calculate eigenvalues, eigenvectors, and matrix inverses from those things. If PLA had a larger team of active developers who wanted to participate in this kind of work it would be an easier decision to make, but if it's primarily me and a handful of occasional-contributors, this really isn't a doable task.
My plan for PLA in the near future is this: After the 2.6 release I want to push for a stable release of Kakapo, and then using that I want to cut a release of PLA. From that point forward, PLA will target the 2.6 version of Parrot at least until 2.9 and maybe later. The first release of PLA is going to provide three basic matrix types: A 2D matrix of floats, a 2D matrix of complex numbers, and a 2D matrix of PMCs. These three matrix types will have a large base of common functionality and each type will have some additional functionality too, as required by that type. Everything provided will be well-tested (including error cases, which I haven't exercised nearly enough so far) and some example programs will be provided in both PIR and NQP.
There are a few projects I am envisioning to start in the future that will rely on PLA, so I really am hoping to create a nice, stable, functional release within the next few months. I'll post more information about any other projects as they arise.
elementary linear algebra
#13 LCM and HCF
How to find L.C.M and H.C.F. ? METHOD DEMONSTRATION
#2
This is a calculator for calculating GCD and LCM
#3
math least common multiple
An Inexpensive ACT Calculator & a Potpourri of Links
An Inexpensive ACT Calculator & a Potpourri of Links
For the revision to our ACT book, Preparing for the ACT: Mathematics and Science Reasoning, we purchased a cheap scientific calculator from Casio, the fx-115 ES, and I was amazed at the power of this pocket calculator. The calculator sells for about $20 (or less depending o! n where you purchase it), and can:
- work with percents, fractions, and mixed numbers
- simplify radical expressions
- express answers in terms of pi
- do operations with complex numbers
- find permutations and combinations
- perform one-variable statistics (including frequency tables)
- solve one variable equations
- do operations with matrices
- calculate integrals
- do all the usual things that a scientific calculator can do (like radicals, powers, trig functions, etc.)
To top it off, the darn thing runs on solar power! The geek in me can’t help but say, “Wow!” It’s proof that technology can be cheap and powerful (I’m talking to you, Apple.) I called the ACT twice just to confirm that this calculat! or is allowed on the test.
!
Note/Warning: The ACT is a timed test – you have an average of 1 minute per question. This means two things: (1) Buying a calculator that you are not accustomed to using right before the test is a big mistake. (2) While using an advanced calculator can be helpful in the classroom, you probably won’t have time to use the advanced features on the ACT test. The Casio fx-115 ES’s features that will be useful on the ACT are fractions, mixed numbers, simplifying radical expressions, expressing answers in terms of pi, and operations with complex numb! ers.
To conclude today’s post, here is the promised potpourri of links:
- Engineers beat math PH.D’s in math contest. The contest: The Netflix Prize. Make Netflix’s movie recommendation system more accurate by 10%. The math: statistics.
- We’re all probably going to speak Chinese one day. A group of computer science students from China created one of the most awesome pieces of software I’ve seen in a long time: PhotoSketch (see the video below). It takes a hand-drawn sketch tagged with the name of the object and turns it into a real-world photo. It works in one of those “Why didn’t I think of that?” ways – the software does a web search based on the tags ! and chooses pictures that match the sketch. The best matches a! re then combined together and the user chooses the best looking image. The results are pretty amazing – check out the video below.
- The Making of a Mathlete. PBS is going to air a documentary about the International Math Olympiad. Need I say more? No, really, it actually looks pretty exciting.
PhotoSketch: Internet Image Montage from tao chen on Vimeo.
How to simplify radicals with fractions
Solving Differential Equations with Mathematica - Part I: Time Series
(* Define control parameters *)Voila!
\[Sigma] = 3; \[Beta] = 1; \[Rho] = 10;
(* Define initial conditions for later use *)
x0 = 0; y0 = 1; z0 = 1;
(* Define interval of integration *)
Tend = 20 \[Pi];
(* Lump the initial conditions in one variable *)
initialConditions = {x[0] == x0, y[0] == y0, z[0] == z0};
(* Lump the Lorenz equations in one variable *)
LorenzEquations = {x'[t] == \[Sigma] (y[t] - x[t]),
y'[t] == \[Rho] x[t] - x[t] z[t] - y[t],
z'[t] == x[t] y[t]! - \[Beta] z[t],
initialConditions};
(* Use NDSolve to integrate the Lorenz equations *)
s1 = NDSolve[LorenzEquations, {x[t], y[t], z[t]}, {t, 0, Tend}, MaxSteps -> \[Infinity]];
(* Plot the solution *)
Plot[Evaluate[x[t] /. s1], {t, 0, Tend}, PlotRange -> All]
Note that \[Sigma] will automatically convert to the greek symbol sigma. The same applies for the rest. You can also generate the greek letters by pressing escape, typing a letter on the keyboard, and then pressing escape. For example, escape, s, escape will turn into sigma.
Going back to the previous code, the two important statements are the NDSolve[] and the Plot[Evaluate[]].
In the first one, we are solving the Lorenz equations for x[t], y[t], and z[t] from t = 0 to t = Tend with an infinite number of time steps (MaxSteps->Infinity).
As for the Plot[Evaluate[]], the "x[t] /. s1" means replace all x[t] with the data contained in s1, which holds the results of the numerical integration. One could have also chosen to plot y[t] or z[t].
For first or higher order ODEs, it is advisable to get rid of all derivatives by definig them as new variables. This will be helpful for phase space diagrams to be discussed in the next article. For example, if you have the following system (Ueda's oscillator)
it can be converted to
Download Mathematica notebook [right click / save as]
Cite as:
Saad, T. "Solving Differential Equations with Mathematica - Part I: Time Series". Weblog entry from Please Make A Note. http://pleasemakeanote.blogspot.com/2008/05/solving-differential-equations-with.html
solve a homogeneous differential equation
Happy Thanksgiving!
geometry theorems and constructions
Students Interpretations
Check out examples of works that the students have created over the past Sacred Geometry class sessions. I've kept their works as anonymous to respect their privacy. Enjoy!
Geometry Constructions
Allysa's Scribepost for November 30, 2009
Number Line:
One pattern we saw was the spaces between the perfect squares, it went up by 2 every time. The spaces were: 3, 5, 7, and 9.
To fill in the spaces, we had to use fractions.
To convert them into a decimal, we just used a calculator.
e.g. 1 divided by 3 = 0.333333333
Then we added the whole number before the decimal.
1, 1.33, 1.66, 2.
For 5 spaces it would be:
1 1/5, 1 2/5, 1 3/5, 1 4/5, 1 5/5
HOMEWORK:
Find the decimals for 5 spaces.
Sorry, I put it up late.
THANKS FOR READING & PLEASE COMMENT ! :D
prime factorization chart
Work Problem 5
The following was a question a anonymous visitor asked: Jim can fill a pool carrying buckets of water in 30 minutes. Sue can do the same job in 45 minutes. Tony can do the same job in 1 ½ hours. How quickly can all three fill the pool together?
Work problem 5 solution here
algebra 1 answers
Area of a Triangle
You are probably familiar with one formula for finding the Area of a Triangle:
Area = 1/2 (base)(height)
Compare this to finding the area of a rectangle:
The area of the rectangle is equal to the product of (base) x (height)..... (or length x widt! h). However, by drawing a diagonal within the rectangle which joins two opposite corners, you can see that each newly-formed triangle is equal to half of the area of the original rectangle. Therefore, the Area of a Triangle is one-half the area of the rectangle, as shown by this triangle area formula. Even if you are looking at a triangle that doesn't immediately look like it is half of a rectangle, this formula still applies.
To prove it, you can draw a line in to represent the height, as I have shown here,! thus creating two smaller triangles, and you can rearrange th! em to se e that they indeed are equal to the area of half a rectangle:
That is one way to find the Area of a Triangle. However, if instead of base and height measurements, you are given lengths of sides or angles, this method won't work for you. In this case, you need to use a trig equation to solve for the Area of a Triangle.
Let's start with the first equation we had above, and modify it. By the standard trig identities, we can show that:
height = (a)(SinC)
So substituting that into our formula:
Area = 1/2(base)(height)
Area = 1/2(b)(a)(SinC)
And this is the trig formula for solving the Area of a Triangle!
Area = (1/2)abSinC
You can use this to find the Area of a Triangle where you know any two sides and the angle between them! It's that easy!
Area of a Triangle
Andrew Ardito: Soul of the Albany Area Math Circle
Andrew is a homeschooled student from Coxsackie whose accomplishments go well beyond mathematics. By the end of his junior year of high school, he had already reached National AP Scholar status by getting high! scores on nine AP exams, and he took five additional AP exams in his senior year. Those 14 AP exams covered a wide variety of disciplines including literature, art history, US and world history, economics and government, as well as mathematics and all the natural sciences. He also managed to find time to study four years of Russian during high school. He went into special depth in mathematics and physics, taking several advanced physics classes at SUNY Albany as well as even more advanced math classes at SUNY Albany and RPI, including number theory, topology, analysis, and a graduate class in combinatorics. His many honors also include a National Merit Scholarship and a Robert C. Byrd Scholarship. In addition, Andrew has won numerous awards for chess, as well as coaching a chess club for younger students.
Andrew started out as one of the youngest members of our first middle school math circle, back in 2003. He eagerly moved on to greater and greater challeng! es, taking over as captain of our high school math circle in h! is sopho more year of high school. He also student-coached MATHCOUNTS teams throughout his high school years. In his junior year, he helped spur a reinvorigation of our middle school math circles, and he leaves a remarkable legacy of inspiring leadership behind, as he has worked with so many students who have already demonstrated great promise as mentors themselves.
He is an outstanding example of the guiding principle of our math circle: the best way to learn is to share what you think you already know with someone else, because explaining it to them deepens your own understanding.
Andrew has accumulated an extraordinary number of accomplishments during his time as a member of our math circle, both in mathematics and in other areas.
A partial list of his many honors in mathematics include:
☞ National MATHCOUNTS contestant all three years as a member of the New York State team in middle school. As a 7th and 8th grader, he won the New York State championship and made the National Countdown Round in both years, placing sixth on writtens in 2005 and and 2nd place in the country at MATHCOUNTS Nationals Countdown in 2006. He also provided team leadership to two Chapter-winning teams, one of which came in 3rd place in the state.
☞ As a high school student, Andrew coached many successful MATHCOUNTS teams, including several teams that won Chapter championships and one team that won a state championship. He also coached many students who won high individua! l honors at Chapter and State MATHCOUNTS contests, including q! uite a n umber of Chapter and State CountDown participants and National MATHCOUNTS contestants. His mentorship and enthusiastic example has also inspired many younger students to embrace some exciting and challenging high school contests while still in middle school, including AMC10/12, HMMT, and NYSML/ARML.
The photo above shows Gili Rusak, a 7th grader at Shaker Junior High, and Aniket Tolpadi, an 8th grader at Iroquois Middle School, working with Andrew in a middle sch! ool math circle meeting. Both Gili and Aniket qualified for the American Invitational Math Exam (AIME) this year. Gili was the only 7th grader in the state to qualify for this exam, while Aniket was one of only four 8th graders in the state to do so.
☞ Andrew has earned many outstanding honors in high school math contests: he began taking the AMC high school contests in sixth grade and has won many honors in that series: he qualified for the AIME seven times, for the USA Math Olympiad five times. In 2007, he qualified for the Math Olympiad Summer Program; in 2008, he was the top scorer in the state on the AMC12B, and he also won an AMC12 gold medal for his consecutive four years as math circle's high scorer on the 12A and/or 12B.
Andrew has consistently been among Albany Area Math Circle's top three scorers on the AMC12 in every year since he began taking that exam in seventh grade, helping the math circle teams score among the top teams i! n the state each year. In February 2010, in his final attempt! at the AMC12, he contributed to a record best-ever team performance by any Albany Area Math Circle on any AMC math contest--the Albany Area Math Circle AMC12B team of Dave Bieber, Andrew Ardito, and Schuyler Smith placed first in New York State. Indeed their team's performance on that remarkably challenging 2010 AMC12B contest was among the top five team scores in the country. Two younger students whom Andrew has coached in the past both earned top individual honors on honor roll for AMC12A/B: Schuyler Smith tied for high-scorer in the NY-NJ region on that contest, and Matthew ! Babbitt earned honors as the region's top-scoring student in ninth grade or below.
The first time Andrew qualified for the USA Math Olympiad, he had to take the 9-hour contest by himself, because he was the only qualifier in our area that year. However, he has generously shared his olympiad problem-solving skills with other students, and this has resulted in the camaraderie of a growing number of math olympians in our math circle taking the contest together in every year since then. This year's qualification process was exceptionally rigorous and yet our math circle had a record-tying five students taking the USA Math Olympiad, as Andrew was joined by Felix Sun (Shenendahoah HS junior), Matthew Babbitt (homeschool freshman), Schuyler Smith (homeschool junior), and Dave Bieber (Niskayuna HS senior).
There is an ancient metaphor invoked by many philosophers, mathematicians, and scientists that seems especially apt here: "standing on the shoulders of giants."
Andrew once stood on the shoulders of earlier members of Albany Area Math Circle, and he gives special credit to founding member and "soul of the Albany Area Math Circle" award-winner Drew Besse for inspiring and mentoring him.
In turn, Andrew's shoulders have surely provided strong support to launch many other students in our math circle, who in turn launch other students, and so on and on. Although Andrew and Dave will be heading off to college in the fall, our other olympiad veterans, who also include Ashley Cho (a rising senior at Emma Willard) and Jay White (a rising homesc! hool senior), as well as Felix, Matthew, and Schuyler, will surely continue to provide strong mentorship shoulders to future aspiring olympians in our math circle. Andrew has also helped to encourage and inspire a growing number of participants in the USA Physics Olympiad contests: our math circle had a record four students reach the top level of that contest series this year, with Dave, Schuyler, and Gurtej Kanwar (a Bethlehem HS junior) all joining Andrew in taking the USAPHO.
☞ Andrew has consistently risen to the occasion with outstanding and enthusiastic contributions to countless collaborative math team rounds in a variety of venues. He began participating on our teams for NYSML and the Harvard-MIT Math Tournament in middle school. PUMaC, the Princeton University Math Contest, did not start until he was a freshman, but he participated in that contest, studded with a stellar field of many of the strongest math olympians in the country, throughout ! his four years of high school, accruing multiple top 10 honors! as a so phomore, junior, and senior years, including 2nd place rankings in combinatorics (2009) and number theory (2007). His record at ARML is also especially noteworthy: in 2007 and 2010, he made the national tiebreak rounds and received Individual High Scorer of the Meet awards, while in 2008 and 2009, he earned honors as high scorer on the Upstate NY team.
As Team Captain, his leadership contributions to team rounds at NYSML, HMMT, PUMaC, and ARML have been exceptionally imp! ressive. Competing against extraordinarily strong teams from all over the state, the country, or even the world, he has led our math circle teams to many great rankings. Special highlights for teams he has led as captain include 3rd place rank at NYSML 2009--and "Most Improved" team award, 2nd place rank at NYSML 2010, 1st place rank for NYSML local in 2009-2010, a top 10 rank at HMMT 2010, and 3rd place Power Round at PUMaC in January 2009.
The crowning accomplishment for Albany Area Math Circle during Andrew's time as captain came in the national/international ARML Power Contest in 2009-2010. That contest is a collaborative round in which all members of the math circle were able to work together, and it's key for the captain to find ways to make the most effective use of the collective talents of rookies as well as veterans. There are two rounds to that contest each year, one in November and one in February. There is extremely challenging competition fro! m math circles and magnet school programs all over the country! and the world. The top 10 teams are honored with plaques at ARML each June. Our math circle had done well in the past, but we had never made that top 10 list before. Andrew clearly had been thinking hard about how to draw on his years of experience and knowledge of other students' strengths to do well this year. In November, the team was thrilled to take eighth place on the honors list, and in February, they were even more ecstatic to learn they'd placed second on the February round, resulting in a fourth place rank overall for the year. The photo below shows happy representatives of the Upstate NY Math team from the Albany Area Math Circle accepting the plaque at the Penn State ARML awards ceremony on behalf of all members of the Albany Area Math Circle.
As I write these words in August 2010, Andrew is at PROMYS, a summer program designed for students with extreme talent and passion for mathematics. Andrew won the ARML Scholarship to attend PROMY! S in 2008 as a student, and returned as an advanced student th! e follow ing year. This summer, he is a member of the instructional staff at PROMYS, working as a counselor.
Next month, Andrew will go off to Princeton, where he plans to study math. Dave Bieber will be heading to Princeton as well, with plans to study computer science. (For more about Dave, see this post.) Dave and Andrew have been the heart and soul of Albany Area Math Circle since Beth Schaffer and Drew Besse graduated in 2007.
Our math circle will miss them both, but we hope they'll come back to visit during school breaks, as Beth and Drew and many of our alumni do. And our math circle students will look forward to seeing them when the t! eam travels to Princeton for PUMaC in November.
And, if Dave and Andrew miss our math circle, Princeton now has one too. Our loss is their gain. Well, no, not really--it is not a zero-sum game, but rather a positive-sum game. Our intersecting circles enrich us all.
And, of course, both Andrew and Dave have left our math circle with a remarkable legacy--their friendly encouraging spirits and their passionate enthusiasm for challenge and their generous willingness to share their prodigious talents and their remarkable ability to help others work well together will remain behind in the hearts and souls of all the enthusiastic younger students with whom they have worked.
area of a circle