Monday, May 31, 2010

Why Study Math?

Why Study Math?
Math is often referred to as a universal language. Discover reasons for studying math with tips from a mathematics tutor in this free video on math lessons.

Expert: Fernando Millan
Filmmaker: Paul Muller


Mathematic Skills for Grades 3-6

  1. A+ Math  Play exciting games like Matho and Hidden Picture (+ -x /)  Also has great flashcards
  2. Builder Ted In this interactive game, students must order decimals
  3. Comparing Big Numbers
  4. Comparing fractions and decimals - Practice converting one to the other in this interactive game.
  5. Concentration Games from Harcourt Brace: Fractions/Decimal Equivalents, Fractions/Decimal Equivalents II, Fractions/Decimal Equivalents III, Equivalent fractions, Decimals, and Percents.
  6. Cash Out You're the cashier at this crazy store. You need to give change to the customers buying things. Try to sell as many items as possible before the time runs out.
  7. Change Maker Get as much money in your piggy bank as possible, by figuring out the correct change. From FunBrain
  8. Count the Goodies - This multiplication activity from Harcourt School’s Mighty Math Calculating Crew asks you to multiply using regrouping. Three problems at a time are presented.
  9. Decimal Squares Add, subtract and multiply decimals in a multitude of games. New!
  10. Divisibility Rules Simplify your math!
  11. Division Machine This activity has three levels of division for students to practice. New!
  12. Disaster Math - Six sets of interactive word problems: Earthquake Math, Hurricane Math, Tornado Math, Wild Fire Math, Winter Storm Math, and Flood Math.
  13. Fraction Shapes Explore geometric models of fractions and discover relations among them.
  14. Fractions, Decimals or Percent - two are given, you supply the third
  15. Fresh Baked Fractions Click on the fraction that is not equal to the others. From FunBrain. Easy to Super Brain
  16. Game Bone Can you find the 10 hidden bones on the 1 - 100 number square in less than a minute ? (Grade 3)
  17. Genius Boxing Punch your genius opponent by completing the number sentence using <, >, or =. (Grade 3) New!
  18. Guess the Number Developing skills in halving and estimation
  19. Interactive Activities Number and Operations
  20. I Know That!  Online Multimedia Fraction Games, Registration is free!
  21. Hooda Math Hooda Math is all about making math fun and making math easy. Teacher and Student Recommended
  22. Johnnie's Math Page Multiplication Activities
  23. Magic Math Market Discover the magic of two important mathematical concepts: fractions and decimals. (Grades 3-5)
  24. Math 5 Live Lessons> Place Value, Multiples, Factors, Primes, Fractions, Decimals, Multiplication and Division (Grade 5)
  25. Math Baseball Math practice from FunBrain (+ - x /)
  26. Math Run How fast is your brain? A simple brain training game for everyone. Begins with simple addition facts and gets progressively more difficult. New!
  27. Maths: Fractions - Revisewise-Interactive guide to fractions
  28. Maths: Number Patterns - Revisewise-Interactive guide to patterns
  29. Multiplying Fractions Practice multiplying fractions. (Grades 4-6)
  30. Number Cop Choose your game mode: select a number to practice multiples of that number, or select 'Primes' or 'Perfect Squares'. In each game type you will have a total of 50 numbers to test.
  31. National Library of Virtual Manipulatives Number & Operations (Grades 3-5)
  32. National Library of Virtual Manipulatives Number & Operations (Grades 6-8)
  33. One False Move Step into the wrong room and you'll be doomed. You are given a table of numbers.
    Click on the numbers in order from lowest to highest (or highest to lowest). (Easy-Medium-Hard)
  34. Penguin Waiter Game Figure out the tip from FunBrain
  35. Percent Problems - a one-player, or two-player Jeopardy style game from Quia
  36. Place Value Darts Use known number facts and place value to multiply and divide integers, including by 10 and then 100 (whole- number answers). (Grades 5-6)
  37. The Place Value Game -A math game requiring logic, strategy and a little luck!
  38. Place Value Puzzler Choose Place Value. Easy · Medium · Hard · Super Brain.
  39. Place Value Up to 1000 Can you put your knowledge to the test to beat Robbie the Robot? (Grade 3)
  40. Power Football   All operations with decimals. From FunBrain
  41. QUIA Math Journey  An online math activity. (+ - x /,rounding)
  42. Rounding-Fun Brain gives games (Easy, Medium, Hard, Super Brain)
  43. Rounding Off Round off numbers to significant figures, decimal places or the nearest 10, 100 or 1000. From BBC
  44. Snork's Long Division Game A step by step long division game that allows students to practice their long division skills
  45. Soccer Shootout - Multiplying Fractions from FunBrain
  46. Solving Percent Problems Using a Pyramid Grid - an interactive lesson
  47. Spy Guys Interactive Lessons>Decimals, Percents, Integers, Prime Factorization, Fractions, Problem Solving Bank (Grade 6)
  48. Tic Tac Toe Squares Get three X's in a row before the computer gets three O's in a row. From FunBrain (+ - x /) Easy to Super Brain
  49. Visual Fractions This Fractions is a collection of activities that use pie charts, number lines, and other graphics to illustrate fraction operations.
  50. Wicked Interactive Play these exciting mathematic games to improve your basic fact recall
  51. Interactive games to help teach the multiplication facts.
  1. Beat the Clock Interactive time game
  2. Gamequarium: Math Games-Measurement
  3. Math 5 Live - Lessons> Area, Perimeter, Volume (Grade 5)
  4. Measurement Equivalents - Match game with standard equivalent measurements, such as pounds, ounces, pints, cups, etc.
  5. Measurement Game - Practice using a ruler, in inches and centimeters. FunBrain
  6. National Library of Virtual Manipulatives Measurement Manipulatives (Grades 3-5)
  7. National Library of Virtual Manipulatives Measurement Manipulatives (Grades 6-8)
  8. Spy Guys Interactive - Lessons>Area, Perimeter, Volume
  9. What Time Will It Be?
  1. Algebra Planet Blaster - Practice your math and defend your planet!
  2. Angles - Interactive game to help recognize and label acute, obtuse, and right angles.
  3. Catch the Fly - [all 4 quadrants are used] Use the keyboard to enter the x and y values of an ordered pair to help the fly catch a bug. No score is kept, each question is essentially a one question game
  4. Cool Math - Pre Algebra has a ton of really easy to follow lessons and examples that will make you a successful pre-algebra game
  5. Create your own pattern - Drag shapes to create a pattern. Instructions on how to use this page in your classroom.
  6. Elapsed Time on a Clock - From Harcourt School Publishers
  7. Gamequarium: Math Games-Algebra
  8. Interactive Activities Algebra
  9. Maths: Shapes - Revisewise-Recognize properties and names of shapes
  10. Measuring Angles - Use a virtual protractor for making measurements.
  11. Measuring Angles - Investigating angles and the use of a protractor
  12. National Library of Virtual Manipulatives Algebra Manipulatives (Grades 3-5)
  13. National Library of Virtual Manipulatives Algebra Manipulatives (Grades 6-8)
  14. Number Cracker - guess what number comes next in the pattern
  15. Spy Guys Interactive - Lessons>Summarizing and Extending Patterns
  16. What's the Point - Find the x-y point on the grid.  Funbrain
  1. Data Interpretation Games - Numerous activities on using and interpreting data - Activities include bar, pie and line graphs, data collecting and much more
  2. Data Picking - While Class 8H pose for a photo, click on each student to collect data to produce a frequency table.  Next, select a chart or graph that matches your data.
  3. Graphing-Grapher - interactive column graph maker, students can change values and labels
  4. Graphing-Creating your own Graph It's easy to create and even print your own graphs.
  5. Graphing - Virtual Manipulatives Bar Chart, Histogram, Pie Chart, Venn Diagrams
  6. Interactive Activities Probability
  7. Math 5 Live - Lessons> Displaying Data, Probability (Grade 5)
  8. Mode, Mean, Median - Interactive lesson followed by activities - (UK measurements)
  9. National Library of Virtual Manipulatives Data Analysis & Probability Manipulatives (Grades 3-5)
  10. National Library of Virtual Manipulatives Data Analysis & Probability Manipulatives (Grades 6-8)
  11. Pattern Quest By using clues, and eliminating wrong guesses, the player attempts to deduce a pattern in the fewest number of guesses.
  12. Reading Charts and Graphs - students answer questions about various graphs
  13. Train Race This game can be used to extend a basic understanding of the median, mean and range. From BBC
  14. Spy Guys Interactive - Lessons>Graphs, Probability
  15. What Are Your Chances? Gain a little knowledge about probability.
  16. Bar Chart create your own bar graph using your data.
General Math Resources
A+ MATH This Web site has flashcards, math problems, and games -- all with the intent of increasing mathematical knowledge.
BBC Education Maths File   ReviseWise
Beacon Learning Center The resources posted in the Beacon database are products of professional development activities teaching a standards-based planning model.
Brainchild - twenty-four questions, followed by practice on any incorrect responses
Brain Teasers Solve these weekly brain teaser math puzzles. Brain teasers are arranged by grade level and include hints in case you need some help. From Houghton Mifflin's Education Place
Cool Math Cool Math is "designed for the pure enjoyment of mathematics." This interactive site features games, puzzles, calculators, and lesson plans.
Cyberchase Based on the award-winning TV show, this site teaches kids that math is everywhere, everyone can be good at it, and it's fun! The site includes more than 40 interactive games; as well as printable activities, Web adventures, e-cards; and much more.  Provides numerous online language arts and math activities for students in grades K-8. In addition, the site provides printable work sheets.
Edustock A ThinkQuest page designed to teach what the stock market is. It includes: tutorials on the stock market, how to pick good stocks, information on a select group of companies, and it provides a FREE 20 minute delayed Stock market simulation on the World Wide Web.
FunBrain-Numbers A variety of games related to numbers and mathematics
Hooda Math Hooda Math is all about making math fun and making math easy. New and Recommended!!
I Know That-Math Online Multimedia Educational Math Games.  Registration is free!
Internet4Classrooms  Math Standards for 3rd Grade  4th Grade  5th Grade  6th Grade
Interactive Websites: Math  Provides standards-based cross curricular web resources designed to enhance online learning opportunities. These sites interact with the user usually through either a text-based or graphical user interface.
Math Advantage The activities at this site were created to accompany the Harcourt’s Math Advantage textbook series. The activities, however, relate to basic math concepts and can be used successfully with any Math program.  Kids will enjoy practicing math skills with these fun activities.
Maths File Game Show Topics covered: Algebra, Numbers, Percentages and Fractions From BBC
Maths: ReviseWise Number, Data Handling, Shape, Space & Measure, Mental Maths. From BBC
Math Playground  Features games, puzzles and hundreds of interactive word problems for elementary school students includes Math Word Problems
NCTM Illuminations  From the National Council of Teachers of Mathematics includes lesson plans, video vignettes of teaching and learning, interactive lessons for students, and an interactive version of NCTM math standards.. From Marco Polo. A free math activity site that has a huge variety of flash-based math games. Beyond the games, they also have very good descriptions of mathematical concepts 
Think Math from Harcourt School Publishers New
Visual Math This site features a free online interactive tutorial for pre-algebra students that includes games, puzzles, and animations that emphasize learning concepts by visualization.
Woodlands Math Zone Several fun online interactive activities here to help improve your mental maths skills. These pages are aimed at 7 -11 year olds. You must have Java and Flash installed to play these games.

Sunday, May 30, 2010

Double Division with 3 Digit Divisors 612416/983

Method of Vietnamese Diagram: Double the Divisor 3 Times

Method of International Diagram: Double the Divisor 3 Times

Method of Vietnamese Diagram:  Double the Divisor 5 Times
It's Easy!
Step 1 - Double, double, double.
Step 2 - Subtract off multiples.
Step 3 - Add up your answer."

Photos source: Nguyen Thi Lan Phuong,

9th Vietnam Mathematical Olympiad 1970

A1.  ABC is a triangle. Show that sin A/2 sin B/2 sin C/2 < 1/4.  

A2.  Find all positive integers which divide 1890·1930·1970 and are not divisible by 45. 

A3.  The function f(x, y) is defined for all real numbers x, y. It satisfies f(x,0) = ax (where a is a non-zero constant) and if (c, d) and (h, k) are distinct points such that f(c, d) = f(h, k), then f(x, y) is constant on the line through (c, d) and (h, k). Show that for any real b, the set of points such that f(x, y) = b is a straight line and that all such lines are parallel. Show that f(x, y) = ax + by, for some constant b.  

B1.  AB and CD are perpendicular diameters of a circle. L is the tangent to the circle at A. M is a variable point on the minor arc AC. The ray BM, DM meet the line L at P and Q respectively. Show that AP·AQ = AB·PQ. Show how to construct the point M which gives BQ parallel to DP. If the lines OP and BQ meet at N find the locus of N. The lines BP and BQ meet the tangent at D at P' and Q' respectively. Find the relation between P' and Q'. The lines DP and DQ meet the line BC at P" and Q" respectively. Find the relation between P" and Q". 

B2.  A plane p passes through a vertex of a cube so that the three edges at the vertex make equal angles with p. Find the cosine of this angle. Find the positions of the feet of the perpendiculars from the vertices of the cube onto p. There are 28 lines through two vertices of the cube and 20 planes through three vertices of the cube. Find some relationship between these lines and planes and the plane p. 


Put x = A/2, y = B/2. We have sin C/2 = sin(90o-x-y) = cos(x+y). So we need to show that sin x sin y cos(x+y) < 1/4, or (cos(x-y) - cos(x+y) )cos(x+y) < 1/2, or 2 cos(x-y) cos(x+y) < 1 + 2 cos2(x+y). But 2 cos(x-y) cos(x+y) ≤ cos2(x+y) + cos2(x-y) ≤ 1 + cos2(x+y) < 1 + 2 cos2(x+y) (since 0 < x,y < 90o

A2. Answer: k·2a7b193c197d, where k = 1, 3, 32, 33, 5, 3·5, a = 0, 1, 2, or 3, b = 0 or 1, c = 0 or 1, d = 0 or 1 (192 solutions in all) 

1890 = 2·335·7, 1930 = 2·5·193, 1970 = 2·5·197 (and 193 and 197 are prime). So 1890·1930·1970 = 2333537·193·197.

Saturday, May 29, 2010

8th Vietnam Mathematical Olympiad 1969

1.  A graph G has n + k points. A is a subset of n points and B is the subset of the other k points. Each point of A is joined to at least k - m points of B where nm < k. Show that there is a point in B which is joined every point in A. 

2.  Find all real x such that 0 < x < π and 8/(3 sin x - sin 3x) + 3 sin2x ≤ 5.  

3.  The real numbers x1, x4, y1, y2 are positive and the real numbers x2, x3, y3, y4 are negative. We have (xi - a)2 + (yi - b)2 ≤ c2 for i = 1, 2, 3, 4. Show that a2 + b2 ≤ c2. State the result in geometric language. 

4.  Two circles centers O and O', radii R and R', meet at two points. A variable line L meets the circles at A, C, B, D in that order and AC/AD = CB/BD. The perpendiculars from O and O' to L have feet H and H'. Find the locus of H and H'. If OO'2 < R2 + R'2, find a point P on L such that PO + PO' has the smallest possible value. Show that this value does not depend on the position of L. Comment on the case OO'2 > R2 + R'2.  


2. Answer: π/2
We have 3 sin x - sin 3x = 4 sin3x. Put s = sin x. Then we want 2/s3 + 3s2 ≤ 5. Note that since 0 < x < π we have s positive. But by AM/GM we have 1/s3 + 1/s3 + s2 + s2 + s2 > 5 with equality iff s = 1, so we must have sin x = 1 and hence x = π/2. 

3. Stated geometrically, the result is: if a disk includes a point in each quadrant, then it must also include the origin. We use the fact that a disk is convex. Let Pi be the point (xi,yi). The segment P1P2 must intersect the positive x-axis. By convexity, the point of intersection, call it X, must lie in the disk. Similarly, P3P4 must intersect the negative x-axis at some point Y, which must be in the disk. Then all points of the segment XY are in the disk and hence, in particular, the origin. 


Find the value of 'x' in the expression

If  a = xy / (x + y) and b = yz / (y + z) and c = zx / (z + x). Know that a, b and c are not equal to zero, find the value of x in terms of a, b and c.

Adding Fractions with Different Denominators

Adding Fractions with Different Denominators

How to Add Fractions with different denominators:
  • Find the Least Common Denominator (LCD) of the fractions
  • Rename the fractions to have the LCD
  • Add the numerators of the fractions
  • Simplify the Fraction

Adding Fractions with Unlike Denominators

Adding Fractions with Different Denominators (No LCD)