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

 

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Mathematic Skills for Grades 3-6

NUMBER OPERATIONS AND CONCEPTS

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. www.multiplication.com Interactive games to help teach the multiplication facts.

MEASUREMENT

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? - 

ALGEBRA

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 student.active 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

DATA ANALYSIS AND PROBABILITY

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.
Dosity.com  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.
NumberNut.com 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.

Math Word Problems

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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, http://321math.blogspot.com

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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. 

Solutions

A1.

Put x = A/2, y = B/2. We have sin C/2 = sin(90^o-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 < 90^o. 

A2. Answer: k·2^a7^b193^c197^d, 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) 

Solution

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.

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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 (x_i - a)² + (y_i - 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.  

Solutions

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 P_i be the point (x_i,y_i). 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. 

Source: http://321math.blogspot.com

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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.

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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)

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4th Vietnam Mathematical Olympiad 1965

1.  At time t = 0, a lion L is standing at point O and a horse H is at point A running with speed v perpendicular to OA. The speed and direction of the horse does not change. The lion's strategy is to run with constant speed u at an angle 0 < φ < π/2 to the line LH. What is the condition on u and v for this strategy to result in the lion catching the horse? If the lion does not catch the horse, how close does he get? What is the choice of φ required to minimise this distance?

2.  AB and CD are two fixed parallel chords of the circle S. M is a variable point on the circle. Q is the intersection of the lines MD and AB. X is the circumcenter of the triangle MCQ. Find the locus of X. What happens to X as M tends to (1) D, (2) C? Find a point E outside the plane of S such that the circumcenter of the tetrahedron MCQE has the same locus as X.

3.  m an n are fixed positive integers and k is a fixed positive real. Show that the minimum value of x1m + x2m + x3m + ... + xnm for real xi satisfying x1 + x2 + ... + xn = k occurs at x1 = x2 = ... = xn.

Source: Nguyễn Thị Lan Phương, http://321math.blogspot.com

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3rd Vietnam Mathematical Olympiad 1964

1.  Find cos x + cos(x + 2π/3) + cos(x + 4π/3) and sin x + sin(x + 2π/3) + sin(x + 4π/3).
2.  Draw the graph of the functions y = | x2 - 1 | and y = x + | x2 - 1 |. Find the number of roots of the equation x + | x2 - 1 | = k, where k is a real constant.
3.  Let O be a point not in the plane p and A a point in p. For each line in p through A, let H be the foot of the perpendicular from O to the line. Find the locus of H.
4.  Define the sequence of positive integers fn by f0 = 1, f1 = 1, fn+2 = fn+1 + fn. Show that fn = (an+1 - bn+1)/√5, where a, b are real numbers such that a + b = 1, ab = -1 and a > b.  Solutions 1. Using cos(A+B) = cos A cos B - sin A sin B, we have cos(x + 2π/3) = -(1/2) cos x + (√3)/2 sin x, cos(x + 4π/3) = -(1/2) cos x - (√3)/2 sin x. Hence cos x + cos(x + 2π/3) + cos(x + 4π/3) = 0. Similarly, sin(x + 2π/3) = -1/2 sin x + (√3)/2 cos x, sin(x + 4π/3) = -1/2 sin x - (√3)/2 cos x, so sin x + sin(x + 2π/3) + sin(x + 4π/3) = 0. 2. Answer 0 for k < -1 1 for k = -1 2 for -1 < k < 1 3 for k = 1 4 for 1 < k < 5/4 3 for k = 5/4 2 for k > 5/4 It is clear from the graph that there are no roots for k < -1, and one root for k = -1 (namely x = -1). Then for k > -1 there are two roots except for a small interval [1, 1+h]. At k = 1, there are 3 roots (x = -2, 0, 1). The upper bound is at the local maximum between 0 and 1. For such x, y = x + 1 - x2 = 5/4 - (x - 1/2)2, so the local maximum is at 5/4. Thus there are 3 roots at k = 5/4 and 4 roots for k ∈ (1, 5/4). 3. Answer: circle diameter AB, where OB is the normal to p Let B be the foot of the perpendicular from O to p. We claim that the locus is the circle diameter AB. Any line in p through A meets this circle at one other point K (except for the tangent to the circle at A, but in that case A is obviously the foot of the perpendicular from O to the line). Now BK is perpendicular to AK, so OK is also perpendicular to AK, and hence K must be the foot of the perpendicular from O to the line. 4. Put a = (1+√5)/2, b = (1-√5)/2. Then a, b are the roots of x2 - x - 1 = 0 and satisfy a + b = 1, ab = -1. We show by induction that fn = (an+1 - bn+1)/√5. We have f0 = (a-b)/√5 = 1, f1 = (a2-b2)/√5 = (a+1 - b-1)/√5 = 1, so the result is true for n = 0, 1. Finally, suppose fn = (an+1 - bn+1)/√5 and fn+1 = (an+2 - bn+2)/√5. Then fn+2 = fn+1 + fn = (1/√5)(an+1(a+1) - bn+1(b+1) ) = (an+1a2 - bn+1b2)/√5, so the result is true for n+1.

Source: http://321math.blogspot.com

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2nd Vietnam Mathematical Olympiad 1963

1.  A conference has 47 people attending. One woman knows 16 of the men who are attending, another knows 17, and so on up to the last woman who knows all the men who are attending. Find the number of men and women attending the conference.

2.  For what values of m does the equation x2 + (2m + 6)x + 4m + 12 = 0 has two real roots, both of them greater than -2.

3.  Solve the equation sin3x cos 3x + cos3x sin 3x = 3/8.

4.  The tetrahedron SABC has the faces SBC and ABC perpendicular. The three angles at S are all 60o and SB = SC = 1. Find its volume.

5.  The triangle ABC has perimeter p. Find the side length AB and the area S in terms of ∠A, ∠B and p. In particular, find S if p = 23.6, A = 52.7 deg, B = 46 4/15 deg. Solutions 1. Suppose there are m women. Then the last woman knows 15+m men, so 15+2m = 47, so m = 16. Hence there are 31 men and 16 women. 2. Answer: m ≤ -3 For real roots we must have (m+3)2 ≥ 4m+12 or (m-1)(m+3) ≥ 0, so m ≥ 1 or m ≤ -3. If m ≥ 1, then -(2m+6) ≤ -8, so at least one of the roots is < -2. So we must have m ≤ -3. The roots are -(m+3) ±√(m2+2m-3). Now -(m+3) ≥ 0, so -(m+3) + √(m2+2m-3) ≥ 0 > -2. So we need -(m+3) - √(m2+2m-3) > -2, or √(m2+2m-3) < -m-1 = √(m2+2m+1), which is always true. 3. Answer: 7½o + k90o or 37½o + k90o We have sin 3x = 3 sin x - 4 sin3x, cos 3x = 4 cos3x - 3 cos x. So we need 4 sin3x cos3x - 3 sin3x cos x + 3 sin x cos3x - 4 sin3x cos3x = 3/8 or 8 sin x cos x(cos2x - sin2x) = 1, or 4 sin 2x cos 2x = 1 or sin 4x = 1/2. Hence 4x = 30o + k360o or 150o + k360o. So x = 7½o + k90o or 37½o + k90o.

Source: Nguyễn Thị Lan Phương, http://321math.blogspot.com

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1st Vietnam Mathematical Olympiad 1962

1.  Prove that 1/(1/a + 1/b) + 1/(1/c + 1/d) ≤ 1/(1/(a+c) + 1/(b+d) ) for positive reals a, b, c, d.
2.  f(x) = (1 + x)(2 + x2)1/2(3 + x3)1/3. Find f '(-1).
3.  ABCD is a tetrahedron. A' is the foot of the perpendicular from A to the opposite face, and B' is the foot of the perpendicular from B to the opposite face. Show that AA' and BB' intersect iff AB is perpendicular to CD. Do they intersect if AC = AD = BC = BD?
4.  The tetrahedron ABCD has BCD equilateral and AB = AC = AD. The height is h and the angle between ABC and BCD is α. The point X is taken on AB such that the plane XCD is perpendicular to AB. Find the volume of the tetrahedron XBCD.
5.  Solve the equation sin6x + cos6x = 1/4. Solutions 1. A straightforward, if inelegant, approach is to multiply out and expand everything. All terms cancel except four and we are left with 2abcd ≤ a2d2 + b2c2, which is obviously true since (ad - bc)2 ≥ 0.  2. Differentiating gives f '(x) = (2 + x2)1/2(3 + x3)1/3 + terms with factor (1 + x). Hence f '(-1) =31/221/3.  3. Let the ray AB' meet CD at X and the ray BA' meet CD at Y. If AB' and A'B intersect, then X = Y. Let L be the line through A' parallel to CD. Then L is perpendicular to AA'. Hence CD is perpendicular to AA'. Similarly, let L' be the line through B' parallel to CD. Then L' is perpendicular to BB', and hence CD is perpendicular to BB'. So CD is perpendicular to two non-parallel lines in the plane ABX. Hence it is perpendicular to all lines in the plane ABX and, in particular, to AB. Suppose conversely that AB is perpendicular to CD. Consider the plane ABY. CD is perpendicular to AB and to AA', so CD is perpendicular to the plane. Similarly CD is perpendicular to the plane ABX. But it can only be perpendicular to a single plane through AB. Hence X = Y and so AA' and BB' belong to the same plane and therefore meet. 4. Put a = sin2x, b = cos2x. Then a and b are non-negative with sum 1, so we may put a = 1/2 + h, b = 1/2 - h. Then a3 + b3 = 1/4 + 3h2 ≥ 1/4 with equality iff h = 0. Hence x is a solution of the equation given iff sin2x = cos2x = 1/2 or x is an odd multiple of π/4.  Source: Nguyễn Thị Lan Phương, http://321math.blogspot.com

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Why Mathematics is so great?

What's correlation between the structure of the mathematical theory and object structure? My ideas about mathematics:

•  Sign language of mathematically is a system non-contradiction.
•  Mathematical language is a language system form of symbolic.
•  Mathematics is a scientific inference, the type of theoretical knowledge.
•  System of mathematical objects are determined a priori object class but applied mathematics achievement test pilot first.

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Dictionary of Classical and Theoretical Mathematics

Title: Dictionary of Classical and Theoretical Mathematics

Author: Catherine Cavagnaro, William T. Haight, II

Publisher: © 2001 by CRC Press LLC

Price: $58.95

Product Description: http://www.amazon.com/Dictionary-Classical-Theoretical-Mathematics-Comprehensive/dp/1584880503

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English Vietnamese Mathematics Dictionary

Title: English Vietnamese Mathematics Dictionary

Authors: Chính Đức Phan, Khanh Minh Lê, Lập Tấn Nguyễn, Thịnh Đình Lê, Thúy Công Nguyễn, Văn Bác Nguyễn.

Url: http://vuimahoc.com/english-vietnamese-mathematics-dictionary.pdf

Size: 1.126 KB

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Report on Fundamental Lemma

Title: Report on Fundamental Lemma
Author: Châu Bảo Ngô, School of mathematics, Institute for Advanced Study, Princeton, NJ 08540 USA.
Url: http://www.math.ias.edu/~ngo/cdm.pdf

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Class Seminar: Application of Mathematics?

In what ways do mathematics affect our lives? (Consider things like  human-based computing, statistics and probability, applied mathematics). When can information be considered property that can be protected by formal laws?

What do visual elements (pattern, shape, size, proportion, texture, color, and proximity of buildings) contribute to the identity of a city? 

References:

• http://en.wikipedia.org/wiki/Applied_mathematics
• http://www.ams.org/notices/200111/rev-blank.pdf

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Numbers and Counting through Ten

Counting to ten with numbers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Counting to ten with words:

one, two, three, four, five, six, seven, eight, nine, ten

Counting to ten with objects:

first ten (10) numbers and the quantities they represent

Counting allows us to know the total number of things or objedts in a group. In order to do so, we separate the items from the group one by one, and we the next larger number to each item removed, till none is left and the total number is discovered.

In other words, counting is answering: how many? Beforehand, in order to count, we must know the unique numbers which identify each quantity of things.

Source: http://321math.blogspot.com

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English - Vietnamese Math Glossary: A

English - Vietnamese Math Glossary: A

about	khoảng chừng
above	ở trên
asolute value	giá trị tuyệt đối, trị số tuyệt đối
accurate	chính xác
accurately label work	công việc có nhãn hiệu chính xác

act it out	làm
acute angle	góc nhọn
acute triangle	tam giác nhọn
add	cộng, tính cộng
addend	phần hay số được cộng thêm vào
addition	phép cộng, tính cộng, cộng, toán cộng
addition fact	cơ sở lập luận của phép cộng
addition sentence	mệnh đề phép cộng
addition sign	dấu cộng
additive inverses	phần nghịch đảo tính cộng
after	sau, sau khi
afternoon	buổi trưa
algebra	đại số
algebraic expression	biểu thức đại số
algebraic patterns	khuôn thức đại số
algebraic relationship	mối liên hệ đại số
algebraic relationships	các sự liên hệ đại số
algebraically	có tính chất đại số
alike	giống nhau
all	tất cả
all together	chung tất cả
almost	gần, hầu như
amount	số lượng
analog clock	đồng hồ có kim chỉ giờ và phút
analyze	phân tích
angle (∠)	góc
angle adjacent	góc kề
answer	đáp số, kết quả, trả lời
ante meridian (a.m.)	trước giờ ngọ (trước 12 giờ trưa)
application	sự áp dụng
apply	áp dụng
approach	giải (bài toán), đạt tới (kết quả)
appropriate mathematical language	từ toán học thích hợp
organize work	xếp đặt bài toán, công việc
arc	cung
area	diện tích
argument	lập luận, bàn luận
argument conjecture counterexample	dẫn chứng dựa trên lập luận phỏng đoán
arithmetic (numeric) expression	biểu thức số học
arithmetic expression	biểu thức toán học
arrange	xếp đặt, sắp xếp
array	mảng, chuỗi số sắp theo thứ tự
as long as	miễn là, với điều kiện là
associative property	đặc tính liên kết
attribute	liên hệ, trực thuộc
autumn (fall)	mùa thu
average	trung bình
axis (axes)	trục

Source: http://321math.blogspot.com

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How to Make Addition - Subtraction - Multiplication - Division War?

Game: Addition War

Goal: 1 – The learner will read write, model, and compute with rational numbers

Materials:  directions; a deck of cards with 4 each of the numbers 1 through 10 (available on-site); paper (preferably graph paper); pencil

Procedure:    In problem solving, it might be helpful to think about solutions and answers as two different things. An answer is the final result to a problem, while a solution presents both the answer and the strategy by which it was found.

- Follow the directions on the worksheet, playing several rounds with the student.

- This activity is excellent for basic practice until student commits the basic facts  to memory.

Materials  1 deck of cards with 4 each of the numbers l through 10

Number of players 2-4

Object of the game To collect the most cards.

Directions Shuffle the cards and place the deck number-side down on the 
playing surface.

Each player turns over 2 cards and calls Out the sum of the 2 numbers. The player with the largest sum wins the round and takes all the cards. In case of a tie for the largest sum. each tied player turns over 2 more cards and calls ouT the sum. The player with the highest sum wins the round and takes all the cards from both plays.

Answers can be checked with an Addition Table or with a calculator.

Play continues until there are too few cards left for each player to have another turn. The player who took the most cards wins. Or, players may toss a penny to determine whether the player with the most or the fewest cards wins.

Variation Each player turns over 3 cards and finds the sum.

Advanced version Players turn over 4 cards, form two 2-digit numbers, and find the sum. Players should consider how they form their numbers since different arrangements have different sums. For example, a player turns over 2, 5, 7, and 4. 74 +  52 has a greater sum than 25 + 47. 

Game: Subtraction War

Goal: 1 – The learner will read write, model, and compute with rational numbers

Materials:    directions; a deck of cards with 4 each of the numbers 1 through 10 (available on-site); paper (preferably graph paper); pencil

Objective(s):     The learner will compute with rational numbers (Goal 7).

Materials:  directions; a deck of cards with 4 each of the numbers 1 through 10 (available on-site); paper (preferably graph paper); pencil

Procedure:    In problem solving, it might be helpful to think about solutions and answers as two different things. An answer is the final result to a problem, while a solution presents both the answer and the strategy by which it was found.

- Follow the directions on the worksheet, playing several rounds with the student.         

- This activity is excellent for basic practice until student commits the basic facts to memory.

Materials  1 deck of cards with 4 each of the numbers l through 10

Number of players 2-4

Object of the game To collect the most cards.

Directions Shuffle the cards and place the deck number-side down on the playing surface.


Each player turns over 3 cards, finds the sum of any two of the numbers, then finds the difference between the sum and the third number. The player with the largest difference takes the cards.

Example:

A 4, 8, and 3 are turned over. There are 3 combinations that will result in a positive number.
              4 +    8 = 12: 12     - 3     = 9
    3 + 8 = 11: 11 - 4 = 7   
    3 + 4 = 7  :   8 - 7 = 1
Advanced version  Players turn over cards, form two 2-digit numbers, and find the difference. Players should consider now they form their numbers. 75 - 24  has a greater difference than 57 - 42.

Game: Multiplication War

Goal: 1 – The learner will read write, model, and compute with rational numbers

Materials:  directions; a deck of cards with 4 each of the numbers 1 through 10 (available on-site); paper (preferably graph paper); pencil

Procedure:    In problem solving, it might be helpful to think about solutions and answers as two different things. An answer is the final result to a problem, while a solution presents both the answer and the strategy by which it was found.

- Follow the directions on the worksheet, playing several rounds with the student.         

- This activity is excellent for basic practice until student commits the basic facts to memory.

Materials  1 deck of cards with 4 each of the numbers l through 10

Number of players 2-4

Object of the game To collect the most cards.

Directions Shuffle the cards and place the deck number-side down on the playing surface.

The game is played the same way as Addition Top-it,  except that players find the product of the numbers instead of the sum. The player with the largest product wins the round and takes all the cards.

Answers can be checked with a Multiplication Table or with a calculator.

Variation:  Players turn over 3 cards. form a 2-digit number and multiply by the remaining number.

Game: Division War

Goal: 1 – The learner will read write, model, and compute with rational numbers

Materials:  directions; a deck of cards with 4 each of the numbers 1 through 10 (available on-site); paper (preferably graph paper); pencil

Procedure:    In problem solving, it might be helpful to think about solutions and answers as two different things. An answer is the final result to a problem, while a solution presents both the answer and the strategy by which it was found.

- Follow the directions on the worksheet, playing several rounds with the student.

- This activity is excellent for basic practice until student commits the basic facts to memory.

Materials  1 deck of cards with 4 each of the numbers l through 10

Number of players 2-4

Object of the game To collect the most cards.

Directions Shuffle the cards and place the deck number-side down on the playing surface.

Each player turns over 3 cards and uses them to generate division problems as follows.

Choose 2 cards to form The dividend. Use the remaining card as the divisor.

Divide and drop the remainder. The player with the largest quotient wins the round and takes all the cards.

Advanced version Turn over 4 cards and choose three of them to form a 3-digit number. Divide it by the remaining number. The arrangement of the numbers may result in a greater quotient. For example: 462/5 is greater than 256/4, but 654/2 is even greater.

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Arithmetics Skills Test: Time, Patterns, Multiplication (Basic Facts)

Skills:  Time, Patterns, Multiplication (basic facts)

1. David James and Pitter Pan began watching a movie at 3:30.  The movie ended at 4:45.  How long was the movie? Look at a clock with hands if you need help figuring this out.

______ hours and ______  minutes long

2. Leah’s dog, Belle, buried 5 bones in the backyard on Monday.  On Tuesday, she buried 7 bones in the  backyard.  On Wednesday, she buried 9 bones.  If the pattern continues, how many bones will Belle bury on Saturday?   Make a table to help you find your answer.

Look at a clock with hands if you need help figuring this out.

3. John had two dozen fishing lures.  His father had four dozen lures. How many lures did they have in all? Show your work and label your answer.

Make a table to help you find your answer.

4. There were 8 cars in the parking lot.  Each car had 4 tires.   How many tires were in the parking lot? Show your work and label your answer.

Show your work and label your answer.

5. There were six ice skaters on the ice rink.  Each skater had two skates on.  How many skates were there in all?

Show your work and label your answer.

Sourse: http://321math.blogspot.com 

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How to make Long Division with Remainders?

When we are given a long division to do it will not always work out to a whole number.

Sometimes there will be numbers left over. These are known as remainders.

e.g: 435 ÷ 25 

	4 ÷ 25 = 0 remainder 4	The first number of the dividend is divided by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 0 = 0	The answer from the first operation is multiplied by the divisor. The result is placed under the number divided into.
	4 – 0 = 4	Now we take away the bottom number from the top number.
		Bring down the next number of the dividend.
	43 ÷ 25 = 1 remainder 18	Divide this number by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 1 = 25	The answer from the above operation is multiplied by the divisor. The result is placed under the last number divided into.
	43 – 25 = 18	Now we take away the bottom number from the top number.
		Bring down the next number of the dividend.
	185 ÷ 25 = 7 remainder 10	Divide this number by the divisor.
		The whole number result is placed at the top. Any remainders are ignored at this point.
	25 × 7 = 175	The answer from the above operation is multiplied by the divisor. The result is placed under the number divided into.
	185 – 175 = 10	Now we take away the bottom number from the top number.
		There is still 10 left over but no more numbers to bring down.
		With a long division with remainders the answer is expressed as 17 remainder 10 as shown in the diagram

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Teaching Long Division and Double Division

"Teaching Double Division can help in teaching long division by reinforcing the principles of division and giving students success with a less frustrating alternative.

 

Double Division does not depend on memorizing the multiplication facts or estimating how many times one number goes into another. It may take 50% longer, but it is far less frustrating and probably easier to understand than Long Division.

 

It's Easy!

Step 1 - Double, double, double.

Step 2 - Subtract off multiples.

Step 3 - Add up your answer."

References:

• http://www.doubledivision.org
•  http://en.wikipedia.org/wiki/Long_division

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