May 31, 2010

May 30, 2010

9th Vietnam Mathematical Olympiad 1970 Problems


9th Vietnam Mathematical Olympiad 1970 Problems

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(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) 

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. 

Source: http://www.kidsmathbooks.com

8th Vietnam Mathematical Olympiad 1969 Problems


8th Vietnam Mathematical Olympiad 1969 Problems

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.  

May 29, 2010

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.
Fun math books for kids (read aloud to ages 9 and up)




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)

May 27, 2010

4th Vietnam Mathematical Olympiad 1965 Problems

4th Vietnam Mathematical Olympiad 1965 Problems


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://www.kidsmathbooks.com

May 26, 2010

3rd Vietnam Mathematical Olympiad 1964 Problems

3rd Vietnam Mathematical Olympiad 1964 Problems



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://www.kidsmathbooks.com

2nd Vietnam Mathematical Olympiad 1963 Problems

2nd Vietnam Mathematical Olympiad 1963 Problems


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://www.kidsmathbooks.com

1st Vietnam Mathematical Olympiad 1962 Problems

1st Vietnam Mathematical Olympiad 1962 Problems


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

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.

Dictionary of Classical and Theoretical Mathematics

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

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.


Report on the Fundamental Lemma by Ngô Bảo Châu

Title: Report on the Fundamental Lemma
Author: Châu Bảo Ngô, School of mathematics, Institute for Advanced Study, Princeton, NJ 08540 USA.
Download: http://www.kidsmathlearning.com/Report-on-the-Fundamental-Lemma-by-Ngo-Bao-Chau.pdf

External
Note
Amazon Books

On Central Critical Values of the Degree Four L-Functions for GSp(4): The Fundamental Lemma (Memoirs of the American Mathematical Society) - Mass Market Paperback (July 1, 2003) by Masaaki Furusawa and Joseph A. Shalika.



Representations of Fundamental Groups of Algebraic Varieties (Lecture Notes in Mathematics) - Paperback (Feb. 3, 2000) by Kang Zuo.


May 24, 2010

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

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

May 23, 2010

English - Vietnamese Math Glossary: A

English - Vietnamese Math Glossary: A


aboutkhoảng chừng
above ở trên
asolute valuegiá trị tuyệt đối, trị số tuyệt đối
accuratechính xác
accurately label workcông việc có nhãn hiệu chính xác




act it outlàm
acute anglegóc nhọn
acute triangletam giác nhọn
addcộng, tính cộng
addend phần hay số được cộng thêm vào
additionphép cộng, tính cộng, cộng, toán cộng
addition factcơ sở lập luận của phép cộng
addition sentencemệnh đề phép cộng
addition signdấu cộng
additive inversesphần 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


May 22, 2010

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.

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.


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 

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

How to do 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. Taking an example similar to that on the Long Division page it becomes more clear: 435 ÷ 25. If you feel happy with the process on the Long Division
page you can skip the first bit.
 
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

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."
Adapted from http://www.doubledivision.org
References:
  • http://www.doubledivision.org
  •  http://en.wikipedia.org/wiki/Long_division
More books about long division for kids

Mathimagination Series: Book A, beginning multiplication and division; Book B, operations with whole numbers; Book C, number theory, sets and number bases; Book D, fractions; Book E, decimals and percentDecimals and Percentages With Pre- And Post-Tests: Place Value, Addition, Subtraction, Multiplication, Division

Kids Math Games - Math Fun for Kids

   1. K-6 Math Practice - IXL: www.ixl.com/math
   2. Cool Math: www.coolmath4kids.com
   3. Education resource for K-8 Kids and Teachers: www.funbrain.com
   4. Dr Mike's Math Games for Kids: www.dr-mikes-math-games-for-kids.com

Beginning Math Activities                
  1.   BBC Education's  Number Time Games

    1. Simple Add and Subtract

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    2. Test the Toad  ( + and -  problems with numbers 1-15)
    3. Little Animals Activity Center ( interactive number games for addition and subtraction)
  2. Number Recognition/Sequencing


    1. Dart Board Level 1 (recognize numbers 1-9)
    2. Find the Number  (Recognition activity for numbers 1-10)
    3. Mend the Number Square ( Place missing numbers on 1-100 grid)
    4. Count Along to 100
    5. Find One More  (matching game)
    6. Numberjacks Snap (matching game digits 1-5)
  3. Snakes and Ladders  (1 player or 2 player activity.. works like chutes and ladders)
  4. Puzzler Jigsaw  (3 levels of jigsaw puzzles:  4pieces, 9 pieces, 16 pieces)
  5. Shaper Japer Dominoes
  6. Watch Addem (Number Movies)
  7. Number Songs  

  •  FunBrain Activities

    1. Number Recognition/Sequencing


      1. Bunny Count ( count and match numbers and characters)
      2. One False Move (sequence numbers from lowest to highest)
      3. Guess the Number (guess number with high low clues)
    2. Math Brain Activities (25 Board Games to teach skills)

  • Funschool: Preschool

    1. Fishin'  Mission (count the number and fish and put them in the net)
    2. Stacker  (stack the blocks in sequence order)
    3. Connect the Numbers (from smallest to largest order)

  • Learning Planet Activities

    1. Number Recognition/Counting/Sequencing


      1. Count Your Chickens ( counting activity)
      2. 1 2 3 Order  (what comes next... up to 10)
      3. Number Train ( make a train and count the cars)

  • IXL Math Practice Activities (click on grade school level)

  • Base 10 Count  (group ones into ten... see what number is made)

  • Primary Math Games Index

  • I Know That

  • Teach R Kids

  • KidPort Math Activities by Grade:

    1. Kindergarten
    2. First
    3. Second
    4. Third

  • Count Us In Games

  • Buzzing with Shapes 

  • Shaper Japer Dominoes  

  • Parent Help Sites


    1. At Home with Math
    2. I Know That
    3. Figure This

  • Britannica Math Learning Activities

  • Learning about Time



  • 2.  Facts / Whole Numbers / Number Line                                             
    1. Addition and Subtraction Activities


      1. Addition Surprise 
      2. Can You Balance?
      3. Addition Fact Challenge 
      4. Elementary Math Games Addition
      5. 3 Digit Addition Activity 
      6. Addition of 3 Digits (using base ten blocks)
      7. Subtraction Fact Challenge 
      8. Elementary Math Games Subtraction
      9. 3 Digit Subtraction Activity 
      10. Dart Board Game (BBC simple addition.. click on control panel to change levels)
      11. Little Animals Activity Center ( interactive number games for addition and subtraction)
      12. Add It Up ( add up number tiles to get target number)
      13. Primary Math Games Index
    2. Addition or Subtraction or Multiplication or Division


      1. Spacey Math by Planet Learning ( fact [ +, -, *, / ] drill)
      2. Interactive Multiplication Games (14 different games to help with multiplication facts)

        1. Flash Fun (practice on own)
        2. Quick Flash (computer quizzes you on level chosen)
        3. Multiflyer
      3. Math Mayhem by LearningPlanet ( +, -, *, / timed practice)
      4. Math Mastery CyberChallenge (+, -, *, / timed practice)
      5. Mad Math Minute ( +, - *, / online quizzes)
      6. Multiplication Matho (race for time as you practice your facts)  
      7. Basic Multiplication Fact Challenge
      8. Basic Division Fact Challenge
      9. Break the Code (basic fact recall)
      10. Elementary Math Games Multiplication
      11. A+ Math Games  ( basic operation activities)
      12. Coloring Book Math ( color online doing math problems)
      13. Math 4 Kids  
      14. Math Activities at Syvum (Helps build basic skills)
      15. Mighty Math Calculating Crew... Multiplication Practice
      16. A+ Math Flashcards( math flashcards and math games for basic operations)
      17. Schoolhouse Flash (click on green start button bottom right of page to practice math facts... + - * /)
      18. I Know That
      19. MathO
      20. Broken Calculator
      21. Arithmetic Four
      22. Printable Math Drill Sheets  


        1. Free Math Worksheet Generator
        2. Math Worksheet Creator (Super Kids)
        3. Printable Math Fact Drill Sheets (Donna Young)
        4. Math Worksheets (Click on red links on right side of page)
        5. Math Slice
      23. Free Online Math Tests (click on purple banner at top of page)
    3. Funbrain Activities with + - * / and whole numbers


      1. Cookie Dough  ( read and write numbers on a check)
      2. Math Baseball  ( fact [ +, -, *, / ] drill)
      3. Math Baseball for 2 people ( fact [ +, -, *, / ] drill)
      4. Math Car Racing ( fun way for fact +, -, * / drills)
      5. Power Football (DECIMAL  fact [ +, -, *, / ] drill)
      6. Tic-Tac-Toe Squares ( practice with + - * / squares, cubes )
      7. Change Maker ( work with $1 or $100, pick coins / bills for change)
    4. Watch Addem (Number Movies)


    3.   Factors/ Multiples/ Primes/ Powers/ Triangular Numbers
    1. Product Game  (ILLUMINATIONS NCTM) [ game of skill with factors and multiples] 
    2. Factor Game  (ILLUMINATIONS NCTM) [ game of skill with factors]
    3. Grid Game by BCC Maths File (find factors, multiples, primes, triangular numbers, and/or powers on a grid)
    4. Multiplication Mystery
    5. Multiple Matrix
    4.  Fractions
    1. What is a Fraction?




      1. Who Wants Pizza (a fun way to learn about fractions)
      2. WebMath:  Get a Fraction  (get a visual picture)
      3. Math Matching...  Fractions to Models
      4. Mystery Picture with Fraction Word Problems (by Dositey)
    2. Equivalent Fractions




      1. Fraction Frenzie Activity by LearningPlanet ( match equivalent fractions before time runs out)
      2. Fraction Game Tool (Illuminations NCTM)[ move game pieces to other side of board) 
      3. Mighty Math Matching... Equivalent Fractions
      4. Fresh Baked Fractions Activity by Funbrain( find fraction that is not equivalent to the rest)
      5. Soccer Shootout  Activity  by Funbrain( addition and subtraction of fractions)
    3.  Fraction/Decimal Relationships




      1. Math Matching... Fractions and Decimal Equivalents
      2. Math Matching... MORE Fractions and Decimal Equivalents
      3. Math Matching... MORE Fractions and Decimal Equivalents
      4. Math Matching... Fractions, Decimal, % Equivalents
      5. Salon Snap Activity by BBC Maths Files ( match fractions to decimals and %)
      6. Double Fun Match Activity (Click on Fraction/Decimal)  by Funbrain (Match Fractions to Decimal Equivalent)
      7. Fraction Decimal Conversion ( JAVA games matching fractions to equivalent decimals)
    4. Finding a Least Common Multiple (LCM)
    5. Fractions
    6. All About Fractions by AAA Math (Explanation, interactive practice and challenge games about fractions)
    7. Visual Fractions  
    8. I Know That: Fractions Workshop  
    9. I Know That:  Fishy Fractions

    5.  Geometry/ Area / Perimeter
    1. General Geometrical Terms




      1. Geometry Flashcards by APlusMath ( match picture to name)
      2. Geometric Terms (JAVA games with common geometric terms)
      3. Online Activities for Geometrical Terms  (from Thinkquest.. Nuts about Geometry)
      4. All About Geometry  by AAA Math ( Explanation, interactive practice and challenge games about geometry)
    2. Polygons/ Shapes /Nets




      1. Shape Database by Ambleweb
      2. Online Pattern Blocks 




        1. Designer Fractions Activity
      3. Interactives: 3 D Shapes 
      4. Nets




        1. Mighty Math.....Nets
        2. Patterns for Solids Figures (Nets)
    3. Angles




      1. Angles (WiseUp from BBC) 
      2. Angles and Terms
      3. What's My Angle (protractor activities by Ambleweb)
      4. Angle Activities by Ambleweb
      5. Measuring Angles by KidPort
      6. Banana Hunt
      7. Introduction to Angles ( JAVA games with common terms related to angles)
    4. Area/ Perimeter




      1. Math Playground: Area and Perimeter




        1. Area and Perimeter Problems to solve
      2. Shape Surveyor  Activity  by Funbrain( solve simple area/perimeter problems)
      3. Mrs Glosser's lessons on Area and Perimeter of Polygons (lessons and activity)




        1. Perimeter of Polygon (Lesson with activity)
        2. Area of Rectangles and Squares (Lesson with activity)
        3. Area of Parallelogram (Lesson with activity)
        4. Area of Triangle (Lesson with activity)
        5. Area of Trapezoid (Lesson with activity)
    5. Symmetry/ Rotations/ Reflections/ Transformations




      1. Symmetry Online Activity
      2. Perfectly Symmetrical.... Rotations
      3. Bathroom Tiles by BBC Maths File ( move bathroom tiles by rotating, reflecting, and translating the patterns) Challenging!!
    6. About Space Activities (from the Annenberg Teacher Project)




      1. I Took a Trip on a Train
      2. Plot Plans and Silhouette
      3. Shadows
    7. About Shape Activities (from the Annenberg Teacher Project)




      1. Quilts
      2. Taxicab Treasure Hunt
    8. Tangrams




      1. Tangrams by KidsCom
      2. Tangrams
      3. Tangrams (Online Tangram activities)
    9. Tessellations 




      1. What are Tessellations by Cool Math
      2. What is a Tessellation?
      3. Totally Tessellated
      4. Tessellation Tool
    10. Challenge of the 7 Cups Activity (Geometry of the Ancients) Challenging!!

    6.  Graphing/ Data Collections/ Coordinate Points
    1. Create a Graph Activity (online activity... create bar graph, pie chart, line graph)
    2. Data Picking Activity by BBC Maths File ( make frequency chart and graph data)
    3. Make a BAR Graph
    4. Flash Demos for Understanding Statistics 
    5. Planet Hop Activity by BBC Maths File ( find coordinates of object & graph equation) Challenging!!
    6. What's the Point? Activity by Funbrain (graphing with coordinate points)
    7. Funbased Learning Graphing Games




      1. Easy Version Graph Game
      2. Medium Version Graph Game
      3. Hard Version Graph Game
    8. Looking for the Top Quark Game  (graphing game)
    9. Lunar Adventure Activity by LearningPlanet (graph coordinates to capture asteroids) Challenging!!
    7.  Integers
    1. Line Jumper Activity by Funbrain (solve + and - INTEGER problems using the NUMBER LINE)
    2. Builder Ted Activity by BBC Maths File (activity that paces integers in numerical order)
    3. Mystery Picture with Integers (by Dositey)

    8.    Online Logic Puzzles/Games
    1. Add It Up
    2. Battleship
    3. Peg Solitare
    4. Cut The Knot
    5. Cryptogram Game (decode messages... letters have been replace with other letters of the alphabet)
    6. Figure This (NCTM)
    7. Interactive Fun Puzzles (many logic puzzles for both younger and older children)
    8. Marbles and Row Puzzle
    9. Mahjongg Puzzle
    10. Math Puzzles (from Discovery School)
    11. Triple Quintzee (like yahtzee)
    12. Math Games (by Math Playground)
    13. Logic Games (by Math Playground)

    9.  Mean, Mode, Median
    1. Weather Today by BBC Education (Activity, Fact Sheet, ONLINE test)
    2. Mrs Glosser's lesson on Statistics....Range, Mean, Mode, Median (online lesson and activity)




    3. Train Race Activity  by BBC Maths File ( need to use own calculator with this)
    4. AAA Statistics

    10.  Measurement/ Weight
    1. Matching Customary Measures
    2. Measure It Activity by Funbrain ( read a ruler in centimeters or inches)
    3. Animal Weigh In Activity by BBC Maths File ( activity that balances weights on a scale)
    11.  Missing number in the series / Patterns
    1. Patterns (BBC Wisenut) 
    2. Number Cracker   Activity by Funbrain (guess  next number in the SERIES)
    3. Geometry Pattern  Activity
    4. Number Pattern  Activity
    5. Crazy Pattern Machine
    6. Pattern Games Activity ( find missing band member to complete the pattern)

    12.   PreAlgebra/ Algebra Activities
    1. Late Delivery Activity by BBC Maths File ( find value of an expression )
    2. Equation Match Activity by BBC Maths File ( matching game with = equations)
    3. Evaluating an Equation with 1 variable  by AAA Math (Explanation, interactive practice and challenge games about equations)
    4. Function Machines:




      1. Function Machine
      2. Whole Number Cruncher Machine
      3. Number Cruncher Machine
    5. Order of Operations




      1. Matho 2
      2. Order of Operations (division and addition.. by Dosity)
      3. Order of Operations Activity
      4. Math Frog Activity
      5. Order of Operations  (Mrs Glosser's online lesson and activity)
      6. Operation Order Activity by Funbrain (complete equations using 3 numbers)
    6. Solving Equations:




      1. Webwise... CLICK ON....Solving Equations with BALANCE STRATEGY
      2. Webwise... CLICK ON....Solving Equations with BALANCE STRATEGY Game  (keeps points)
      3.  Virtual Manipulative's Balance Scale ( to help solve equations)
      4. Algebra Four
    7. Guess the Number(s)




      1. Weigh the Wangdoodle  (by Math Playground)




        1. Weigh the Wangdoodle Level 1 (addition)
        2. Weigh the Wangdoodle Level 2 (multiplication)
        3. Weigh the Wangdoodle Level 3 (multiplication)
      2. Funbrain Magician  Activity
    8. Virtual Manipulative's  Algebra Tiles 
    9. Mrs Glosser's lesson...Topics in PreAlgebra (online lesson and activity)




      1. Writing Algebraic Equations for word sentences 
    10. Solving An Equation  by AAA Math (Explanation, interactive practice and challenge games about equations)
    11. Patterns, Functions, and Algebra (Learner.org)
    12. Algebra Webquests




      1. Algebra Quest
      2. March Madness
    13. Absurd Math.. Interactive pre-algebra problem solving games Challenging




      1. Nomean City  Activity
      2. Dr. Plenobius Activity
      3. The 7th Floor Activity
      4. Airtight College Activity

    13.  Place Value
    1. Place Value Pirates
    2. Wise Up... Our Number System ( BBC)
    3. Me and My Math... click on Numbers and Place Value
    4. Frog Palace ( PLACE VALUE activity for ones, tens, hundreds place)
    5. WebMath: Placevalue
    6. Place Value Game
    7. All About Place Value  by AAA Math ( Explanation, interactive practice and challenge games about place value)
    14.  Probability
    1. Fish Tank Activity by BBC Maths File ( guess probability of catching fish in the tank)
    2. Mrs Glosser's lesson on Probability (online lessons and activity)
    3. What Are Your Chances?  (National Center for Education Statistics) 

    15.  Rounding
    1. Rounding Off Numbers Activity by BBC Maths File
    2. Estimation by Rounding Activity
    3. Place Value Puzzler Activity by Funbrain (type in rounded version of number)
    4. Rounding Flashcards
    5. Rounding Decimals to Whole Numbers Game
    6. Hangman Rounding
    16.  Games and Mind Bender Puzzles
    1. Battleship
    2. Cryptogram Game (decode messages... letters have been replace with other letters of the alphabet)
    3. Enchanted Mind Java Puzzles




      1. Pentominos
      2. Tangrams
      3. Pyramid Puzzle
      4. Peg Solitaire
      5. Mastermind
      6. Knight's Tour
    4. Flash Games
    5. FunBrain Number Games
    6. Fun Math Memory Game
    7. Marbles and Row Puzzle
    8. Mahjongg Puzzle
    9. Math Brain Games
    10. Mega-Math Activities   
    11. Mental Math Tricks
    12. Peg Solitare
    13. Puzzle Picnic
    14. Reversi  
    15. Simon Says Puzzle (Click on color sequence. See how long you can go )
    16. Soma Cube Fun ( active Soma Cube puzzles)
    17. Su Doku puzzles
    18. Who wants to be a Mathionaire? 
    19. Yahtzee Game 


    17.  Metric
    1. Math Matching... Metric Equivalents  
    2. Math Matching... MORE Metric Equivalents  
    3. Metric Matters (Thinkquest tutorial about metric system)
    4. Measure It Activity by Funbrain ( read a ruler in centimeters)
    5. Metrics:  British and English conversions
    18 Money
    1. Learning to Use Money
    2. WebMath: Help Me Count U.S. Coins
    3. Change Maker Activity by Funbrain ( work with $1 or $100, pick coins / bills for change)
    4. Counting Change Game
    5. H.I.P Pocket Change ( by US Mint Dept)  GREAT SITE!!!




      1. Games
      2. Cartoons
      3. Time Machine
      4. Coin News
      5. Teacher Help
    6. Money by AAA Math 
    7. Fleet Kids ( Activities about money management, investing, etc.)
    8. Moneyopolis ( financial planning game )

    19.  Math in Daily Life
    1. Math in Daily Life
    20Math Problems of the Week/ Word Problems
    1. Word Problems for Kids
    2. Word Problems (by Math Playground))
    3. Open Ended Math Problems  
    4. Brain Teasers ( Grades 3,4,5,6,7+)
    5. Math Statistics Teaser Challenges
    6. Aunty Math Problems
    7. Figure This  ( Math index by National Science Foundation & US Dept. of Ed)
    8. Figure This (problems for Middle School students )

    21.  Math questions and answers
    1. Ask Dr. Math
    23.  Vocabulary
    1. Math Dictionary for Kids
    2. Harcourt Math Vocabulary
    24.  Everyday Math Resources
    1. Everyday Math SAMPLE Online Games 

    1. AAA Teacher Resource Page
    2. Algebra Help
    3. Blue Webn Math
    4. Digital Math Filing Cabinet
    5. Education Planet Math Lesson Links
    6. Illuminations  by NCTM  


      1. Interactive Math Tools
      2. Selected Web Resources
      3. Internet-Based Lessons
    7. Math.com 


      1. Math.com for Teachers 
      2. Math.com for Parents
    8. Math Forum  (Math Web Units and lesson plans and more)

    9. Printable Math Drill Sheets  


      1. Math Worksheet Creator (Super Kids)
      2. Printable Math Fact Drill Sheets (Donna Young)
      3. Math Slice
    10. Math Libs
    11. Mrs. Glosser's Math Goodies
    12. Math Ideas (Math ideas/lessons for a variety of topics)
    13. Math Playground
    14. NCTM
    15. NZ Maths Home Page (lesson plan help from New Zealand)
    16. Plane Math Lesson Help  
    17. Principles and Standards for School Mathematics (NCTM)
    18. National Library of Virtual  Manipulatives 
    19. Scholastic Math Activities indexed by topics
    20. Shape and Space in Geometry (from the Annenberg Teacher Project)
    21. Teach R Kids 
    22. Online Activities by Shodor
    23. Free Online Math Tests (click on purple banner at top of page)
    24. Online Worksheets (click on red links of right side of page) 

    Long Division Books

    Fun Math Games for Kids