Math Glossary

Math Glossary


Absolute Value Definition: Always a positive number, refers to the distance of a number from 0, the distances are positive.

Examples: You will use this term to refer to the distance of a point or number from the origin (zero point) of a number line. The symbol to show the absolute value is two vertical lines: | -2 | = 2.











Acute Angle Definition: The measure of an angle with a measure between 0° and 90° or with less than 90° radians.

Also Known As: A positive angle that measures less than 90°




Addend Definition: A number which is involved in addition. Numbers being added are considered to be the addends.

Also Known As: 3 + 2 = 4 The three and the two are the addends.
In the expression a + b + c the a, b, and c are also referred to as the addends.

Examples: What is the missing addend? 3 + __ = 5?
What two addends when added will give a sum of 4? The answer could be 1 + 3, or 2 + 2, or 4 + 0.



Algebra - A branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants.

Algorithm - A step-by-step problem-solving procedure for solving computational mathematical problems.

Angle - Angles are formed by two rays that begin at the same point.

Arc - A section or portion of the circumference of the circle.


Area - The space measured in square units that any 2 dimensional shape or polygon occupies.


Base - The bottom of a shape, solid or three dimensional object. The base is what the object 'rests' on.

Base 10 - the numbering system in common use, in which each place to the left or right of the decimal represents a power of 10.

Bar Graph - A visual representation of horizontal and vertical bars or lines to represent data.

Bell Curve - The shape of the graph that indicates the normal distribution.

Binomial - A polynomial equation with two terms usually joined by a plus or minus sign.

Bisect - To divide into two equal parts. The bisector can be the point or the line that divides the object or shape into two equal or congruent parts.

Box and Whisker Plot/Chart - A graphical representation of data that spots differences in distributions. Plots the ranges of data sets.



Calculus - The branch of mathematics involving derivatives and integrals. The study of motion in which changing values are studied.

Capacity - The amount a container will hold.

Centimeter - A measure of length. 2.5cm is approximately an inch. A metric unit of measurement.

Circumference - The complete distance around a circle or a square.

Chord - The segment which joins two points on a circle.

Coefficient - A factor of the term. x is the coefficient in the term x(a + b) or 3 is the coefficient in the term 3y.

Common Factors - A factor of two or more numbers. A number that will divide exactly into different numbers.

Complementary Angles - The two angles involved when the sum is 90°.


Composite Number - A composite number has at least one other factor aside from its own. A composite number cannot be a prime number.

Cone - A three dimensional shape with only one vertex, having a circular base.

Conic Section - The section formed by the intersection of a plane and a cone.

Constant - A value that doesn't change.

Coordinate - The ordered pair that states the location on a coordinate plane. Used to describe location and or position.

Congruent - Objects and figures that have the same size and shape. The shapes can be turned into one another with a flip, rotation or turn.

Cosine - The ratio of the length (in a right triangle) of the side adjacent to an acute angle to the length of the hypotenuse

Cylinder - A Three dimensional shape with a parallel circle and each end and joined by a curved surface.


Decagon - A polygon/shape that has ten angles and ten straight lines.

Decimal - A real number on the base ten standard numbering system.

Denominator - The denominator is the bottom number of a fraction. (Numerator is the top number) The Denominator is the total number of parts.

Degree - The unit of an angle, angles are measured in degrees shown by the degree symbol: °

Diagonal - A line segment that connects two vertices in a polygon.

Diameterl - A chord that passes through the center of a circle. Also the length of a line that cuts the shape in half.

Difference - The difference is what is found when one number is subtracted from another. Finding the difference in a number requires the use of subtraction.

Digit - Digits are making reference to numerals. 176 is a 3 digit number.

Dividend - The numer that is being divided. The number found inside the bracket.

Divisor - The number that is doing the dividing. The number found outside of the division bracket.


Edge - A line that joins a polygon or the line (edge) where two faces meet in a 3 dimensional solid.

Ellipse - An ellipse looks like a slightly flattened circle. A plane curve. Orbits take the form of ellipses.

End Point - The 'point' at which a line or a curve ends.

Equaliteral - All sides are equal.

Equation - A statement showing the equality of two expressions usually separated by left and right signs and joined by an equals sign.

Even Number - A number that can be divided or is divisible by 2.

Event - Often refers to the outcome of probability. Answers questions like 'What is the probability the spinner will land on red?'

Evaluate - To calculate the numerical value.

Exponent - The number that gives reference to the repeated multiplication required. The exponent of 34 is the 4.

Expressions - Symbols that represent numbers or operations. A way of writing something that uses numbers and symbols.


Face - The face refers to the shape that is bounded by the edges on a 3 dimensional object.

Factor - A number that will divide into another number exactly. (The factors of 10 are 1, 2 and 5).

Factoring - The process of breaking numbers down into all of their factors.

Factorial Notation - Often in combinatorics, you will be required to multiply consecutive numbers. The symbol used in factorial notation is ! When you see x!, the factorial of x is needed.

Factor Tree - A graphical representation showing the factors of a specific number.

Fibonacci Sequence - A sequence whereby each number is the sum of the two numbers preceding it.

Figure - Two dimensional shapes are often referred to as figures.

Finite - Not infinite. Finite has an end.

Flip - A refllection of a two dimensional shape, a mirror image of a shape.

Formula - A rule that describes the relationship of two or more variables. An equation stating the rule.

Fraction - A way of writing numbers that are not whole numbers. The fraction is written like 1/2.

Frequency - The number of times an event can happen in a specific period of times. Often used in probability.

Furlong - A unit of measurement - the side length of one square of an acre. One furlong is approximately 1/8 of a mile, 201.17 meters and 220 yards.


Geometry - The study of lines, angles, shapes and their properties. Geometry is concerned with physical shapes and the dimensions of the objects.

Graphing Calculator - A larger screen calculator that's capable of showing/drawing graphs and functions.

Graph Theory - A branch of mathematics focusing on the properties of a variety of graphs.

Greatest Common Factor - The largest number common to each set of factors that divides both numbers exactly. E.g., the greatest common factor of 10 and 20 is 10.

Hexagon - A six sided and six angled polygon. Hex means 6.

Histogram - A graph that uses bars where each bar equals a range of values.

Hyperbola - One type of conic section. The hyperbola is the set of all points in a plane. The difference of whose distance from two fixed points in the plane is the positive constant.

Hypotenuse - The longest side of a right angled triangle. Always the side that's opposite of the right angle.


Identity - An equation that is true for values of their variables.

Improper Fraction - A fraction whereby the denominator is equal to or greater than the numerator. E.g., 6/4

Inequality - A mathematical equation containing either a greater than, less than or not equal to symbols.

Integers - Whole numbers, postive or negative including zero.

Irrational - A number that cannot be represented as a decimal or as a fraction. A number like pi is irrational because it contains an infinite number of digits that keep repeating, many square roots are irrational numbers.

Isoceles - A polygon having two sides equal in length.


Kilometer - A unit of measure that equals 1000 meters.

Knot - A curve formed by an interlacing piece of spring by joining the ends.

Like Terms - Terms with the same variable and the same exponents/degrees.

Like Fractions - Fractions having the same denominator. (Numerator is the top, denominator is the bottom)

Line - A straight inifinite path joining an infinite number of points. The path can be infinite in both directions.

Line Segment - A straight path that has a beginning and an end - endpoints.

Linear Equation - An equation whereby letters represent real numbers and whose graph is a line.

Line of Symmetry - A line that divides a figure or shape into two parts. The two shape must equal one another.

Logic - Sound reasoning and the formal laws of reasoning.

Logarithmn - A power to which a base, [actually 10] must be raised to produce a given number. If nx = a, the logarithm of a, with n as the base, is x.


Mean - The mean is the same as the average. Add up the series of numbers and divide the sum by the number of values.

Median - The Median is the 'middle value' in your list or series of numbers. When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two.

Midpoint - A point that is exactly half way between two set points.

Mixed Numbers - Mixed numbers refer to whole numbers with fractions or decimals. Example 3 1/2 or 3.5.

Mode - The mode in a list of numbers refers to the list of numbers that occur most frequently. A trick to remember this one is to remember that mode starts with the same first two letters that most does. Most frequently - Mode.

Monomial - An algebraic expression consisting of a single term.

Multiple - The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 2)

Multiplication - Often referred to as 'fast adding'. Multiplication is the repeated addition of the same number 4x3 is the same as saying 3+3+3+3.


Natural Numbers - Regular counting numbers.

Negative Number - A number less than zero. For instance - a decimal .10

Net - Often referred to in elementary school math. A flattened 3-D shape that can be turned into a 3-D object with glue/tape and folding.

Nth Root - The nth root of a number is the number needed to multiply by itself 'n' times in order to get that number. For instance: the 4th root of 3 is 81 because 3 X 3 X 3 X 3 = 81.

Norm - The mean or the average - an established pattern or form.

Numerator - The top number in a fraction. In 1/2, 1 is the numerator and 2 is the denomenator. The numerator is the portion of the denominator.

Number Line - A line in which points all correspond to numbers.

Numeral - A written symbol referring to a number.


Obtuse Angle - An angle having a measure greater than 90° and up to 180°.

Obtuse Triangle - A triangle with at least one obtuse angle as described above.

Octagon - A polygon with 8 sides.

Odds - The ratio/likelihood of an event in probability happening. The odds of flipping a coin and having it land on heads has a 1-2 chance.

Odd Number - A whole number that is not divisible by 2.

Operation - Refers to either addition, subtraction, multiplication or division which are called the four operations in mathematics or arithmetic.

Ordinal - Ordinal numbers refer to the position: first, second, third etc.

Order of Operations - A set of rules used to solve mathematical problems. BEDMAS is often the acronym used to remember the order of operations. BEDMAS stands for 'brackets, exponents, divison, multiplication, addition and subtraction.

Outcome - Used usually in probability to refer to the outcome of an event.


Parralellogram - A quadrilateral that has both sets of opposite sides that are parallel.

Parabola - A type of curve, any point of which is equally distant from a fixed point, called the focus, and a fixed straight line, called the directrix.

Pentagon - A five sided polygon. Regular pentagons have five equal sides and five equal angles.

Percent - A ratio or fraction in which the second term on denominator is always 100.

Perimeter - The total distance around the outside of a polygon. The total distance around is obtained by adding tegether the units of measure from each side.

Perpendicular - When two lines or line segments intersect and form right angles.

Pi p - The symbol for Pi is actually a greek letter. Pi is used to represent the ratio of a circumference of a circle to its diameter.

Plane - When a set of points joined together form a flat surface, the plan can extend without end in all directions.

Polynomial - An algebraic term. The sum of 2 or more monomials. Polynomials include variables and always have one or more terms.

Polygon - Line segments joined together to form a closed figure. Rectangles, squares, pentagons are all examples of polygons.

Prime Numbers - Prime numbers are integers that are greater than 1 and are only divisible by themselves and 1.

Probability - The likelihood of an event happening.

Product - The sum obtained when any two or more numbers are multiplied together.

Proper Fraction - A fraction where the denominator is greater than the numerator.

Protractor - A semi-circle device used for measuring angles. The edge is subdivided into degrees.


Quadrant - One quarter (qua) of the plane on the cartesian coordinate system. The plane is divided into 4 sections, each section is called a quadrant.

Quadradic Equation - An equation that can be written with one side equal to 0. Asks you to find the quadratic polynomial that is equal to zero.

Quadrilateral - A four (quad) sided polygon/shape.

Quadruple - To multiply or to be multiplied by 4.

Qualitative - A general description of properties that cannot be written in numbers.

Quartic - A polynomial having a degree of 4.

Quintic - A polynomial having a degree of 5.

Quotient - The solution to a division problem.


Radius - A line segment from the center of a circle to any point on the circle. Or the line from the center of a spere to any point on the outside edge of the sphere. The radius is the distance from the center of a circle/sphere to the outside edge.

Ratio - The relation between to quantities. Ratios can be expressed in words, fractions, decimals or percents. E.g., the ratio given when a team wins 4 out of 6 games can be said a 4:6 or four out of six or 4/6.

Ray - A straight line with one endpoint. The line extends infinitely.

Range - The difference between the maximum and the minimum in a set of data.

Rectangle - A parallelogram which has four right angles.

Repeating Decimal - A decimal with endlessly repeating digits. E.g., 88 divided by 33 will give a 2.6666666666666

Reflection - A mirror image of a shape or an object. Obtained from flipping the image/object.

Remainder - The number that is left over when the number cannot be divided evenly into the number.

Right Angle - An angle that is 90°.

Right Triangle - A triangle having one angle equal to 90°.

Rhombus - A parallelogram with four equal sides, sides are all the same length.


Scalene Triangle - A triangle with 3 unequal sides.

Secant

Sector - An area between an arc and two radiuses of a circle. Sometimes referred to as a wedge.

Slope - The slope shows the steepness or incliine of a line, determined from two points on the line.

Square Root- To square a number, you multiply it by itself. The square root of a number is the value of the number when multiplied by itself, gives you the original number. For instance 12 squared is 144, the square root of 144 is 12.

Stem and Leaf - A graphic organizer to organize and compare data. Similar to a histogram, organizes intervals or groups of data.

Subtraction - The operation of finding the difference between two numbers or quantities. A process of 'taking away'.

Supplementary Angles - Two angles are supplementary if their sum totals 180°.

Symmetry - Two halves which match perfectly.


Tangent - When an angle in a right angle is X, the tangent of x is the ratio of lengths of the side opposite x to the side adjacent to x.

Term - A part of an algebraic equation or a number in a sequence or a series or a product of real numbers and/or variables.

Tessellation - Congruent plane figures/shapes that cover a plane completely without overlapping.

Translation - A term used in geometry. Often called a slide. The figure or shape is moved from each point of the figure/shape in the same direction and distance.

Transversal - A line that crosses/intersects two or more lines.

Trapezoid - A quadrilateral with exactly two parallel sides.

Tree Diagram - Used in probability to show all of the possible outcomes or combinations of an event.

Triangle - Three sided polygon.

Trinomial - An algebraic equation with 3 terms - polynomial.


Unit - A standard quantity used in measurement. An inch is a unit of length, a centimeter is a unit of length a pound is a unit of weight.

Uniform - All the same. Having the same in size, texture, color, design etc.


Variable - When a letter is used to represent a number or number in equations and or expressions. E.g., in 3x + y, both y and x are the variables.

Venn Diagram - A Venn diagram is often two circles (can be other shapes) that overlap. The overlapping part usually contains information that is pertinent to the labels on both sides of the Venn diagram. For instance: one circle could be labeled 'Odd Numbers', the other circle could be labeled 'Two Digit Numbers' the overlapping portion must contain numbers that are odd and have two digits. Thus, the overlapping portions shows the relationship between the sets. (Can be more than 2 circles.)

Volume - A unit of measure. The amount of cubic units that occupy a space. A measurement of capacity or volume.

Vertex- A point of intersection where two (or more) rays meet, often called the corner. Wherever sides or edges meet on polygons or shapes. The point of a cone, the corners of cubes or squares.


Weight - A measure of how heavy something is.

Whole Number - A whole number doesn't contain a fraction. A whole number is a positive integer which has 1 or more units and can be positive or negative.


X-Axis - The horizontal axis in a coordinate plane.

X-Intercept - The value of X when the line or curve intersects or crosses the x axis.

X - The roman numeral for 10.

x - A symbol most often used to represent an unknown quantity in an equation.

Y-Axis - The vertical axis in a coordinate plane.

Y-Intercept - The value of y when the line or curve intersects or crosses the y axis.

Yard- A unit of measure. A yard is approximately 91.5 cm. A yard is also 3 feet.

How To How to Find Greatest Common Factors

Factors are numbers that divide evenly in a number. The greatest common factor of two or more numbers is the largest number that can divide evenly into each of the numbers. Here, you will learn how to find factors and greatest common factors.
You will want to know how to factor numbers when you are trying to simplify fractions.


Factors of the number 12
You can evenly divide 12 by 1, 2, 3, 4, 6 and 12.
Therefore, we can say that 1,2,3,4,6 and 12 are factors of 12.
We can also say that the greatest or largest factor of 12 is 12.


Factors of 12 and 6
You can evenly divide 12 by 1, 2, 3, 4, 6 and 12.
You can evenly divide 6 by 1, 2, 3 and 6.
Now look at both sets of numbers. What is the largest factor of both numbers?
6 is the largest or greatest factor for 12 and 6.


Factors of 8 and 32
You can evenly divide 8 by 1, 2, 4 and 8.
You can evenly divide 32 by 1, 2, 4, 8, 16 and 32.
Therefore the largest common factor of both numbers is 8.


Multiplying Common PRIME Factors
This is another method to find the greatest common factor. Let's take 8 and 32.
The prime factors of 8 are 1 x 2 x 2 x 2.
Notice that the prime factors of 32 are 1 x 2 x 2 x 2 x 2 x 2.
If we multiply the common prime factors of 8 and 32, we get:
1 x 2 x 2 x 2 = 8 which becomes the greatest common factor.


Both methods will help you determine the greatest common factors (GFCs). However, you will need to decide which method you prefer to work with. I have discovered that most of my students prefer the first method. However, if they're not getting it that way, be sure to show them the alternative method.

Multiplication Tricks

Math Tricks to Learn the Facts (Multiplication)

Free Multiplication Worksheets in PDF and MS Word here.
Learn the Multiplication Tables in 21 days - more here.
Word Problems 1st to the 6th grade. Children should be able to solve these!

More and more in my teaching career, I'm seeing that children no longer memorize their multiplication tables. With the math curriculum as extensive as it is, teachers cannot afford to take the time to ensure that students learn the basic facts. Parents are partners in the process and will have greater opportunities for their children to succeed in math if they support the learning of the basics at home. Work with your children to ensure that they do not fall between the cracks. Help your children learn the facts. There are many tricks to teach children multiplication facts in mathematics. Some tricks that I used to use in my classroom are listed here. If you know of some that I may have missed, drop into the forum and let everyone know. I'll add them to this list as I see them.

The 9 Times Quickie

Hold your hands in front of you with your fingers spread out.
For 9 X 3 bend your third finger down. (9 X 4 would be the fourth finger etc.)
You have 2 fingers in front of the bent finger and 7 after the bent finger
Thus the answer must be 27
This technique works for the 9 times tables up to 10.
The 4 Times Quickie

If you know how to double a number, this one is easy.
Simply, double a number and then double it again!
The 11 Times Rule #1

Take any number to 10 and multiply it by 11.
Multiply 11 by 3 to get 33, multiply 11 by 4 to get 44. Each number to 10 is just duplicated.
The 11 Times Rule #2

Use this strategy for two digit numbers only.
Multiply 11 by 18. Jot down 1 and 8 with a space between it. 1 --8.
Add the 8 and the 1 and put that number in the middle: 198
Deck 'Em!

Use a deck of playing cards for a game of Multiplication War.
Initially, children may need the grid (below) to become quick at the answers.
Flip over the cards as though you are playing Snap.
The first one to say the fact based on the cards turned over (a four and a five = Say "20") gets the cards.
The person to get all of the cards wins!
Children learn their facts much more quickly when playing this game on a regular basis.
Seeing the Patterns

Use a multiplication grid or let your students/children create one.
Look carefully at all of the patterns, especially when the numbers correspond with the facts e.g., 7X8 and 8X7 = 56
Let students/children practice the 'fast adding' which is what multiplication is.
When students can count by 3s, 4s, 5s 6s, etc. they will automatically know their multiplication tables.

Divisibility Tricks

Divisibility Math Tricks to Learn the Facts (Divisibility)


Dividing by 2

All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8.
Dividing by 3

Add up all the digits in the number.
Find out what the sum is. If the sum is divisible by 3, so is the number
For example: 12123 (1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is too!

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Dividing by 4

Are the last two digits in your number divisible by 4?
If so, the number is too!
For example: 358912 ends in 12 which is divisible by 4, thus so is 358912.


Dividing by 5

Numbers ending in a 5 or a 0 are always divisible by 5.


Dividing by 6

If the Number is divisible by 2 and 3 it is divisible by 6 also.


Dividing by 7 (2 Tests)

Take the last digit in a number.
Double and subtract the last digit in your number from the rest of the digits.
Repeat the process for larger numbers.
Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

NEXT TEST
Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary
Add the products.
If the sum is divisible by 7 - so is your number.
Example: Is 2016 divisible by 7?
6(1) + 1(3) + 0(2) + 2(6) = 21
21 is divisible by 7 and we can now say that 2016 is also divisible by 7.


Dividing by 8

This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number.
Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.


Dividing by 9

Almost the same rule and dividing by 3. Add up all the digits in the number.
Find out what the sum is. If the sum is divisible by 9, so is the number.
For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!
Dividing by 10

If the number ends in a 0, it is divisible by 10.

Math Riddles

Why should you never mention the number 288 in front of anyone?
Because it is too gross (2 x 144 - two gross).


Which weighs more? A pound of gold or a pound of feathers?
Both weigh the same.


How is the moon like a dollar?
They both have 4 quarters.


What is alive and has only 1 foot?
A leg.

Why did the dentist need to know math?
Because they may need to do a square root canal.


What do you get if you add two apples and three apples?
A high school math problem!


Why did the amoeba flunk the math test?
Because it multiplied by dividing.


What makes arithmetic hard work?
All those numerals you have to carry.


What are ten things you can always count on?
Your fingers.




What kind of pliers do you use in arithmetic?
Multipliers.




How does a cow add?
With a cow-culator.




When do giraffes have 8 feet?
When there's two of them.


How many eggs can you put in an empty basket?
Only one, after that the basket is not empty.






What coin doubles in value when half is deducted?
A half dollar.


What is the difference between a new penny and an old quarter?
24 cents.


If you can buy eight eggs for 26 cents, how many can you buy for a cent and a quarter?
8.


Where can you buy a ruler that is 3 feet long?
At a yard sale.


If there were 9 cats on a bridge and one jumped over the edge, how many would be left?
None - they are copycats.


If you take three apples from five apples, how many do you have?
You have three apples.





What has 4 legs and only 1 foot?
A bed.


How many times can you subtract 6 from 30?
Once; after that it is no longer 30 (Don't try this in a test!)


If one nickel is worth five cents, how much is half of one half of a nickel worth?
$0.0125


How many 9's between 1 and 100?
20.


Which is more valuable - one pound of $10 gold coins or half a pound of $20 gold coins?
One pound is twice of half pound.


It happens once in a minute, twice in a week, and once in a year. What is it?
The letter 'e'.


How can half of 12 be 7?
Cut XII into two halves horizontally. You get VII on the top half.


When things go wrong, what can you always count on?
Your fingers.






Why are diapers like 100 dollar bills?
They need to be changed.


A street that is 40 yards long has a tree every 10 yards on both sides. How many total trees on the entire street?
10, 5 on each side.


What goes up and never comes down?
Your age.


What did one math book say to the other math book?
Wow, have I got problems!

Why didn’t the Romans find algebra challenging?
Because X was always 10.





How do you expand (a + b)2?
(a + b) 2


How would you prove that 2 = 1?
If a = b (so I say) [a = b]
And we multiply both sides by a
Then we'll see that a2 [a2 = ab]
When with ab compared
Are the same. Remove b2. OK? [a2− b2 = ab − b2]

Both sides we will factorize. See?
Now each side contains a − b. [(a+b)(a − b) = b(a − b)]
We'll divide through by a
Minus b and olé
a + b = b. Oh whoopee! [a + b = b]

But since I said a = b
b + b = b you'll agree? [b + b = b]
So if b = 1
Then this sum I have done [1 + 1 = 1]
Proves that 2 = 1. Q.E.D.


(Just in case you're wondering - the above proof is incorrect because in step 5, we divided by (a - b) which is 0 since a = b)

What kind of tree does a math teacher climb?
Geome-tree.



Why was the obtuse angle upset?
Because he was never right.


What's a polygon?
Polly gone - A dead parrot.




Which triangles are the coldest?
Ice-sosceles triangles.





What's the best dessert in the Math Teacher's Café?
A slice of chocolate pi.

How many calories are there in that slice of chocolate pi?
Approximately 3.142


What is 8 divided in two parts?
Vertically it is 3; horizontally it is 0.

Maths Mnemonics

Pre-Algebra
Arithmetic (Spelling)

Spelling of Arithmetic:
A Rat In The House May Eat The Ice-Cream.


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Arithmetic Properties

Commutative Property
The commuting distance is the same in either direction, from home to work or work to home.
The Commutative Property tells us that: a + b = b + a or a × b = b × a

Associative Property
To associate with people is to group up with them.
The Associative Property are about grouping:
(a + b) + c = a + (b + c), or (a × b) × c = a × (b × c)

Distributive Property
To distribute something is to give it to everyone.
The Distributive Property “gives” whatever’s outside the parentheses to everything inside: a (b + c)=ab + ac



Dividing One Fraction With Another

Keep the first fraction, Change the sign from divide to multiply, Flip the last fraction.

Kentucky Chicken Fried
Kangaroo Candy Flowers
Koalas Chasing Ferrets


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Division Algorithm (How To Do Long Division)

Divide - Multiply - Subtract - Compare - Bring Down

Does McDonalds Serve Cheese Burgers?
Does My Sister Cook Bananas?

Divide - Multiply - Subtract -Compare -Remainder

Does McDonald's Sell Cheeseburgers Raw?


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Feet in a Mile

5280 feet = 1 mile
5 Tomatoes
5 to (m)ate oe(s)
5 2 8 0 -- or 5,280 feet in a mile






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Greater Than or Less Than?

< or > ?
Which means greater than? Which means less than? How can you tell?
The alligator has to open its mouth wider for the larger number.


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Metric Unit Prefixes

Metric Units of Measure in Order

* Kilo
* Hecto
* Deca
* Units [meter, liter, gram]
* Deci
* Centi
* Milli


King Hector Doesn't Usually Drink Cold Milk.
King Henry Danced [Merrily / Lazily / Grandly] Drinking Chocolate Milk. For the standard units you can insert [Merrily] for meter, [Lazily] for liter, or [Grandly] for gram.


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Mode & Median

Mode
The mode is the value there are the most of

“MOde” and “MOst” have the same starting 2 letters.

The mode is the Most Occuring Data Entity.

Median
The median splits the data down the middle, like the median strip in a road.






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Which is the Numerator, which is the Denominator?

Think “Notre Dame” (N before D)
nUmerator Up, Denominator Down
Nice Dog (N before D)


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Multiplying Signed Numbers
Analogies can help here.

From sports:
we = pos, win = pos, they = neg, lose = neg, good = pos, bad = neg
we win = good (pos × pos = pos)
we lose = bad (pos × neg = neg)
they win = bad (neg × pos = neg)
they lose = good (neg × neg = pos)

From friendship:
friend = pos, enemy = neg
My friend's friend is my friend (pos × pos = pos)
My friend's enemy is my enemy (pos × neg = neg)
My enemy's friend is my enemy (neg × pos = neg)
My enemy's enemy is my friend (neg× neg = pos)

From life:
good = pos, bad = neg

A good thing happening to a good person is good. (pos × pos = pos)
A good thing happening to a bad person is bad. (pos × neg = neg)
A bad thing happening to a good person is bad. (neg × pos = neg)
A bad thing happening to a bad person is good. (neg× neg = pos)


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Order of Operations
The order of mathematical operations:
* Parentheses
* Exponents
* Multiplication/Division (left to right)
* Addition/Subtraction (left to right).

PEMDAS Mnemonics
1. Please
Excuse
My Dear
Aunt Sally

2. Please
Educate
My Daughters
And Sons


3. Pursuing
Education
Means Dedication
And Study


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Roman numerals

Roman numerals: I =1, V = 5, and X = 10.
I View Xrays.

Roman Numerals: I =1, V = 5, X = 10, L = 50 , C = 100 , D = 500 and M = 1000.
I Viewed Xerxes Loping Carelessly Down Mountains.
I Value Xylophones Like Cows Dig Milk


Roman numerals: L = 50 , C = 100 , D = 500 and M = 1000.
Lucy Can't Drink Milk.



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Use the first letter of each word to help you remember the integers zero through ten.

"Zowie! Only time travelers forge forward." said Sam, entering next Tuesday.

(Thanks Amy for the contribution)
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Speed, Distance and Time

Remember the formula triangle
D = distance traveled
S = Average speed
T = Time taken


You can use the formula triangle in the following way:
If you need to find D, cover up D in the triangle and you get S × T

If you need to find S, cover up S in the triangle and you get

If you need to find T, cover up T in the triangle and you get

Math Tricks and puzzles




One Equals Zero!


The following is a "proof" that one equals zero.

Consider two non-zero numbers x and y such that

x = y.
Then x2 = xy.
Subtract the same thing from both sides:
x2 - y2 = xy - y2.
Dividing by (x-y), obtain
x + y = y.
Since x = y, we see that
2 y = y.
Thus 2 = 1, since we started with y nonzero.
Subtracting 1 from both sides,
1 = 0.
What's wrong with this "proof"?

Presentation Suggestions:
This Fun Fact is a reminder for students to always check when they are dividing by unknown variables for cases where the denominator might be zero.

The Math Behind the Fact:
The problem with this "proof" is that if x=y, then x-y=0. Notice that halfway through our "proof" we divided by (x-y).


Memorizing Pi
The digits of Pi are fascinating. As the ratio of the circumference of a circle to its diameter, Pi has such a fundamental definition, and yet this ratio is irrational and so its decimal expansion never repeats. It is easy to be mesmerized by the digits of the decimal expansion:

3.14159265358979323846264338327950288419716939937510...

and many people have tried to memorize digits of pi for fun.
One fun way to memorize the first few digits is to use sentence mnemonics for pi--- phrases in which the number of letters of each successive word corresponds to a digit of pi. Here are some well-known pi mnemonics:

"Wow! I made a great discovery!" (3.14159...)

"Can I have a small container of coffee?" (3.1415926...)

"How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics." (3.14159265358979...)


Kaprekar's Constant
Take any four digit number (whose digits are not all identical), and do the following:

Rearrange the string of digits to form the largest and smallest 4-digit numbers possible.
Take these two numbers and subtract the smaller number from the larger.
Use the number you obtain and repeat the above process.
What happens if you repeat the above process over and over? Let's see...
Suppose we choose the number 3141.
4311-1134=3177.
7731-1377=6354.
6543-3456=3087.
8730-0378=8352.
8532-2358=6174.
7641-1467=6174...
The process eventually hits 6174 and then stays there!

But the more amazing thing is this: every four digit number whose digits are not all the same will eventually hit 6174, in at most 7 steps, and then stay there!

Presentation Suggestions:
Remember that if you encounter any numbers with fewer than has fewer 4 digits, it must be treated as though it had 4 digits, using leading zeroes. Example: if you start with 3222 and subtract 2333, then the difference is 0999. The next step would then consider the difference 9990-0999=8991, and so on. You might ask students to investigate what happens for strings of other lengths or in other bases.

The Math Behind the Fact:
Each number in the sequence uniquely determines the next number in the sequence. Since there are only finitely many possibilities, eventually the sequence must return to a number it hit before, leading to a cycle. So any starting number will give a sequence that eventually cycles. There can many cycles; however, for length 4 strings in base 10, there happens to be 1 non-trivial cycle, and it has length 1 (involving the number 6174).


Mind-Reading Number Trick
Think of a number, any positive integer (but keep it small so you can do computations in your head).

1. Square it.
2. Add the result to your original number.
3. Divide by your original number.
4. Add, oh I don't know, say 17.
5. Subtract your original number.
6. Divide by 6.

The number you are thinking of now is 3!

How did I do this?

Presentation Suggestions:
Ham it up with magician's patter. Step 4 could be anything you want---someone's age, or their favorite number--- just ask the crowd for suggestions. (This will change the final outcome of Step 6, but see below for how.)

The Math Behind the Fact:
Clearly no matter what you start with, the answer should come out the same (zero wasn't allowed because of Step 3). We can see why this trick works by using a little bit of high school algebra! If you follow the instructions starting with the variable X instead of an actual number, you will see that X is eliminated by Step 5.

Using this idea, you can make up your own mental math trick right on the spot! (Just don't do anything too obvious, like tell people to add 5, subtract their original number, and say "the number you are thinking of is 5".)

Magic 1089
Here's a cool mathematical magic trick. Write down a three-digit number whose digits are decreasing. Then reverse the digits to create a new number, and subtract this number from the original number. With the resulting number, add it to the reverse of itself. The number you will get is 1089!

For example, if you start with 532 (three digits, decreasing order), then the reverse is 235. Subtract 532-235 to get 297. Now add 297 and its reverse 792, and you will get 1089!

Presentation Suggestions:
You might ask your students to see if they can explain this magic trick using a little algebra.

The Math Behind the Fact:
If we let a, b, c denote the three digits of the original number, then the three-digit number is 100a+10b+c. The reverse is 100c+10b+a. Subtract: (100a+10b+c)-(100c+10b+a) to get 99(a-c). Since the digits were decreasing, (a-c) is at least 2 and no greater than 8, so the result must be one of 198, 297, 396, 495, 594, 693, 792, or 891. When you add any one of those numbers to the reverse of itself, you get 1089!


Red-Black Card Trick
Here's a pretty easy card trick that you can do that can also be pretty surprising. Here's how the trick you do will appear to others:

Take a deck of cards, and give it to a spectator and ask her to shuffle the deck and return it to you face down. You take the cards, and (with a little showmanship but without looking at the fronts of the cards) separate them into two piles, and then say "Just by feeling the redness or blackness of the cards with my fingers, I've made two piles so that the number of red cards in the first pile is the number of black cards in the second pile."

Have your spectator turn over the cards and verify!

Presentation Suggestions:
Your spectator can shuffle the cards as many times as she likes--- it won't matter! When she gives the cards to you, all you are really doing (though don't make it obvious) is counting the cards into two piles so that there are 26 cards in each pile.

The Math Behind the Fact:
The reason this trick works is simple... if the number of red cards in the first and second piles is R and S, and the number of black cards in the first and second piles is A and B, then we know that R+S=26 (since the total number of red cards is 26) and S+B=26 (since the total number of cards in the second pile is 26). These two equations can be subtracted from one another to show that R-B=0, or R=B.


Red-Black Pairs Card Trick
Here's a terrific mathematical card trick that will impress your friends. When you do this trick, the effect of the card trick will look like this:

You have a deck of cards, and you ask for a volunteer who knows how to do a riffle shuffle. You then cut the deck and then give the volunteer the halves of the deck and ask him to do one riffle shuffle and return the deck to you. Now say "There's no way I could know anything about the deck right now, right? Well, I was born with the amazing ability to feel the redness and blackness of cards with my fingertips. However, my talent is not that refined. I can only feel red and black cards in pairs." As you say this, put the deck of cards behind your back (so that you cannot see them) and then, at regular intervals, you fish around in the deck and pull out pairs of cards and show them to the audience. These pairs will all have exactly one black and one red card!

Presentation Suggestions:
Before performing the trick, order the deck alternating colors, all the way through, red-black-red-black-... etc. (When you flash the deck before their eyes, they really won't notice this pattern if you do it quickly.)

After this, there is really only one thing you need to remember to ensure that the trick works: you must cut the deck (not the spectator), and you must do it in such a way that the bottom of each half of the deck is a different color. Then, no matter how the spectator riffle shuffles the deck, the cards will always drop in red-black or black-red pairs. See below for explanation.

Then, all you have to do after the deck is returned and you put it behind your back is to pull out the top 2 cards. It will be either red-black or black-red! Then pull out the next 2 cards, which again will be red-black or black-red. You can continue in this fashion to the end of the deck, if you like!

Of course, you should make it look as if you are trying really hard to find the cards (even though what you are really doing is very easy). Spectators will wonder if you are pulling one card off the top and one card off the bottom; but you can pull the deck out and show them that this is not the case.

The Math Behind the Fact:
The reason the trick works at the point of the riffle shuffle is both simple and stunning: if you cut the deck so that the cards at the bottom of each half are different colors, then the first card that gets "dropped" in the shuffle will be a different color then the second card that gets dropped, no matter which half of the deck they come from. As an example, if the first card that gets dropped is black, then after that both halves will have red cards at the bottom, so no matter which card falls next it will be red! After this, both halves again have different colored cards at bottom and we are back to the situation at start. So all the cards will fall off in either red-black or black-red pairs.

The message of this trick is that one shuffle is not enough to randomize a deck of cards-- you really can know something about the deck after one shuffle... but only if you stack the deck in a particular way first!


Binary Card Trick
You put a deck of cards in your pocket, and invite anyone in the audience to call out a number between 1 and 15. Then you reach into your pocket, you take out a set of cards whose sum is the number that was called!

How can you perform this magic trick?

The Math Behind the Fact:
This mathematical magic trick can be found in the reference and is based on the properties of binary numbers. Every number between 1 and 15 has a unique representation as a sum of some collection of the numbers 1, 2, 4, and 8. (To see which collection, just take the given number and successively subtract the largest number of 1, 2, 4, and 8 that is less than the given number. That number is part of your collection. The subtraction yields a new number; now repeat the process with this number, over and over, until you get 0.) The collection of numbers you obtain reveals the binary decomposition of the given number into sums of powers of two (in contrast to the usual representation of a number into sums of powers of ten).

Now before the trick starts, pick an Ace, 2, 4, and 8 and put them on top of the deck, and then put the deck in your pocket.

Then when a number between 1 and 15 is called out, take the binary decomposition of the number, and use that to determine which of the first four cards you will pull out. No one needs to know that you never need to use the other cards!


Leapfrog Addition
Here's a nice mathematical magic trick based on properties of the Fibonacci sequence.

Give your friend a card with ten blank lines, numbered 1 through 10. Have your friend think of two numbers between 1 and 20 and write them down on the first 2 lines of the card. Now in each of the successive lines, have your friend write the sum of the previous two lines. For instance, in line 3, write the sum of lines 1 and 2. In line 4, write the sum of lines 2 and 3, etc. until finally in line 10, your friend has written the sum of lines 8 and 9.

Ask your friend to total the numbers. If you've practiced the Multiplication by 11 Fun Fact, you'll be able to tell your friend the total faster than she can add the numbers (because the total will be just 11 times the number in line 7). Also, you can announce the quotient of line 10 divided by line 9... to 2 decimal places, it will be 1.61!

Let's do an example. Suppose your friend tells gives you the numbers 3 and 7. Her card will then have these numbers:

7
3
10
13
23
36
59
95
154
249
The total is 649 (which is just 11 times 59, do this in your head with the Multiplication by 11 Fun Fact.
The quotient 249/154 is 1.61 (to 2 decimal places).
Presentation Suggestions:
Write down the quotient 1.61 on another card, and place it in an envelope before the start of your magic trick. Then you can have your friend open that envelope after she has computed the quotient.

The Math Behind the Fact:
The trick works for the following reason. If the number X is in line 1, and the number Y is in line 2, then the number X+Y will be in line 3, the number (X+Y)+Y=(X+2Y) will be in line 4, and so on. Continuing, you will find that line 7 contains (5X+8Y), line 9 contains (21X+34Y), and line 10 contains (55X+88Y), which is indeed just 11 times line 7.

For the ratio of line 10 divided by line 9, we appeal to a property of "adding fractions badly": for positive numbers A, B, C, D where (A/B) < (C/D), it is a neat fact that the fraction you get by "adding badly": (A+C)/(B+D), must lie in between the values (A/B) and (C/D). So the ratio (55X+88Y)/(21X+34Y) must lie in between (55X/21X)=1.615... and (88Y/34Y)=1.619...

An even more stunning fact is that if you continue this leapfrog procedure with many more lines, then the ratio of successive lines will approach the golden ratio: 1.6180339... (If you know some linear algebra, this follows because the leapfrog procedure can be written as a 2-dimensional linear system of equations, and the largest eigenvalue of this system happens to be the golden ratio.)

This magic trick may be found in the delightful book in the reference.


Squares Ending in 5
Give me any 2 digit number that ends in 5, and I'll square it in my head!
452 = 2025
852 = 7225, etc.

There's a quick way to do this: if the first digit is N and the second digit is 5, then the last 2 digits of the answer will be 25, and the preceding digits will be N*(N+1).

Presentation Suggestions:
After telling the trick, have students see how fast they can square such numbers in their head, but doing several examples.

The Math Behind the Fact:
You may wish to assign the proof as a fun homework exercise: multiply (10N+5)(10N+5) and interpret! The trick works for larger numbers, too, although it may be harder to do this in your head. For instance 2052 = 42025, since 20*21=420. Also, you can combine this trick with other lightning arithmetic tricks. So 1152 = 13225, because 11*12 = 132, using the Multiplication by 11 trick.

The reference also contains more secrets of fast mental calculations.


Fibonacci Number Formula
The Fibonacci numbers are generated by setting F0=0, F1=1, and then using the recursive formula


Fn = Fn-1 + Fn-2

to get the rest. Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... This sequence of Fibonacci numbers arises all over mathematics and also in nature.
However, if I wanted the 100th term of this sequence, it would take lots of intermediate calculations with the recursive formula to get a result. Is there an easier way?

Yes, there is an exact formula for the n-th term! It is:


an = [ Phin - (phi)n ]/Sqrt[5].

where Phi=(1+Sqrt[5])/2 is the so-called golden mean, and phi=(1-Sqrt[5])/2 is an associated golden number, also equal to (-1/Phi). This formula is attributed to Binet in 1843, though known by Euler before him.
The Math Behind the Fact:
The formula can be proved by induction. It can also be proved using the eigenvalues of a 2x2-matrix that encodes the recurrence. You can learn more about recurrence formulas in a fun course called discrete mathematics.

Six Degrees of Separation
The word graph has two different meanings in mathematics. One involves plotting the domain and range of a function, and another is used to model relationships between discrete objects.

In this definition, a graph is any set of vertices (dots) in which some pairs of vertices are connected by edges (lines). Often the lines are used to represent relationships between objects (represented by dots).

For example, we can construct a graph in which the vertices represent the people in this class, and we'll draw edges between any two people who mutually know one other. We can measure the "distance" between two vertices A and B by the least number of edges that one has to cross to get from A to B in the graph.

Here's a popular question: what is the minimum distance between any two people in the world, using the graph above?

It is popularly believed that the number is 6 or less for any pair of people. You may have heard the term "six degrees of separation". In fact, in the U.S. it is probably easy to get to anyone using a chain of 3 or fewer people... try using your mayor, congressman, or college professors as intermediate points!

Presentation Suggestions:
If your students like this concept, you can also mention that there is a similar concept of "Erdos number", which is the length of the chain in the graph where edges represent the relations "co-authored a paper with" and distance is measured from a famous (prolific!) number theorist named Paul Erdos. The website in the reference contains a wealth of interesting information about this relation.

The Math Behind the Fact:
Graph theory is an branch of mathematics that is very useful in computer science. You can get an introduction to graph theory in a course on discrete mathematics.


Successive Differences of Powers
List the squares:

0, 1, 4, 9, 16, 25, 36, 49, ...
Then take their successive differences:
1, 3, 5, 7, 9, 11, 13, ...
Then take their successive differences again:
2, 2, 2, 2, 2, 2, ...
So the 2nd successive differences are constant(!) and equal to 2.
OK, now list the cubes, and in a similar way, keep taking successive differences:

0, 1, 8, 27, 64, 125, 216, 343, 512, ...
1, 7, 19, 37, 61, 91, 127, 169, ...
6, 12, 18, 24, 30, 36, 42, ...
6, 6, 6, 6, 6, 6, ...
Gee, the 3rd successive differences are all constant(!) and equal to 6.
What happens when you take the 4th successive differences of 4th powers? Are they constant? What do they equal? (They're all 24.) And the 5th successive differences of 5th powers?
Aren't derivatives similar to differences? What do you think happens when you take the n-th derivative of xn?

Presentation Suggestions:
Have students do these investigations along with you. If you assign the n-th derivative of xn on a previous homework, then you can make the connection between the two right away.

The Math Behind the Fact:
This pattern may seem very surprising. It can be proved by induction. Taking differences is like a discrete version of taking the derivative, where the space between successive points is 1.

This idea has a very practical application: given a sequence generated by an unknown polynomial function, you use the calculation of successive differences to determine the order of the polynomial! Then use the first N terms of the sequence with the first N terms of the polynomial to solve for the generating function.


Birthday Problem

How many people do you need in a group to ensure at least a 50 percent probability that 2 people in the group share a birthday?

Let's take a show of hands. How many people think 30 people is enough? 30? 60? 90? 180? 360?

Surprisingly, the answer is only 23 people to have at least a 50 percent chance of a match. This goes up to 70 percent for 30 people, 90 percent for 41 people, 95 percent for 47 people. With 57 people there is better than a 99 percent chance of a birthday match!

Presentation Suggestions:
If you have a large class, it is fun to try to take a poll of birthdays: have people call out their birthdays. But of course, whether or not you have a match proves nothing...

The Math Behind the Fact:
Most people find this result surprising because they are tempted to calculate the probability of a birthday match with one particular person. But the calculation should be done over all pairs of people. Here is a trick that makes the calculation easier.

To calculate the probability of a match, calculate the probability of no match and subtract from 1. But the probability of no match among n people is just

(365/365)(364/365)(363/365)(362/365)...((366-n)/365),
where the k-th term in the product arises from considering the probability that the k-th person in the group doesn't have a birthday match with the (k-1) people before her.
If you want to do this calculation quickly, you can use an approximation: note that for i much smaller than 365, the term (1-i/365) can be approximated by EXP(-i/365). Hence, for n much smaller than 365, the probability of no match is close to


EXP( - SUMi=1 to (n-1) i/365) = EXP( - n(n-1)/(2*365)).

When n=23, this evaluates to 0.499998 for the probability of no match. The probability of at least one match is thus 1 minus this quantity.
For still more fun, if you know some probability: to find the probability that in a given set of n people there are exactly M matches, you can use a Poisson approximation. The Poisson distribution is usually used to model a random variable that counts a number of "rare events", each independent and identically distributed and with average frequency lambda.

Here, the probability of a match in a given pair is 1/365. The matches can be considered to be approximately independent. The frequency lambda is the product of the number of pairs times the probability of a match in a pair: (n choose 2)/365. Then the approximate probability that there are exactly M matches is:

(lambda)M * EXP(-lambda) / M!
which gives the same formula as above when M=0 and n=-365.


Largest Known Primes


Since there are infinitely many primes, what are the largest primes that we know of?

The largest known primes are ones of the form (2m - 1). The reason is that there exist efficient ways to test whether such numbers are prime. Primes of this type are called a Mersenne primes.

the largest known primes are
26,972,593 - 1
23,021,377 - 1
22,976,221 - 1
The largest is over 2 million digits long! These primes were all discovered in the last 3 years; the search for large primes has accelerated with the help of several hundred people across the internet in a project called GIMPS [the Great Internet Mersenne Prime Search]. For more on this, see the URL in the reference.

Presentation Suggestions:
Ask students to guess how large those numbers are, before you tell them.

The Math Behind the Fact:
As it turns out, knowing large primes is very important in cryptography. Being able to factor large numbers is "equivalent" to being able to crack codes, and typical codes that are nearly impossible to break are ones which depend on knowing a large number that is almost prime.


Zero to the Zero Power


It is commonly taught that any number to the zero power is 1, and zero to any power is 0. But if that is the case, what is zero to the zero power?

Well, it is undefined (since xy as a function of 2 variables is not continuous at the origin).

But if it could be defined, what "should" it be? 0 or 1?

Presentation Suggestions:
Take a poll to see what people think before you show them any of the reasons below.

The Math Behind the Fact:
We'll give several arguments to show that the answer "should" be 1.

The alternating sum of binomial coefficients from the n-th row of Pascal's triangle is what you obtain by expanding (1-1)n using the binomial theorem, i.e., 0n. But the alternating sum of the entries of every row except the top row is 0, since 0k=0 for all k greater than 1. But the top row of Pascal's triangle contains a single 1, so its alternating sum is 1, which supports the notion that (1-1)0=00 if it were defined, should be 1.
The limit of xx as x tends to zero (from the right) is 1. In other words, if we want the xx function to be right continuous at 0, we should define it to be 1.
The expression mn is the product of m with itself n times. Thus m0, the "empty product", should be 1 (no matter what m is).
Another way to view the expression mn is as the number of ways to map an n-element set to an m-element set. For instance, there are 9 ways to map a 2-element set to a 3-element set. There are NO ways to map a 2-element set to the empty set (hence 02=0). However, there is exactly one way to map the empty set to itself: use the identity map! Hence 00=1.
Here's an aesthetic reason. A power series is often compactly expressed as

SUMn=0 to INFINITY an (x-c)n.

We desire this expression to evaluate to a0 when x=c, but the n=0 term in the above expression is problematic at x=c. This can be fixed by separating the a0 term (not as nice) or by defining 00=1.


Scrabble quiz


Question: What positive number, when spelled out, has a Scrabble score equal to that integer?

Answer: Twelve = 12 points in Scrabble


Four Fours Problem



Here's a challenge that you may wish to try: can you express all the numbers from 1 to 100 using an arithmetic combination of only four 4's?

The operations and symbols that are allowed are: the four arithmetic operations (+,x,-,/), concatenation (44 is ok and uses up two 4's), decimal points (using 4.4 is ok), powers (using 44 is ok), square roots, factorials (using 4! is ok), and overbars for indicating repeating digits (e.g., writing .4 with an overbar would be a way of expressing 4/9). Ordinary use of parentheses are allowed. No digits other than 4 are allowed.

This problem is sometimes called the four fours problem. I'm not sure where it first originated but it was popularized by Martin Gardner, among others.

Presentation Suggestions:
This puzzle makes an excellent extra credit problem. Or, you might suggest it as a joint project for a whole class to work on: have them post solutions on a bulletin board as they find them. For a computer science course it makes a nice programming exercise in the language prolog.

The Math Behind the Fact:
Actually, all the numbers less than 113 can be constructed in this fashion. While I won't spoil the fun and tell you the answers, let me just say (from experience) that the hardest numbers to express in four 4's are the numbers 69 and 73. These require especially clever combinations of the operations above.

A difficult (and as far as I know unsolved) mathematical challenge is to prove that the number 113 cannot be constructed using these operations.

It should be noted that there are many versions of this problem that have floated around, differing only in the sets of operations that are allowed. (For instance, 113 can be done if you allow arccos as a function.)



Pi Approximations


Pi is the ratio of the circumference of a circle to its diameter. It is known to be irrational and its decimal expansion therefore does not terminate or repeat. The first 40 places are:

3.14159 26535 89793 23846 26433 83279 50288 41971...

Thus, it is sometimes helpful to have good fractional approximations to Pi. Most people know and use 22/7, since 7*Pi is pretty close to 22. But 22/7 is only good to 2 places. A fraction with a larger denominator offers a better chance of getting a more refined estimate. There is also 333/106, which is good to 5 places.

But an outstanding approximation to Pi is the following:

355/113
This fraction is good to 6 places! In fact, there is no "better approximation" among all fractions (P/Q) with denominators less than 30,000. [By "better approximation" we mean in the sense of how close Q*Pi is to P.]
Presentation Suggestions:
Have people verify that 355/113 is a good rational approximation. You can also point out that 355/113 is very easy to remember, since it consists of the digits 113355 in some order!

The Math Behind the Fact:
The theory of continued fractions allows one to find good rational approximations of any irrational number. This is covered in an introductory course on number theory!



Multiplication by 11

Multiplication by 11 is easy! To multiply by a 2-digit number add the two digits and place the sum in between!


25 x 11 = 275
31 x 11 = 341
57 x 11 = 627 <-- you need to carry the 1!

What about a 3-digit number? Can you figure out what's going on here?


253 x 11 = 2783
117 x 11 = 1287
532 x 11 = 5852
267 x 11 = 2937

Presentation Suggestions:
Do examples! What you notice is that multiplication by 11 can be done quickly with numbers of any length by starting with the first and last digits (they remain the same, unless there is a carry) and then inserting the sums of adjacent pairs of digits sequentially in between. For example, 253 x 11 begins with a 2 (like 253 does), then the next digit is 2+5=7, the next digit is 5+3=8, and the last digit is 3 (like 253 has). So the product is 2873. Remember to carry if necessary. So 267 x 11 starts with a 2 (like 267 does), the next digit is 2+6=8, the next digit is 6+7=13 (oops! carry the 1 back to the previous digit, leaving a 3 in this place), then the last digit is 7 (just like 267 has). Thus the product is 2937.

The Math Behind the Fact:
Long multiplication reveals why the trick works... you end up adding adjacent digits. Based on this, can you figure out a snappy rule for Multiplication by 111?


Multiplication by 111


If you liked the Fun Fact Multiplication by 11, you'll enjoy seeing how to take that idea one step farther. Here's a quick way to multiply by 111.

To multiply a two-digit number by 111, add the two digits and if the sum is a single digit, write this digit TWO TIMES in between the original digits of the number. Some examples:


23x111= 2553
41x111= 4551

The same idea works if the sum of the two digits is not a single digit, but you should write down the last digit of the sum twice, but remember to carry if needed. So

57x111= 6321

because 5+7=12, but then you have to carry the one twice.
If the number you are multiplying by 111 is a three-digit number, say ABC, then the answer will have five digits (though it may be six if there is a carry involved):
the first digit is A,
the second digit is A+B,
the third digit is A+B+C,
the fourth digit is B+C,
the fifth digit is C.
Again, you must remember to carry if any of these sums is more than one digit. Thus 123x111=13653, 241x111=26751, and for an example where carrying is needed, 352x111=39072. (Because of the carries, it may be easier to do the sums and write the answer down from right to left.)

Presentation Suggestions:
Do the Multiplication by 11 Fun Fact first.

The Math Behind the Fact:
Multiply using the traditional (long) method for multiplication, and you will find that the above shortcut works because it is doing exactly the same sums that you would have to do using the traditional method for multiplication.



Squaring Quickly


You may have seen the Fun Fact on squares ending in 5; Here's a trick that can help you square ANY number quickly.

It's based on the algebra identity for the difference of squares, but with a twist! Can you figure it out?

542 = 50 * 58 + 42 = 2916.
422 = 40 * 44 + 22 = 1764.
372 = 34 * 40 + 32 = 1369.
You have to pretty proficient at multiplying one digit numbers by two digit numbers in your head to do this trick well. But if you master this, then you can build upon it in some amazing ways:
1162 = 100 * 132 + 162 = 13,200 + 256 = 13,456.
Thinking CREATIVELY about everything you learn, no matter how trivial it may seem, will allow you to find some really clever applications!

Presentation Suggestions:
If you practice this a LOT beforehand, you can start off by asking students to name any 2-digit number and you will do it in your head quickly. Then tell them the trick. But only do this with a LOT OF PRACTICE!

The Math Behind the Fact:
If you look closely, we are using the identity:


a2 = (a-b)(a+b) + b2.
The reference contains more ideas for doing fast mental calculations.


Squares Ending in 5


Give me any 2 digit number that ends in 5, and I'll square it in my head!
452 = 2025
852 = 7225, etc.

There's a quick way to do this: if the first digit is N and the second digit is 5, then the last 2 digits of the answer will be 25, and the preceding digits will be N*(N+1).

Presentation Suggestions:
After telling the trick, have students see how fast they can square such numbers in their head, but doing several examples.

The Math Behind the Fact:
You may wish to assign the proof as a fun homework exercise: multiply (10N+5)(10N+5) and interpret! The trick works for larger numbers, too, although it may be harder to do this in your head. For instance 2052 = 42025, since 20*21=420. Also, you can combine this trick with other lightning arithmetic tricks. So 1152 = 13225, because 11*12 = 132, using the Multiplication by 11 trick.



Multiplying Complementary Pairs
Quick! What's 23 x 27?


621
There's a trick to doing this quickly. Can you see a pattern in these multiplications?


42 x 48 = 2016
43 x 47 = 2021
44 x 46 = 2024
54 x 56 = 3024
64 x 66 = 4224
61 x 69 = 4209
111 x 119 = 13209

In each pair above, the numbers being multiplied are complementary: they are the same number except for the rightmost digit, and the rightmost digits add to 10.

The trick to multiplying complementary pairs is to take the rightmost digits and multiply them; the result forms the two rightmost digits of the answer. (So in the last example 1 x 9 = 09.) Then take the first number without its rightmost digit, and multiply it by the next higher whole number; the result forms the initial digits of the answer. (So in the last example: 11 x 12 = 132. Voila! The answer is 13209.)

The Math Behind the Fact:
This trick works because you are multiplying pairs of numbers of the form 10*N+A and 10*(N+1)-A, where N is a whole number and A is a digit between 1 and 9. A little algebra shows their product is:


100*N*(N+1) + A*(10-A).
The first term in the sum is a multiple of 100 and it does not interact with the last two digits of sum, which is never more than two digits long.


Difference of Squares

You have all learned that

a2 - b2 = (a + b)(a - b)

But perhaps you haven't thought about how to use this to do fast mental calculations! See if you can guess how this trick can help you do the following in your head:


43 x 37
78 x 82
36 x 24

Let's do the first one. 43 x 37 = (40 + 3)(40 - 3) = 402 - 32 = 1600 - 9 = 1591.

Practice these, and you'll be able to impress your friends!



Visual Multiplication with Lines



Here's a way to multiply numbers visually!



Suppose you want to multiply 22 by 13. Draw 2 lines slanted upward to the right, and then move downward to the right a short distance and draw another 2 lines upward to the right (see the magenta lines in Figure 1). Then draw 1 line slanted downward to the right, and then move upward to the right a short distance and draw another 3 lines slanted downward to the right (the cyan lines in Figure 1).

Now count up the number of intersection points in each corner of the figure. The number of intersection points at left (green-shaded region) will be the first digit of the answer. Sum the number of intersection points at the top and bottom of the square (in the blue-shaded region); this will be the middle digit of the answer. The number of intersection points at right (in the yellow-shaded region) will be the last digit of the answer.

This will work to multiply any two two-digit numbers, but if any of the green, blue, gold sums have 10 or more points in them, be sure to carry the tens digit to the left, just as you would if you were adding.

Presentation Suggestions:
First do simple examples like the one above; then try a problem that involves a carry, such as 21 x 34.

The Math Behind the Fact:
The method works because the number of lines are like placeholders (at powers of 10: 1, 10, 100, etc.), and the number of dots at each intersection is a product of the number of lines. You are then summing up all the products that are coefficients of the same power of 10. Thus the in the example


22 x 13 = ( 2*10 + 2 ) * ( 1*10 + 3 ) = 2*1*100 + 2*3*10 + 2*1*10 + 2*3 = 286.
The diagram displays this multiplication visually. In the green-shaded region there are 2*1=2 dots. In the blue-shaded region there are 2*3+2*1=8 dots. In the gold-shaded region there are 2*3=6 dots. This method does exactly what you would do if you wrote out the multiplication the long way and added the columns!

The method can be generalized to products of three-digit numbers (or more) using more sets of lines (and summing the dot groupings vertically and remembering to carry when needed). It can also be generalized to products of three-numbers using cubes of lines rather than squares! (Of course, it gets pretty unwieldy to use the method at that point.)

By the way, for the specific problem 22 x 13 there is actually another way to do it using lightning arithmetic; can you figure out how?




Making Magic Squares





A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Ever wonder how to construct a magic square?

A silly way to make one is to put the same number in every entry of the matrix. So, let's make the problem more interesting--- let's demand that we use the consecutive numbers.

I will show you a method that works when N is odd. As an example, consider a 3x3 magic square, as in Figure 1. Start with the middle entry of the top row. Place a 1 there. Now we'll move consecutively through the other squares and place the numbers 2, 3, 4, etc. It's easy: after placing a number, just remember to always move:

1. diagonally up and to the right when you can,
2. down if you cannot.
The only thing you must remember is to imagine the matrix has "wrap-around", i.e., if you move off one edge of the magic square, you re-enter on the other side.
Thus in Figure 1, from the 1 you move up/right (with wraparound) to the bottom right corner to place a 2. Then you move again (with wraparound) to the middle left to place the 3. Then you cannot move up/right from here, so move down to the bottom left, and place the 4. Continue...

It's that simple. Doing so will ensure that every square gets filled!

Presentation Suggestions:
Do 3x3 and 5x5 examples, and then let students make their own magic squares by using other sets of consecutive numbers. How does the magic number change with choice of starting number? How can you modify a magic square and still leave it magic?

The Math Behind the Fact:
See if you can figure out (prove) why this procedure works. Get intuition by looking at lots of examples!

If you are ready for more, you might enjoy this variant: take a 9x9 square. You already know how to fill this with numbers 1 through 81. But let me show you another way! View the 9x9 as a 3x3 set of 3x3 blocks! Now fill the middle block of the top row with 1 through 9 as if it were its own little 3x3 magic square... then move to the bottom right block according to the rule above and fill it with 10 through 27 like a little magic square, etc. See Figure 2. When finished you'll have a very interesting 9x9 magic square (and it won't be apparent that you used any rule)!