A, B, C's of Math

1Lameez Parthab

A, B, C’s of Math

Special thanks goes out to Mr. Debnath, my math teacher. Without him, this book wouldn’t have been created.

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TABLE OF CONTENTS

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LETTER A – ALGEBRAIC EQUATION

A mathematical equation is a phrase made up of various numbers and/or variables that are linked together by operations to equal other numbers and/or variables. An algebraic equation is a mathematical equation that must contain a variable. A variable is a symbol that is used to represent a number in an expression or equation.

Example of mathematical equation:

7 – 3 = 4 or 2 + 9 = 11Example of algebraic expression:

7x - 3 = 8 or 2 + 9y = 56As a result of variables being included in these expressions,

they would be called algebraic expressions whereas the first examples do not have variables and would not be considered algebraic expressions. Algebraic expressions can include more than one variable.

Example using word problem:

If Kate has 27 dollars, how many 3 dollar rulers can she buy?

The algebraic expression that should be used is:

3x = 27If we were to solve the equation, we would get 9 which is

now the amount of rulers Kate can buy.*See page 26 for more on variables

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LETTER B – BISECTOR

A bisector is a straight line that divides a geometrical figure into two equal parts. There are different types of bisectors such as an angle bisector and a line bisector.

Example of line bisector:

Example of angle bisector:

In both examples, the red line is the bisector. The two divided parts are shown as equals.


Lucy and Kyle both want to sit on a bench that is 3 meters wide. Find the bisector to determine how much room each person should get.

The bisector is at 1.5 meters so therefore you know Lucy and Kyle both get 1.5 meters of bench room.

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LETTER C – COLLINEAR

Collinear is when two or more points on a graph can be connected together with a linear line.

Example of collinear points:

P1, P2, and P3 are examples of collinear points because they all lie on the same linear line which would be line L.

This graph is collinear because the points can be connected by one linear line.


Nathan needs to go to the mall and the market. The mall, the market, and his house are collinear and the mall in between the market and his house. If Nathan walks in a collinear line from his house to the mall and then to the market, will the route he take be the shortest?

The answer is yes he will be taking the shortest route because we know that collinear lines are straight and therefore, the shortest.

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LETTER D – DEGREE OF A TERM

The degree of a term is the sum of all the variables exponents in that term.

Example of the degree of a term:

7x2y3 the degree of this term would be 5

x2 y3

2+3=5

If a variable doesn’t have an exponent, it is considered to have the exponent of 1.

Example of the degree of a term where variable has no exponent:

7xy3 the degree of this term would be 4

x1 y3

1+3=4Do not confuse the degree of a term to the degree of a

polynomial. The degree of a polynomial is just the degree of the highest term.

In the first example, the degree of a polynomial would be 3 because it is the highest exponent.

*See page 8 for more on exponents*See page 26 for more on variables

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LETTER E – EXPONENT

An exponent is shown as a smaller number at the top right of a larger number. It can also be shown as a variable. It shows how many times to multiply a number by its base.

Examples of exponents:

73 this can also be written as 7 x 7 x 7

The answer would be 343

36 this can also be written as 3 x 3 x 3 x 3 x 3 x 3

The answer would be 729.Exponents can also be represented as a variable. Though,

the amount of times to multiply a number by its base will be unknown.

Example of exponent represented as variable:

4x or 3y


Layla has a blank sheet of paper which she folds in half. After opening it, she sees she had formed two rectangles. Then she folds in in half a second time. Opening it again, she sees that she has formed four rectangles. Once again, she folds it a third time only to open it and see eight rectangles. Using an exponential equation, determine how many rectangles she will have after folding it six times.

The answer is that she will have sixty-four rectangles. The expression used would be 2x, x being the amount of times the paper is folded.

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LETTER F – FRACTION

A fraction is a ratio of numbers. They are put on top of one another and are separated by a line underneath. The top number is called the numerator while the bottom number is called the denominator. Fractions can be used to represent any number as long as the denominator is not 0.

Examples of fractions:

There are two types of fractions; improper and proper. A proper fraction is when the numerator is smaller than the denominator making the fraction as a whole lower than 1. An improper fraction is when the numerator is larger than the denominator making the fraction as a whole higher than 1.

Improper fractions: Proper Fractions:

=Example using word problem:

Bruce ate two slices of pizza out of nine slices. What fraction of pizza slices are left?

The answer is 7/9 because 9/9 - 2/9 = 7/9.

LETTER G – GREATEST COMMON FACTOR

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The greatest common factor would be the largest number or variable that two or more terms have in common. If none of the terms have anything in common, then the greatest common factor would be considered 1 or nothing.

Example of the greatest common factor:

8x2 + 16x3 – 64x4 the greatest common factor would be 4x2 because each term can be divided by it.

Example of when there is no common factor:

7x2 + 19y3 – 64z the greatest common factor would be 1 or it can be said that there is no common factor.

In the first example, 2x is also a common factor for the each term but it is not the largest number possible and so therefore it is not the greatest common factor.


Shyann wants to plant 54 tulips and 27 daffodils. She would like to plant them in rows that have the same amount of flowers and have only one type of plant. What is the greatest number of trees she can plant in each row?

The answer is 27 flowers because 27 is the greatest common factor between 54 and 27.

LETTER H – HYPOTENUSE

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A hypotenuse is the side that is opposite to the right angle of a right-angled triangle.

Example of hypotenuse:

The triangle must be a right-angled triangle. There are many ways to find the length of the hypotenuse such as measuring it with a ruler or using Pythagoreans Theorem.


A ladder is placed on the ground, leaning against a vertical wall that is 12 meters high. If the bottom of the ladder is 5 meters away from the wall, how long is the ladder?

The answer using Pythagoreans Theorem is 13 meters long.

(122 + 52 = 132)

*See page 20 for more on Pythagoreans Theorem

LETTER I – ISOSCELES TRIANGLE

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All the angles are less than 90o

An isosceles triangle is a triangle that has at least two equal sides. There are many types of isosceles triangles such as isosceles acute, obtuse, and right triangles.

Example of isosceles triangle:

The two sides that are equal have one marking line while the side that is different has two marking lines.

An isosceles acute triangle is an isosceles triangle that has all its angles less than 90o.

An example of an isosceles acute triangle is:

An isosceles obtuse triangle is an isosceles triangle that has one side greater than 90o.

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This one angle is greater than 90o

There is a 90o angle shown

An example of an isosceles obtuse triangle is:

An isosceles right triangle is an isosceles triangle that has one angle of 90o.

An example of an isosceles right triangle is:


A poster is in the shape of an isosceles triangle and has a base that of 2 inches shorter than either of the two equal sides. If the perimeter of the triangle is 28 inches, what is the length of the equal sides?

x + x + (x - 2) = Px + x + x - 2 = 283x = 28 + 2x = 10

Therefore, the equal sides are 10 inches long.

LETTER J – JUNCTION

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A junction is the intersecting point of two or more lines. When graphing two lines on a coordinated graph, if the two lines intersect, this would be known as a junction.

Example of junction:

Example of junction in a linear graph:


Julie and Sara want to race. They start at the same point but because Sara is faster than Julie, she gives Julie a head start. Julie goes at a speed of 5 miles/hour while Sara goes at a speed of 7 miles/hour. If Sara gives Julie a ten minute head-start, how long will it take for Sara to catch up to Julie? (The junction point)

Looking at a graph made from this data, we see that they meet at the junction point which is exactly 25 minutes.

*See page 16 for more on linear equations

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Junction point

Junction point

LETTER K – KITE

A kite is a quadrilateral that has two pairs of adjacent sides. This means that there are two pairs of two sides that are next to each other and that have the same lengths. A rhombus is not considered a kite.

Example of two types of kites:

The two pairs of adjacent sides are shown using marking lines. The equation to find the area is base x length.


Laura wants to build a kite but only has a 58 cm long and wide sheet of cloth. She wants to build the kite with a base of 7 cm and a length of 8 cm. Will Laura have enough material for it?

The answer is yes because 7 x 8 is less than 58 cm.

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LETTER L – LINEAR EQUATION

A linear equation is an equation in which the highest degree term is 1. A linear equations’ table would have the variables increasing or decreasing in a constant rate.

Examples of linear equations:

6 = xx + y = 0

A linear graph is represented as a normal coordinate graph where the line drawn out of all the plotted points is a straight line. This linear line can go in any direction as long as it is straight.

Example of linear graph:

or Example using word problem:

Riya has 19$ and wants to purchase some erasers and pencils are 3$ and erasers are 2$. Create a linear equation to represent how much of each she can buy.

The answer would be 19 = 3x + 2y, ‘x’ representing the cost of pencils and ‘y’ representing he cost of erasers.

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LETTER M – MONOMIAL

A monomial is a polynomial with only one term. A polynomial is a mathematical expression or phrase that contains one or more terms. The terms have exponents that are whole numbers and the coefficients are real numbers.

Examples of monomials:

8x or 17 or 42

Mono- can means one. In the word monomial, it means one term. Therefore, binomial means two terms as bi- means two. Then there is trinomial which means three terms and lastly, there is polynomial which is many terms but can also be represented as one term.

Example of binomials:

3x + 7y

Example of trinomials:

5y2 - 9 + 2yExample using word problem:

Taylor wants to buy some apples. The apples are 4$ each. How much will he spend if he wants to buy ‘n’ amount.

The answer is a monomial; 4n.

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LETTER N – NEGATIVE INTEGERS

A negative integer is any number lower than 0. On a number line, a negative integer can be found anywhere to the left of 0. Negative integers are normally shown as a number with a negative sign in front of it.

Examples of negative integers:

-3 or -15 or -23,375

Examples of negative integers on a number line:

As shown, every number on the left side of 0 is a negative integer. Though the numbers seem to be increasing, they are actually getting smaller.


Iroh owes his mom 7$, his brother 13$, and his friend 18$. He makes 30$ by selling lemonade at a lemonade stand. If he uses that money to pay back his debts, will he have made a profit or still be in depth.

7 + 13 + 18 = 3830 - 38 = -8

He is still in debt by 8 dollars. We know this because the answer provided was a negative integer.

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LETTER O – OPPOSITE ANGLES

An opposite angle is when two equal angles are formed after two lines intersect. No matter how two lines intersect, as long as there is one intersecting point, the opposite angles will always be equal.

Examples of opposite angles:

As shown, angles 1 and 3 are equal and angles 2 and 4 are equal.


Tom has a math problem. The problem is shown below. In the picture below, find x. Its opposite angle is shown as 85o and so x has to be 85o as well.

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LETTER P – PYTHAGOREAN THEOREM

Pythagorean Theorem is used to find the length of the hypotenuse. This is because the square of Side A plus the square of Side B is equal to the square of the hypotenuse. When the square of the hypotenuse is square rooted, the result is the hypotenuses length.

The equation of Pythagoreans Theorem would be:

a2 + b2 = c2

Example of Pythagoreans Theorem would be:

*See page 11 for more on hypotenuse

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a2 + b2 = c2

32 + 42 = c2

9 + 16 = c2

25 = c2

5 = c

Therefore, c = 5 which means the hypotenuse is 5 units.

LETTER Q – QUADRATIC EQUATION

The main quadratic equations:

ax2 + bx + c = 0‘a’ cannot be 0. A quadratic equation can be any equation

that has an exponent higher than 1 in it. Therefore, the highest degree term is above 1. A quadratic equations’ table would have the variables increasing or decreasing at an increasing or decreasing rate.

An example of a quadratic equation:

2x2 + 4x + 9 = 0A quadratic graph is represented as a normal coordinate

graph where the line drawn out of all the plotted points is a curved line. This line is known as a parabola.

Example of linear graph:

or

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LETTER R – RECIPROCALS

Reciprocals are two numbers that multiply together to get 1. For fractions, you can flip the denominator and numerator of the fraction to get its reciprocal.

Examples of reciprocals:

2 and 1/23/4 and 4/3

2 x 1/2 = 13/4 x 4/3 = 1

1/3 x 3= 1Examples of fraction reciprocals:

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LETTER S – SQUARE ROOTS

A square root is a number which when multiplied by itself produces a given number.

A square root’s symbol is as shown:

Examples of square roots:


Noreen has a square blanket of 1600 cm2. What is one of the side lengths?

The answer is 40 cm because the square root of 1600 is 40.

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(3 x 3 = 9)

(4 x 4 = 16)

(9 x 9 = 81)

LETTER T – TABLE

When given an equation or graph, you can create a table out of the data given. A table normally consists of two variables or more that are in relation to one another. Then quantitative data about the relationship is recorded. A table is basically a set of data that is organized into rows and columns.

Examples of tables:

X Y1 2

2 4

3 6

Distance (km) 50 60 75

Time (hour) 1.5 1.8 2.25

The data on table can be made up of constant data or random numbers that when plotted on a graph, don’t have any patter-like structure.

Example of table with no equation:

X Y5 2

7 6

10 8

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Variables

DataEquation: y=2x

X Y1 7

2 11

3 3

X Y2 16

4 2

6 -1

LETTER U – UNLIKE TERMS

Unlike terms are just terms that have different variables or the same variables with different exponents.

Example of unlike terms:

3x and 7y3x and 7xy3x2 and 7x3

In the first example, the two variables are ‘x’ and ‘y’ and so they are unlike terms. In the second example, even though the 7 has ‘x’, the full term sans the 7 is ‘xy’ and so it is not similar to the ‘x’ with a coefficient of 3. In the last example, the variables are both ‘x’ but because they have different exponents, they are considered unlike terms.


Joe has three dollars and seven quarters. He wants to buy as many 50 cent erasers as he can. Create an equation to express this relation.

The equation would be 3x + 0.25y = 50c. This equation would have unlike terms because all the variables are different and represent different things.

*See page 26 for more on variables

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LETTER V – VARIABLE

A variable is a letter or symbol that is representing a number or quantity and can vary between different numbers. There are numerous amounts of variables.

An example of variables in expressions:

6x - 7y62r x 12kz

Variables are also used in equations such as:

4n – 3m = 128h

As shown above, variables can be used as exponents or put together to show that once those two variables are given numbers, they can be multiplied together.


Janice has 400g of dog food. Her dog only eats ‘x’ grams of food each day. Create an equation to represent the relationship between the amount of food the dog eats and the number of days the food can last.

The answer is 400/x=y, and ‘x’ represents the amount of food while ‘y’ represents the number of days.

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LETTER W – WHOLE NUMBERS

Whole numbers are similar to negative integers but include 0 and positive numbers. Whole numbers are any numbers that aren’t fractions or decimals. They are also known as natural numbers or counting numbers because they are simple numbers people use almost every day.

Examples of whole numbers:

32624-12-260

Whole numbers are the basic numbers that people normally start of learning before continuing with more advanced math.


Harry has 3 tomatoes. If he eats 2, how many will he have left?

The answer to this basic math question using whole numbers is 1 because 3 - 2 = 1.

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LETTER X – X-AXIS

The x-axis is the horizontal number line found on a coordinate grid. It is normally used as a reference line so that you know where to plot your points.

Similar to the x-axis is the y-axis. The y-axis is the vertical number line you find on a coordinate grid. Both the x-axis and the y-axis intercept at the coordinates (0, 0)

Examples of the x-axis and y-axis:

or

These graphs are normally used to represent statistics or quantities. These graphs can hold numerous amounts of data as well as linear information and quadratic information.

*See page 16 for more on linear equations*See page 21 for more on quadratic equations

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Letter Y – Y-InterceptThe y-intercept of a graph can be found on the y-axis. It is

the point where the plotted line intercepts with the y-axis. If the line is linear, there is a possibility of zero, one, or infinite amounts of y-intercepts. If it isn’t linear than the y-intercept can possible be zero, one, two, many, or infinite.

Similar to the y-intercept is the x-intercept. The concept is similar to the y-intercept except that the x-intercept occurs on the x-axis.

Example of y-intercepts:

or

Example of x-intercept:

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Image to the leftThe yellow parabola had one x-intercept while the purple parabola didn’t have any x-intercepts. The green parabola had two x-intercepts.

Letter Z – Zero PrincipleThe zero principle is when you have two opposite numbers

that when added together, they equal 0. These two numbers will be the same number though one will be positive and one will be negative.

Examples of the zero principle:

6 + (-6) = 03 + (-3) = 0

27 + (-27) = 0Example using balls and number line:

If there were two red, negative balls and two blue, positive balls added together, they would cross each other out.

+

On the number line, we start at negative two and add two. The result is 0.


Zahra has twenty dollars. She owes her friend twenty dollars. If she pays her debt, how much money will she have left?

The answer is 0 because 20 - 20 = 0.

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