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A, B, C's of Math -- Algebra May Work Sample
Transcript of A, B, C's of Math -- Algebra May Work Sample
Definition: A number that indicates the steepness of a line
T is for Transformations
Definition: Operations that change the form or position of a figure
X is for X-Intercept
L is for Linear Systems of Equations
Definition: A group of two or more equations that have the same set of variables.
A, B, C's of Math
M is for Midpoint
Definition: The point at which a line intersects the x-axis
May Work Sample
B is for Binomial
Definition: A polynomial with two terms.
I am in the 8th grade.
My math teacher is Mrs. Troxell.
My favorite part of math class this year is going to the lessons in Class Connect because they’re so fun.
The most important math skill I learned this year is learning how to solve for and simplify variables, because it is one of the most crucial skills needed to learn about all the other concepts in algebra.
A is for Absolute Value
Definition: The distance a number is away from zero.
The absolute value symbol is represented by two bars enclosing a term or expression. The outcome is positive, whether the expression inside is positive or not.
| x | = x ; | -x | = x
The absolute value bars can function as grouping symbols, shown in the example below:
2 + | -4 + 3 |
2 + | -1 |
2 + 1
A polynomial is a sum of terms involving nonnegative integer powers of a variable. A binomial is a polynomial that has two terms.
3x + 2 is a binomial.
To multiply two binomials together, you can use a method called FOIL to help you, as shown in the example below:
FOIL stands for First, Outside, Inside, Last. This refers to the terms inside each of the binomials.
If we have the following binomial...
(a + b)(c + d)
A and C are the "first" terms, since they are both the first terms in each binomial.
A and D are the "outside" terms, since those two are on the outside of the group.
B and C are the "inside" terms. They are the innermost terms in the two binomials.
B and D are the "last" terms. They are both the second term in their respective binomial.
To multiply the binomials together, multiply the first terms together, then the outside terms, then the inside, then the last.
(a + b)(c + d)
ac (first terms); ad (outside terms); bc (inside terms); bd (last terms)
ac + ad + bc + bd
Related Vocab: Absolute Value Function, Positive Number, Negative Number, Number Line, Order of Operations
Related Vocab: Polynomial, Trinomial, Term, FOIL, Factoring
C is for Contrapositive
Definition: A conditional statement that both switches and negates the hypothesis and the conclusion of the original conditional statement
Related Vocab: Conditional Statement, Hypothesis, Conclusion, Inverse, Converse
There are many different forms of a conditional statement. There is the inverse, which negates both the hypothesis and the conclusion of the conditional statement. There is the converse, which switches the hypothesis and conclusion around. There is no guarantee that the inverse and converse is true. The contrapositive of a conditional statement, however, is always true as long as the original statement is true.
The contrapositive combines both the inverse and the converse; the contrapositive both negates and switches the hypothesis and conclusion.
Conditional Statement: If a shape has four sides, it is a quadrilateral.
Contrapositive: If a shape is not a quadrilateral, it does not have four sides.
D is for Discriminant
Definition: A number that can be calculated from any quadratic equation
Related Vocabulary: Quadratic Equation, Quadratic Formula
The discriminant of a quadratic equation can be calculated by using part of the quadratic formula:
Discriminant = b^2 - 4ac
It's a good idea to calculate the discriminant before solving a quadratic equation because the discriminant can tell you a number of things:
If the discriminant is negative, the equation has no real-number solutions.
If the discriminant is positive and is a perfect square, the equation has two rational roots.
If the discriminant is positive but is not a perfect square, the equation has two irrational roots.
If the discriminant equals zero, the equation has one rational solution.
It is also helpful in graphing quadratic equations.
If the discriminant is positive and is a perfect square, the equation has two rational x-intercepts.
If the discriminant is positive and is not a perfect square, the equation has two irrational x-intercepts.
If the discriminant is zero, the parabola only intercepts the x-axis at one point.
If the discriminant is negative, the parabola does not intercept the x-axis at all.
List the number and type of solutions for the equation.
4x^2 + 20x + 25
Discriminant = b^2 - 4ac
b^2 - 4ac
20^2 - 4(4)(25)
400 - 4(4)(25)
400 - 16(25)
400 - 400
The equation has one rational solution.
E is for Equation
Definition: A mathematical statement that uses the sign =. It shows that two expressions are equal.
Related Vocab: Variable, Inequality, Transformation, Inverse Operation
There are two parts to a power: the base and the exponent.
In this example, 2 is the base, which tells us what to multiply by. 3 is the exponent, which tells us how many times to multiply the base by.
If you have two terms that have the same variable base and multiply them together, you multiply the coefficients together, then add the exponents together.If two terms with the same variable bases are divided by one another, divide the coefficients by each other and subtract the exponents. If an exponent is raised to another exponent, the two are multiplied together.
When a base has a negative exponent, it is equal to the following:
A power with a negative exponent is equal to a fraction with one as the numerator. For the denominator, the negative exponent is turned into a positive one, while the base stays the same.
This is useful to know, especially when simplifying expressions, like the next problem shows:
Simplify the following expression:
F is for Function
Definition: A correspondence between two sets--the domain and the range--that assigns to each member of the domain exactly one member of the range
Related Vocabulary: Domain, Range, Relation
A relation is a set of ordered pairs. A function is a relation that assigns each member of the domain (set of x-coordinates) to one element of the range (set of y-coordinates).
Functions can be written in many ways, such as in tables, arrow diagrams, or a set of ordered pairs in a graph. However, they are most often written in equations. This can be done in either arrow notation or in function notation.
To express an equation as a function equation, replace the dependent variable y with the notation f(x). Sometimes, the variables that stand for the dependent variable are different (for example, g(x) is also used, but it means the same thing).
There are different kinds of functions, and their graphs depend on the equation.
A special type of function called a direct linear variation has the general equation:
y = kx
K is the constant of variation, which tells you how quickly the dependent (output) variable increases or decreases as the independent variable (input) changes. It does not equal zero. The graph for direct linear variations is a straight line.
Absolute value functions, functions that include absolute value bars, have graphs that are shaped like a V.
Another type of function called an inverse variation has the equation:
y = k/x
This type of equation is graphed with a hyperbola.
Quadratic variation is another type of relationship between x and y. This function can be written with the general equation:
y = kx^2
The graph of a quadratic variation is made with a parabola.
Example Problem: If y varies directly with x^2, and y = 16 when x = 2, find the specific equation.
General Equation: y = kx^2
To find k, we need to substitute in the given values of y and x.
y = kx^2
16 = k(2^2)
16 = k(4)
16/4 = k
4 = k
Now we can write the specific equation:
y = 4x^2
G is for GCF
Definition: Greatest Common Factor; the greatest number that can evenly divide into two or more numbers
Related Vocabulary: Factor, Factoring, Prime Factorization
There are many ways to find the GCF of two or more numbers; one way is to list all the factors of the numbers and then see which ones are common. Another way is to find the prime factorization of both of the numbers, and then see which of the prime numbers are common, and how many are common. Yet another way is to use divisibility rules to calculate the GCF.
The GCF is especially useful in factoring expressions, as shown in the problem below:
3x^3 + 27x^2 + 9x
The GCF of the coefficients is 3, so we can factor that out. The GCF of the variables is x. We end up factoring 3x out of the expression.
3x(x^2 + 9x + 3)
H is for Hypotenuse
Definition: The longest side of a right triangle, opposite to the right angle.
Related Vocabulary: Pythagorean Theorem, Right Triangle, Legs
The hypotenuse is the longest side of a right triangle, opposite to the right angle. The other two shorter sides are called the legs. If you have the lengths of two of the legs, you can solve for the length of the hypotenuse.
a^2 + b^2 = c^2 is the Pythagorean Theorem.
In the Pythagorean Theorem, the variables a and b stand for the side lengths of the two legs. It does not matter what order they are put in. The variable c stands for the length of the hypotenuse.
You can modify the Pythagorean Theorem to find the length of one of the legs, as well. As long as the triangle is a right triangle and two of the side lengths are already defined, you can solve for the last length with the Pythagorean Theorem.
Example Problem: Find the length of the hypotenuse of a right triangle if the side lengths equal 6 and 8.
a^2 + b^2 = c^2
6^2 + 8^2 = c^2
36 + 64 = c^2
100 = c^2
c = 10
I is for Inequality
Definition: A mathematical sentence built from expressions using one or more of the symbols <, >, ≤, or ≥
Related Vocabulary: Greater Than, Greater Than or Equal to, Less Than, Less Than or Equal to, Conjunction, Disjunction
You can graph an inequality on a number line or coordinate graph. For an inequality that has only one variable:
Graph on a number line.
Use a closed circle if the inequality has the greater than or equal to sign or the less than or equal to sign.
Use an open circle if the inequality has the greater than sign or the less than sign.
Shade the number line to the left of the dot if the variable is less than or less than or equal to the other side of the inequality.
Shade the number line to the right of the dot if the variable is greater than or greater than or equal to the other side of the inequality.
To graph an inequality with two variables:
Graph the inequality on the coordinate plane.
Use a dotted line if the inequality uses the greater than sign or the less than sign.
Use a solid line if the inequality uses the greater than or equal to sign or the less than or equal to sign.
Shade above the line if y is greater than or greater than or equal to the other side of the inequality.
Shade below the line if y is less than or less than or equal to the other side of the inequality.
Example Problem: Graph y ≤ 2x - 3
This inequality has two variables, so we need to graph it on a coordinate plane.
This inequality uses a greater than or equal to sign, so the line will be solid.
The x-intercept is -3, and the slope is 2.
Y is less than or equal to the other side of the inequality, so we should shade below the line.
Based on this information, we can create the following graph.
J is for Justification
Definition: Evidence which helps draw conclusions.
Related Vocabulary: Proof, Conclusion, Data, Postulate, Theorem, Conjecture
Proofs are needed to prove algebraic statements to be true. They are constructed using two columns. One lists statements that lead to the statement you want to prove, and the other column lists reasons that justify the statements in the first column. The reasons that justify the statements can be properties, definitions, or theorems.
As you can see in this example problem, the statements on the left are justified by the reasons on the right.
K is for Kilo-
Definition: A prefix meaning 1000
Related Vocabulary: Kilogram, Kilometer, Metric System
Kilo- is a prefix for many units of the metric system. It means 1000; a kilogram is equal to 1000 grams, and a kilometer is equal to 1000 meters. Knowing this is especially useful when converting between units.
Example Problem: Convert 25000 mg into kg.
25000/1000/1000 = 0.025
25000 mg = 0.025 kg
Related Vocab: Substitution Method, Linear Combination Method, Graph, Variable
A system of equations is made up of two or more equations that have the same set of variables. The solution to a system of equations is the point at which all the lines intersect. The solution can be found in many different ways.
Graphing the lines is one way to solve a system of equations. Once all the lines are graphed, you can find the point at which they intersect. This method is tedious and time-consuming, however.
Another way to find the solution is to combine both of the equations. Only do this if one of the variables can easily cancel with the other.
Another way to find the solution is to use the substitution method. With this method, you solve for one of the variables in one of the equations. This leaves you with one of the variables on one side of the equation, and an expression equal to the variable on the other side. Substitute this expression in place of the variable that it is equal to in the other equation. This leaves you with an equation with like terms that can be simplified. Repeat this process, finding the value of variables and then substituting them into other equations, to find the solution.
3a + 6b = 12; a + 9b = 18
3a + 6b = 12
3a = -6b + 12
a = -2b + 4
a + 9b = 18
-2b + 4 + 9b = 18
7b + 4 = 18
7b = 14
b = 2
a + 9b = 18
a + 9(2) = 18
a + 18 = 18
a = 0
a = 0; b = 2
a + b = 10; -a + b = 8
a + b - a + b = 10 + 8
b + b = 18
2b = 18
b = 9
a + b = 10
a + 9 = 10
a = 1
a = 1; b = 9
Definition: A point exactly halfway between two other points.
Related Vocabulary: Line Segment, Midpoint Formula
To calculate the midpoint between two points, use the following formula:
Example Problem: Find the midpoint between (6, 10) and (8, 2).
N is for Negative Exponent
Definition: An exponent that is a negative number.
Related Vocabulary: Power, Base, Exponent, Fraction
O is for Order of Operations
Many equations have one or more variables. We can solve for these variables. To do this, we need to use inverse operations to get the variable on one side of the equation.
Example Problem: Find the value of x in the equation 2(3x - 7) + 4 (3 x + 2) = 6 (5 x + 9 ) + 3
6x - 14 + 12x + 8 = 30x + 54 + 3
18x - 14 + 8 = 30x + 54 + 3
18x - 6 = 30x + 54 + 3
18x - 6 = 30x + 57
18x - 63 = 30x
-63 = 12x
-21/4 = x
Definition: A set of rules that clarifies in what sequence certain operations should be performed.
Related Vocab: Grouping Symbols, Parentheses, Brackets, Multiplication, Division, Addition, Subtraction.
To correctly simplify an expression, follow the order of operations. You can use the mnemonic PEMDAS to help you remember the sequence in which to perform operations; P for parentheses is first. Simplify expressions in the innermost grouping symbols first. Then, simplify any terms with exponents. Next, multiply and divide from left to right. Last, add and subtract from left to right.
Example problem: Simplify the expression 16 – 3(8 – 3)^2 ÷ 5
16 - 3(5)^2 ÷ 5
16 - 3(25) ÷ 5
16 - 75 ÷ 5
16 - 15
P is for Polynomial
Definition: A sum of terms involving nonnegative integer powers of a variable
Related Vocab: Monomial, Binomial, Trinomial, Factoring, Term, Degree
To factor polynomials, first factor out the greatest monomial factor, if possible. Then, look for a difference of squares.
To factor a difference of squares, check to see if the two terms are squares. Then, use the following rule to factor out two binomials from the difference of squares:
x^2 - y^2 = (x + y)(x - y)
Also check if there is a perfect square trinomial.
To see if a trinomial is a perfect square trinomial, check if the equation is in the form
ax^2 + bx + c
Next, check if both the first and last terms are perfect squares. Then, check if the absolute value of b is equal to twice the product of the square root of the first and last terms. If they are, then you can factor out two identical binomials, one of the terms inside being the square root of the first term, and the other term being the square root of the last term.
If a trinomial is not a square, look for a pair of two binomial factors. If a polynomial has four or more terms, look for a way to group the terms in pairs or in a group of three terms that is a perfect square trinomial. Make sure that any binomials and trinomials are prime.
Example problem: Factor completely.
32x^4 - 2y^8
2(16x^4 - y^8)
2(4x^2 + y^4)(4x^2 - y^4)
2(4x^2 + y^4)(2x + y^2)(2x - y^2)
Q is for Quadratic Equation
Definition: An equation that has a degree of 2.
Related Vocab: Quadratic Formula, Polynomial, Trinomial, Factoring
A quadratic equation is usually written in the form
ax^2 + bx + c = 0
Here, x is the variable, and a, b, and c are all constants.
There are a few ways to factor a quadratic equation:
If a equals one, find the factors of c that add to equal b. The resulting binomials will be in the form
(x __)(x __)
If a does not equal one, multiply a and c together. Then, find the factors of the resulting product that add to make b. Then, in place of the term bx, insert two new terms with the variable x that have the two terms that add to make b as coefficients. You can then factor until you have two binomials left.
The quadratic formula is a foolproof way to find the value of x. It is the following equation:
Example Problem: Solve for t in the equation -5t^2 + 14t + 3 = 0
-5t^2 + 14t + 3 = 0
The product of a and c here is -15, and the factors of -15 that add up to b, 14, are 15 and -1.
-5t^2 + 15t - t + 3 = 0
(-5t^2 + 15t) + (-t + 3) = 0
5t(-t + 3) + (-t + 3) = 0
(-t + 3)(5t + 1) = 0
-t + 3 = 0 5t + 1 = 0
-t = -3 5t = -1
t = 3 t = -1/5
R is for Radical
Definition: An expression that uses a root, shown with √
Related Vocab: Square Root, Cube Root, Radicand, Index
The radical symbol shows that the square root of the expression inside, called the radicand, should be calculated. Sometimes, there is a small number somewhat resembling an exponent hovering to the left of the radical symbol. This number, called the index, indicates the degree of the radical. For example, if the index is 3, the radicand would be calculated to the cube root. If there is no index, the radical symbol is assumed to be a square root symbol.
Radicals are somewhat like variables. They can be added and subtracted as long as the radicals are the same. When multiplying radicals together, the coefficients should be multiplied, and then the radicands should be multiplied. When dividing radicals, divide the coefficients and then divide the radicands.
To simplify a radical, take the prime factorization of the numbers inside the symbol. All the squares (or cubes if there is a cube root, and etc.) can be taken out of the radical and shown as a coefficient. Make sure to only show one of the numbers in the pair, and not both. For instance, if we were trying to simplify the square root of 8, we would find the prime factorization of it, which is 2^3. Since we can take one pair of 2s out, 2 becomes the coefficient, and whatever is left over stays inside the radical. The simplified version of √8 is 2√2.
If a number has an exponent that is a fraction, the number becomes the radicand in a radical. The denominator of the fraction turns into the radical's index. The radical is raised to a power, which is the numerator of the fraction (if it is not 1; anything raised to the power of 1 equals itself).
Example Problem: Simplify the radical √150
The prime factorization of 150 is 2 * 3 * 5^2
Since there are a pair of 5s, we can take that out of the radical. It becomes the coefficient, while the 2 and 3 stay in the radical.
The answer is 5√6.
Related Vocab: Linear Equation, Graph, Y-Intercept, Slope-Intercept Form
A linear equation can be written in the slope-intercept form
y = mx + b
In this equation, b is the y-intercept, and m is the slope of the line.
The slope can be calculated with two points. The slope is the rise/run ratio.
Example Problem: Find the slope of a line if two of its points are (10, 2) and (6, 1)
Slope = 4
Related Vocab: Translations, Dilations, Reflections, Rotations
There are four standard types of transformations: translations, dilations, reflections, and rotations.
A translation is a transformation in which a point or figure is moved to another location. Note that this does not affect the shape or size of the figure; the preimage and the result are congruent.
A dilation is a transformation that changes the size of a figure by a scale factor. If the scale factor is less than one, the figure gets smaller. If it is greater than one, the figure grows larger. The preimage and the result are similar.
A reflection is a transformation that reflects a point or figure across a line. The figures are still congruent, but they may be in different positions.
A rotation is a transformation in which a point or figure pivots around a center point. The center point may be the origin, or it can be another point. Rotations are measured in degrees; a quarter-turn is a 90 degree rotation, a half-turn is a 180 degree rotation, etc.
The points before the transformation is performed are marked with letters or variables. After the transformation is done, however, the corresponding point is marked with an apostrophe. For example, if point B was translated, the resulting point would be B', read as "B prime".
Which of the following transformations is illustrated by the graph at the right?
The answer is translation; the figure was translated three units up, and five units to the left.
U is for Undefined Slope
Definition: The slope of a vertical line.
Related Vocabulary: Slope, X-Coordinate
The slope of a vertical line is called an "undefined slope". The formula for slope is rise/run, or the vertical change over the horizontal change. Since there is no horizontal change in a vertical line, the denominator is zero, making the slope undefined.
Example Problem: What is the equation for this line?
The answer is x = 2. Since all the points in the line all share one x-coordinate, we can write this line's equation as x = 2.
V is for Variable
Definition: A letter or symbol representing an unknown quantity.
Related Vocab: Equation, Inequality, Function, Expression
Variables appear in equations, inequalities, functions, and many more. They stand for unknown values. Sometimes, that value is given so you are able to simplify the expression, but most other times, you need to solve for the variable.
To do this, use inverse operations to rewrite the equation, making the variable alone on one side.
Terms that have variables can be added, subtracted, multiplied, and divided. They can only be added or subtracted if the terms are like terms, meaning that their variables (both the base and the exponent) are the same. When this happens, the coefficients of the variables are added or subtracted, and the variable ending stays the same.
When terms are multiplied together, it doesn't matter whether the variables are the same or not; they are still able to be multiplied. The coefficients are multiplied together. If the terms have a like variable (it doesn't matter whether their exponents are different or not), you can add the exponents of the variable bases together.
When dividing terms, the coefficients are divided. If the terms have like variable bases, subtract their exponents.
W is for Whole Number
Definition: The set of numbers consisting of zero and all the positive integers
Related Vocab: Real Numbers, Rational Numbers, Irrational Numbers, Integers, Natural Numbers
Example Problem: List what categories the number 18 falls under.
18 is a real number. It is a rational number because it can be expressed as a fraction, 18/1. It is also an integer, a whole number, and a natural number.
Related Vocab: Line, Graph, Discriminant, Y-Intercept
You can find some hints about the x-intercept of a parabola by finding the discriminant of the quadratic equation.
If the discriminant is positive and is a perfect square, the parabola has two rational x-intercepts.
If the discriminant is positive but is not a perfect square, the parabola has two irrational x-intercepts.
If the discriminant is zero, the parabola has one square root.
If the discriminant is negative, the parabola does not intersect the x-axis.
By finding the root of the equation, you can find the x-intercepts of the parabola.
List the number and type of solutions for the equation.
y = x^2 + 4x + 7
Discriminant = b^2 - 4ac
b^2 - 4ac
4^2 - 4(1)(7)
16 - 4(1)(7)
16 - 4(7)
16 - 28
The parabola does not intersect the x-axis. It does not have a x-intercept.
Y is for Y-Intercept
Definition: The point at which a line intercepts the y-axis
Related Vocab: Slope-Intercept Form, Line, Graph, Slope, X-Intercept
When an equation for a line is written in slope-intercept form,
y = mx + b
m equals the slope, and b equals the y-intercept. This is all you need to graph the line; first, plot the y-intercept on the coordinate plane. Then, you can use the slope provided in the equation to find the next point.
Example Problem: Plot the line of the equation y = 2x + 3
Using the y-intercept, we can plot the point (0, 3) on the graph. Then, we use the slope to figure out the other points.
Z is for Zero Slope
Definition: The slope of a horizontal line
Related Vocab: Line, Graph, Slope
Example problem: Write the equation for this line:
The slope of a horizontal line is called "zero slope". The formula for slope is rise/run, or the vertical change over the horizontal change. Since there is no vertical change in a horizontal line, the numerator is zero, and anything with a numerator of zero equals zero.
The answer is y = 2. Since the y value of all the points of the line are the same, you can simply write this equation as y = 2.