Elwood Middle School Math
Resource Book
Rational Numbers
Natural Numbers (Counting Numbers): 1, 2, 3, …….infinity ()
Whole Numbers: 0, 1, 2, 3, ….
Integers: positive and negative whole numbers: …-3, -2, -1, 0, 1, 2, 3, … Integers are used in temperatures, the stock market, golf, and football. Note: 0 is not positive or negative.
Absolute Value: the distance a number is from zero on a number line. Symbol for Absolute Value: |x|
Ex. |5| = +5 |-7| = +7 |0| = 0
Opposite: two numbers that are the same distance from zero on a number line, but on different sides.
Ex. 2 and –2 -15 and 15 0 and 0
Comparing Integers: <, >, <, >
A positive is bigger than a negative.
A big positive is bigger than a small positive. A small negative is bigger than a big negative. Ex. 4 > -3
+4 > +3 -3 > -4
Integers on a Number Line: A number line starts with smaller numbers on the left and gets larger as it goes to the right.
Notes:
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Rational Numbers
How to Add 2 Integers:
1. If the signs are the same, add the absolute value of the numbers and keep the sign. 2. If the signs are different, subtract the absolute value of the numbers and keep the
sign of the number that has the larger absolute value.
Ex. Same Signs Different Signs
+ 5 + (+6) = +11 - 5 + (+6) = +1
-5 + (-6) = -11 +5 + (-6) = -1
Note: On a number line, if you are adding a positive number, you move to the right that many places. If you are adding a negative number, you move to the left that many places.
Ex. 0 + -2 means start at 0 and move to the left 2 places. You end up at -2.
Adding Fractions and Mixed Numbers:
Find a common denominator by finding the LCM of the denominators.
Make equivalent fractions using the denominators.
Add the numerators.
Keep the denominators.
Add the whole numbers
Simplify the fraction.
Ex. 7 ½ = 2/4 +4 ¾ = ¾
11 5/4 = 1 ¼ + 11 = 12 ¼
Notes:
Rational Numbers
How to Subtract 2 Integers:
1. Change the subtraction symbol to an addition symbol.
2. Change the sign on the number following the subtraction symbol (Note: if there is no sign, the number is positive).
3. Follow the rules for addition (see above).
Ex. +6 – (+10) +6 + (-10) = -4 -2 – (-8) -2 + (+8) = +6 -5 – 12 -5 + (-12) = -17
Note: On a number line, if you are subtracting a positive number, you move to the left that many places. If you are subtracting a negative number, you move to the right that many places.
Ex. 0 - -2 means start at 0 and move to the right 2 places. You end up at 2.
Formula for the distance between two rational numbers:
For two rational numbers p and q, the distance between p and q is |p-q|.
Subtracting Fractions and Mixed Numbers:
Find a common denominator by finding the LCM of the denominators.
Make equivalent fractions using the denominators.
If you need to borrow, borrow one from the whole number and put it in front of the fraction. Make that mixed number an improper fraction.
If there is no fraction, borrow one from the whole number and make that one into the denominator over the denominator.
Subtract the numerators.
Keep the denominator.
Subtract the whole numbers.
Simplify the fraction. 6
Ex. 7 ½ = 1 2/4 = 6/4 -4 ¾ = ¾ = ¾ 2 ¾
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Rational Numbers
How to Multiply and Divide 2 Integers:
1. If the signs of the 2 numbers are the same, multiply/divide the absolute values of the 2 numbers and your answer is positive.
2. If the signs of the 2 numbers are the different, multiply/divide the absolute values of the 2 numbers and your answer is negative.
Ex. Signs Same Signs Different + 5 + 3 = +15 +3 -5 = -15
-12
-2 = +6 -12
+2 = -6Powers
When raising a number to a power, write the problem out, then solve.
Hint: When raising a negative number to a power, if the power is even, your answer is positive and if the power is odd, your answer is negative.
Ex. (-3)2 = -3 • -3 = +9 (-3)3 = -3 • -3 • -3= -27
Note: If the negative number is not in parentheses, follow the order of operations. Raise the number to a power and then apply the negative.
Ex. -22 = -(2 • 2) = -4
To Multiply Mixed Numbers:
Change Mixed numbers to improper fractions.
Simplify vertically or diagonally.
Multiply across.
Simplify.
To Raise a Mixed Number to a Power:
Change mixed numbers to improper fractions.
Raise the numerator to the power.
Raise the denominator to the power.
Simplify.
Ex. 3 ½ • 1 ¼ = Ex. (3 ½)2 = 72 = 49 = 12 ¼ 7 • 5 = 35 = 4 ⅜ 22 4
2 4 8
Rational Numbers
To Divide Mixed Numbers:
Change mixed numbers to improper fractions.
Take the reciprocal (flip) of the second number.
Change division to multiplication.
Simplify vertically or diagonally.
Multiply across.
Simplify.
Ex. 3 ½ ÷ 1 ¼ = 7 ÷ 5 =
2 4
7 • 4 = 28 = 28 10 = 2
4 5 2 5 10
Order of Operations: PEMDAS
Parentheses or Absolute Value (work from the inside out)
Exponents
Multiplication OR Division: whichever comes first, left to right
Addition OR Subtraction: whichever comes first, left to right
Ex. 2 + (52 – 2 2) = 2 + (25 – 2 2) = 2 + (25 – 4) = 2 + 21 = 23
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Expressions
Variable: a symbol or letter used to represent a number.
Expression: a mathematical phrase made up of a variable or combinations of variables and/or numbers and operations.
Use the following words to translate expressions.
Words that mean:
Add Subtract Multiply Divide
Augment Decreased by Double (times 2) Divided by Enlarge Depreciate Multiply by Half (divide by 2) *Sum *Difference *Product *Quotient Exceeds Diminish Quadruple (times 4) Ratio
**Greater than Drop Triple (times 3) **Divided into Grow **Less than Twice (times 2) Divide
Increased by Lose Thrice (times 3) Third (divide by 3) **Larger than Loss Times Quarter (divide by 4) **More than Lower Times by **Into
Plus Minus Of Rise Take away
**Longer than Shorter
Total **Smaller than Gain **From
**Take away from
Powers
Square (to the second power) Cube (to the third power) Square root (√ )
Notes:
* Used for the words sum, difference, product, quotient. Means to put the expression in parentheses.
Ex. The sum of 5 and a number x is written (5 + x).
** Used with the words from, into, to, and than. Means that the order of the parts of the expression are switched.
Ex. 4 more than a number x is written x + 4.
Expressions
How to Evaluate an Expression:
1. Substitute known values for each variable. 2. Evaluate using order of operations.
Ex. a = -2, b = 3, c = -1
4a + 6b – 2 = c + 2
4 -2 + 6 3 – 2 = -1 + 2
-8 + 18 – 2 = -1 + 2
10 – 2 = -1 + 2
8 = 8 1
Notes: ab means a • b
b a
means a ÷ b
Function: a relation in which each input has exactly one output according to a specified rule.
Ex. If # means to add 6, then 5# means 5 + 6 = 11 where 5 is the input and 11 is the output.
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Expressions
Monomial: a real number, a variable, or a product of real numbers and/or one or more variables. Ex. 5, 5x, 5xy, 5x2, 5x2y
Polynomial: a monomial or the sum or difference of monomials. Ex. 5x2 + 5x2y + 6x + 6
Binomial: A polynomial made up of 2 monomials. Ex. 5x2 + 5x2y
Trinomial: A polynomial made up of 3 monomials. Ex. 5x2 + 5x2y + 6
How do you evaluate polynomials?
1. Substitute the value for the variables into the polynomial. 2. Evaluate using the order of operations.
Ex. Evaluate 5x2 + 5x2y + 6x + 6 if x = 2 and y = 3
5x2 + 5x2y + 6x + 6=
5 22 + 5 22 3 + 6 2 + 6= 5 4 + 5 4 3 + 6 2 + 6= 20 + 20 3 + 12 + 6 = 20 + 60 + 12 + 6 = 80 + 12 + 6 = 92 + 6 = 98
Expanded Form: An expression that is written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers is said to be in expanded form. A single number, variable, or a single product of numbers and/or variables is also considered to be in expanded form, such as 324, 3x, 3x + 6, etc.
Standard Form: An expression that is in expanded form where all like-terms have been collected is said to be in standard form.
IMPORTANT: An expression in standard form is the equivalent of what is traditionally referred to as a “simplified” expression. This curriculum does not utilize the term
“simplify” when writing equivalent expressions, but rather asks you to “put an expression in standard form” or “expand the expression and combine like terms.” However, you must know that the term “simplify” will be seen outside of this curriculum and that the term is directing you to write an expression in standard form.
Expressions
Like Terms: Like terms have the same variables raised to the same power. For example, 5x and 6x are like terms but 5x and 5y are not because the variable is
different. Likewise, 5x and 5x2 are not like terms because the variable x is to a different power in each monomial.
How do you add or subtract monomials? 1. Add or subtract the constant.
2. Keep the variable and the powers the same.
Note: Monomials must be like terms in order to add or subtract them. If there is no coefficient, put a 1 in front of the variable. x becomes 1x.
Ex. 5x2y + 6x2y = 11x2y
How do you add or subtract polynomials? (Also called Combining Like Terms) 1. Distribute the number outside the parentheses to all numbers inside the
parentheses, including negatives. Remember, a negative sign really means to multiply by –1.
2. Line up the polynomials by like terms and do the addition or subtraction. 3. When you are done, there should not be any like terms left.
Remember, x means 1x1
Ex. (3x2 + 4x2y – 6) + (5x2 - 3x2y + 8) = 3x2 + 4x2y – 6
+ 5x2 - 3x2y + 8 8x2 + 1x2y + 2
Ex. (3x2 + 4x2y – 6) - (5x2 - 3x2y + 8) = 3x2 + 4x2y – 6
-5x2 + 3x2y - 8 Note that the negative sign was distributed to all -2x2 + 7x2y -14 items inside the parentheses by multiplying by –1.
Ex. 3x2 + 4x2y – 6 - 5x2 - 3x2y + 8 = x2: 3x2 - 5x2 = -2x2
x2y: + 4x2y - 3x2y = 1x2y #: -6 + 8 = 2
-2x2 +1x2y + 2
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Expressions
How do you divide monomials? 1. Divide the constants
2. Subtract the exponents of the same variable in each monomial. Remember, anything to the zero power is 1.
6x5y2 ÷ 3x2y = 2x3y
How do you divide polynomials by monomials? 1. Find the greatest common factor of all terms. 2. Divide the constants.
3. Subtract the exponents of the same variable in each monomial. Remember, anything to the zero power is 1.
5x2 + 6x3 = 5x2 + 6x3 = 5x2-1 + 6x3-1 = 5x + 6x2 x x x
(The GCF is x.)
43a7b9c9 + 45acd9 = 43a7b9c9 + 45acd9 = 43b9c8 + 3d8 15a8cdk 15a8cdk 15a8cdk 15adk a7k
Factoring: to write a number or an expression as a multiplication problem.
To factor a polynomial whose terms have a common monomial factor:
1. Find the greatest monomial that is a factor of each term of the polynomial. (GCF of the coefficients, smallest power for each variable.)
2. Divide the polynomial by the monomial factor. The quotient is the other factor. 3. Express the polynomial as the indicated product of the two factors.
Ex. 6c3d – 12 c2d2 + 3cd
(1) 3cd is the greatest common factor of 6c3d, 12 c2d2,and 3cd (2) To find the other factor, divide 6c3d – 12 c2d2 + 3cd by 3cd.
Answer 6c3d – 12 c2d2 + 3cd = 3cd( 2c2 – 4cd + 1)
Ex. 8ab + 4ad – 2ac + 6a = 2a (4b + 2d – c + 3)
7a + 7b = 7(a + b)
3x2 + 5x = x(3x + 5)
Hint: if you are unsure of your final answer, distribute as a multiplication problem to check that you get the original polynomial.
Equations
Equation: a mathematical sentence with an equal sign.
An equation is true if the answer is correct. Ex. 1 + 2 = 3 An equation is false if the answer is incorrect. Ex. 1 + 2 = 8
An equation is open if you can't tell if the answer is correct or incorrect because of a variable. Ex. x + 10 = 19
Inverse: the inverse of an operation is used to "undo" an operation. The inverse of addition is subtraction.
The inverse of subtraction is addition. The inverse of multiplication is division. The inverse of division is multiplication.
The inverse of squaring a number is taking the square root of a number. The inverse of taking the square root of a number is squaring a number.
Solution: the solution of an equation is the number that replaces a variable to make an equation true.
Ex. In the equation x + 10 = 19, x = 9 is a solution because when x = 9, 9 + 10 = 19 is a true equation.
Solving One-Step Equations: There are two sides to one step-equations: the side with the variable and the side without the variable.
1. Do the inverse operation to the number that is on the side of the equation with the variable. You should have a solution to your equation where the variable equals a number.
2. Check your equation.
a. Rewrite the original equation.
b. Substitute in your solution for the variable. Put a tiny question mark over your equal sign.
c. Solve your equation. Both sides should be equal. If so, put a tiny checkmark over the equal sign. If not, go back and retry solving the equation again.
Ex. x + 9 = 10 x – 8 = 20 15 = 3x 7 = 97 - 9 - 9 + 8 + 8 3 3
x = 1 x = 28 5 = x x = 63
check: check: check: check:
x + 9 = 10 x – 8 = 20 15 = 3x = 9
1 + 9 = 10 28 – 8 = 20 15 = 35 = 9 10 = 10 20 = 20 15 = 15
Page 13 Equations
Solving Two-step Equations: In a two-step equation, there are two operations done to the variable. In order to get the variable alone, we must undo both of these operations. 1. Do the inverse of the addition or subtraction to both sides of the equation.
2. Do the inverse of the multiplication or division to both sides of the equation. 3. Check your equation.
a. Recopy the original equation.
b. Substitute in your solution for the variable. Make sure you have a question mark over the equal sign.
c. Solve your equation using the order of operations. This will take two steps. The first should have a question mark over the equal sign. The second should have a check over the equal sign.
Ex. Ex.
Notes:
___________________________________________________________ 2x + 6 = 12
- 6 - 6 2x = 6 2 2 x = 3
Check: 2x + 6 = 12
2(3) + 6 = 12
6 + 6 = 12
12 = 12
+6 +6 7 x = 17 • 7 7
x = 119
Check:
17 – 6 = 11
11 = 11
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Equations
Graphing on a Number Line
A number line can be used to show the solution to equations and inequalities. It is very useful when solving an inequality because in an inequality there is not just a single numerical answer.
Draw a number line with the number and at least 2 numbers in either direction. Then draw an open or closed circle and shade in the appropriate direction(s). Use a solid dot for =, , and and an open dot for <, >, and .
Examples of solutions graphed on a number line:
x = 3
x > 2
x -2
x < 4
x 0
-1 0 1 2 3 4 5 6
A solid dot on
3.
-1 0 1 2 3 4 5 6
An open dot on
2, shaded to the
right (greater
than).
-5 -4 -3 -2 -1 0 1 2
A solid dot on -2,
shaded to the right
(greater than).
-1 0 1 2 3 4 5 6
An open dot on
4, shaded to the
left (less than).
-5 -4 -3 -2 -1 0 1 2
Page 15 Equations
Solving and Graphing Inequalities:
1. To solve an inequality you must use all of the same procedures and rules used for solving equations.
2. If you are multiplying or dividing by a negative number, you must change the direction of the inequality sign.
3. You must then graph the solution on a number line. 4. Check a number that is shaded, but not on the circle.
Ex. Check 2x + 17 > 21 2x + 17 > 21 -17 -17 2(5) + 17 > 21 2x > 4 10 + 17 > 21
2 2 27 > 21
x > 2
Notes:
-1 0 1 2 3 4 5 6
√ ?
Equations
In addition to using inverse operations to solve basic equations, there are other skills and procedures needed to solve more complex equations. Three types of more difficult equations are equations with parentheses, equations with more than one variable term, and equations with the variable on both sides of the equation.
Equations with More Than One Variable Term (Combining Like Terms)
1. Before using inverse operations, combine all of the like terms into one term. To do this for variables, add the coefficients in front of the variable and keep the variable and power the same.
2. Follow the rules to solve the resulting two-step equation. 3. Check following the rules to check an algebraic equation.
Ex. x + 4 + 3x + 1 – 6x = 17 -2x + 5 = 17 - 5 - 5 -2x = 12
-2 -2 x = -6
Check:
x + 4 + 3x + 1 – 6x = 17 -6 + 4 + 3 • -6 + 1 – 6•-6 = 17 -6 + 4 + -18 + 1 - -36 = 17 -2 + -18 + 1 - -36 = 17 -20 + 1 - -36 = 17 -19 - -36 = 17 17 = 17
Notes:
Combine x + 3x – 6x = -2x Combine 4 + 1 = 5
2-step equation – solve with inverse operations
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Page 17 Equations
Equations with Parentheses (Equations using the Distributive Property)
1. Before using inverse operations (remember in PEMDAS, parentheses go first) the number outside of the parentheses should be multiplied by each term inside of the parentheses (the distributive property).
2. Combine like terms.
3. Follow the rules to solve the resulting two-step equation. 4. Check following the rules to check an algebraic equation.
Ex. 4(x – 3) + 5 = 21 4x – 12 + 5 = 21 4x – 7 = 21 + 7 +7
4x = 28 4 4 x = 7
Check:
4(x – 3) + 5 = 21 4(7 – 3) + 5 = 21 4 • 4 + 5 = 21 16 + 5 = 21
21 = 21
Notes:
4 x = 4x and 4 -3 = -12 (use distributive property) Combine like terms (-12 + 5 = -7) 2-step equation – solve with inverse operations
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Equations
Equations with the Variable on Both Sides of the Equal Sign (Equation)
1. As with numerical terms, inverse operations can be used on variable terms. Subtract one variable term from both sides, so that it is eliminated from one side and appears only on the other side.
2. Follow the rules to solve the resulting two-step equation. 3. Check following the rules to check an algebraic equation.
Ex. 2x + 15 = 6x – 13 -2x -2x 15 = 4x – 13 +13 +13
28 = 4x 4 4 7 = x
Check:
2x + 15 = 6x – 13 2 • 7 + 15 = 6 • 7 – 13 14 + 15 = 42 – 13
29 = 29
Notes:
Subtracting 2x from both sides results in a 2- step equation.
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Page 19
Applications of Equations
When using an equation to solve a word problem, it is important (and helpful!) to set up your variables before writing your equation. A variable or variable
expression is usually assigned to the value(s) that are asked for. Different types of word problems often have their own type of setup.
Consecutive Integer Problems
Use the following setup for a consecutive integer problem. Let x = the first integer
x + 1 = the second integer x + 2 = the third integer x + 3 = the fourth integer
etc. Use this pattern for as many integers as the problem asks for.
Use the following setup for a consecutive even or consecutive odd integer problem Let x = the first even/odd integer
x + 2 = the second even/odd integer x + 4 = the third even/odd integer x + 6 = the fourth even/odd integer
etc. Use this pattern for as many integers as the problem asks for.
Remember that consecutive even integer problems and consecutive odd integer problems are set up the same way. If the first integer (x) is odd then by adding 2 you will get to the next odd integer.
Ex. Find 3 consecutive odd integers whose sum is –27. Let x = the first odd integer
x + 2 = the second odd integer x + 4 = the third odd integer x + x + 2 + x+ 4 = -27
3x + 6 = -27 -6 -6 3x = -33
3 3
x = -11 therefore x + 2 = -9, x + 4 = -7
Applications of Equations
Ex. Find four consecutive integers where three times the smallest is nine more than twice the largest.
Let x = the first integer
x + 1 = the second integer x + 2 = the third integer x + 3 = the fourth integer 3x = 2(x + 3) + 9
3x = 2x + 6 + 9 3x = 2x + 15 -2x -2x
x = 15 therefore x + 1 = 16, x + 2 = 17, and x + 3 = 18
The four consecutive integers are 15,16, 17, and 18.
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Applications of Equations Perimeter Problems
In perimeter problems the setup should cover each side of the polygon. Sometimes all of the sides are the same (like a square) so only one variable expression will be
needed, and sometimes you will need to set up 2 or more variable expressions for different sides.
A picture may be useful to help you visualize the problem.
The equation will require that you add up all of the sides and set it equal to a given amount for the perimeter.
Ex. The three sides of an equilateral triangle are represented by x + 2. If the perimeter of the triangle is 30 feet, find x.
Let x + 2 = each side of the triangle
In a rectangle the length is 3 more than twice the width. The perimeter is 54 cm. Find each dimension.
Let x = the width 2x + 3 = the length
Notes:
x + 2
x + 2 x + 2
(x + 2) + (x + 2) + (x + 2) = 30 3x + 6 = 30
-6 -6 3x = 24
3 3 x = 8 feet
2x + 3
2x + 3
x x
(2x + 3) + x + (2x + 3) + x = 54 6x + 6 = 54
-6 -6 6x = 48
6 6
x = 8 and 2(8) + 3 = 19
Applications of Equations Money Problems
Most money problems involve two items, their quantities, and their costs.
It is helpful to use two variables (one for each item) even though the problem will give you the value of one of them.
The equation will involve multiplying the cost by the quantity of each item, adding them together and making it equal to the total.
Ex. Jon bought 4 footballs and 3 hats for $77. If the hats cost $15 each, how much did the footballs cost?
Let f = the cost of a football h = the cost of a hat 4f + 3h = 77 (remember h = 15) 4f + 3(15) = 77
4f + 45 = 77 - 45 -45 4f = 32
4 4
f = 8 Each football costs $8.
Maria bought some notebooks for $1.50 each and some markers for $2 each. If she bought 8 notebooks and spent a total of $26, how many markers did she buy?
Let n = the number of notebooks bought (n = 8) m = the number of markers bought
1.50n + 2m = 26 1.5(8) + 2m = 26 12 + 2m = 26 -12 -12
2m = 14 2 2
m = 7 She bought 7 markers.
Page 23
Applications of Equations
General Problems and Inequalities
To write equations from word problems:
1. Define the variable-choose a variable to represent one of the unknowns. Often, this is the first letter of the word you are looking for. This is called a let statement since you “Let x = an unknown”
2. Use the variable to write an equation. 3. Solve the equation.
4. Check.
5. Write a complete sentence.
Ex. Six less than the number of cookies is 18. Let c = number of cookies
c – 6 = 18 + 6 + 6 c = 24
Check. c – 6 = 18 24 – 6 = 18 18 = 18
The number of cookies is 24.
To solve one-step inequality word problems:
1. Let statement. 2. Write inequality. 3. Solve.
4. Graph.
5. Check solution that answers the question. 6. Write a sentence.
Notes:
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Ratios andProportions
Ratio: a comparison between two numbers. Ratios can be written using the word “to”, a colon (:), or as a fraction. It should be simplified. Order is important.
Ex. You have 10 boys and 5 girls in a class. A ratio of boys to girls could be written as 2:1, 2 to 1, or . A ratio of girls to boys could be written as 1:2, 1 to 2, or ½.
Proportion: a proportion is two equal ratios. To see if two ratios make a proportion, cross-multiply and see if the results are equal.
Ex. 100 100 Since the cross-products are equal, these two ratios are proportions.
To solve proportions algebraically: 1. Cross-multiply.
2. Divide both sides by the number with the variable. 3. Check.
a. Re-copy the original proportion.
b. Substitute the solution for the variable. Put a question mark over the equal sign. c. Cross-multiply. The cross-products should be equal. Put a check over the equal
sign.
Ex. = Check: =
40 = 100x = 100 100
.4 = x 40 = 40
Notes:
2 1
100 4
25 1
100 4
10
x 100 4
10 x 100 4
10 .4 ?
Page 25
Proportions
Rate: a comparison of two quantities with different units.
Ex. You drive 50 miles in 2 hours. A rate would be 25 miles per hour.
Unit Rate: a unit rate compares a quantity to one. Ex. You read 2 pages per one minute.
To find a unit rate, you can either simplify or you can set up a proportion comparing the rate to 1.
Ex. You can read 60 pages in 30 minutes. Therefore you read 2 pages per one minute.
= =
60 = 30x = 30 30 2 = x 60 = 60
Unit Price: the price for one unit of an item. This is used to find the better buy. To find unit price, you can simplify or you can set up a proportion where one of the units is 1. The better buy costs less for one.
Ex. 12 ounces cost $6.00. What is the unit price?
= =
6 = 12x = 12 12 .50 = x 6 = 6 Therefore, one ounce is $0.50.
Constant of Proportionality: When a proportion is represented by the equation x = ky, k is the constant of proportionality.
Ex. If a pound of bananas costs $0.39, the equation c = 0.39b can be used to relate the cost to the pounds of bananas with 0.39 being the constant of proportionality.
Proportionality:
To check proportionality in a table, divide y by x in each row. Each quotient should be the same number, k, the constant of proportionality. Double check that this will work for 0,0).
To check proportionality in an equation, the equation should be y = kx.
To check proportionality in a graph, the graph should go through the point of origin and it should be linear.
Notes: 60 30 x 1 √ 60 30 x 1 60 30
? 2 1 12 6 √ 12 6 1 x 12 6
? 1 .5 1
Proportions
Scale Drawings: a drawing is called a scale drawing if the drawing is in proportion to the actual object.
To find the scale: create a ratio and simplify. Ex. 5 in.: 15 ft. The scale is 1 in.: 3 ft.
To find a measurement:
1. Set up a proportion with one side being the scale and the other side being the drawing.
Real life drawing = Scale drawing Real life actual figure Scale actual figure 2. Solve and check algebraically.
3. Write a sentence.
Ex. On a map, the scale is 1 cm. : 5 km. You measure the distance you need to drive as 4 cm. on the map. How far must you drive?
4 = 1 Check: 4 = 1
x 5 x 5
1x = 20 4 = 1
1 1 20 5
x = 20 20 = 20
You must drive 20 km.
Notes:
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Page 27 Percents
Converting Between Fractions and Decimals
To change a fraction to a decimal use either of the following methods: 1) Divide the numerator by the denominator
2) Change the fraction to an equivalent fraction with one of the following denominators: 10, 100, 1000, 10,000, etc. Use the new numerator in the appropriate place value after the decimal point.
Ex. ¾ = 3 4 = 0.75 or ¾ = 75/100 = 0.75
To change a decimal to a fraction:
1) Write the decimal as a fraction over the correct place value of the last digit. 2) Simplify if possible.
Ex. 0.05 = 5/100 = 1/20 0.0025 = 25/10,000 = 1/400
Repeating decimals to fractions
1. If 1 digit repeats, let 10n = the repeating decimal, but with the decimal moved over 1 place to the right. 2. If 2 digits repeat, let 100n = the repeating decimal, but with the decimal moved over 2 places to the right. 3. Let 1 n = the repeating decimal.
4. Subtract your let statements. 5. Solve for n.
_ __
Example: .3 _ .63 __ Let 10n = 3.3 Let 100n = 63.63 -Let 1n = .3 -Let 1n = .63 9n = 3 99n = 63 9 9 99 99
n = 3/9 = 1/3 n = 63/99 = 7/11
Percents
Forms of percents:
1. To change a decimal into a percent, move the decimal point two spaces to the right, and put the percent sign on. "Right on."
Ex. .65 = 65% .8 = 80% 5 = 500%
2. To change a percent to a decimal, move the decimal point two spaces to the left and take the percent sign off. "Left off." Note: If the percent has a fraction in it, change the fraction to a decimal before you begin.
Ex. 5% = .05 10% = .10 .6% = .006
7½% = 7.5% = .075
3. To change a fraction to a percent, make the fraction into a decimal by dividing the denominator into the numerator, then move the decimal two places to the right and put the percent sign on. (Make sure when you divide, you go out two decimal places and put the remainder over the denominator.)
Ex. ¾ 3 ÷ 4 = .75 = 75%
7 ½ 1 ÷ 2 = .5 , so 7 ½ = 7.50 = 750%
1/3 1 ÷ 3 = .331/3 = 331/3 %
4. To change a percent to a fraction, make the percent into an improper fraction if necessary, (if it is a decimal, change it to a fraction) and multiply by 1/100.
Ex. 7% = 7 1 = 7 1 100 100
7.5% = 7 ½ % 7 ½ = 15 1 = 15 = 3 2 100 200 40
Page 29 Percents
Solving percent equations:
1. You must translate the word problems into a proportion, using the following: Is = %
Of 100
2. Solve the proportion algebraically.
Ex. What is 50% of 6? x = 50
6 100 100x = 300
100 100 x = 3
10% of what is 6? 6 = 10
x 100 10x = 600
10 10 x = 60
What percent of 40 is 80? 80 = x
40 100 40x = 8000
40 40 Note, you must now add a percent sign. x = 200%
To solve % problems by proportion 1. Create a proportion is = % of 100
2. Fill in the numbers you know. The unknown is x. 3. Solve.
Percents
Percent of change: The percent of change is the percentage something has increased or decreased by. It is also sometimes referred to as rate, such as markup rate or discount rate.
To find the percent of change:
original number – new number (highest # - lowest #) = decimal original number
Change your decimal answer to a percent.
Ex. You had an 80 on your first math test this year and a 100 on your second test. Find the percent of increase.
100 – 80 = 20 =.25 = 25% 80 80
An item was marked for 100 dollars and it is now on sale for 80 dollars. Find the discount rate.
100 – 80 = 20 = .2 = 20% 100 100
Note how changing the original number changed the answer to the problem.
Page 31 Percents
Interest rate: the percentage paid on money when it is in use by someone else, for example as a loan or at a bank.
Interest: the amount of money paid when it is in use by someone else.
To find simple interest: Interest = principalratetime I =prt
p = principal, the amount of money r = interest rate, change this to a decimal t = time in years
To find the total:
Total = interest + principal
Ex. You take out a $1000 loan for a car. You are paying 8% interest for 2 years. What is interest on your loan and what is the total amount you are paying back at the end of the two years?
I = prt
I = 1000.082
I = $160 Your interest is $160.
Total = P + I
Total = $1000 + $160
Total = $1160 You will pay back a total of $1160.
To find interest when the time is given in months:
The formula is similar, but you must divide your answer by 12 because there are 12 months in a year.
I = prt 12
Ex. You take out a $1000 car loan at an 8% interest rate, but will pay it back in 6 months. How much interest will you pay?
I = prt 12
I = $1000.086 12
I = $40
Percents
Wholesale price: the price a store buys the item for.
Markup rate: the percentage a store uses to markup the price of an item from the wholesale price to the price they sell it for.
Markup: the amount a store marks up an item from wholesale price to the price they sell it for.
Markup = markup rate (as a decimal) wholesale price.
Retail price: the price a store sells an item for. Retail price = wholesale price + markup
Discount rate: the percentage a store uses to discount an item from the retail price to the sale price.
Discount: the amount of money a store takes off an item to sell it at sale price. Discount = retail price discount rate (as a decimal)
Sale price: the price an item is sold for after the discount is taken.
Sale price = retail price – discount
Tax rate: the percentage added to the price of an item for the state or federal government.
Commission: the amount earned by a sales person as a percentage of the sales price.
Tax: the amount of money added to the price of an item that is paid to the state or federal government.
Tax = tax rate (as a decimal) price of item
Total price: price of an item after markup, discount, and tax. Total price = tax + price
Ex. A $50 item is marked up 200%. What is the retail price of the item? $50 2 = $100
$50 + $100 = $150
A $50 is discounted 10%. What is the sale price of the item?
Page 33
Gratuity (Tip): The percentage of the cost of a service paid to the server. Gratuity = Percentage as a decimal • cost
Total = cost + gratuity
Gratuity is usually 15%, or 20% for very good service or on a large party.
Percents
Percent Error
Absolute Error: Given the exact value, x, of a quantity and an approximate value, a, of the quantity, the absolute error is |a-x|.
% error = (|Your Result - Accepted Value|) x 100
Accepted Value
A student made a mistake when measuring the volume of a big container. He found the volume to be 65 liters.
However, the real value for the volume is 50 liters. What is the percent error?
Percent error = |65 – 50| = 15 = .3 = 30% 50 50
Notes:
Page 35 Percents
Estimating and Mental Math with Percents
Percents can easily be estimated by using multiples of 10% and 1%, which are both easily determined by sliding the decimal point.
When taking 10% of any number, simply slide the decimal point one place to the left.
10% of 350 is 35 10% of 27.9 is 2.79 10% of 4500 is 450
When taking 1% of any number, simply slide the decimal point two places to the left.
1% of 350 is 3.5 1% of 27.9 is 0.279 1% of 4500 is 45
To estimate values other than 1% or 10% create multiples of 1 and 10 and add them together.
20% of $550 is the same as 10% of $550 + 10% of $550 = $55 + $55 = $110
3% of 2200 is the same as 1% of 2200 + 1% of 2200 + 1% of 2200 = 22 + 22+ 22 = 66
Estimation can also be done by comparing to the “benchmark” percents of 25%, 50% and 75%.
48% of 84 is close to 50% of 84 which is 42
26% of 60 is close to 25% of 60 which is 15.
Two-Dimensional Geometry
Polygon: A closed plane figure such that no two segments with common endpoints are collinear and segments intersect only at endpoints.
Congruent: Equal measure. The symbol for congruent is .
Regular Polygon: A regular polygon has all equal sides and all angles are congruent.
Examples:
Names of Polygons:
Name Number of Sides Sum of Interior Angles
Triangle 3 180°
Quadrilateral 4 360°
Pentagon 5 540°
Hexagon 6 720°
Heptagon 7 900°
Octagon 8 1080°
Nonagon 9 1260°
Decagon 10 1440°
To find the number of degrees in any polygon: Use the formula (number of sides – 2) • 180
Ex. If a figure has 12 sides, the number of degrees= (12-2) • 180=10 • 180=1800°
Notes:
Page 37
Two-Dimensional Geometry
Triangle Properties
Classification by Angle:
Acute: all angles are less than 90 degrees.
Right: 1 right angle.
Obtuse: 1 angle is greater than 90 degrees.
Classification by Sides:
Scalene: no congruent sides.
Isosceles: 2 congruent sides called legs. The non-congruent side is called the base.
Equilateral: (equiangular) all three sides are congruent. All 3 angles are also congruent.
To solve triangle equations:
Set an equation where the angles sum to 180 degrees.
Combine like terms. Solve.
Evaluate to find all angles.
Check the angles sum to 180 degrees.
Conditions that Determine Unique Triangles: 1. Three sides condition
2. Two sides and included angle condition 3. Two angles and included side condition
4. Two angles and the side opposite a given angle
5. Two sides with the non-included angle 90 degrees or greater condition
Two-Dimensional Geometry
Circle: A circle is a two dimensional figure with all points equidistant from a center point, P. The circle is the actual line; the rest is the interior of the circle. A circle is named by its center point. A circle has 360°.
Chord: A chord is a line segment with endpoints on the circle.
Diameter: A diameter is a chord going through the center of the circle. It is the longest chord. There are an infinite number of diameters in a circle.
Radius: A radius is a line segment with one endpoint on the center of the circle and the other endpoint on the circle. There are an infinite number of radii in a circle. A radius is half of a diameter. A diameter is twice a radius.
To find a radius or a diameter: d = 2r
Multiply a radius by 2 to get the diameter.
Ex. If radius = 6 inches, diameter = 2 • 6 = 12 inches.
r = ½ d
Divide a diameter by 2 to get the radius.
Ex. If diameter = 9 meters, radius = 9
2 = 4.5 metersArc: An arc is a piece of a circle. You name it with two points on a circle. Arc AC is
Page 39
Two-Dimensional Geometry
Circumference: The distance around a circle. To find the circumference, you use the formula C =
π
d where C stands for circumference, d stands for diameter andπ
is pi. You can also use the formula C = 2π
r where C stands for circumference,π
is pi, and r stands for radius. You must include the units in your answer.Note: Without the use of a calculator, you must leave your answers in terms of
π
. If you can use a calculator, you must use theπ
key on the calculator, and then follow the directions in terms of rounding. You CANNOT use an approximation of pi.Ex. Find the circumference of a circle with a diameter or 7 meters. Leave your answer in terms of pi.
C=
π
d C=π
•7 C = 7π
m.Ex. If you have a radius, you must double it before you begin the problem to find the diameter. Round your answer to the nearest tenth.
Find the circumference of a circle with a radius of 3 inches. If the radius is 3 inches, the diameter is 6 inches.
C=
π
d C = 6•π
C = 18.84955592 inches C = 18.8 in.
Two-DimensionalGeometry
Area: The amount of space inside a figure measured in square units.
To find the area of a figure: 1. Copy the formula
2. Substitute known values 3. Calculate.
Area of Rectangles and Parallelograms
The formula for the area of a parallelogram is Area = base • height or A = bh.
Remember in an ordinary parallelogram, although the sides do not form right angles, the base and the height must form right angles.
Ex.
Altitude: drawn at a right angle to the base.
Notes:
b = 18 cm.
h = 8 cm. A = bh
A = 18 8
A = 144 square cm.
b = 3.9 mi.
h = 0.65 mi.
A = bh
Page 41
Two-DimensionalGeometry
Area of Triangles
Area of a Triangle = ½ • base • height or A = ½ bh = bh 2
**Remember, the base and the height must be perpendicular, so unless you have a right triangle, you may not use 2 sides. Choose one side to be the base and the height will be a segment connecting the other vertex to the base at a right angle.
Ex:
Area of a Trapezoid
Area of a Trapezoid = the average of the bases • the height or A =½(b1 + b2)h
The 2 parallel sides of the trapezoid are the 2 bases and a perpendicular line connecting them is the height.
Since one base is too short and one base is too long, we average the bases before multiplying by the height.
Ex.
Notes:
5 cm.
17 in.
8 in. 9.5 cm.
A = ½bh
A = ½ 9.5 5 A = 23.75 sq. cm.
A = ½bh A = ½ 8 17 A = 68 sq. in.
10.6 ft 7.8 ft
9 ft
A = ½(b1 + b2)h
A = ½ (7.8 + 10.6) 9 A = ½(18.4) 9
Two-Dimensional Geometry
Area of Circles
Pi (π): The value you get when you divide circumference by diameter. Pi is a non-terminating decimal, so it never ends and it never repeats.
Area of a Circle = Pi • radius • radius or A = r2
Note: Without the use of a calculator, you must leave your answers in terms of π. If you can use a calculator, you must use the π key on the calculator, and then follow the directions in terms of rounding. You CANNOT use an approximation of pi.
If given the diameter of the circle, divide by 2 to get the radius. Round your answer to the
nearest hundredth.
Notes:
12 m
Leave your answer in terms of pi.
A = r2
A = 12 12 A = 144 sq. m
20 m
r = d/2 A = r2
r = 20/2 A = 10 10
Page 43
Two-Dimensional Geometry
To find the radius or diameter given the circumference: 1. Use the formula C = πd or C = 2πr.
2. Substitute the numbers you have for circumference and your approximation of pi. 3. Divide on both sides of the equation by your approximation of pi to get the diameter. (If using C = 2πr, you must also divide by 2.)
4. If you want the radius, divide the diameter by 2.
Ex. Find the radius of a circle with a circumference of 15.7 ft. C = πd
15.7 = 3.14d 3.14 3.14 5 ft. = d
r = d÷2 r = 5 ÷ 2 r = 2.5 ft.
To find the radius or diameter given the area: 1. Use the formula A = r2.
2. Substitute the the numbers you have for area and your approximation of pi. 3. Divide on both sides of the equation by your approximation of pi.
4. Take the square root of your answer to get the radius 5. If you want the diameter, multiply the radius by 2.
Ex. The area of a circle is 314 sq. m. What is the radius? A = r2
314 = 3.14r2 3.14 3.14 √100 = 10 m. = r
Three-Dimensional Geometry
3-Dimensional Figures and Properties (Space Figures)
POLYHEDRON: The general term for all 3-dimensional figures made up of polygons. Polyhedrons consist of faces (sides), edges (where 2 faces meet), and vertices
(corners).
There are 2 main types of 3-dimensional figures that we will be studying; they are prisms and pyramids.
PRISM: A prism is a polyhedron with two congruent, parallel surfaces called bases. A prism is named after the polygon that is its base. The 2 bases are connected by rectangles.
examples:
EDGES- where 2 faces meet. They form a line
segment.
VERTICES-corners
This shape has 6 faces (sides of the figure). All of the faces of this polyhedron are rectangles.
Page 45
Three-Dimensional Geometry
PYRAMIDS: A polyhedron with one base and triangular faces that meet at a point. The pyramid is named after the polygon that is its base.
Examples:
Net: cut out used to form 3-D figure.
Notes:
TRIANGULAR PYRAMID
Three-Dimensional Geometry
Surface Area of Prisms and Pyramids
Surface Area: the sum of the areas of the faces of a 3-dimensional figure.
To determine the surface area of a prism or pyramid you must find the area of each individual face. The sum of these areas is called the surface area. Be aware that many of the faces will be congruent. Sometimes drawing a 2-dimensional net will be helpful. Ex.
Notes:_____________________________________________________ The surface area of this prism
would be found by adding the area of the two triangles to the area of each of the other 3 rectangles.
NET for TRIANGULAR PRISM
5 cm. 7.5 cm
5 cm.
Area of triangles A = ½bh
A = ½ 4 5
A = 10 2 triangles 10•2 =20 sq. cm.
Area of rectangles A = bh
A = 5 7.5 A = 37.5
3 congruent rectangles 37.5 3 = 112.5 sq. cm.
TOTAL SURFACE AREA = 20 + 112.5 SA = 132.5 square cm.
5 cm.
Page 47
Surface Area of a Rectangular Prism: You can use the formula SA=2lw + 2lh + 2wh where l is length, w is width, and h is height.
Ex. SA = 2 • 12 • 2 + 2 • 12 • 5 + 2 • 2 • 5 SA = 48 + 120 + 20
SA = 188 sq. cm.
Notes:_____________________________________________________
5 cm. 12 cm.
Three-Dimensional Geometry
Volume of Prisms
Volume: the measure of the inside of a 3-dimensional figure, measured in cubic units.
To determine the volume of a prism or a cylinder you must multiply the area of the base of the object by the distance between the bases, called the height.
Volume = Base area • height or V = Bh
Ex. The base is a rectangle, so B = lw V = lwh
V = 6 • 8 • 10 V = 480 cu. m.
Volume of Pyramids = ⅓ the volume of the corresponding prism.
V = ⅓lwh V = ⅓•30•10•5 V = 500 cu. m.
Notes:
h = 10 m.
l = 6 m.
w = 8 m.
w = 5 m.
Page 49
Two-Dimensional Geometry
Point: Represents an exact location in space. A point is named with a capital letter. A is called point A.
Line: An infinite collection of points. A line continues forever in both directions without end. Two points make a line.
A B named AB (line AB) or BA (line BA).
Ray: A part of a line with one endpoint that continues forever in one direction.
A B named AB (ray AB) where the first letter is the endpoint.
Line Segment: A part of a line with two endpoints.
A B named AB (line segment AB) or BA (line segment BA).
Plane: A flat surface with no thickness that continues forever in all directions.
A B
Named ABCD or plane m.
Parallel lines: Lines that lie in the same plane and don’t intersect. You must be told
that they are parallel either with markings on the lines or the symbol for parallel, ||. >
>
Perpendicular lines: Two lines that intersect to form a 90° angle. You must be told that they are perpendicular either with a marking on the lines or the symbol for
perpendicular, .
Skew Lines: Two lines not in the same plane that don’t intersect.
Oblique Lines: Lines that intersect, but not at a 90° angle.
Notes:
C D
Two-Dimensional Geometry
Angle: Two rays with a common endpoint. The endpoint is called the vertex. You can name an angle using the vertex if the angle stands alone, or using the vertex as the middle of the three letters.
Angle Classifications:
1. Acute Angle: An acute angle measures less than 90°.
2. Right Angle: A right angle measures 90°.
3. Obtuse Angle: An obtuse angle measures more than 90°, but less than 180°.
4. Straight Angle: A straight angle measures 180°.
5. Reflex Angle: A reflex angles measures more than 180°, but less than 360°.
Adjacent Angles: Angles that share the same vertex and a common side, but have no interior points in common.
Angles 1
A
B Called Angle A,
Angle BAC, or Angle CAB. Ray
Interior of angle Ray
Page 51 1
2
1 2
a d c b
Two-Dimensional Geometry
Angle Pairs:
1. Complementary Angles: Two angles whose measures add up to 90°. To find the complement of an angle, subtract the measure of the angle from 90°.
2. Supplementary Angles: Two angles whose measures add up to 180°. To find the supplement of an angle, subtract the measure of the angle from 180°.
3. Vertical Angles: Opposite angles formed by intersecting lines. Vertical angles have equal measures.
Solving angle pair equations:
1. To solve angle pair equations, you must set up the equation to mathematically represent what you know about that angle pair. For complementary angles, the two angles will add up to 90°. For supplementary angles, the two angles will add up to 180°. For vertical angles, the two angles will be equal.
2. Combine like terms. 3. Solve.
4. Find all angles by evaluating. 5. Check.
Angles one and two are complementary because they form a 90° angle. Therefore if angle one measures 40°,
then angle two measures 90°- 40°, which equals 50°.
Angles one and two are supplementary because they form a 180° angle, or a straight line. Therefore if angle two measures 50°, then angle one measures 180°- 50°, which equals 130°.
Two-Dimensional Geometry
Ex. Two angles are complementary. Angle one is 2x + 5. Angle 2 is 3x + 5.
2x + 5 + 3x + 5 = 90 5x + 10 = 90 - 10 - 10 5x = 80 5 5 x = 16
2x + 5 = 2•16 + 5 = 32 + 5 = 37° 3x + 5 = 3•16 + 5 = 48 + 5 = 53°
Check: 37 + 53 = 90√
The problem would be similarly solved if the angles were supplementary, but they would add up to 180°.
Ex.
Since these are vertical angles, we know that they have equal measures. Therefore we set up the equation where the two angles are equal to each other.
3x + 100 = 5x + 80 -3x -3x 100 = 2x + 80 -80 -80 20 = 2x 2 2 10 = x
Check:
3x + 100 = 5x + 80
3x + 100
Page 53
Qualitative Data: Not numeric, for example, eye color. Quantitative Data: Numeric, for example, ages.
You can find mean, median, mode, and range for Quantitative Data.
Statistics: Ways to organize data. It has many uses is sports, grades, and advertising.
Mean: Average. To find the mean, find the sum of your data and divide by the number of pieces of data.
Ex. 70, 100, 90, 80, 60
70 + 100 + 90 + 80 + 60 = 400 = 80
5 5
Median: The middle piece of data. To find the median, you must put your data in numerical order and then cross out one number from each end in pairs until you have one or two pieces of data left in the middle. If you have two pieces of data, your median is the mean of those 2 numbers.
Ex. 70, 100, 90, 80, 60 Ex. 10, 80, 90, 85, 90, 70 60, 70, 80, 90, 100 10, 70, 80, 85, 90, 90
80 80 + 85 = 165 = 82.5
2 2
Mode: The piece of data that occurs the most often. You can have one mode, more than one mode, or no modes.
Ex. 70, 100, 90, 80, 60: no mode
Ex. 10, 80, 90, 85, 90, 70: 90 is the mode
Ex. 10, 80, 90, 80, 90: 80 and 90 are the modes.
Measures of Central Tendency: Mode, Median, and Mean.
Range: The numerical difference between the highest piece of data and the lowest piece of data.
Ex. 10, 100, 90, 80, 60. 100 – 10 = 90 The range is 90.
Statistics
Collecting Data: Data can be collected a number of ways.
1. A written questionnaire or list of questions in which a person can check one of several categories as an answer or fill in some written response.
2. An interview, either in person or by telephone, in which answers are given verbally and responses are recorded by the person asking the questions.
3. A log or a diary, in which a person records information on a regular basis.
Page 55
Statistics
To make a box and whisker plot:
1. Find the median. This is the Second Quartile. Half of the information is below this number.
2. Find the median of the lower half. This is the First Quartile. 25% of the information is below this number.
3. Find the median of the upper half. This is the Third Quartile. 75% of the information is below this number.
4. Draw lines above all 3 quartiles. Connect these into one big box.
5. Draw a dot above the lowest number. This is the Lower Extreme. No information is below this number. Connect this to the box with a line.
6. Draw a dot above the highest number. This is the Upper Extreme. 100% of the information is below this number. Connect this to the box with a line.
The Interquartile Range is the difference between the third quartile and the first quartile.
An outlier is more than 1.5 times the value of the interquartile range beyond either quartile.
Mean Absolute Variation: Find the absolute values of the differences between each value and the mean. Then find the mean of those differences. This is also called variability or mean absolute deviation.
How to Compare Populations:
Both sets are symmetric
Neither set is symmetric
One set is symmetric
Measure of
Center Mean Median Median
Measure of Variation
Mean Absolute Variation
Interquartile Range
Interquartile Range
Statistics
Misleading Statistics:
1. If you have a piece of data that is very far from the other pieces, it could throw off the mean.
2. You can change the vertical scale to change the way a graph looks. 3. The vertical scale could start above zero, without the break mark.
How to Choose an Appropriate Graph:
Table- Shows all of the exact data. It is not visual.
Histogram- Bar graph where the numerical data is organized in intervals. The bars touch.
Line Graph- Shows the relationship between two items.
Bar Graph- Displays the data from a table. It shows comparisons and sometimes predictions. It looks like a histogram, but the bars don’t touch.
Circle (Pie) Graph- Shows percentages.
Scatter plot- Allows comparisons between two variables on the same graph. It looks like a line graph, but the points aren’t connected with a line.
Pictograph- Shows a visual representation of the data.
You can use the results of a survey to make a prediction: Set up a proportion as part over whole.
OR
Multiply the percent as a decimal by the total number.
Page 57 Statistics
Sample: a portion of the items to be counted.
Fair Sampling: You must be careful when choosing samples.
1. The sample must be fair, to reflect the entire population being studied.
Ex. In order to know what an apple pie tastes like, it is not necessary to eat the entire pie. A sample, or piece of pie, would be a fair way of knowing how the pie tastes. However, eating only the crust or only the apples would be unfair since these samples would not truly tell us what the entire pie tastes like.
2. The sample must contain a reasonable number of items being tested or counted. Ex. If a medicine is generally effective, it must work for many people. The sample tested cannot include only one or two patients.
3. Patterns of sampling or random selection should be used in a study.
Ex. The manufacturer of batteries might set up a pattern to test every 1000th battery to come off the assembly line. He may also select the batteries to be tested at random.
Biased versus unbiased samples:
Unbiased Samples:
1. Simple Random Sample: Each item or person in the population is as likely to be chosen as any other. Ex. Picking names from a hat.
2. Systematic Random Sample: The items are selected according to a specific time or item interval. Ex. Every 20th person is chosen from an alphabetical list.
Biased Samples:
1. Convenience Sample: Members of a population that are easily accessible are chosen. Ex. 1 HR out of the school.
2. Voluntary Response Sample: Only those who want to participate. Ex. Students who want to complete an online survey.
Probability
Event: An event is an action.
Outcome: The result of an event.
Random Experiment: A fair game. All outcomes are equally likely.
Ex. 1. When you roll a six-sided die that has the numbers 1, 2, 3, 4, 5, and 6, each number had an equal chance of occurring. Therefore this is a random experiment.
2. When you roll a six-sided die that has the numbers 1, 1, 1, 2, 3, and 4, this is not a random experiment because the number 1 has a greater chance of occurring.
Sample Space: The list of all possible outcomes.
Ex. The sample space for rolling a six-sided die is SS: {1,2,3,4,5,6}.
The sample space for days of the week you have school is SS: {Monday, Tuesday, Wednesday, Thursday, Friday}.
The sample space for flipping a coin is SS: {H,T}.
The sample space for flipping a coin twice is SS: {HH, HT, TH, TT}.
Tree Diagram: A list of all possible outcomes where the first event is listed and the following events are listed off the branches of the previous event.
Ex. The tree diagram for flipping a coin twice.
H
T
If you have three shirts, a red, a blue and an orange, and two pairs of pants, jeans and khakis, the tree diagram would be:
R
B H T
T H
Page 59 Probability
Probability: The likelihood that a certain event will occur. Probability of an event, notated
P(e)=the number of favorable outcomes (outcomes you want) the number of possible outcomes
Probability can be expressed as a fraction, decimal, or percent. Ex. If P(e) = ½, this can also be expressed as 50% or 0.5.
0≤P(e)≤1. In other words, probability must fall between 0 and 1. You can’t have less than a 0% chance of something happening, and you can’t have greater than a 100% chance of something happening. A 0% chance is impossible and a 100% chance is certain.
Ex. If you roll a die, what is the probability of getting a 1? because there is one 1 out of six numbers.
Ex. If you roll a die, what is the probability of getting an odd number? ½ because there are 3 odd numbers out of 6, which simplifies to ½.
Ex. In the spinner below, what is the probability of getting an A? ½ B? ¼
The probability of something happening + the probability it does not happen = 1. Ex. When you spin the spinner above, P(B) = ¼. The probability you do not get a B = P(A) + P(C) =¾. ¼ + ¾ = 1
P(and, then) means multiplication.
P(or) means addition.
Ex. Find P(1 or 3) when you roll a die. P(1 or 3)=P(1) + P(3)= + = = Ex. Find P(1 on the first roll of a die and a 3 on the second roll)
P(1 then 3) = • =
Notes:______________________________________________________
A
C B
Probability
Theroretical versus Empirical (Experimental) Probability:
Theoretical probability: is the chance of something happening based on what should happen.
For example, if you roll a die, the chance of each number coming up is .
Empirical, also known as experimental, probability: is the chance of something happening based on an experiment that was already conducted.
For example, if you roll a die 10 times and get the results 1, 2, 1, 4, 1, 3, 1, 4, 1, 5, the empirical probability of getting a 1 is ½ because it occurred 5 times out of 10. The empirical probability of getting a 6 is 0 because it didn’t occur at all.
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Counting Principle: The number of outcomes for an event with 2 or more stages equals the product of the number of outcomes for each stage.
Ex. You have 12 shirts and 4 pairs of jeans. How many outfits do you have?
12 • 4 = 48 outfits. What’s the probability you choose the red shirt with the white jeans?
Ex. There are 3 ways of driving from Elwood to Greenlawn, 2 ways of driving from Greenlawn to Northport, and 5 ways of driving from Northport to Huntington. How many different routes can you take from Elwood to Huntington?
3 • 2 • 5 = 30 different routes.
Notes:
Probability
Compound Event: A combination of 2 or more events.
Independent Events: Events in which the outcome of the first event has no effect on other events. For example, when you roll a die and then roll it again, the outcome of the first roll has no effect on the outcome of the second roll. To find the probability of
independent events, just multiply the probability of each event. Ex. You roll a die, find P(odd, odd) = ½ • ½ = ¼
P(1, odd) = • =
-They often call this with replacement because if you take an item and then replace it before taking another item, the outcome of the first draw has no effect on the outcome of the second draw.
Ex. You have a bag with 3 red marbles, 2 blue marbles, and 1 green marble. You pull a marble out of the bag, replace it, then pull a second marble.
Find P(red, green) = • =
P(blue, blue) = • =
Dependent Events: Events in which the outcome of the second event depends on the outcome of the first event. For example, if you pull a blue marble out of a bag and don’t replace it, this effects what you can pull out of the bag on the second draw. For this reason, dependent events are often called without replacement. To find the probability of dependent events, you must find the probability of the first event, adjust what you have left to use, and then find the probability of the second event. You then take the product of these probabilities.
Ex. You have 10 girls names and 10 boys names in a hat. You pull a name and throw it out before pulling another name. Find
P(girl, boy) = • = . After we pulled a girl’s name from the hat, we had 10 boys
names left, but only 19 names total left in the hat.
P(girl, girl) = • = . After we pulled a girl’s name from the hat, we had 9 girls names
left, and only 19 names total left in the hat.
P(boy, boy, boy) = • • = .
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Coordinate Geometry
Cartesian Plane (Coordinate Plane or Grid): the perpendicular intersection of two number lines, which allows for finding an exact location in two-dimensional space.
Ordered Pairs: numbers assigned to a point on a coordinate plane. (x, y)
Quadrants: one-fourth of the Cartesian Plane labeled Roman Numeral I, II, III, and IV, as above.
How to Graph a Point on the Coordinate Grid: Each point on the grid is assigned an x value and a y value (x, y). The x value indicates the horizontal placement and the y value indicates the vertical placement.
Ex. (2, 8) is 2 units right of the origin and eight units up from the origin. (See above)
(-4, 5) is 4 units to the left of the origin and five units up from the origin. (See above)
(0, -6) is on the y-axis and six units below the origin. (See above)
Domain or Abscissa The x-coordinate Range or Ordinate The y-coordinate
Notes:
y-axis
x-axis Origin (0,0)
(2,8)
(-4,5)
