Download Math information – notes from class OPERATIONS: ADD

OPERATIONS:




Math information – notes from class
ADD - SUM
SUBTRACT - DIFFERENCE
MULTIPLY – PRODUCT
DIVISION – QUOTIENT
ESTIMATION:
Think rounding Rounding rules: 5 and up round up
4 and lower stays the same
- estimate to make Math easier & quicker
- use when we don’t need precise totals
Estimating sum or difference – rule: round to the same place value
Estimating product – rule: round to the leading digit (number)
Estimating quotient – rule: round the divisor & then find compatible number
INTERGERS negative and positive numbers that represent real life amounts


positive – gain, deposit, above
negative – loss, debit, withdrawal, below
Absolute value – how far away from “0” the integer is
Opposite – two numbers that are the same amount from “0”
Adding integers:


same sign – 1. add the absolute value of the integers 2. Use the common sign
different signs – 1. subtract the smaller absolute value from the larger absolute value.
2. use the sign from the bigger absolute value
Subtracting integers:

1. change the subtraction to addition 2. change sign of the integer that follows. 3.
Follow rules for adding integers
Multiplying / dividing integers: same sign will be positive different sign will be
negative



positive positive = positive
negative positive = negative
negative negative = positive
ORDER of OPERATIONS – Parenthesis
PATTERNS –
Exponents
Multiplication/Division left to right
Addition/ Subtraction left to right
 Numbers going down - subtract & divide
 Numbers going up – add & multiply
Example: 256, 128, 64, 32 decreasing so you would use subtraction or division
32, 64, 128, 256 increasing so you would use addition or multiplication
NUMERICAL EXPRESSION – includes numbers and operations
(numbers
sentence)
- do not have = signs
15-8
1679 x 2345
VARIABLE – a letter that represents one or more numbers
- do not use “o” as a variable (can be confused with zero)
VARIABLE EXPRESSION – includes variables (letters), operations, numbers
7m +13
134- 7t
TERMS- parts of the expression
 LIKE TERMS – identical variable parts ex. – 6b 4b
 CONSTANT TERMS – your numbers that do not change
 Simplify expressions by combining the like terms
 Example – 2d + 4 + 6d + 12
8d + 16
EQUATION – two expressions separated by an equal sign
-
solution for an equation is the number substituted for the
variables
- What number minus 8 equals 4?
- b–8=4
+8 +8
b= 12
EVALUATE – solve the problem
 Determine the operation
 Get the variable by itself
 Do the inverse (opposite) operation
 What you do to one side of the = sign, you MUST do to the other side of the =
IS – equal, equal to, a total
PROPERTIES
DISTRIBUTIVE- use with multiplication distribute a(b + c) = ab+ ac
COMMUNATIVE- use with use with multiplication and addition #’s have moved
a+b+c=c+a+b
ASSOSCIATE - use with use with multiplication and addition
(a + b) +c = a + (b + c)
associate
FUNCTION – relationship between numbers


input (x)  output (y)
function rule y = x + 5
DATA – information that we gather
Ways to represent data: makes data easier to read & understand
 pie chart & circle graph
 pictograph
 frequency table
 line plot
 line graph
 bar graph
 histogram
 double bar graph
FREQUENCY TABLE - how often something happens
LINE PLOT-
LINE GRAPH - shows change over time
BROKEN SCALE – use when the data starts at a large number
BAR GRAPH - comparisons of specific numbers
INCREMENTS - the number you go up by
HISTOGRAM – differs from bar graph because the bars touch
DOUBLE BAR GRAPH – compares two pieces of data Must include a key
CIRCLE GRAPH - percentage (%) of the whole (total number)
AVERAGES – all of the following are averages
 mean – add up all the numbers and divided by the number of numbers added
7, 1, 2, 6, 1, 7 7+1+2+6+1+7=24 24/6 = 4 4 is the mean
 median – the middle number after you order the numbers from least to
greatest. When it is an even number of numbers you must add the 2 middle
number together and divide by 2. This will then be the median.
1, 1, 2, 6, 7, 7 2+6 =8 8/2= 4
4 is the median
 mode – the number that occurs most often; you may have more than one
mode and may be no mode; the mode must occur at least twice. 1 and 7 are
modes for the above set of numbers
GEOMETRY – Chapter 9 in textbook
 line – extends without end in two opposite directions; 0 end points
 ray – has one endpoint and extends without end in one direction.
 segment – has 2 endpoints;
 parallel lines – lines that never meet
 intersecting lines – meet at a point
 perpendicular lines – lines that meet at right angles
ANGLES



made by combining 2 rays
VERTEX- point where lines meet
Measured in degrees
VERTICAL ANGLE - angles that are opposite and equal
COMMPLEMENTARY ANGLES – 2 angles that make 90 ̊
SUPPLEMENTARY ANGLES – 2 angles that make 180 ̊
180 ̊ = a straight line
CLASSIFYING TRIANGLES
 Size : Scalene - no equal sides
Isosceles - at least 2 equal sides
Equilateral – all sides are equal


Angles: Acute – has three acute angles
Right – has a right angle
Obtuse – has one obtuse angle
All angles of a triangle = 180 ̊
PARALLELOGRAM – 2 pairs of parallel sides
POLYGONS – a shape with many sides


TRIANGLE – 3 sides
QUADRILATERAL – 4 sides Angles all add up to 360 ̊
 REACTANGLE – 4 right angles, parallelogram
 TRAPEZOID – 1 pair of parallel sides
 RHOMBUS – 4 equal sides; parallelogram
 SQUARE – 4 equal sides; 4 right angles; parallelogram
POLYGONS –
REGULAR POLYGONS – all equal sides and all equal angles
IRREGULAR POLYGONS – not the same
 TRIANGLE- three sides
 PENTAGON – 5 sides
 HEXAGON – 6 sides
 OCTOGAN – 8 sides
 DECAGON – 10 Sides
DIAGONALS- segment that connects 2 vertice
TO FIND THE TOTAL DEGREES OF A POLYGON:
 Take the number of sides and subtract 2 (n-2)
example: hexagon 6 – 2 = 4 triangles 4 × 180 ̊̊ = 720 ̊
SIMILAR AND CONGRUENT FIGURES
 CONGRUENT – same shape and same size
 SIMILAR - same shape but different size
CORRESPONDING PARTS - parts of polygons that match
LINE OF SYMMETRY – divides a figure into 2 parts that match exactly
AREA- The amount of surface covered by a figure. Area is measured in square units
such as
square feet (ft²) or square meters (m²)
PERIMETER- The distance around a figure.
FORMULAS TO FIND AREA & PERIMETER:
RECTANGLE - A = l · w
PARALLELOGRAM –
P = 2(l +w)
area = base · height
A=b·h
TRIANGLE – Area = ½ · base · height A = b · h ÷ 2
Height is formed at a right angle
CIRCLES –Has no straight lines.
RADIUS – the distance from the center to any point on the circle.
DIAMETER – The distance across the circle through its center
The diameter is twice the radius.
CIRCUMFERENCE – The distance around the circle.
We use pi (3.14) when we calculate the area of a circle.
PI - How many times the diameter goes around the circumference of a circle; ratio
of the diameter to the circumference of a circle
VALUE of PI – 3.14
A = π r²
symbol for PI – π
C= πd or 2πr
CLASSIFYING SOLIDS
SOLID- closed figure that is 3 –dimensional
SPHERE – all points on the sphere are the same distance form the cente
 CONE – one vertex and a circular base
 PRISM – a solid with 2 parallel bases that are congruent polygons
Base names the solid
 PYRAMID – solid made up of polygons The base can be any polygon and
names the pyramid. The other polygons are triangle and meet a common
vertex.
 CYLINDER – 2 bases that are congruent and parallel circles.
FACES – sides of a figure
VERTICES – point where edges meet
EDGES – segments where faces meet
 Dotted line in a drawing indicates the edge you can not really see
SURFACE AREA - sum of all the areas of all the faces
 Find the area of each face & add together
 SA = 2(lw) + 2(lh) + 2(hw)
VOLUME – the amount that would fit inside
v=l ·w · h answers are always in cubic units (units ³)
PRIME FACTORIZATION – a number written as the product of prime numbers
Think of FACTOR TREES
The prime factorization of 54 is 2 x 3 x 3 x 3 or 2 x 3 
FACTORS – Two numbers multiplied together to make another number.
Ex. – 6 x 4 = 24
6 and 4 are factors of 24
DIVISIBILITY RULES:
 A number is divisible by 2 if it is an even number.
 A number is divisible by 5 if it ends with a 5 or a 0.
 A number is divisible by 10 if it ends with a 0.
PRIME NUMBER - has only 2 factors – one and itself ;
2 is the smallest prime number; one is NOT a prime number as it has only 1 factor.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …
COMPOSITE NUMBER – have three or more factors
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, …
GREATEST COMMON FACTOR (GCF) – The highest number that divides
exactly into two numbers. Will be smaller than your numbers ex. GCF of 18 &
12 is 6 6 is smaller than 12 or 18.
LEAST COMMON MULTIPLE (LCM) – The smallest (non-zero) number that
is a multiple of two or more numbers. Will be bigger than your numbers ex.
LCM of 40 & 32 is 160 160 is bigger than 42 or 32
Finding the GCF using the list method:
The factors of 12 are 1, 2, 3, 4, 6 and 8.
The factors of 18 are 1, 2, 3, 6, 9 and 18.
The common factors of 12 & 18 - 1, 2, 3, 6
The GCF of 12 and 18 is 6.
Finding the LCM using the list method:
The LCM of 2, 3, 4, & 6 is 12
Finding the GCF & LCM using the Venn diagram:
1. Make factor trees to find the prime factorization.
2. Complete the Venn diagram using the prime factorization.
GCF – multiply the numbers in the middle 2 ·2 · 3 = 12
LCM – multiply across the Venn diagram 2 · 2 · 2 ·2 · 3 · 3 · 5 = 720
FRACTIONS – part of the whole
Numerator - top number
Denominator- number on the bottom of the fraction
Equivalent fraction – fractions that are equal
Mixed numbers – whole number and a fraction
Improper fraction - Fraction in which the numerator is larger than the
denominator
Converting fractions –
DECIMALS- a number that uses a decimal point followed by digits as a way of
showing less than one.
Repeating decimals –having a pattern of one or more digits repeated
indefinitely.
Bar on is only over the digits that repeat.
Converting fractions to decimals: divide the numerator by the denominator
Converting decimals to fractions: make sure fraction is in lowest terms
Ones you should know:
¼ = .25 ½ = .50 ¾ = .75 think of money – quarters.
RATIO: Compare the number of one thing to the number of another.
Must be in lowest form
RATE: a ratio of 2 measurements with different units
Unit Rate- when the denominator is 1. The amount for one unit.
Use this to compare
PROPORTIONS – when an equation that shows 2 ratios are equivalent
Use cross product to determine if they are equivalent.
PERCENTAGES - per 100
100% means
all. Example:
100% of 80 is 100/100 × 80 =
50% means
half
50% of 80 is 50/100 × 80 =
80
Example:
40
5% means 5/100ths. Example:
5% of 80 is 5/100 × 80 =
4
How to find a percentage of a number:
Converting percents to decimals and fractions:
COORDINATE PLANES:
Quadrants – the four regions of a coordinate plane.
The vertical line is called the Y axis.
The horizontal line is called the X axis.
The axes intersect at the origin.
ORDERED PAIRS –
The first number tells you how many units to move to the left or right.
The second number tells you how many units to move up or down.
example: (4, -2) go to right 4 and then down 2. (see below)
TRANSFORMATION: When a figure moves on a coordinate plane.
We have an original figure, after the transformation we have an image.
Image- The new figure after an transformation.
labeled as : original
image
A
A’
B
B’
Translation (slide) All points of a figure move the same number of units and in the
same direction.
Reflection (flip) The figure is flipped over either the X-axis or the Y-axis. It must be
same distance from axis. You must state line of reflection when describing.
When reflecting over the X-axis, the Y value of the ordered pair will have an
opposite sign.
When reflecting over the Y-axis, the X value of the ordered pair will have an
opposite sign.
Example: A (-2, 1)
A’ (-2, -1)
B (2, 4)
B’ (2, -4)
C ((4, 2)
C’ (4, -2)
Rotation (turn)- where a figure is turned about a given point
center of rotation – origin
angle of rotation- 90 ̊, 180 ̊, 270 ̊, 360 ̊
direction of rotation- clockwise or counter clockwise
Dilation (get bigger or smaller)
PROBABILITY –
outcomes – possible result of an experiment
events – collection of outcomes
favorable outcomes – the outcome you are looking to happen
probability – chance or likelihood that an event may happen
P (event) = number of favorable outcomes
number of possible outcomes
list as percent, fraction, or decimal
When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6.
The probability of any one of them is 1/6.
Example: there are 5 marbles in a bag: 4 are blue, and 1 is
red. What is the probability that a blue marble will be
picked?
Number of ways it can happen: 4 (there are 4 blues)
Total number of outcomes: 5 (there are 5 marbles in total)
4
So the probability =
= 0.8 or 80%
5
Two types of probability:
Theoretical – the probability is based on what in theory should happen
Experimental – probability is based on repeated trials of an experiment
Independent event - if one event does not affect the likelihood the other event will
occur.
Dependent event – if the one event affects the other event
Combinations – when order does not matter
Permutations – when order matters
TREE DIAGRAM - list of possible outcomes