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Math information – notes from class

OPERATIONS:

• 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

Exponents

Multiplication/Division left to right

Addition/ Subtraction left to right

PATTERNS –

• 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 associate

(a + b) +c = a + (b + c)

FUNCTION – relationship between numbers

• input (x) ( output (y)

• function rule y = x + 5

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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

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LINE PLOT-

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LINE GRAPH - shows change over time

BROKEN SCALE – use when the data starts at a large number

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BAR GRAPH - comparisons of specific numbers

INCREMENTS - the number you go up by

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HISTOGRAM – differs from bar graph because the bars touch

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DOUBLE BAR GRAPH – compares two pieces of data Must include a key

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CIRCLE GRAPH - percentage (%) of the whole (total number)

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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;

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• parallel lines – lines that never meet

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• intersecting lines – meet at a point

• perpendicular lines – lines that meet at right angles

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ANGLES

• made by combining 2 rays

• VERTEX- point where lines meet

• Measured in degrees

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VERTICAL ANGLE - angles that are opposite and equal

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COMMPLEMENTARY ANGLES – 2 angles that make 90 ̊

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SUPPLEMENTARY ANGLES – 2 angles that make 180 ̊

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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 ̊

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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

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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 ̊

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SIMILAR AND CONGRUENT FIGURES

• CONGRUENT – same shape and same size

• SIMILAR - same shape but different size

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CORRESPONDING PARTS - parts of polygons that match

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LINE OF SYMMETRY – divides a figure into 2 parts that match exactly

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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 P = 2(l +w)

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PARALLELOGRAM – area = base · height A = b · h

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TRIANGLE – Area = ½ · base · height A = b · h ÷ 2

Height is formed at a right angle

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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.

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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 symbol for PI – π

A = π r² C= πd or 2πr

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CLASSIFYING SOLIDS

SOLID- closed figure that is 3 –dimensional

SPHERE – all points on the sphere are the same distance form the cente

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• CONE – one vertex and a circular base

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• PRISM – a solid with 2 parallel bases that are congruent polygons

Base names the solid

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• 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.

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• CYLINDER – 2 bases that are congruent and parallel circles.

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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

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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)

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VOLUME – the amount that would fit inside

v=l ·w · h answers are always in cubic units (units ³)

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PRIME FACTORIZATION – a number written as the product of prime numbers Think of FACTOR TREES

[pic]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:

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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.

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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

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Equivalent fraction – fractions that are equal

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Mixed numbers – whole number and a fraction

Improper fraction - Fraction in which the numerator is larger than the denominator

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Converting fractions –

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DECIMALS- a number that uses a decimal point followed by digits as a way of showing less than one.

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Repeating decimals –having a pattern of one or more digits repeated indefinitely.

[pic]Bar on is only over the digits that repeat.

Converting fractions to decimals: divide the numerator by the denominator

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Converting decimals to fractions: make sure fraction is in lowest terms

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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.

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Must be in lowest form

RATE: a ratio of 2 measurements with different units

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Unit Rate- when the denominator is 1. The amount for one unit.

Use this to compare

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PROPORTIONS – when an equation that shows 2 ratios are equivalent

Use cross product to determine if they are equivalent.

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PERCENTAGES - per 100

[pic] 100% means all. Example:

100% of 80 is 100/100 × 80 = 80

[pic] 50% means half Example:

50% of 80 is 50/100 × 80 = 40

[pic] 5% means 5/100ths. Example:

5% of 80 is 5/100 × 80 = 4

How to find a percentage of a number:

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Converting percents to decimals and fractions:

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COORDINATE PLANES:

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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)

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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.

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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.

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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

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Dilation (get bigger or smaller)

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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

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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)

|So the probability =   |4|  = 0.8 or 80% |

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| |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

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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

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Combinations – when order does not matter

Permutations – when order matters

TREE DIAGRAM - list of possible outcomes

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