ARITHMETIC
§ Math facts 12x12
§ Know all immediately on sight.
§ Quickness with math facts leaves more time for complex
questions.
§ Decimals
§ Add/Subtract
§ Stack with decimal point aligned
§ Multiply
§ Multiply normally, count decimals and place it where needed
§ Divide
§ Move decimals first, then divide normally
§ Negative numbers
§ Mentally: always imagine a number line!
§ Subtracting a negative is a double ( - - ) so is the same as
adding a positive ( + + )
§ Add/Subtract with negatives. Eg. 45 - 90:
§ 2 Steps
§ 1) Determine whether answer will be positive or negative
§ If positive number is bigger answer = positve
§ If negative number is bigger answer = negative
§ 2) Subtract the smaller number from the larger to get
magnitude of answer
§ 45 - 90, 90 is bigger, so result will be negative. 90-45 =
45. So, -45.
§ Multiplying and dividing:
§ even number of minus signs = positive
§ odd number of minus signs = negative
§ Exponents with negative numbers
§ - y^2 and (-y)^2 are different
§ Even exponents always positive (will always have even number
of minus signs)
§ Odd can be negative (will have odd number of minus signs)
§ Negative exponents mean the number is on the other side of
fraction
§ 2^(-2) = (1/2)^2 = 1/4
§ Percentages
§ Understand for example: 100% increase means double
§ Percent of a percent: 20% of 25% of 100 = 5
§ Reverse to 5 is 20% of 25% of what number?
§ Powers
§ Just repeated multiplication -- Always come back to that!!
§ Squares/Square Roots
§ Know common squares:
§ Math Facts through 12, 2*2, 3*3, etc
§ 20*20 = 400, 13*13 = 169
§ Common Roots:
§ 4, 9 ,16, 25, 36, 49, 64, 81, 100, 121, 144, 169
§ Cubes
§ Know common:
§ 2*2*2 = 8
§ 3*3*3 = 27
§ 4*4*4 = 64
§ 5*5*5 = 125
§ Know the powers of 2: 2^3=8, 2^4=16, 2^5=32, etc
§ Negative exponents just flip to the other side of fraction:
2^(-2) = 1/4
§ When multiplying like powers, add exponents: 2^3 * 2^5 = (2
* 2 * 2) * (2 * 2 * 2 * 2 * 2) = 2^8
§ With an exponent raised to an exponent, multiply them:
(2^2)^3 = (2^2) * (2^2) * (2^2) = (2 * 2 ) * (2 * 2 ) * (2 * 2 ) = 2^6
§ Anything to the zeroth power is one: x^0 = 1
§ This includes zero: 0^0 = 1
§ This includes negative numbers: (-5)^0 = 1
§ PEMDAS - Order of operations
§ Distributive property!!!
§ Scientific Notation
§ Rewrite number so that it is in the form #.### * 10^i
§ There should be one number to the left of the decimal point
§ For positive powers of ten, the decimal point moves to the
right
§ For example: 3,454 = 3.454 * 10^3
§ You just move the decimal point to the right one place for
each power of 10. Here, we have 10 to the power of 3, so, move the decimal
point over 3 places.
§ The exponent on 10 can be negative
§ With negative powers of ten, the decimal place moves to the
left.
§ For example: 0.003454 = 3.454 * 10^(-3)
FRACTIONS
§ Know simple as decimals: 1/4 (0.25), 1/2 (0.5), 3/4 (0.75),
1/3 (0.33), 2/3 (0.67), 1/8 (0.125), 3/8 (0.375), 5/8 (0.625), 7/8 (0.875)
§ Shaded area of shape
§ Usually requires finding two or more areas and subtracting
§ Common denominator
§ Least Common Multiple
§ Take larger of the two, look at multiples
§ LCM of 4 & 6:
§ 6 is larger
§ Multiples of 6 in order are: 6, 12, 18, 24....
§ note the number itself counts. The LCM of 4 and 12 is 12.
§ The LCM can be one of the two numbers
§ 4 doesn't go into 6, but
§ 4 does go into 12, so
§ 12 is the least common multiple
§ When multiple is divisible by other number, you're done
§ Can do prime factorization method as well
§ Multiplying the two denominators always works, but requires
more simplifying later
§ With 4 & 6 as the denominators, we could have multiplied
them together and used 24
§ But using 12 means less need to simplify later
§ Adding/Subtracting
§ Once both fractions have been converted to have the same
denominator, simply add or subtract
§ Multiplying
§ simply multiply across
§ factor like terms to simplify reducing later
§ Dividing
§ Flip second and multiply
§ How many times does the second number fit into the first
§ This is obvious with whole numbers: 2 fits into 10 five
times: 10/2 = 5
§ Harder with fractions: (2/3) / (1/4) means: 1/4th fits into
2/3rds how many times?
§ (2/3) / (1/4) = (2/3) * (4/1) = 8/3 = 2 2/3
§ But why?
§ convert to like fractions: (8/12) / (3/12)
§ 3/12ths goes into 8/12's easily 2 times, with 2/12ths left
over
§ Now the hard part: 2/12ths is 2/3rds of 3/12ths, so another
2/3rds of 3/12 can go into it
§ together you have 2 and 2/3rds
§ Converting mixed fraction before X or /
§ When multiplying or dividing mixed fractions, you need to
convert to improper first.
§ (1 1/4) * (3 7/8) = (5/4) * (31/8) = 155/32 = 4 27/32
§ Or you'd have to use FOIL: (1 + 1/4) * (3 + 7/8) = 1*3 +
1*7/8 +3*1/4 +1/4 * 7/8) = 3 + 7/8 + 3/4 + 7/32 = 3 + 28/32 + 24/32 + 7/32 =
3+59/32 = 4 27/32
§ Factoring like terms - Reducing
§ Changing units
§ 1 yd * (3 feet/1 yd) * (12 inches/1foot) = 36 inches
NUMBER THEORY
§ Whole Numbers
§ All numbers zero and up that do not have fractional part
§ 0, 1, 2, 3, ...
§ Positive numbers
§ 1, 2, 3, ...
§ Zero is neither positive nor negative
§ Negative numbers
§ ...-3, -2, -1
§ Zero is neither positive nor negative
§ Real numbers
§ All numbers
§ Positive and negative and zero
§ Even/Odd
§ Negative numbers can also be even or odd
§ Divisible evenly by 2
§ Zero is even
§ Integers
§ All numbers, + and - with no fractional part: ...-2, -1, 0,
1, 2...
§ Digit: integers 0-9.
§ Absolute value - how far a number is from zero. Essentially
the number stripped of the minus sign
§ Defined by: square root of the square of a number. The
square kills the minus sign, the square root returns it to the magnitude of the
original number.
§ Zero
§ Zero is not + or -
§ Zero is even
§ Can not divide by zero
§ Primes
§ Only factors are 1 and itself
§ Know common primes (up to 50?)
§ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
§ All the numbers which aren't part of 12x12 math facts.
§ 1 is NOT a prime number (by definition of the term)
§ Factors
§ Like fractions, are smaller than the number you begin with
§ All (non-zero) whole numbers which can be evenly divided
into number
§ Example 45 >> 1 * 45, 3 * 15, 5 * 9, Factors = 1, 3,
5, 9, 15, 45
§ Prime factorization
§ Break the number into its factors, keep breaking it down
until everything is a prime.
§ 45 = 5 * 9.
§ 5 is already prime, but 9 is not:
§ 45 = 5 * 3 * 3
§ Factor tree
§ The factorization includes repeats: 45 = 3 * 3 * 5 so: 3, 3,
5.
§ Least Common Multiple
§ Method:
§ Do prime factorization
§ factorization includes repeats: 45 = 3 * 3 * 5 so: 3, 3, 5.
§ Select factors so that each compound number can be created
using the factors
§ Factors can be used multiple times: 45 = 3 * 3 * 5, 36 = 2 *
2 * 3 * 3. Both have 3 * 3, so you only need to use those once. Then, you need
to 5 from 45 and the 2 * 2 from the 36: LCM = 2 * 2 * 3 * 3 * 5 = 180
§ Greatest Common Factor
§ Method:
§ Do prime factorization
§ Select all factors that numbers have in common.
§ Factors can only be used once.
§ Repeats must be included (if include in both factorizations)
§ 45 = 3 * 3 * 5, 36 = 2 * 2 * 3 * 3.
§ They only have 3 * 3 in common, so
§ GCF = 3 * 3 = 9
§ "Between" means "between" - it does not
include the edges! So, if they say integers between 1 and 10, they mean 2
through 9.
GEOMETRY
§ Rectangles
§ Area
§ Perimeter
§ Can calculate by adding all, or adding 2 adjacent sides and
multiplying by 2
§ A cut out perimeter is the same as the outer.
§ Example: Sara walks around the outside of 4 blocks, 2 blocks
east, two blocks south, two blocks west, and two blocks north. Mary zig-zags: 2
blocks east, 2 blocks south, one block west, one block north, one block west, one
block north. They travel the same distance.
§ Angles add up to 360
§ Triangles
§ Area = 1/2 B * H
§ Finding the height is the tricky part!
§ Select one side as base
§ Draw perpendicular line upwards until line is a tall as the
triangle (think about measuring yourself with a book, measure the height of a
triangle the same way.) That's the height
§ Types of triangles: right, acute, obtuse, isosceles,
equilateral
§ All angles add up to 180
§ Right triangles
§ a^2 + b^2 = c^2
§ 3 - 4 - 5 triangles, or 6 - 8 - 10, or 9 - 12 - 15, etc.
§ 45-degree triangles -
§ both non-right angles the same
§ both side lengths the same (not hypotenuse)
§ other two angles add up to 90.
§ Angles
§ complimentary angles
§ supplementary angles
§ Circles
§ Diameter - all the way across
§ Radius - half-way across
§ Area
§ Circumference
§ Rectangular Prisms
§ Volume
§ Surface Area
§ Opposite sides have same area.
§ Can compute as (2 * side1) + (2 * side2) + (2 * side 3), or
§ 2 * (side1 + side2 + side3)
§ The surface area of a cube 6 units on a side is the same as
its volume.
§ Volume = 6 * 6 * 6 = 216
§ Surface area = 6 sides * (area of a side) = 6 * ( 6 * 6 ) =
6 * 6 * 6 = 216
§ 8 Vertices - 12 edges
§ Net folding - Spacial
§ Similar shapes
§ Transforms
§ Reflection
§ Rotation
§ Slide
§ Names of common shapes:
§ Trapesoid
§ Paralellogram
§ Rhombus
§ Cone
§ Pyramid
§ Rectangular pyramid
§ Cylinder
DATA
§ Graphs
§ Bar
§ Line
§ Stem and Leaf
§ Whisker
§ Scatter
§ Ordinals always walk before you fly: (x, y)
§ Best-fit line
§ Mean, Median, Mode, Range
§ If a number is added/subtracted, how does it change?
LINEAR EQUATIONS
§ Solving simple: 3y + 4 = 10
§ Positive vs Negative slope
§ Greater or lesser slopes
§ y-intercept
§ Ordinals always walk before you fly: (x, y)
§ Calculating slope from line
§ Visually: over one, up two
§ Pick two points use those with m = Rise/Run
§ same point must come first on both top and bottom: (y1 - y2)
/ (x1 -x2)
§ Calculating slope from two ordinals
§ m = Rise/Run
§ y = mx + b, m = slope, b = intercept
§ Finding slope of equation, must eliminate y's coefficient
and isolate it on other side of equals
§ Perpendicular lines
§ Slope is opposite: 2/3 >>> -3/2
PROBABILITY
§ How many combinations
§ Multiply options:
§ 1 6-sided die and 2 4-sided = 6 * 4 * 4 = 96 combinations
§ Probability on one pull. Even on die = 3/6 or 1/2
§ Probability on multiple independent pulls, even on 2 die:
1/2 * 1/2 = 1/4
§ Probability when each pull removes an option from the next
pull
§ 5 marbles in bag, each pull reduces number left: 1/5 * 1/4 *
1/3 * 1/2
§ Complimentary events
§ Chance of rolling 1 on a 6-sided die = 1/6th -- chance of
not rolling a 1 = 5/6ths
RATIOS & PROPORTIONS
§ Shapes
§ Similar Triangles
§ Counts
§ Ducks/Geese
§ Solving
§ Set up as double fraction: x/y = a/b
§ Always match up the like terms: ducks/geese = ducks/geese
§ One of the numbers is missing
§ multiply the two numbers that diagonally cross the empty,
and divide by the one across from empty.
RATE
INEQUALITIES
§ Answer is a range: x > 0 etc.
§ Multiplying or dividing equation by negative flips the sign
§ -x > 1, divide both sides by negative 1: x < -1
§ All other algebra works fine
MEASUREMENT
§ Metric
§ Prefixes
§ milli = 1/1000
§ centi = 1/100
§ (100 cents in a dollar)
§ deci = 1/10
§ kilo = 1000
§ (so big it will kill you)
§ English
§ Length
§ inch
§ foot = 12 inches
§ yard = 3 feet = 36 inches
§ mile = 5,280 feet = 63,360 inches
§ Volume
§ ounce
§ cup = 8 ounces
§ pint = 2 cups = 16 ounces
§ quart = 2 pints = 32 ounces
§ gallon = 4 quarts (think 4 quarters in dollar) = 128 ounces
§ Weight
§ ounce
§ pound = 16 ounces
No comments:
Post a Comment