Download Chapter 1 – Exponents and Measurement Exponents – A shorthand

Chapter 1 – Exponents and Measurement
Exponents – A shorthand notation for multiplying the same number by itself several times
53 = 5 x 5 x 5
86 = 8 x 8 x 8 x 8 x 8 x 8
(32)2 = 32 x 32 = 3 x 3 x 3 x 3
What if the exponent is zero? Anything to the zero power is one.
50 = 1
4540 = 1
Exponents and Place Value
3,465,277 = the sum of ….
3 x 106 = 3,000,000
4 x 105 = 400,000
6 x 104 = 60,000
5 x 103 =
5,000
2
2 x 10 =
200
7 x 101 =
70
7 x 100 =
7
Scientific Notation – A shorthand for large numbers.
It's easier to write 4.6 x 1021 than 4,600,000,000,000,000,000,000
From ordinary notation to scientific notation:
- Move the decimal until the number is between 1 and 10.
- How many times did you move the decimal?
2345 → 2.345 x 103
From scientific notation to ordinary notation:
- Move the decimal over the number of times represented by the exponent.
5.43 x 105 → 543,000
Exponents and Metric System – The metric system is based on powers of 10.
1 m = 100 cm (102 cm) = 1000 mm (103 mm)
1 km = 1000 m (103 m)
1 L = 1000 mL (103 mL)
1 g = 1000 mg (103 mg)
1 kg = 1000 g (103 g)
Rounding – What are you rounding to? (Tens? Tenths?) Look at that digit and then to the right.
If five or greater → round up.
If less than five → round down.
945.753
To the nearest ten → 950
To the near tenth → 945.75
Chapter 2 – Basic Operations with Decimals
Definitions
variable – a letter used to represent a known or unknown number
solution – the unknown number (or numbers) that makes an equation or inequality true
7 x a = 28 → solution: a is 4 (a = 4)
a – 3 > 10 → solution: a is any number greater than 13 (a > 13)
Properties of Addition and Multiplication
1) commutative property
order doesn't matter
3+6=6+3
3x6=6x3
2) associative property
grouping doesn't matter
(3 + 6) + 10 = 3 + (6 + 10)
(3 x 6) x 10 = 3 x (6 x 10)
3) identity property
any number added to zero remains the same
3+0=3
any number multiplied by one remains the same
3x1=3
4) zero property
any number multiplied by zero is zero
Distributive Property (you will see this again – many times!)
a x (b + c) = (a x b) + (a x c)
5 x (3 +2) = (5 x 3) + (5 x 2)
[geometric picture]
3x0=3
Basic Operations with Decimals
addition and subtraction
→ line up the decimals
→ add in the missing zeros
multiplication
→ ordinary multiplication (don't forget to shift)
→ count up decimal digits
division
→ add in zeros when needed
→ if divisor is a decimal, move the decimal in both the divisor and dividend
→
REMEMBER – Ask yourself whether the answer seems reasonable!
Chapter 3 – Divisibility and Fractions
Definitions
factor –
a number that divides evenly into the given number (4 is a factor of 16)
multiple –
a number that is the product of a given number and some other number
(72 is a multiple of 8)
prime –
a number with only 2 factors (1 and itself)
composite –
a number that is not prime (it is composed of more than 2 factors)
Divisibility Tests
Divisible by...
2
5
10
3
6
9
4
8
Prime Factorization
for example...
Test
even
ends in 0 or 5
ends in 0
sum of the digits can be divided by 3
can be divided by both 2 and 3
sum of the digits can be divided by 9
the last 2 digits can be divided by 4
the last 3 digits can be divided by 8
90 = 9 x 10 = 3 x 3 x 2 x 5 = 2 x 32 x 5
64 = 8 x 8 = 4 x 2 x 4 x 2 = 2 x 2 x 2 x 2 x 2 x 2 = 26
Method of Finding the Greatest Common Factor (GCF)
- do prime factorization of both numbers
- circle the factors that are common to both
Method of Finding the Least Common Multiple (LCM)
- do prime factorization of both numbers
- list each unique factor
- how many times should each unique factor be listed?
the greatest number of times it appears in one of the factorizations
Fractions
Equivalent Fractions
Create equivalent fractions by multiplying or dividing by 1.
Cross-Multiplication – If two fractions are equal, their cross products are equal.
Lowest Terms – A fraction may be reduced to its lowest terms by dividing top and
bottom by GCF
Mixed Numbers
Improper Fraction → Mixed Number
- divide
- place remainder over the old denominator
Mixed Number → Improper Fraction
- numerator: multiply whole number by denominator, then add numerator
- denominator: the same old denominator
Comparing and Ordering Fractions
To compare or order two fractions that do not share a common denominator:
- change each fraction to an equivalent fraction with a common denominator
- the common denominator will be the LCM of the two denominators
Fractions and Decimals
Fractions → Decimals
- long division
Decimals → Fractions
- a decimal is a fraction with a denominator that is a power of 10
Chapter 4 – Operations with Fractions
Basic Operations with Fractions
Addition and Subtraction
Warning!!! You can only add and subtract fractions that have the same denominator!!!
Common Denominator
- add or subtract the numerators
Different Denominators
- find the Least Common Denominator (i.e. LCM)
- rewrite each fraction with an equivalent fraction with the LCD
- add or subtract
Mixed Numbers
Method #1
- convert all numbers into fractions
- convert the answer back into a mixed number
Method #2
- add or subtract the whole number portions and the fractions portions separately
- addition may require carrying
(i.e. from the fractions portion to the whole number portion)
- subtraction may require borrowing
(i.e. from the whole number portion to the fraction portion)
Multiplication
- multiply numerators, multiply denominators
- mixed numbers:
· turn mixed numbers into fractions
· multiply
· turn the fraction back into a mixed number
Division
- turn fraction division problems into fraction multiplication problems
- reciprocal
two numbers are reciprocals if their product is 1
simply flip the fraction
2
5
→
5
2
7
2
→
2
7
3 →
1
3
1
5
3
(remember 3 is the same as 1 )
→5
dividing by a number is the same as multiplying by its reciprocal
2
5
divided by
7
2
2
=5 *
2
7
- mixed numbers:
same as above: turn the mixed numbers into fractions
Chapter 5 – Basic Operations with Negative Numbers
Absolute Value
a number's distance from zero on the number line
|5| = 5
|-9| = 9
Basic Operations with Negative Numbers
Addition
no matter what the starting point, positive or negative,...
adding a positive number → move up the number line (to the right)
adding a negative number → move down the number line (to the left)
in general...
positive + positive = positive
6 + 4 = 10
negative + negative = negative
-6 + -4 = -10
positive + negative → subtract the two numbers and take the sign of the larger number
6 + -4 = 2
-6 + 4 = -2
Subtraction
turn subtraction problems into addition problems
i.e. to subtract an integer, add its opposite (first number stays the same, second number changes)
6 - -4 =
6 + 4 = 10
-6 - -4 =
-6 + 4 = -2
6- 4 =
6 + -4 = 2
-6 - 4 =
-6 + -4 = -10
If negative numbers are confusing you, think about temperature
temperature was -10º and went up 15º
→ -10 + 15 = 5
temperature was - 20º and went up 15º
→ -20 +15 = -5
the difference between 15º and -10º
→ 15 - -10 = 25
the difference between -5º and -10º
→ -5 - -10 = 5
Multiplication
positive * positive = positive
negative * negative = negative
positive * negative = negative
Division
same rules as for multiplication
Coordinate Plane
[draw]
remember: given an ordered pair (x,y)...
x → left/right
y → up/down
Chapter 6 – Solving Equations
Order of Operations
1. operations inside parentheses
2. operations above or below a division bar
3. simplify expression with exponents
4. multiplication and division (from left to right)
5. addition and subtraction (from left to right)
Solving Algebraic Equations
method
- turn a given equation into a series of equivalent equations
reasoning
- if you have an equality, you have the same number on both sides of the equal sign
- if you add, subtract, multiply, or divide the same number to both sides of the equal
sign, the numbers will change but they will still be equal numbers
goal
- have the unknown variable on one side of the equal sign
- the other side of the equal sign will be the solution
Chapter 7 - Geometry
Basic Terms
line – a set of points that extend infinitely in two directions
ray – a part of line that extends infinitely in one direction from a single point
collinear – means given points are on the same line
coplanar – means given lines are n the same plane
vertex – the corner of an angle
Different Types of Angles
angle measures
straight -
180º (a straight line)
obtuse -
more than 90º
right -
90º
acute -
less than 90º
angle relationships
adjacent – next to each other
complementary – add up to 90º
supplementary – add up to 180º
congruent – same measure
vertical – (to angle 1)
corresponding – (to angle 1)
alternate interior – (to angle 1)
alternate exterior – (to angle 1)
[put in picture, and angle numbers above]
Different Types of Lines
parallel – two lines in the same plane that never intersect
perpendicular – two lines that cross at a right angle
Triangles
by side length
scalene – no sides the same
isosceles – two sides the same
equilateral – three sides the same
by angle
acute – 3 acute angles
obtuse – 1 obtuse angle
right – 1 right angle
Polygons
formula relating number sides to the sum of the interior angles
sum of interior angles = (number of sides – 2) * 180º
S = 180(n-2)
Quadrilaterals
parallelograms – opposite sides parallel and congruent, opposite angles congruent
rectangles – four right angles
rhombus – four congruent sides
square – four right angles, four congruent sides
trapezoid – only one pair of parallel sides
isosceles trapezoid – non-parallel sides congruent
Perimeter
polygon – add up the sides
circle – the perimeter of a circle is called the circumference
the circumference is π times as long as the diameter (the distance across the circle)
C = πd
the radius of a circle is half the diameter (the distance from the center to the edge)
C = 2πr
Area
rectangle:
A = bh
parallelogram: A = bh
1
bh
2
1
b b h
2 1 2
triangle:
A=
trapezoid:
circle:
A=
A = πr2
Chapter 8 – Ratios and Proportions
Ratio – a comparison of measure
e.g. student-teacher ratio (how many students are there compared to the number of teachers?)
reduce fraction to lowest terms
remember: you must use the same units of measurement to make the comparison
Proportion – equivalent ratios
e.g. given a student-teacher ratio of 10:1, 100 students must have 10 teachers
scale drawing
if a boat is drawn to scale, the ratio of measurements on the boat will equal the ratios on
the drawing
e.g. 10 ft: 1 inch scale, means a 100 foot boat will be 10 inches long on paper
similar figures
geometric shapes that have congruent angles and proportional sides
e.g. these two triangles (sides 3-4-5 and 6-8-10, congruent angles) are similar,
their sides are proportional (6:3, 8:4, 10:5 – a ratio of 2:1)
Trigonometric Ratios
all right triangles that have the same angles are similar – the ratios of their sides are constant
tangent of A: tan A =
opposite
adjacent
opposite
sine of A: sin A = hypotenuse
adjacent
cosine of A: cos of A = hypotenuse