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Transcript
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