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N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet Notes on Summer Review Topics οΆ Required Basic Math Skills: ο Order of Operations ο§ Expressions inside Grouping Symbols (parenthesis), [brackets], |absolute value bars|, βπππ ππππ πππππ, etc., if possible. ο ο§ Exponents ο§ Multiplication or Division as encountered left to right ο§ Addition or Subtraction as encountered left to right The Distributive Property π(π + π) = ππ + ππ (can be applied across parenthesis or brackets, but not across absolute value bars or radical signs) ο§ βππ(π β π) = βπππ + ππ οΆ Algebra ο Solving Equations ο§ Step 1 - Are there Grouping Symbols? If so, apply the Distributive Property. ο§ Step 2 - Are there Fractions or Decimals? If so, multiply by the common denominator or power of 10 needed to clear (remove) the fraction or decimal. ο§ Step 3 - Simplify each side of the equation independently (combine like terms). ο§ Step 4 - Move the Variables to one side of the equation: ο· Add or Subtract to move the variable term with the smaller coefficient. ο§ Step 5 - Add or subtract to Move the Constant to the other side of the equation. ο§ Step 6 - Multiply or divide to Remove the variable termβs Coefficient. ο§ Unusual results: ο· True statement => indicates Infinite Solutions ο· False statement => indicates No Solution H.S. Math Summer Notes Review page 1 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο Solving Inequalities ο§ ο§ Remember how to read inequality signs: ο· < βis less thanβ (looks like a tilted βLβ for Less than) ο· β€ βis less than or equal toβ (looks like a tilted βLβ for Less than) ο· > βis greater thanβ ο· β₯ βis greater than or equal toβ Which signs are graphed shaded? ο· Since the included section on a graph is shaded, inequalities that are strictly βgreater thanβ or βless thanβ donβt include the boundary and are, therefore, open circles. ο· Since the included section on a graph is shaded, inequalities that are βor equal toβ include the boundary and are, therefore, shaded circles. ο§ What is the difference between solving an equation and solving an inequality? ο· You either solve the inequality so that the variable term is positive then there is no difference between solving equations and solving inequalities, ο· or, you solve the inequality without regard to the sign of the variable term then you must remember every time you multiply or divide by a negative number, you must reverse the inequality sign. ο Graphing Linear Equations ο§ Cartesian coordinates use coordinates in the form (x,y) where the x-axis is horizontal and the y-axis is vertical II I III IV x ο§ The quadrants in a Cartesian coordinate graph start in the upper right with first quadrant, continuing counter-clockwise through fourth quadrant. ο§ y Slope is represented as m. given the points (π₯1 , π¦1 ); (π₯2 , π¦2 ) π ππππ β πππ π πβππππ ππ π¦ Ξπ¦ π¦2 β π¦1 β β β ππ’π πβππππ ππ π₯ Ξπ₯ π₯2 β π₯1 H.S. Math Summer Notes Review page 2 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο§ The slope-intercept form of a line is where m is the slope and π = ππ + π, (0, π) is the y-intercept. ο· Plot the y-intercept on the graph. ο· Use the slope to plot the line. Write the slope as a fraction. The numerator (top) of the slope fraction gives the rise of the line (or drop if negative), the denominator (bottom) of the slope fraction gives the run to the right of the line (or left if negative). Plot a minimum of three points if possible. ο· If the equation is not already in slope-intercept form, solve the equation for y (get y isolated on one side of the equation) to change it into slope-intercept form. Use inverse operations to isolate y. ο§ A line crosses the x-axis when y = 0, so set y to zero in the equation to find the x-intercept. A line crosses the y-axis when x = 0, so set x to zero in the equation to find the y-intercept. ο Writing Linear Equations ο§ Write a linear equation of a line in slope-intercept form π = ππ + π: Follow the following steps to write a linear equation when given two points. ** (if you are given the slope and one point, begin at step 3) ο· step 1 - Write out the points. ο· step 2 - Find the slope using the two points. ο· step 3 - Substitute the slope into y-intercept form of the equation. ο· step 4 - Substitute in the x & y value from one of the points you were given. (It does not matter which point you choose) ο§ ο· step 5 - Solve for b. ο· step 6 - Place b into the equation you wrote in step 3. Write a linear equation of a line in point-slope form π β ππ = π(π β ππ ) Follow the following steps to write a linear equation when given two points. ** (if you are given the slope and one point, begin at step 3) ο· step 1 - Write out the points. ο· step 2 - Find the slope using the two points. ο· step 3 - Substitute the slope into the point-slope form of the equation. ο· step 4 - Substitute in the π₯1 & π¦1 value from one of the points you were given. (It does not matter which point you choose) ο§ Vertical lines have a slope that is undefined. ο§ Horizontal lines have a slope of zero. ο§ Parallel lines have the same slope. ο§ Perpendicular lines have negative reciprocal slopes. H.S. Math Summer Notes Review page 3 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο Solving Systems of Equations: (revisited in HS Math) The solution to a system of equation makes both equations true. Think of solving systems of equations as if you had two definitions that need to be true at the same time. ο Solving a System of Equations by Graphing. To solve a system by graphing, just graph each equation on the same coordinate plane. Read off your solution which is the intersection point, if any. Note the three possibilities: ο· Case 1 - Intersecting Lines β results in one point of intersection. ο· Case 2 - Parallel Lines β results in one point of intersection. ο· Case 3 - Coincident Lines (the same line graphed twice) β results in one point of intersection. ο§ Solving a System of Equations by Substitution. Think of the equations in your system as definitions that must be true at the same time. Graphically, this is the intersection of the linear equations. ο· Step 1 - Solve one equation for a single variable. ο· Step 2 - Substitute that alternate definition in for the solved variable into the other equation. ο· Step 3 - Solve the new combined equation. ο· Step 4 - Take your solution and substitute it into one of the original equations to solve for the second variable. ο· Step 5 - Place your solutions for both variables into the other original equation to check your answer. ο§ Solving a System of Equations by elimination. ο· Step 1 - Align your equations, placing your variables in alphabetical order, 1st variable ο· 2nd variable equal sign constant Step 2 - Multiply by the least common denominator or power of ten to eliminate any fractions or decimals. ο· Step 3 - Look to see if you can now add to eliminate a variable. ο· Step 4 - If not, multiply by whatever is appropriate to eliminate a variable. ο· Step 5 - Add the equations together to eliminate one variable. ο· Step 6 - Solve the equation for the remaining variable ο· Step 7 - Substitute your solution into One of the Original Equations. ο· Step 8 - Solve the equation for the other variable. ο· Step 9 - Substitute both of your solutions into the Other Original Equation to check your work. ο Properties of Exponents: ο§ Remember that exponents are a shortcut way of writing repeated multiplication. πππ π ππ₯ππππππ‘ = πππ€ππ πππ π₯3 = π₯ β π₯ β π₯ H.S. Math Summer Notes Review page 4 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο§ Multiplying Two or More Expressions with an Exponent: If the two quantities raised to an exponent have the sane base, then π₯ 2 β π₯ 3 β (π₯ β π₯)(π₯ β π₯ β π₯) = π₯ 5 (the exponents are added) ο§ Raising Expressions with an Exponent to an Exponent: Remember the definition of an exponent: (π₯ 3 )2 β (π₯ β π₯ β π₯)(π₯ β π₯ β π₯) = π₯ 6 (the exponents are multiplied) ο Scientific Notation: Scientific notation is used to make sense of and easily calculate with extremely large or extremely small numbers. ο§ Scientific notation is a number expressed in the form π × 10π where a is a number 1 β€ π < 10 and π is an integer, in other words, a is a number with exactly one non-zero digit before the decimal ο§ Adding or Subtracting Numbers in Scientific Notation: ο· Step 0) adjust the numbers so the exponents match (it is best to use the larger one) ο· step 1) add or subtract the coefficients ο· step 2) keep the same exponent ο· step 3) combine new coefficient with exponent and adjust the answer to proper scientific notation ο· ο· Example 1: 8 x 105 + 3 x 105 ο¨ step 1) 8 + 3 = 11 ο¨ step 2) power of ten = 105 ο¨ step 3) 11 x 105 = 1.1 x 106 Example 2: 8.000 x 105 - 3 x 102 ο¨ step 0) 8.000 x 105 - 0.003 x 105 ο¨ step 1) 8.000 - 0.003 = 7.997 ο¨ step 2) power of ten = 105 ο¨ step 3) 7.997 x 105 H.S. Math Summer Notes Review page 5 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο§ Multiplying Numbers in Scientific Notation ο· First: Multiply the real number portions. ο· Second: Combine the exponential portions. ο· Third: Check that the answer is in scientific notation. ο· Example: (5.2 × 10β7 )(4.3 × 10β3 ) (5.2 × 4.3)(10β7 × 10β3 ) 55.9 × 10β10 is not in scientific notation. Rewrite 55.9 (5.59 × 101 ) × 10β10 answer: 5.59 × 10β9 ο§ Dividing Numbers in Scientific Notation ο· First: Divide the real number portions. ο· Second: Combine the exponential portions. ο· Third: Check that the answer is in scientific notation. ο· Example: 2.5 × 10β7 4.0 × 10β3 2.5 × 10β7 × 103 4.0 0.625 × 10β4 0.625 × 10β10 is not in scientific notation. Rewrite 0.625 (6.25 × 10β1 ) × 10β4 answer: 6.25 × 5 οΆ Geometry ο Special Angle Pairs ο§ Complementary Angles Two angles whose measures sum to 90o. The angles do not need to be adjacent. ο§ Supplementary Angles Two angles whose measures sum to 180o. The angles do not need to be adjacent. ο§ Adjacent Angles Two angles which share a common ray (side). H.S. Math Summer Notes Review page 6 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο§ Vertical Angles The non-adjacent angles formed by intersecting lines. β π & β π or β π& β π ο§ A Linear Pair of Angles Two adjacent angles whose non-adjacent sides form opposite rays (make a line). 2 1 3 β π & β π or β π& β π or β π & β π or β π& β π ο ο 4 Parallel Lines and Transversals (revisited in HS Math) ο§ Skew lines are non-coplanar lines that do not intersect. ο§ Parallel lines are coplanar lines that do not intersect. Similarity ο§ Similar triangles, denoted by the symbol ~, have exactly the same shape but not necessarily the same size. ο§ ο§ Two triangles are similar if ο· All pairs of corresponding angles are equal. ο· All pairs of corresponding sides are proportional. Example 1: A tree casts a 14 ft. shadow at the same time that a 5 ft. man casts a 2 ft. shadow. How tall is the tree? Set up a proportion: ο§ ππππππ ππππ ππ π πβ² β ππβ² = πβ² giving you a 35 ft. tree Example 2: How long is the shadow of a 17 meter tree if a 28 meter building casts a 34 meter shadow? Set up a proportion: ππππππ ππππ ππ β ππ π π ππ π = ππ π giving you a πππ ππ π shadow οΆ Statistics ο Measures of Center ο§ Mean: The mean of a numbered set of data is the average value. ο§ Median: The median of a numbered set of data is the middle number when the data is arranged in numerical order. ο§ ο Mode: The mode of a numbered set of data is the value that occurs most often. Measure of Spread H.S. Math Summer Notes Review page 7 rev. 9-Feb-16 N Butterworth BRMS & VMMS Gifted High School Level Math - Summer Work Packet ο§ Range: The range of a numbered set of data is the difference between the highest value and the lowest value. οΆ Probability ο Factorial, n! example: 10! = 10 β 9 β 8 β 7 β 6 β 5 β 4 β 3 β 2 β 1 0! Is defined as 1 ο Permutations (order is important), nPr, ο§ Order makes a difference (gold, silver, bronze medals) i.e.: {Smith, Jones, Brown} is distinct from {Smith, Brown, Jones} If all n items are chosen for sample, then there are π! permutations. ο§ If there are fewer than n items chosen for each sample - read as permutations of n items taken r at a time π! (π β π)! ο Combinations (order is unimportant), nCr ο§ Order makes no difference (members of a committee) i.e.: {Smith, Jones, Brown} is equivalent to {Smith, Brown, Jones} If all n items are chosen for sample, then there is only 1 combination. ο§ If there are fewer than n items chosen for each sample - read as combinations of n items taken r at a time π! (π π! β π)! ο Probability of Independent Events: Independent events are like coin tosses or dice rolls, one outcome is not affected by another outcome. Just multiply the individual probabilities. ο§ The probability of flipping a coin and getting heads both times is π(π»|π») = β β ο§ The probability of drawing a heart, then a spade from a deck of cards with replacement (I put the first card drawn back into the deck) is π(π»|π) = β β ο 1 1 1 4 4 16 1 1 1 2 2 4 Probability of Dependent Events: Dependent events are like drawing cards from a 52-card deck without replacement. If you now have one fewer cards in the deck, then the number of possible outcomes drops to 51. ο§ The probability of drawing a heart, then a spade from a deck of cards without replacement (I keep the first card drawn out of the deck) is π(π»|π) = β 1 13 4 51 β H.S. Math Summer Notes Review page 8 13 204 rev. 9-Feb-16