PF M 251
Math 251!
Advanced Math Strategy! Prerequisite: Math score of 600+ on most recent prac7ce exam
Course Description
This micro-course will tackle some more challenging math problem-types (such as Functions with graphs, Multiple Functions, Absolute Value, Combination/Permutation, Probability) and review how students can use the two “nonmath” approaches of Backsolving and Plug and Chug to maximize score potential, along with reviewing when it's ok to skip a question.!
Math 251
2
Absolute Value Problems
Absolute Value Problems! ! Identify:! Any problem that uses the | … | absolute value bars.! ! Set Up:! |x+3|=7 Create two equations:! ! ! ! !
x+3=7 x=4
x + 3 = –7 x = –7 + –3 x = –10
! Make Sure:! Be careful with negatives. If the question is an INEQUALITY, don’t forget to FLIP THE DIRECTION of the inequality sign for the negative case. ! ! Execute:! Solve the two equations for the variable.!
Math 251
3
Absolute Value Problems
Example:! What are all values of x for which |2x – 3| < 7 ? (A) x < – 2 or x > 5 (B) x < – 2 (C) x > 5 (D) – 5 < x < 2 (E) – 2 < x < 5
Math 251
4
Function Problems
Function Problems – Graphs! ! Identify:! Whenever they give you a function problem that involves a graph.! ! Set Up:! The input to the function is also an x coordinate on the graph.! The output of the function is also a y coordinate on the graph.! ! Make Sure:! Whenever they tell you “y = f(x)” [or “y = h(x),” etc.], they ARE NOT giving you a new function – they are just explaining that on the graph, “y-values are equal to f(x)-values.”! ! Execute:! FIND the value they are asking about.! FIND the value that goes with it.!
Math 251
5
Function Problems
Example:! y y = h(x) 2 O -2 -1
1 1
2
x
The graph of y = h(x) is shown above. If h(m) = 2, which of the following is a possible value of m? (A) –1.5 (B) –0.5 (C) 1 (D) 1.5 (E) 2
Math 251
6
Function Problems
Function Problems – Multiple Functions! ! Identify:! When they give you more than one function, and you have to combine them.! ! Set Up:! Find the FUNCTION, then find the QUANTITY being plugged in. ! ! Make Sure:! Be careful to solve the functions one at a time, beginning with the innermost parentheses.! ! Execute:! PLUG the QUANTITY into the FUNCTION. SOLVE by combining simultaneous equations.!
Math 251
7
Function Problems
Example:! For which of the following functions is f(–1) > f(1)? (A) f(x) = 1 (B) f(x) = 3(x – 2) (C) f(x) = 7 – 3x (D) f(x) = 5x2 (E) f(x) = 2 – x2
Math 251
8
Combination & Permutation Problems
Combination & Permutation Problems! ! Identify:! Look for the phrase “how many different [combinations, arrangements, assignments, etc.].”! ! Set Up:! Determine which function to use on your calculator:! Combination = nCr! Permutation = nPr! ! Make Sure:! Make sure you input numbers into the calculator function in the right way.! ! Execute:! Get the answer from your calculator, then verify that it makes sense.!
Math 251
9
Combination & Permutation Problems
Example:! Nick is trying to decide what suit, dress shirt, and tie he will wear to a job interview. If he has 3 different suits, 5 different dress shirts, and 4 different ties, how many distinct outfits could Nick potentially wear to his interview? (A) 12 (B) 30 (C) 32 (D) 60 (E) 120
Math 251
10
Probability Problems
Probability Problems! ! Identify:! They will usually mention “probability,” often in reference to containers of items.! ! Set Up:! List the elements you know, and plug them into the probability formula:! ! Desired Outcomes Probability = ! Total Outcomes ! Make Sure:! Make sure you don’t forget any possibilities. Answer the question they are asking.! ! Execute:! Just do the math.!
Math 251
11
Probability Problems
Example:! Eight coworkers are involved in an office holiday gift exchange. Two of these coworkers are male, and the rest are female. If everyone puts their names into a hat, what is the probability that the first name drawn from the hat will belong to one of the female coworkers? 1 4 1 (B) 3 1 (C) 2 2 (D) 3 3 (E) 4 (A)
Math 251
12
Review Questions If there are 20 athletes who enter a marathon, how many different combinations of 1st, 2nd, and 3rd place finishers could there be? (A) 5814 (B) 6840 (C) 7980 (D) 6859 (E) 8000
Math 251
13
Review Questions For all numbers a and b, let the operation ¤ be defined by a ¤ b = 2ab – 4a. If a and b are positive integers, which of the following can be equal to zero? I. a ¤ b II. (a – b) ¤ b III. b ¤ (a – b) (A) (B) (C) (D) (E)
I only II only III only I and II only I, II, and III
Math 251
14
Review Questions
There are 360 members in the university band. One of these members is to be selected at random to be a band representative. If the probability that a woodwind player will 3
be selected is 8 , how many woodwind players are in the band?
Math 251
15
Review Questions
The total daily cost in dollars of making x pounds of gourmet cheese is c(x) =
300x − 50 + h , where h is a constant. If 10 x
pounds of the cheese were produced for a total cost of 350 dollars, then what is the value of h? (A) 15 (B) 25 (C) 35 (D) 45 (E) 55
Math 251
16
Review Questions Mark rolls a six-sided die twice. If a represents his first roll, and b represents his second roll, what is the probability that a > b?
Math 251
17
Review Questions Let the function f (x) = 2x2 – 4x + 3. If q is a positive number such that f (2q – 1) = f (q), what is the value of q?
Math 251
18
Review Questions
A
B
C
D
E
F
The figure above shows the top view of an open box that is divided into 6 compartments. The compartments labeled A and D have the same area, while the area of each of the compartments B, C, E, and F is three times that of compartment A. If a small rock is dropped into the box at random, what is the probability that the rock will end up in compartment E?
Math 251
1 6 3 (B) 14 1 (C) 7 1 (D) 12 1 (E) 14 (A)
19
Review Questions The function f is defined by f(x) = (x – 1)2. If f(a) = a2, what is the value of a? (A) -1 (B) -
1 2
(C) 0 1 2 (E) 1 (D)
Math 251
20
Review Questions V, W, X, Y, Z Consider all possible three-letter arrangements formed from the letters listed above. If no letter can appear more than once in any one arrangement, how many different arrangements are possible using these letters? (A) 12 (B) 24 (C) 30 (D) 60 (E) 120
Math 251
21
Advanced Math Strategy! Prerequisite: Math score of 600+ on most recent prac7ce exam
Course Description
This micro-course will tackle some more challenging math problem-types (such as Functions with graphs, Multiple Functions, Absolute Value, Combination/Permutation, Probability) and review how students can use the two “nonmath” approaches of Backsolving and Plug and Chug to maximize score potential, along with reviewing when it's ok to skip a question.!
Math 251
2
Absolute Value Problems
Absolute Value Problems! ! Identify:! Any problem that uses the | … | absolute value bars.! ! Set Up:! |x+3|=7 Create two equations:! ! ! ! !
x+3=7 x=4
x + 3 = –7 x = –7 + –3 x = –10
! Make Sure:! Be careful with negatives. If the question is an INEQUALITY, don’t forget to FLIP THE DIRECTION of the inequality sign for the negative case. ! ! Execute:! Solve the two equations for the variable.!
Math 251
3
Absolute Value Problems
Example:! What are all values of x for which |2x – 3| < 7 ? (A) x < – 2 or x > 5 (B) x < – 2 (C) x > 5 (D) – 5 < x < 2 (E) – 2 < x < 5
Math 251
4
Function Problems
Function Problems – Graphs! ! Identify:! Whenever they give you a function problem that involves a graph.! ! Set Up:! The input to the function is also an x coordinate on the graph.! The output of the function is also a y coordinate on the graph.! ! Make Sure:! Whenever they tell you “y = f(x)” [or “y = h(x),” etc.], they ARE NOT giving you a new function – they are just explaining that on the graph, “y-values are equal to f(x)-values.”! ! Execute:! FIND the value they are asking about.! FIND the value that goes with it.!
Math 251
5
Function Problems
Example:! y y = h(x) 2 O -2 -1
1 1
2
x
The graph of y = h(x) is shown above. If h(m) = 2, which of the following is a possible value of m? (A) –1.5 (B) –0.5 (C) 1 (D) 1.5 (E) 2
Math 251
6
Function Problems
Function Problems – Multiple Functions! ! Identify:! When they give you more than one function, and you have to combine them.! ! Set Up:! Find the FUNCTION, then find the QUANTITY being plugged in. ! ! Make Sure:! Be careful to solve the functions one at a time, beginning with the innermost parentheses.! ! Execute:! PLUG the QUANTITY into the FUNCTION. SOLVE by combining simultaneous equations.!
Math 251
7
Function Problems
Example:! For which of the following functions is f(–1) > f(1)? (A) f(x) = 1 (B) f(x) = 3(x – 2) (C) f(x) = 7 – 3x (D) f(x) = 5x2 (E) f(x) = 2 – x2
Math 251
8
Combination & Permutation Problems
Combination & Permutation Problems! ! Identify:! Look for the phrase “how many different [combinations, arrangements, assignments, etc.].”! ! Set Up:! Determine which function to use on your calculator:! Combination = nCr! Permutation = nPr! ! Make Sure:! Make sure you input numbers into the calculator function in the right way.! ! Execute:! Get the answer from your calculator, then verify that it makes sense.!
Math 251
9
Combination & Permutation Problems
Example:! Nick is trying to decide what suit, dress shirt, and tie he will wear to a job interview. If he has 3 different suits, 5 different dress shirts, and 4 different ties, how many distinct outfits could Nick potentially wear to his interview? (A) 12 (B) 30 (C) 32 (D) 60 (E) 120
Math 251
10
Probability Problems
Probability Problems! ! Identify:! They will usually mention “probability,” often in reference to containers of items.! ! Set Up:! List the elements you know, and plug them into the probability formula:! ! Desired Outcomes Probability = ! Total Outcomes ! Make Sure:! Make sure you don’t forget any possibilities. Answer the question they are asking.! ! Execute:! Just do the math.!
Math 251
11
Probability Problems
Example:! Eight coworkers are involved in an office holiday gift exchange. Two of these coworkers are male, and the rest are female. If everyone puts their names into a hat, what is the probability that the first name drawn from the hat will belong to one of the female coworkers? 1 4 1 (B) 3 1 (C) 2 2 (D) 3 3 (E) 4 (A)
Math 251
12
Review Questions If there are 20 athletes who enter a marathon, how many different combinations of 1st, 2nd, and 3rd place finishers could there be? (A) 5814 (B) 6840 (C) 7980 (D) 6859 (E) 8000
Math 251
13
Review Questions For all numbers a and b, let the operation ¤ be defined by a ¤ b = 2ab – 4a. If a and b are positive integers, which of the following can be equal to zero? I. a ¤ b II. (a – b) ¤ b III. b ¤ (a – b) (A) (B) (C) (D) (E)
I only II only III only I and II only I, II, and III
Math 251
14
Review Questions
There are 360 members in the university band. One of these members is to be selected at random to be a band representative. If the probability that a woodwind player will 3
be selected is 8 , how many woodwind players are in the band?
Math 251
15
Review Questions
The total daily cost in dollars of making x pounds of gourmet cheese is c(x) =
300x − 50 + h , where h is a constant. If 10 x
pounds of the cheese were produced for a total cost of 350 dollars, then what is the value of h? (A) 15 (B) 25 (C) 35 (D) 45 (E) 55
Math 251
16
Review Questions Mark rolls a six-sided die twice. If a represents his first roll, and b represents his second roll, what is the probability that a > b?
Math 251
17
Review Questions Let the function f (x) = 2x2 – 4x + 3. If q is a positive number such that f (2q – 1) = f (q), what is the value of q?
Math 251
18
Review Questions
A
B
C
D
E
F
The figure above shows the top view of an open box that is divided into 6 compartments. The compartments labeled A and D have the same area, while the area of each of the compartments B, C, E, and F is three times that of compartment A. If a small rock is dropped into the box at random, what is the probability that the rock will end up in compartment E?
Math 251
1 6 3 (B) 14 1 (C) 7 1 (D) 12 1 (E) 14 (A)
19
Review Questions The function f is defined by f(x) = (x – 1)2. If f(a) = a2, what is the value of a? (A) -1 (B) -
1 2
(C) 0 1 2 (E) 1 (D)
Math 251
20
Review Questions V, W, X, Y, Z Consider all possible three-letter arrangements formed from the letters listed above. If no letter can appear more than once in any one arrangement, how many different arrangements are possible using these letters? (A) 12 (B) 24 (C) 30 (D) 60 (E) 120
Math 251
21
