Partner with MasteryPrep to increase the number of students scoring Level 3 and Level 4 on the NC Math 1 Assessment.
In just one day, students learn: • Core skills for success in algebra • Pacing and time management • Test-taking and guessing strategies that really work • How to overcome test anxiety and put their best foot forward on test day
Why schedule a NC Math 1 Boot Camp? • Authentic, up-to-date practice questions • Students review exactly what they need in the “final hours” before the test. • Improves student confidence • Easy to schedule, during the school day or on the weekend • Makes test prep fun and less overwhelming for students
Implementation Models • Full-day workshop during school hours • After-school or Saturday programming • Virtual and in-person programs available
Harnett County Schools
End of Course Grade-Level Progress 55 50 45 2016–17 2017–18
40 35 30 25 20 15 Harnett Central High
Overhills High
Western Harnett High
End of Course Boot Camp Success: Harnett County Schools, North Carolina Harnett County Schools serves 20,800 students in North Carolina. 37%
Student Feedback:
of its students belong to racial minorities, and 58% are enrolled in free or reduced lunch. In 2018, MasteryPrep delivered NC Math I EOC Boot Camps to Harnett Central High, Overhills High, and Western Harnett High. These schools saw major improvements in both grade-level
“An easy way to help you study.”
progress and college and career readiness on the NC Math I test.
Here’s what Harnett educators had to say:
“Continue this. It helped people like me who struggle.”
“(MasteryPrep’s instructor) did a GREAT job, and our students feel much more confident about taking the test from attending her boot camp!” - Sara Williamson, Harnett Central High
“Exciting to learn math!!”
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“THIS WAS AMAZING!!!”
Lumberton High School NC Math 1 Proficiency
35 30 Grade-Level Proficient
College Ready
25 20 15 10 2015–16
2016–17
End of Course Boot Camp Success: Lumberton High School, North Carolina Lumberton High School was one of the first North Carolina schools to implement our NC Math I Boot Camp, and has since realized gains on its NC Math I proficiency scores for two consecutive years. Lumberton
2017–18
Student Feedback:
“You need to take this. It’s amazing and will really help you!”
serves 2,114 students, 74% of whom belong to racial minorities, and 99% of whom are eligible for free lunch. The school’s gains in Math proficiency helped to propel their school performance grade from a D to a C!
“Facilitator was wonderful and the students loved him. They were engaged and enjoyed the session.” - Kelsey Cummings, Assistant Principal
“This is a very helpful program that can help you on the final exam.”
“He explained everything in a fun and interesting way!”
“Really cool math teacher. He makes things easier.”
“Excellent!”
Table of Contents
Table of Contents Chapter 1: NC Math 1 Overview .............................................................................................7 Chapter 2: NuMber aNd QuaNtity aNd algebra ...............................................................11 Number aNd QuaNtity aNd algebra Overview ..................................................................12 Plug iN POiNts ON a graPh ..................................................................................................16 distributive PrOPerty: shOw yOur wOrk ...........................................................................18 wOrd PrOblem traNslatiON ...............................................................................................22 Negative ParaNOia ..............................................................................................................25 PrOcess Of elimiNatiON .......................................................................................................26 eQuatiONs Of liNes ..............................................................................................................29 try Numbers ........................................................................................................................32 Chapter 3: FuNCtiONs...............................................................................................................43 fuNctiONs Overview............................................................................................................44 create a visual ...................................................................................................................47 PrOcess Of elimiNatiON .......................................................................................................50 use the aNswer chOices.....................................................................................................51 Plug it iN ..............................................................................................................................54 dON’t OverthiNk it.............................................................................................................. 55 read the QuestiON ..............................................................................................................58 draw it Out ........................................................................................................................ 59 Chapter 4: statistiCs & PrObability ...................................................................................69 statistics & PrObability Overview .....................................................................................70 fiNdiNg PerceNtages ...........................................................................................................74 PrOcess Of elimiNatiON ....................................................................................................... 76 Wrap-Up ......................................................................................................................................83 FUrther praCtiCe .......................................................................................................................87 Practice set ONe ................................................................................................................88 Practice set twO ................................................................................................................91 Practice set three .............................................................................................................94 Practice set exPlaNatiONs.................................................................................................98
Schedule Fill in the times following your instructor’s directions. This is the agenda we will follow throughout the day. There will be breaks throughout the session. Next to each section name in the schedule, you’ll find the corresponding page number where it begins in this workbook.
Time
Section
Page Number
NC Math 1 Overview
7
Number and Quantity and Algebra
11
Break
—
Functions
43
Break
—
Statistics & Probability
69
Boot Camp Wrap-Up
83
NC Math 1 Boot CaMp
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WORKKEYS OVERVIEW
Chapter One: NC Math 1 Overview
Chapter One: NC Math 1 Overview
Chapter 1 NC Math 1 Overview
NC Math 1 Boot CaMp
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Chapter One: NC Math 1 Overview NC Math 1 Overview
What Is End-of-Course Testing? End-of-Course (EOC) testing measures your aptitude in a given subject after you have finished a course. Consider it a subject understanding checkup. Teachers use it to identify both your strengths and areas where improvement is needed. This helps ensure you are on track in developing the knowledge and skills needed for the next grade and, eventually, college and a career. In this Boot Camp, we’ll focus on the three most common concepts in Math 1: Number and Quantity and Algebra, Functions, and Statistics & Probability. Your understanding of each of these concepts will help you pass the NC Math 1 test.
Why Should You Care? •
Your NC Math 1 score can be used for at least 20% of your final course grade, so doing well on this test can boost your GPA.
•
A good NC Math 1 test score is a positive indicator that you are on track for college.
•
Mastering the foundational skills taught in this Boot Camp will help you succeed in more difficult math courses in the future.
•
Put in the effort now and save yourself from repeating a course or taking summer school.
NOTES:
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NC Math 1 Boot CaMp
Chapter One: NC Math 1 Overview NC Math 1 Overview
Orientation The NC Math 1 test is an online test, but a paper option is available for students who need accommodations. There are 50 questions on the test. The three most common conceptual categories tested on the NC Math 1 test are Number and Quantity and Algebra, Functions, and Statistics & Probability. Some Geometry is also tested, but this book focuses on the three most frequently tested question categories. Number and Quantity and Algebra makes up 36–40% of the test. Functions makes up 32–36% of the test. Statistics & Probability makes up 18–20% of the test. Geometry makes up 8–12% of the test. Here is a breakdown of possible ways the three main categories will be covered on the test: Number and Quantity and Algebra tests equations, expressions, and inequalities in the following ways: •
Evaluate problems with one or two variables.
•
Create models to describe real-life situations and relationships.
•
Understand and apply basic mathematical principles.
Functions measures your ability to interpret, understand, and build functions. Statistics & Probability assesses how well you can summarize, represent, and interpret categorical and quantitative data and interpret linear models.
On the NC Math 1 test, every question is worth one point. There will be both a calculator-inactive and a calculator-active section. The calculator-inactive section will have multiple-choice and griddedresponse questions and the calculator-active section will have multiple-choice, gridded-response, and technology-enhanced questions (drag and drop, select multiple choices, and matching questions).
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Chapter One: NC Math 1 Overview NC Math 1 Overview
About This Boot Camp This is a one-day event, preparing you for the most important questions on the NC Math 1 test. This Boot Camp is not meant to be the only form of preparation for the NC Math 1 test. You should also find other practice tests online and ask your teachers for help and for other resources that specifically target the skills needed to do well on this test. This book contains key strategies for taking the test, instructional content, and mini-tests that give you practice with the type of questions you’ll see on test day (plus an explanation for how to solve every question). This book has plenty of places for you to take notes, and we highlight the most important strategies to use so you can continue practicing on your own. This Boot Camp will go by fast! Be ready, take notes, and stay focused!
NOTES:
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NC Math 1 Boot CaMp
Chapter Two: Number and Quantity and Algebra
Chapter 2 Number and Quantity and Algebra
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Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
overview
Number and Quantity and Algebra Overview The Introductory Algebra conceptual category tests your proficiency over a broad range of algebra skills. The skills that will be tested on your exam include but are not limited to the following:
Seeing Structure in Expressions •
Interpret the structure of expressions.
•
Write expressions in equivalent forms to solve problems.
Arithmetic with Polynomials and Rational Expressions •
Perform arithmetic operations on polynomials.
•
Understand the relationship between zeros and factors of polynomials.
Creating Equations •
Create equations that describe numbers or relationships.
Reasoning with Equations and Inequalities •
Understand solving equations as a process of reasoning and explain the reasoning.
•
Solve equations and inequalities in one variable.
•
Solve systems of equations.
•
Represent and solve equations and inequalities graphically.
NOTES:
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NC Math 1 Boot CaMp
Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
overview
What Are Boot Camp Mini-Tests? During this Boot Camp you will take several mini-tests, which are small segments of an Algebra I test. While taking these mini-tests, it’s important to imagine that you are in an actual testing environment. Practice all of the skills that you have learned as you complete the mini-tests. For these mini-tests, you have 8 minutes to answer 5 questions. Your instructor will signal when you are out of time. Try to get through all the questions within the time limit. Unless your instructor has provided you with an answer sheet, circle your answers directly in this book. The real test does not allow the use of cellphones, watches, or computers, so you shouldn’t use them on the mini-tests either.
NOTES:
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Chapter Two: Number and Quantity and Algebra
Number and Quantity and Algebra - Mini-Test One 1.
Which graph is a solution to 2x – 7y > 21? y
A.
y
12 10 8
B.
6
6
4
4
2 –12 –10 –8 –6 –4 –2 –2
C.
12 10 8
2 2
4
6
8 10 12
x
–12 –10 –8 –6 –4 –2 –2
–4
–4
–6 –8
–6 –8
–10
–10
–12
–12
y
y
12 10 8
D.
6
4
4
–12 –10 –8 –6 –4 –2 –2
4
6
8 10 12
2
4
6
8 10 12
x
12 10 8
6 2
2
2 2
4
6
8 10 12
x
–12 –10 –8 –6 –4 –2 –2
–4
–4
–6 –8
–6 –8
–10
–10
–12
–12
Number and Quantity and Algebra - Mini-Test One
14
x
GO ON TO THE NEXT PAGE.
Chapter Two: Number and Quantity and Algebra
2.
3.
What are the solutions to the equation 3x 2 – 27x + 160 = 118? A.
{2, –7}
B.
{–2, 7}
C.
{–2, –7}
D.
{2, 7}
A grocery store purchases crates of oranges. •
Each crate contains 75 oranges.
•
Each crate costs $60.
How much does the grocery store have to charge for each orange to make a profit of $30 per crate?
4.
5.
A.
$0.75
B.
$0.83
C.
$0.90
D.
$1.20
Which expression is equivalent to k 2 – 16? A.
(k – 8)(k + 2)
B.
(k – 8)(k – 2)
C.
(k + 4)(k – 4)
D.
(k – 4)(k – 4)
Which expression is equivalent to (x – 2)(3x 2 – 5x + 9)? A.
3x 2 – 4x + 7
B.
3x 2 – 4x + 9
C.
3x 3 – 5x 2 + 9x – 2
D.
3x 3 – 11x 2 + 19x – 18
Number and Quantity and Algebra - Mini-Test One
15
STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
plug iN poiNts oN a graph
Plug in Points on a Graph If you’re having trouble graphing a line on the coordinate plane, try plugging in points. This is a quick and easy method for solving graph problems. If you forget what each number in the equation of a line represents, then plug in values for x and y to find coordinate points on the line. Let’s take a look at how plugging in points can help you solve a problem on your test. 1.
A.
Which graph is a solution to 2x – 7y > 21? y
y
12 10 8
12 10 8
B.
6
6
4
4
2 –12 –10 –8 –6 –4 –2 –2
C.
2 2
4
6
8 10 12
x
–4
–4
–6 –8
–6 –8
–10
–10
–12
–12
y
y
12 10 8
12 10 8
D.
6
6
4
4
2 –12 –10 –8 –6 –4 –2 –2
16 |
–12 –10 –8 –6 –4 –2 –2
4
6
8 10 12
2
4
6
8 10 12
x
2 2
4
6
8 10 12
x
–12 –10 –8 –6 –4 –2 –2
–4
–4
–6 –8
–6 –8
–10
–10
–12
–12
NC Math 1 Boot CaMp
2
x
Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
plug iN poiNts oN a graph
First, simplify the inequality. 2x – 7y > 21 – 7y > –2x + 21 y<
2 x–3 7
Now, plug in a value for x to find a set of coordinates on the graph of the inequality. Let’s try x = 0. y<
2 (0) – 3 7
y < –3 Therefore, the point (0,–3) is on the graph of the inequality. Since y < –3, all the values of y below this point should be shaded. This makes choice B the only correct answer.
NOTES:
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Chapter Two: Number and Quantity and Algebra
Number and Quantity and Algebra - Mini-Test Two 1.
Three systems of equations are shown in the table below. Write the letter of the choice that describes the number of solutions for each system in the appropriate column in the table below. 3x + 2y = 12 2x + 5y = 19
2.
A.
One solution
B.
No solution
C.
Infinitely many solutions
x+y=6 3x + 3y = 18
3x + y = 22 3x + y = 17
Seven times Hector’s age minus two times Sandra’s age equals 5. Sandra’s age is also three times Hector’s age. How old is Sandra?
3.
4.
An artist spends d days expanding a mural. The existing mural is 6 feet long. Each day she adds 1.5 feet of the expansion. Which equation models the total length (L) of the mural over time? A.
d = 1.5L + 6
B.
d = 1.5L – 6
C.
L = 1.5d + 6
D.
L = 1.5d – 6
Jordan drove a distance represented by the equation 3x + 2. Omar drove a distance represented by the expression 18x + 12. Which of the following describes how the distance Omar drove compares to the distance Jordan drove? A.
The distance Omar drove is 3 times the distance Jordan drove.
B.
The distance Omar drove is 4 times the distance Jordan drove.
C.
The distance Omar drove is 5 times the distance Jordan drove.
D.
The distance Omar drove is 6 times the distance Jordan drove.
Number and Quantity and Algebra - Mini-Test Two
20
GO ON TO THE NEXT PAGE.
Chapter Two: Number and Quantity and Algebra
5.
The formula for the area of a trapezoid is A = bases, and h is the height.
b1 + b2 h, where A is the area, b 1 and b 2 are the lengths of the 2
What is the area of the trapezoid below? b1 = x + 9
h=6
b2 = x – 1 A.
A = 6x + 4
B.
A = 6x + 24
C.
A = 3x 2 + 8x – 9
D.
A = 3x 2 + 24x – 27
Number and Quantity and Algebra - Mini-Test Two
21
STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
worD proBleM traNslatioN
Word Problem Translation The secret to solving a word problem is translating it into math. When translating word problems to algebraic equations, it is important to know which words translate to which operations. is, equal to, is the same as → =
times, product, each, per, of → •
minus, without, less, difference, change → – plus, together, and, combined, both → +
divided into, split between or among, divvied up → ÷
Let’s take a look at how word problem translation can help you solve a problem on your exam.
2.
Seven times Hector’s age minus two times Sandra’s age equals 5. Sandra’s age is also three times Hector’s age. How old is Sandra?
Translate the sentences into algebraic equations. Let variables represent the unknown ages and use the translations above to determine what math operations to use. 7h – 2s = 5 s = 3h Substitute 3h from the second equation for s in the first equation. 7h – 2(3h) = 5 7h – 6h = 5 h=5 Hector is 5 years old. Substitute 5 for h in the second equation. s = 3(5) = 15 Sandra is 15 years old. Remember to translate words into math! Only a one-two punch like this can knock out a word problem.
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NC Math 1 Boot CaMp
Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
MiNi-test explaNatioNs
Mini-Test Explanations MINI-TEST ONE 1. The correct answer is B. Solve the inequality for y. Remember to flip the inequality when dividing by a negative number. 2x – 7y > 21 –7y > –2x + 21 y<
2 x–3 7
The line represented by the linear inequality has a positive slope, so choices A and C are incorrect. The shaded region should be below the line represented by the linear inequality, so choice D is incorrect. The graph in choice B is the correct representation of the inequality. 2. The correct answer is D. Factor the equation. 3x2 – 27x + 160 = 118 3x2 – 27x + 42 = 0 x2 – 9x + 14 = 0 (x – 2)(x – 7) x = 2 and 7 3. The correct answer is D. To make $30 profit, a crate of oranges must be sold for 60 + 30 = $90. Divide $90 by the number of oranges in the crate. 90 = 1.2 75
Each orange must be sold for $1.20 to make a profit of $30 per crate. 4. The correct answer is C. The expression provided is a difference of squares with a2 – b2 = (a + b)(a – b). Therefore, the given expression can be factored to the following: k2 – 16 k2 – 42 (k + 4)(k – 4)
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Chapter Two: Number and Quantity and Algebra NUmber aNd QUaNtity aNd algebra:
MiNi-test explaNatioNs
5. The correct answer is D. Distribute both terms in the first parentheses to all terms in the second parentheses and combine like terms. (x – 2)(3x2 – 5x + 9) 3x3 – 5x2 + 9x – 6x2 + 10x – 18 3x3 – 5x2 – 6x2 + 9x + 10x – 18 3x3 – 11x2 + 19x – 18
MINI-TEST TWO 1. The correct answers are A, C, and B. For the first system of equations, subtract the equations to solve for the variables. 3x + 2y = 12 becomes 6x + 4y = 24 2x + 5y = 19 becomes 6x + 15y = 57 6x + 15y = 57 –(6x + 4y = 24) 0 + 11y = 33 y=3 Plug this value into one of the original equations and solve for x. 3x + 2(3) = 12 3x + 6 = 12 3x = 6 x=2 There is one solution (2, 3). For the second system of equations, notice that the second equation is the first equation multiplied by 3. This means that any values that satisfy the first equation will also satisfy the second equation, so it has infinitely many solutions. For the third system of equations, the expressions on the left side of the equal signs are the same for both equations, but the numbers on the right side of the equal signs are not. Since an expression cannot simultaneously be equal to two different numbers, this system of equations has no solution.
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NC Math 1 Boot CaMp
Chapter Three: Functions
Chapter 3 Functions
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Chapter Three: Functions FUNCtioNs:
overview
Functions Overview The Functions conceptual category tests your proficiency over a broad range of algebra skills. The skills that will be tested on your exam include but are not limited to the following:
Interpreting Functions •
Understand the concept of a function and use function notation.
•
Interpret functions that arise in applications in terms of the context.
•
Analyze functions using different representations.
Building Functions •
Build a function that models a relationship between two quantities.
•
Build new functions from existing functions.
Linear, Quadratic, and Exponential Models •
Construct and compare linear, quadratic, and exponential models and solve problems.
•
Interpret expressions for functions in terms of the situation they model.
NOTES:
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NC Math 1 Boot CaMp
Chapter Three: Functions
Functions - Mini-Test One 1.
2.
ISP (Internet Service Provider) A charges a $15 installation fee and $0.10 per GB of data, x. ISP B charges $0.15 per GB of data and no installation fee. Which function below represents the difference in cost between ISP A and ISP B? A.
f(x) = –0.05x – 15
B.
f(x) = –0.05x + 15
C.
f(x) = 15x + 0.05
D.
f(x) = 15x – 0.05
Remy has a right triangle and a rectangle. • The length of the rectangle is 2 more than its width, w. • The length of the longer leg of the triangle is equal to the rectangle’s length. • The length of the shorter leg of the triangle is twice the width of the rectangle. Which function f(w), represents the combined area of the rectangle and the triangle?
3.
A.
2w 2 + 4w
B.
3w 2 + 6w
C.
w 2 + 4w + 4
D.
w 2 + 6w + 8
Andre is testing two prototype devices for heating small mechanical components. The results are shown in the table below. Time (minutes)
Prototype X Temperature (C°)
Prototype Y Temperature (C°)
0
6
6
1
34
12
2
59
24
3
83
48
4
108
96
5
136
192
Which statement best describes his results? A.
Prototype X resulted in temperature changing at a constant rate.
B.
Prototype Y resulted in temperature changing at an exponential rate.
C.
Both devices resulted in temperature changing at a constant rate.
D.
Both devices resulted in temperature changing at an exponential rate.
Functions - Mini-Test One
45
GO ON TO THE NEXT PAGE.
Chapter Three: Functions
4.
Rayan and Megan are playing a game. •
Rayan and Megan each started with 50 points.
•
At the end of each turn, Rayan’s points increased by 250.
•
At the end of each turn, Megan’s points doubled.
At the end of which turn will Megan have more points than Rayan?
5.
Which choice could be modeled by a linear function? A.
the daily cost, y, to drive a car for x miles at a rate of $2 per mile
B.
the population, y, of bacteria remaining after x days when decay occurs at a rate of 15% each day
C.
the amount of money, y, in a savings account after x years earning 2% interest compounded annually
D.
the distance, y, of a ball rolling for x minutes, if the speed each minute is previous minute
Functions - Mini-Test One
46
1 the speed of the 2
STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Chapter Three: Functions
FUNCtioNs:
proCess of eliMiNatioN
Process of Elimination When you get stuck on a question and need to make a guess, always try to avoid making a random guess when you can. Instead, narrow down the choices to increase your chance of guessing correctly.
1.
The table below shows the weight of an algae bloom after several days of growth. Time (days)
Weight (lb)
2
0.13
3
0.17
4
0.23
5
0.32
6
0.51
What is the average rate of change in weight of the algae bloom from day 2 to day 6? A.
0.032 pound per day
B.
0.095 pound per day
C.
0.190 pound per day
D.
0.380 pound per day
Looking at the table, you can see that the rate of growth increases slightly each day. The lowest growth rate is between the first two days, days 2 and 3, when the plant grows 0.17 – 0.13 = 0.040 lb. The highest growth rate is between the last two days, days 5 and 6, when the plant grows 0.51 – 0.32 = 0.190 lb. Since the question asks for the average rate of change and an average is a way of measuring the “middle” of a set of data, you can eliminate answer choices that do not reflect a number between 0.040 and 0.190 lb per day. Choice A can be eliminated because 0.032 is less than 0.040. Choices C and D can be eliminated because 0.190 and 0.380 are equal to or greater than 0.190. Since choice B, 0.095 lb per week, is the only option that falls between the highest and lowest rates of change, it is the best guess and also the correct answer.
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NC Math 1 Boot CaMp
Chapter Three: Functions FUNCtioNs:
use the aNswer ChoiCes
Use the Answer Choices You have learned that when you get stuck, you can try plugging numbers into the given equation to see what happens. This process is far more effective when you don’t have to pick those numbers yourself. When the answer choices give numbers, use them! 2.
Clayton throws a stone from the edge of a cliff into a lake below. The height of the stone, in meters, is modeled by the function h(s) = –s 2 + 6s + 27, where s represents how far the stone is from the cliff. How far from the cliff will the stone be when it touches the surface of the lake below? A.
0 meters
B.
3 meters
C.
9 meters
D.
27 meters
When the stone touches the surface of the lake, its height, h(s), will be 0. To solve this problem, you would need to set the equation equal to 0 and solve for s. If you struggle with factoring polynomials, though, an easier and faster method would be to plug each of the answer choices into the equation and see which one results in a height of 0. When your numbers come from the answer choices, start in the middle and move to higher or lower numbers if necessary. 0 = –(9)2 + 6(9) + 27 0 = –81 + 54 + 27 0 = –81 + 81 0=0 The stone is 9 meters from the cliff when it touches the surface of the lake.
NOTES:
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Chapter Four: Statistics & Probability
Chapter 4 Statistics & Probability
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| 69
Chapter Four: Statistics & Probability statistiCs & probability:
overview
Statistics & Probability Overview The Statistics & Probability conceptual category tests your proficiency over a broad range of algebra skills. The top skill that will be tested on your exam includes the following:
Interpreting Categorical and Quantitative Data •
Summarize, represent, and interpret data on a single count or measurement variable.
•
Summarize, represent, and interpret data on two categorical and quantitative variables.
•
Interpret linear models.
•
Understand and calculate measures of center and spread.
NOTES:
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NC Math 1 Boot CaMp
Chapter Four: Statistics & Probability
Statistics & Probability - Mini-Test One 1.
A set of seven data points is shown below. 6 4 1 11 10 5 6 Which statement is true if an eighth data point of 60 is added to the data set?
2.
A.
The mean and median will both decrease.
B.
The mean and median will both increase.
C.
The mean will increase, and the median will decrease.
D.
The mean will increase, and the median will stay the same.
Use the table below to answer the question. Photo Booth Price Statistics for Two Companies
Company
Median
Mean
Interquartile range
Memories Made Fun Photos
$80.75 $90.99
$82.00 $60.90
$61.21 $33.50
Which statement about the prices of renting photo booths at Memories Made and Fun Photos is best supported by the data? A.
The fact that the median and mean are very close at Memories Made means that all of the photo booths cost under $85.
B.
Comparing the interquartile range of the companies shows that photo booths at Fun Photos are about half the cost of photo booths at Memories Made.
C.
Comparing the mean prices of both companies shows that most photo booths are more expensive at Memories Made.
D.
Some low-priced photo booths at Fun Photos make the mean lower than the median.
Statistics & Probability - Mini-Test One
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GO ON TO THE NEXT PAGE.
Chapter Four: Statistics & Probability
3.
The table below shows the orders at an ice cream shop during the month of June. Ice Cream Shop Sales - June Chocolate Syrup
Sprinkles
No Topping
Total
Mint Chocolate
101
0
45
146
Café Mocha
89
60
27
176
Cookie Dough
60
44
99
203
Total
250
104
171
525
Which statement about the ice cream shop sales for the month of June is true?
4.
A.
The percentage of mint chocolate sales was less than the percentage of café mocha sales.
B.
The percentage of sales of cookie dough with chocolate syrup was greater than 25% of all sales.
C.
The percentage of sales of a café mocha with no topping was less than 1% of all sales.
D.
The number of ice cream sales with no topping was less than the number of mint chocolate ice cream sales.
In the first eight games of the season, a water polo player scored the following points per game. 1, 6, 0, 2, 4, 1, 0, 5 If she scored 3, 1, 1, and 3 points in each of the next four games, what would happen to the data distribution? A.
The data distribution would become less widely spread and more peaked.
B.
The data distribution would become more widely spread and more peaked.
C.
The data distribution would become more widely spread and less peaked.
D.
The data distribution would become less widely spread and less peaked.
Statistics & Probability - Mini-Test One
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Chapter Four: Statistics & Probability
5.
Rodrigo’s biology class planted tomato seeds as part of an experiment. Each week, a student measured the height of any one of the tomato vines. The data in the scatter plot shows the findings from weeks 3 through 13.
Height (in)
y 80 72 64 56 48 40 32 24 16 8 0
Tomato Vine Growth
1 2 3 4 5 6 7 8 9 10 11 12 13 Number of Weeks
x
Which function, h(x), approximates the height of the tomato vines after x weeks? A.
h(x) = 0.16x + 4.34
B.
h(x) = 0.16x – 7.99
C.
h(x) = 6.02x + 15.79
D.
h(x) = 6.74x – 16.81
Statistics & Probability - Mini-Test One
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Chapter Four: Statistics & Probability statistiCs & probability:
proCess of eliMiNatioN
Process of Elimination When you notice only slight differences between answer choices, this is a good clue that you can use the process of elimination. Take a look at the answer choices in the question below.
4.
In the first eight games of the season, a water polo player scored the following points per game. 1, 6, 0, 2, 4, 1, 0, 5 If she scored 3, 1, 1, and 3 points in each of the next four games, what would happen to the data distribution? A.
The data distribution would become less widely spread and more peaked.
B.
The data distribution would become more widely spread and more peaked.
C.
The data distribution would become more widely spread and less peaked.
D.
The data distribution would become less widely spread and less peaked.
Choices A and B both say the data distribution becomes more peaked, while choices C and D say it becomes less peaked. If you can determine how the data peak changes, you can eliminate two answer choices and guess from the two remaining choices rather than four. Furthermore, choices B and C both say the data distribution becomes more widely spread out, while choices A and D say it becomes less widely spread out. If you can determine how the spread changes, you can also eliminate two choices to make a better guess. If you can only determine the peak or the spread, use that information to make a guess from two answer options, which is always better than choosing from four. However, if you can determine both, you can use the process of elimination all the way down to one remaining answer. The first step is to put the data in numerical order: 0, 0, 1, 1, 2, 4, 5, 6. Then, create a data distribution of the values. Sketching out the distribution, you’ll see the line resembles a bell curve, with a peak around 1 and 2. If the new numbers are added into the data, the data set becomes 0, 0, 1, 1, 1, 1, 2, 3, 3, 4, 5, 6. Adding these values to the data distribution shows that a higher percentage of the data is now concentrated around the peak of the bell curve. This eliminates choices C and D. This higher concentration of data in the middle also means the data values are less spread out. This eliminates choice B and makes choice A the best answer.
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Chapter Four: Statistics & Probability statistiCs & probability:
MiNi-test explaNatioNs
Mini-Test Explanations MINI-TEST ONE 1. The correct answer is D. First, find the median of the data set by listing the numbers in numerical order and finding the middle number. 1, 4, 5, 6, 6, 10, 11 The number in the middle is 6. Add 60 to the data set and find the median again. Because there are an even number of values, the median is the average of the two middle values. 1, 4, 5, 6, 6, 10, 11, 60 The numbers in the middle are 6 and 6, which average to 6. In both cases, the median is 6, so the median stays the same. Next, find the mean of the data set by adding the values together and dividing by the total number of values. (1 + 4 + 5 + 6 + 6 + 10 + 11) ÷ 7 ≈ 6.14 Add 60 to the data set and find the mean again. (1 + 4 + 5 + 6 + 6 + 10 + 11 + 60) ÷ 8 = 12.875 The mean increased from approximately 6.14 to 12.875. Therefore, the mean will increase, and the median will stay the same. 2. The correct answer is D. Using only the table, it is impossible to know the total number of values in the data set used to find the median, mean, and interquartile range for each company. Therefore, there are some generalizations that cannot be made. Choice A is incorrect because the data could have very high prices, well over $85, and still have a median and mean in the low 80s. Choice B is incorrect because the interquartile range is the difference between certain quartile values in the data set and does not give insight about the highest or lowest values themselves. Therefore, the given interquartile ranges show that the upper half of the data is closer in value to the lower half of the data for Fun Photos than for Memories Made, not that the overall pricing is always half as much for Fun Photos as it is for Memories Made. Choice C is incorrect because the mean does not provide evidence about the number of photo booths that are priced higher at Memories Made than at Fun Photos. Choice D is correct because lower values can always lower the mean and cause it to be lower than the median. 3. The correct answer is A. Choice A is correct because the percentage of mint chocolate sales is 146 176 · 100 ≈ 28%, which is less than the percentage of café mocha sales, · 100 ≈ 34%. Choices B, 525 525 C, and D do not accurately describe the data. The percentage of cookie dough with chocolate syrup is 60 27 · 100 ≈ 11%, which is not over 25%. The percentage of café mocha with no topping is · 100 ≈ 5%, 525 525 which is not less than 1% of all sales. The number of ice cream sales with no topping is 171, which is not less than the number of mint chocolate ice cream sales, 146.
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Chapter Four: Statistics & Probability statistiCs & probability:
MiNi-test explaNatioNs
4. The correct answer is A. The first step is to put the data in numerical order: 0, 0, 1, 1, 2, 4, 5, 6. Then, create a data distribution of the values. Sketching out the distribution, you’ll see the line resembles a bell curve, with a peak around 1 and 2. If the new numbers are added into the data, the data set becomes 0, 0, 1, 1, 1, 1, 2, 3, 3, 4, 5, 6. Adding these values to the data distribution shows that a higher percentage of the data is now concentrated around the peak of the bell curve. This higher concentration of data in the middle also means the data values are less spread out. Therefore, choice A is the best answer. 5. The correct answer is D. Sketching a line in the direction of the data shows that a line of best fit would cross the y-axis at a negative value. Since the equations listed in the answer options are in slope-intercept form, this eliminates choices A and C, both of which show a positive value for the y-intercept. Next, examine the slopes of choices B and D. The slope of choice B is 0.16, or
4 . Sketching out a line with this slope, beginning with 25
the first point on the graph, gives a line that falls far below the data points and is therefore unlikely to be the 7 . Sketching a line with this slope, beginning with the 1 first point on the graph, gives a line that is much closer to the location of the data points plotted on the graph, line of best fit. The slope of choice D is 6.74, or around
making choice D the best approximation.
MINI-TEST TWO 1. The correct answer is C. The correlation coefficient of a function is a number between –1 and 1, calculated to represent the linear dependence between two variables. The higher the correlation coefficient, the greater the linear dependence. Since the value is higher for women than men in this data, the correlation between the time spent watching TV and income is stronger for women than for men. 2. The correct answers are Club 1 and $178.80. Find the mean fundraising amount for each club by adding the fundraising totals per club and dividing by the number of fundraisers, 5. The mean fundraising amount for Club 1 is
148 + 200 + 80 + 300 + 166 = $178.80. 5
The mean fundraising amount for Club 2 is
275 + 200 + 75 + 253 + 70 = $174.60. 5
The mean fundraising amount for Club 3 is
60 + 355 + 131 + 220 + 99 = $173.00. 5
The club with the highest average fundraising amount is Club 1, which had a mean fundraising value of $178.80.
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NC Math 1: Wrap-Up
NC math i Wrap-Up
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NC Math 1: Wrap-Up
WORKKEYS OVERVIEW
Wrap-Up
Wrap-Up Remember These Key Test-Taking Techniques •
Plug in Points on a Graph
•
Create a Visual
•
Distributive Property: Show Your Work
•
Use the Answer Choices
•
Word Problem Translation
•
Plug It In
•
Negative Paranoia
•
Don’t Overthink It
•
Process of Elimination
•
Read the Question
•
Equations of Lines
•
Draw It Out
•
Try Numbers
•
Finding Percentages
NOTES:
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NC Math 1: Further Practice
NC math i Further Practice
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NC Math 1: Further Practice FUrther praCtiCe:
praCtiCe set oNe
Practice Set One 1.
2.
Which of the following expressions is equivalent to 5x 2 – (x + 3) 2 + 8x – 12? A.
4x 2 + 2x – 21
B.
4x 2 + 2x – 3
C.
4x 2 + 14x – 21
D.
4x 2 + 14x – 3
The graph represents the change in internal temperature of a pie as it bakes in an oven for one hour. y Pie Internal Temperature
Temperature (°F)
200 150 100 50 0
15
30 45 Time (min)
60
x
Which unit would be appropriate for the rate of change in the graph? A.
degrees hour
B.
degrees minute
C. D.
3.
hours degree minutes degree
Which of the following is a zero of the polynomial expression 4x + 32? A.
–32
B.
–8
C.
4
D.
8
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NC Math 1: Further Practice FUrther praCtiCe: 4.
5.
praCtiCe set oNe
Which equation could be used to find the zeros of the function 15x – 3x – 12? 2
A.
(x – 4)(x + 3) = 0
B.
(x + 5)(x + 3) = 0
C.
(3x + 5)(4x – 3) = 0
D.
(5x + 4)(3x – 3) = 0
Three times Sasha’s age plus two times Paulo’s age equals 52. Paulo’s age is also five times Sasha’s age. How old is Paulo?
6.
Force is related to the mass of two objects by the formula F = •
Gm1 m2 . d2
G is the gravitational constant.
•
m1 and m2 are the mass of two objects.
•
d is the distance between the objects.
Which equation finds d, given F, G, m1, and m2?
7.
A.
d=
F Gm1 m2
B.
d=
Gm1 m2 F
C.
d=
F Gm1 m2
D.
d=
Gm1 m2 F
The table below shows the cost of buying protein bars from a health food store. Protein Bars
30
60
90
120
150
Cost (in dollars)
25
50
75
100
125
What is the meaning of the slope of the linear model for the data? A.
The cost of 5 bars is 1 dollar.
B.
The cost of 5 bars is 6 dollars.
C.
The cost of 6 bars is 1 dollar.
D.
The cost of 6 bars is 5 dollars.
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NC Math 1: Further Practice FUrther praCtiCe:
praCtiCe set oNe
8.
What is the value of the negative zero of the function, g, defined by g(x) = x 2 – 169?
9.
An acrobat’s height as she jumps from a platform above a trampoline, in feet, is modeled by the function h(x) = –x 2 + 5x + 36, where x represents the distance of the acrobat from the platform. How far from the platform will the acrobat be when she reaches the trampoline? A.
0 feet
B.
4 feet
C.
5 feet
D.
9 feet
10. Jessica wants to earn at least $145 dollars from her two jobs next week. She can work 18 hours at most. Her first job pays $8 per hour, and her second job pays $9 per hour. Let b represent the number of hours worked at the first job and s represent the number of hours worked at the second job. Which system of linear inequalities models Jessica’s situation? A.
b + s < 18 8b + 9s > 145
B.
b + s ≤ 18 8b + 9s ≥ 145
C.
b + s ≥ 18 8b + 9s ≤ 145
D.
b + s > 18 8b + 9s < 145
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NC Math 1: Further Practice FUrther praCtiCe:
praCtiCe set explaNatioNs
Practice Set Explanations PRACTICE SET ONE 1. The correct answer is A. Simplify the expression by using FOIL on (x + 3)2, distributing the negative sign, and combining like terms. 5x2 – (x + 3)2 + 8x – 12 5x2 – (x + 3)(x + 3) + 8x – 12 5x2 – (x2 + 6x + 9) + 8x – 12 5x2 – x2 – 6x – 9 + 8x – 12 4x2 + 2x – 21 2. The correct answer is B. In the figure, the y-axis represents temperature (°F) and the x-axis represents time (minutes). The rate of change, or slope, of the linear equation is
degrees change in y or . minute change in x
3. The correct answer is B. Find the zero of the polynomial by setting the expression equal to zero and solving for x. 4x + 32 = 0 4(x + 8) = 0 x+8=0 x = –8 4. The correct answer is D. Factor the function and use the process of elimination to quickly rule out incorrect answer options. Because 15x2 appears in the function, the x terms in the answer choices must multiply to equal this. The only answer choice where this occurs is choice D, where (3x)(5x) = 15x2. 5. The correct answer is 20. Let Sasha’s age be s and let Paulo’s age be p. Use the first sentence to create an equation. Then, use the second sentence to create another equation. 3s + 2p = 52 p = 5s Substitute 5s in for p in the first equation and solve for s.
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NC Math 1: Further Practice FUrther praCtiCe:
praCtiCe set explaNatioNs
3s + 2(5s) = 52 3s + 10s = 52 13s = 52 s=4 Sasha is 4 years old. So Paulo is p = 5(4) = 20 years old. 6. The correct answer is D. Use the formula to solve for d. F=
Gm1m2 d2
Fd = Gm1m2 2
d2 =
Gm1m2 F
d=
Gm1m2 F
7. The correct answer is D. The slope of the linear model is in the form
bars . Take two points from the table, dollars
(25,30) and (50,60), and use the slope formula. m=
60 − 30 30 6 6 bars = = = 50 − 25 25 5 5 dollars
The cost of 6 bars is 5 dollars. 8. The correct answer is –13. The function is a difference of squares. Set the function equal to zero, factor, and solve for x. 0 = x2 – 169 0 = (x + 13)(x –13) x = –13 and 13 The negative zero is –13. 9. The correct answer is D. Set the quadratic function equal to zero. Then, factor and solve for x. –x2 + 5x + 36 = 0 x2 – 5x – 36 = 0 (x – 9)(x + 4) = 0 x = –4, 9 The acrobat cannot reach the trampoline at a negative distance, so ignore x = –4. The acrobat reaches the trampoline when she is 9 feet from the platform.
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