maryland algebra/data analysis - walch 4: graphing linear equations 146 the slope for line a is 1....

EDUCATIONWALCH

Maryland

Algebra/Data AnalysisStudent Resource Book

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Table of Contents

iii

To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

PPaarrtt 11:: AAllggeebbrraa BBaassiiccss Positive and Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Operations with Signed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Multiplying and Dividing Positive and Negative Rational Numbers . . . . . . . . . . . . . 13Algebra Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Exponents and Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

PPaarrtt 22:: SSoollvviinngg EEqquuaattiioonnss aanndd IInneeqquuaalliittiieess Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Solving Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

PPaarrtt 33:: DDaattaa AAnnaallyyssiiss aanndd PPrroobbaabbiilliittyy Frequency Distributions and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Analyzing Data Using Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . 104Analyzing Data with Respect to Measures of Variation . . . . . . . . . . . . . . . . . . . . . . . 109Creating Scatter Plots and Plotting Lines of Best Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 111Experimental and Theoretical Probability and Simulations . . . . . . . . . . . . . . . . . . . 114Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

PPaarrtt 44:: GGrraapphhiinngg LLiinneeaarr EEqquuaattiioonnssThe Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Graphing Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Table of Contents

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Maryland Algebra/Data Analysis Student Resource Book

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PPaarrtt 55:: GGeeoommeettrryyTransformations and Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Measurement: Parts of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Measurement: Finding Circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171Measurement: Area of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173Measurement: Finding Volume and Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Measurement: Volume of a Pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Measurement: Volume of a Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Measurement: Volume of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Surface Area of Cylinders and Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

PPaarrtt 66:: PPoollyynnoommiiaall OOppeerraattiioonnssAdding and Subtracting Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201Multiplying and Dividing Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

PPaarrtt 77:: QQuuaaddrraattiicc EEqquuaattiioonnssQuadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

PPaarrtt 88:: AAppppeennddiicceess Appendix A: Table of Squares and Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Appendix B: Review of Rules and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293Appendix C: Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314Appendix D: Maryland Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Appendix E: Selected Practice Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

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Welcome to the Maryland Algebra/Data Analysis Student Resource Book. This bookwill help you learn how to solve algebraic equations and inequalities, and how toanalyze data. It even includes some geometry. Each lesson builds on what you havealready learned. As you participate in classroom activities and use this book, you willmaster important concepts in algebra and data analysis. This knowledge will help toprepare you for the HSA and for other mathematics assessments and courses.

This book is your resource as you work your way through the Algebra/DataAnalysis course. It includes explanations of the concepts you will learn in class, mathvocabulary and definitions, formulas and rules, real-world examples and questions tothink about, and activities so you can practice the math you are learning. Most of yourassignments will come from your teacher, but this book will allow you to review whatwas covered in class, including terms, formulas, and procedures.

• In PPaarrtt 11:: AAllggeebbrraa BBaassiiccss,, you will learn about negative numbers. You will learnhow to add, subtract, multiply, and divide them. You will also learn how tocombine terms and solve simple equations.

• In PPaarrtt 22:: SSoollvviinngg EEqquuaattiioonnss aanndd IInneeqquuaalliittiieess,, you will learn how to work withalgebraic terms. You will add, subtract, multiply, divide, and simplify terms. Youwill also learn about inequalities, and how to simplify and solve problems usinginequalities.

• In PPaarrtt 33:: DDaattaa AAnnaallyyssiiss aanndd PPrroobbaabbiilliittyy,, you will learn various strategies foranalyzing data, including measures of central tendency, frequency distributions,and measures of variation.

• In PPaarrtt 44:: GGrraapphhiinngg LLiinneeaarr EEqquuaattiioonnss,, you will learn two ways to graph equations for lines on the coordinate plane. You will also learn how to find the slope of a line.

• In PPaarrtt 55:: GGeeoommeettrryy,, you will learn about transformations and measurement,including circumference, area, surface area, and volume of various figures andshapes.

• In PPaarrtt 66:: PPoollyynnoommiiaall OOppeerraattiioonnss,, you will learn about three different types ofalgebraic expressions: monomials, binomials, and trinomials. You will also learnhow to factor different types of algebraic expressions.

• In PPaarrtt 77:: QQuuaaddrraattiicc EEqquuaattiioonnss,, you will learn two ways to solve quadraticequations. You will also learn how to use the quadratic formula to solve wordproblems.

To the Student

To the Student

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Maryland Algebra/Data Analysis Student Resource Book

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Each lesson is made up of short sections that explain important concepts. Each of these sections is followed by a few problems to help you practice what you havelearned. The book also has some special features to make learning easier. “Tips” giveyou hints on ways to master the ideas and facts in the text. “Real Life” sections showyou how the skills you are learning are used in the world outside the classroom.“Think About It” questions ask you to look at algebra in new ways. “Try This” sectionsfeature quick problems that allow you to practice the new techniques you have justlearned. The “Words to Know” section at the beginning of most lessons includesimportant terms introduced in the lesson. Finally, the “Review of Rules and Formulas”at the back of the book includes the rules, formulas, and other important informationintroduced in the book.

As you move through your Algebra/Data Analysis course, you will become a moreconfident and skilled mathematician. We hope this book will serve as a usefulresource as you learn.

PGC MDA PT1 NEW 7/9/08 5:01 PM Page vi

Slope

Goal: To learn to find the slope of a line, and to use slope to graph lines tounderstand ratios and proportional relationships

WWOORRDDSS TTOO KKNNOOWW

rraattiioo the relation between two quantities; can be expressed in words,

fractions, decimals, or percents (Example: The ratio given when

a team wins 4 out of 6 games can be expressed as 4:6, four out

of six, or .)

ssllooppee the steepness of a line expressed as a ratio, using any two pointson the line (Slope is represented by the symbol m.)

uunnddeeffiinneedd ssllooppee the slope of a vertical line

zzeerroo ssllooppee the slope of a horizontal line

Finding Linear SlopeSSllooppee is the steepness of a line. It is the number of y-units a line goes up or down when it moves overone x-unit.

Look at the coordinate plane below.

SlopePART 4 • GRAPHING LINEAR EQUATIONS

145

yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

AABB

4

6

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Part 4: Graphing Linear Equations

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The slope for line A is 1. This means that as the line moves over one space, it movesup one space. For example, as the line moves from 0 to 1 on the x-axis, it rises from 0to 1 on the y-axis. This means that line A has a slope of 1.

The slope for line B is much steeper. As it moves from 0 to 1 on the x-axis, it risesfrom 0 to 3 on the y-axis. This means that line B has a slope of 3.

The symbol for slope is m. For line A, m = 1. For line B, m = 3.

Slope is a ratio. This means that it is a relationship between two things. What twothings about a line must be related to each other to find the line’s slope?

Think about a relationship where when one value changes, another value changesproportionally. For example, if you get $3.50 per day for lunch, then the amount youget for the week is proportional to the number of days that you’ll be buying lunch. Ifthere are 5 school days, then it will be $3.50 � 5, but if there are only 4 days (oneholiday) then it will be $3.50 � 4. The relationship remains constant—$3.50 per day,regardless of the number of days.

TThhee FFoorrmmuullaa ffoorr FFiinnddiinngg SSllooppee

You can find the slope for any line if you know the coordinates of two points on theline. The formula below is the formula for finding slope.

In the formula for finding slope, y2 means the y-coordinate of the second point ofthe line whose slope you are trying to find. In the formula, y1 means the y-coordinateof the first point of the line whose slope you are trying to find. Also, x2 is the x-coordinate of the second point of the line. Finally, x1 is the x-coordinate of the first point of the line.

EExxaammppllee

Use the formula to find the slope of a line with the points (2, 1) and (6, 4).

my y

x x=

−( )−( )

2 1

2 1

11.. Choose which point you want to be point 1 and which point you wantto be point 2. It doesn’t matter which you choose. But it is importantto remember what you have chosen so you don’t mix up the formula.

point 1 = (2, 1)

point 2 = (6, 4)


Slope

147

PPoossiittiivvee aanndd NNeeggaattiivvee SSllooppee

Slope can be positive or negative.

• If a line moves up as it goes from left to right across the page, it has a positive slope.

• If a line moves down as it goes from left to right across the page, it has a negative slope.

Look at the diagram to the right.

• Line A has a positive slope.

• Line B has a negative slope.

my y

x x=

−( )−( ) =

−−

=2 1

2 1

4 1

6 234

3

4

yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

BBnneeggaattiivvee ssllooppee

AAppoossiittiivvee ssllooppee

TTRRYY TTHHIISS

It doesn’t matter which point you call point 1 and which point youcall point 2. Try doing the problem above again. This time, call (6, 4)point 1 and (2, 1) point 2. Is the slope the same?

22.. Put the coordinates in the two ordered pairs into the formula for slope.

33.. The slope of the line is . This means that for every 4 units the line

moves across, it moves 3 units up.

ppoossiittiivvee ssllooppee nneeggaattiivvee ssllooppee



148

To find the slope of line A, take (3, 2) and (5, 4) as points on the line.

The slope of line A is positive.

To find the slope of line B, you could take (1, 0) and (0, 1) as points on the line.

The slope of line B is negative.

m =−( )−( ) = =

4 2

5 322

1

m =−( )−( ) =

−= −

1 0

0 111

1

yy

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-10

-10 10

AA

BB

If a line moves up as it goes from left to right across the page, it has a positiveslope. If a line moves down as it goes from left to right across the page, it hasa negative slope. In the following diagram, Line A has a positive slope, andLine B has a negative slope.


Slope

149

ZZeerroo aanndd UUnnddeeffiinneedd SSllooppee

Sometimes when you calculate the slope of a line, you’ll find that it equals zero. Whatdo you think that means about the line?

• When the slope of a line equals zero, it is a horizontal line.

• Remember the equation of a line is y = mx + b, where m stands for the slope.

• If you use zero for m, you get y = b, which is the standard form for a horizontal line.

• A vertical line does not have a slope. It’s not that you don’t know what the slope is equal to; it doesn’t exist. There isn’t any slope. This is called anuunnddeeffiinneedd ssllooppee..

Draw a horizontal line where the slope = 0. If the y value is the same for everyordered pair on the line, then the slope is 0.

Now draw a vertical line where there is no pattern or progression for the values of xon the line. Therefore, the slope is undefined.

yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

ssllooppee == 00

nnoo ssllooppee

((uunnddeeffiinneedd ssllooppee))

((zzeerroo ssllooppee))



150

GGrraapphhiinngg aa LLiinnee WWhheenn YYoouu KKnnooww tthhee SSllooppee aanndd OOnnee PPooiinntt

The slope of a line tells you how many units the line moves up or down on the y-axisas it moves across the x-axis. You can use the slope of a line to help you graph the lineon a coordinate plane. All you need to know is the line’s slope and the coordinates ofone point on the line. You then use this information to find the coordinates for asecond point on the line. Once you have two points, you can draw the line.

Look at the following examples.

EExxaammppllee 11

Graph a line with a point that has the coordinates (0, –4) and with a slope m = 8.

11.. Graph the point (0, –4) on a coordinate plane.

22.. The slope is a whole number. Make it into an improper fraction byplacing it over 1.

yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

((00,, ––44))

m =81


Slope

151

Note: Use the same steps even if the slope is negative or a fraction (or both).

33.. Find the x-coordinate for the second point on the line. To do this, takethe denominator (the bottom number) of the fraction for the slope.Add it to the x-coordinate of the point you graphed in Step 1.

x = 0 + 1 = 1

44.. Now find the y-coordinate for the second point in the line. To do this,take the numerator (the top number) of the fraction for the slope. Addit to the y-coordinate of the point you graphed in Step 1.

y = –4 + 8 = 4

The coordinates for the second point on the line are (1, 4).

55.. Graph the second point. Draw a straight line through the two points.yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

((00,, ––44))

((11,, 44))



152

EExxaammppllee 22

Graph a line that has the point (–4, 2) and the slope . m = −12

11.. Graph the point (–4, 2) on a coordinate plane.

22.. Find the x-coordinate of the second point. Add the denominator ofthe fraction for the slope to the x-coordinate of the point you graphedin Step 1. Find the y-coordinate of the second point.

Add the numerator of the fraction for the slope to the y-coordinate ofthe point you graphed in Step 1.

x = –4 + 2 = –2

y = 2 + –1 = 1

(–2, 1)

yy

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-10

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((––44,, 22))

33.. Graph the second point.


Slope

153

SSeeee pprraaccttiicceess oonn ppaaggeess 116600––116622..

44.. Draw a straight line through the two points.yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

((––44,, 22)) ((––22,, 11))



160

Finding Linear SlopeFind the slope for the line below. Show your work.

PART 4 • GRAPHING LINEAR EQUATIONSSlope

PRACTICE

m = _________

yy

xx-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

109876543210

-1-2-3-4-5-6-7-8-9

-10

-10 10

((11,, 44))

((00,, ––44))


Practice: Slope

161

Identifying Type of SlopeLook at the slope of each line below. Identify whether it is positive, negative, zero, or undefined.

11.. 33..

22.. 44..


PRACTICE

1

1 2 3 4 50

2

3

4

5

1

1 2 3 4 50

2

3

4

5

–1–2–3–4–5

1

0

2

3

4

5

–1

–1

–2

–2

–3

–3

–4

–4

–5

–5

0

PGC MDA PT4 7/9/08 5:06 PM 1


162

Graphing EquationsChoose the correct answer for each question below.

11.. If y = 0 in the equation below, what does x equal?

4x + 2y = 8

aa.. –1 cc.. –2

bb.. 1 dd.. 2

22.. Which line on the graph below represents the following equation?

–x + 2y = 4

aa.. line A cc.. line C

bb.. line B dd.. line D


PRACTICE

yy

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A

B

C

D


maryland algebra/data analysis - walch 4: graphing linear equations 146 the slope for line a is 1....

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