algteks

7/28/2019 algteks

http://slidepdf.com/reader/full/algteks 1/62

THE CHARLES A. D ANA CENTER

THE UNIVERSITY OF TEXAS AT AUSTIN

7/28/2019 algteks


Algebra TEKS

AssessmentSupplement

a Texas SSI Publication

7/28/2019 algteks


This publication is based upon work supported by the National Science Foundation

under Cooperative Agreement #ESR-9250036. Any opinions, findings, and conclu-

sions or recommendations expressed in this material are those of the author(s) and

do not necessarily reflect the views of the National Science Foundation.

Copyright © 1998 by Texas Statewide Systemic Initiative.

The Charles A. Dana Center, The University of Texas at Austin, Austin, Texas.

Permission to photocopy is granted for educational purposes. Permission must be

sought for commercial use of any or all of this document.

This book was designed and produced by John Budz & Vee Sawyer of

Firefly Multimedia in Austin, Texas ([email protected]).

It was printed by Mpress, Inc. of Austin, Texas.

ii

7/28/2019 algteks


Algebra Action TeamASSESSMENT WORKING GROUP

Barrie Madison, Chair Lewisville ISD

David D. Molina, Texas SSI Staff Liaison The Charles A. Dana Center

Bill Hadley, Consultant Pittsburgh Public Schools

Robbie Bonneville La Joya ISD

Cindy Boyd Abilene ISD

Armando Cisneros Austin ISD

Lucy Flores-Sanchez Edinburg ISD

Susan T. Funk Ysleta ISD

Diana Garcia United ISD

Juan Manuel Gonzalez Laredo ISD

Mary Alice Hatchett Round Rock ISD

Sallie Langseth Deer Park ISD

Lori Mitchell Lewisville ISD

Diane Reed Ysleta ISD

Jane Silvey ESC Region VII

Liz Smith Martinsville ISD

Susan Thomas Alamo Heights ISD

Terry Whistler Austin ISD

iii

7/28/2019 algteks


table of contents

Introduction l v

Texas Essential Knowledge and Skills for Algebra I l viii

Foundations for Functions

b1 Understanding Functions l 2

b2 Properties and Attributes of Functionsl

6

b3 Representing Situations Using Algebra l 10

b4 Using Algebraic Skills to Solve Problems l 14

Linear Functions

c1 Representations of Linear Functions l 18

c2 Slope and Intercepts l 22

c3 Formulating and Solving Equations and Inequalities l 28

c4 Formulating and Solving Systems of Linear Equations l 32

Quadratic and Other Nonlinear Functions

d1 Graphs and Parameters of Quadratic Functions l 36

d2 Solving Quadratic Equations l 40

d3 Non Linear and Non Quadratic Functions l 44

iv

7/28/2019 algteks


introduction

The state goal that all students successfully complete Algebra 1 hastremendous implications and impact on how teachers assess student mas-tery of Algebra 1 summatively as well as day-to-day. Created with “Alge-

bra for All” in mind, the Algebra 1 Texas Essential Knowledge and Skills (TEKS)and their supporting Performance Descriptions have determined the content of the course. The potential importance of the Algebra 1 End-of-Course Exam oncampus/district accountability has focused attention on the assessment of stu-dent performance. The purpose of this document is to provide a model of howto assess the thinking skills addressed in the Texas Essential Knowledge and

Skills in Algebra 1, and in doing so to be confident that students will be suc-cessful on the Algebra 1 End-of-Course Exam.

With increased opportunities for professional development, many teachersare implementing lessons emphasizing concept development and relevance of content. Yet, assessment of these lessons may remain traditionally focused onskills and concepts that may not require students to prove their ability to solveproblems requiring high levels of thinking. This document attempts to bridgethis gap by providing sample assessments that require students not to demon-strate their ability to only manipulate symbols, but also to use and/or apply im-portant mathematical concepts. The assessment items support the implementa-tion of the Texas Essential Knowledge and Skills and their PerformanceDescriptions and are connected to the Algebra 1 End-of-Course Exam. The doc-ument also includes recommendations for optimizing Algebra 1 End-of-Courseresults and a bibliography of additional resources for teachers.

This document was written and tested by the Assessment Working Group of the Texas SSI Algebra Action Team. The Team consisted of classroom teachers,school administrators, regional service center representatives, district facilita-tors, and higher education personnel.

Purpose of the Assessment Document

For teachers, the document provides

• examples of TEKS-based assessment items for use in Algebra 1,

• sample assessment items that can be an integral part of teaching,learning, and student evaluation,

v

7/28/2019 algteks


• sample assessment items that complement and enhance teacher under-standing of the Algebra 1 TEKS,

• sample performance-based assessment items,

• sample assessment items for teachers to use as models in creating similar

assessment items, and

• connections between the Algebra 1 TEKS and the Algebra 1End-of-Course Exam.

Students successful on the sample assessment items in this document

• will demonstrate a deeper understanding of the mathematics,

• will more readily make connections within mathematics and to otherdisciplines,

• will perform better on standardized tests, such as the Algebra 1End-of-Course Exam, and

• will be able to solve a problem in multiple ways.

Document Format

The Assessment Document follows the sequence of the Algebra 1 TEKS. Foreach Knowledge and Skill statement, there is a comprehensive/global assess-ment item. For each Performance Description, there is a shorter item that sup-

ports the type of thinking required for the student to be successful on theKnowledge and Skill problem.

Finally, Algebra 1 End of Course questions from released tests are con-nected to each Knowledge and Skill statement. If students can successfullycomplete the suggested assessments for the complete TEKS statement (Knowl-edge and Skill and the Performance Descriptions), they should be prepared forthe corresponding End-of-Course Exam questions.

Use of the Document

This document is intended to be used by individual teachers and/or cam-pus teams as a model for the kind of assessments that are appropriate in today’sAlgebra 1 classroom. It also can be useful in the professional development of teachers and can be incorporated as part of the appropriate TEXTEAMS trainingsuch as The Algebra Institute.

vi

7/28/2019 algteks


Recommendations for OptimizingEnd-of-Course Results

In the Algebra 1 Classroom,

• use graphing calculators as an integral component of the course.

• teach test-taking skills—especially how to complete grid-in answers.

• use sample assessments from this document as part of ongoing classroomassessment.

• use items from released tests as part of ongoing classroom assessment.

For the testing environment,

• have the students’ algebra teachers administer the examination.

• do not set real or implied time limits.

• have a tutorial session immediately before the test that addresses BIGMATHEMATICAL IDEAS and test-taking tips.

• provide healthy snacks before students enter to take the examination.

• encourage frequent breaks.

• encourage students to answer items in the test booklet and then transferanswers to the bubble sheet.

• have teachers verify that the students have recorded an answer to everyquestion before accepting students’ answer sheets.

Bibliography

Mathematics Assessment, from National Council of Teachers of Mathematics.

Waves of Learning Issue V Academic Assessment, by Carolyn S. Carr, Ph.D., EasternWashington University, from Texas Assocation for Supervision and CurriculumDevelopment.

Assessing Student Outcomes, by Robert J. Marzano, Debra Pickering, and JayMcTighe, from Association for Supervision and Curriculum Development.

vii

7/28/2019 algteks


Texas Essential Knowledgeand Skills for Algebra 1

From Texas Administrative Code

Chapter 111. Texas Essential

Knowledge and Skills in

Mathematics

Subchapter C. High School

§111.32 Algebra 1 (one credit)

(a) BASIC UNDERSTANDINGS.

(1) Foundation concepts for high school mathematics. As presented in Grades K-8,

the basic understandings of number, operation, and quantitative reasoning; pat-

terns, relationships, and algebraic thinking; geometry; measurement; and proba-

bility and statistics are essential foundations for all work in high school mathe-

matics. Students will continue to build on this foundation as they expand their

understanding through other mathematical experiences.

(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical

role in algebra; symbols provide powerful ways to represent mathematical situa-

tions and to express generalizations. Students use symbols in a variety of ways to

study relationships among quantities.

(3) Function concepts. Functions represent the systematic dependence of one quan-

tity on another. Students use functions to represent and model problem situa-

tions and to analyze and interpret relationships.

(4) Relationship between equations and functions. Equations arise as a way of ask-

ing and answering questions involving functional relationships. Students work in

many situations to set up equations and use a variety of methods to solve these

equations.

viii

7/28/2019 algteks


7/28/2019 algteks


(2) The student uses the properties and attributes of functions.

FOLLOWING ARE PERFORMANCE DESCRIPTIONS.

(A) The student identifies and sketches the general forms of linear (y = x )

and quadratic (y = x 2) parent functions.

(B) For a variety of situations, the student identifies the mathematicaldomains and ranges and determines reasonable domain and range

values for given situations.

(C) The student interprets situations in terms of given graphs or creates

situations that fit given graphs.

(D) In solving problems, the student collects and organizes data, makes

and interprets scatterplots, and models, predicts, and makes deci-

sions and critical judgments.

(3) The student understands how algebra can be used to express generalizationsand recognizes and uses the power of symbols to represent situations.


(A) The student uses symbols to represent unknowns and variables.

(B) Given situations, the student looks for patterns and represents gener-

alizations algebraically.

(4) The student understands the importance of the skills required to manipulate

symbols in order to solve problems and uses the necessary algebraic skills re-

quired to simplify algebraic expressions and solve equations and inequalities inproblem situations.


(A) The student finds specific function values, simplifies polynomial ex-

pressions, transforms and solves equations, and factors as necessary

in problem situations.

(B) The student uses the commutative, associative, and distributive

properties to simplify algebraic expressions.

x

7/28/2019 algteks


(c) LINEAR FUNCTIONS:KNOWLEDGE AND SKILLS AND PERFORMANCE DESCRIPTIONS.

(1) The student understands that linear functions can be represented in different

ways and translates among their various representations.


(A) The student determines whether or not given situations can be rep-

resented by linear functions.

(B) The student determines the domain and range values for which lin-

ear functions make sense for given situations.

(C) The student translates among and uses algebraic, tabular, graphical,

or verbal descriptions of linear functions.

(2) The student understands the meaning of the slope and intercepts of linear func-tions and interprets and describes the effects of changes in parameters of linear

functions in real-world and mathematical situations.


(A) The student develops the concept of slope as rate of change and de-

termines slopes from graphs, tables, and algebraic representations.

(B) The student interprets the meaning of slope and intercepts in situa-

tions using data, symbolic representations, or graphs.

(C) The student investigates, describes, and predicts the effects of

changes in m and b on the graph of y = mx + b .

(D) The student graphs and writes equations of lines given characteris-

tics such as two points, a point and a slope, or a slope and y -inter-

cept.

(E) The student determines the intercepts of linear functions from

graphs, tables, and algebraic representations.

(F) The student interprets and predicts the effects of changing slope and

y -intercept in applied situations.

(G) The student relates direct variation to linear functions and solves

problems involving proportional change.

xi

7/28/2019 algteks


(3) The student formulates equations and inequalities based on linear functions,

uses a variety of methods to solve them, and analyzes the solutions in terms of

the situation.


(A) The student analyzes situations involving linear functions and formu-lates linear equations or inequalities to solve problems.

(B) The student investigates methods for solving linear equations and in-

equalities using concrete models, graphs, and the properties of

equality, selects a method, and solves the equations and inequalities.

(C) For given contexts, the student interprets and determines the reason-

ableness of solutions to linear equations and inequalities.

(4) The student formulates systems of linear equations from problem situations,

uses a variety of methods to solve them, and analyzes the solutions in terms of

the situation.


(A) The student analyzes situations and formulates systems of linear

equations to solve problems.

(B) The student solves systems of linear equations using concrete mod-

els, graphs, tables, and algebraic methods.

(C) For given contexts, the student interprets and determines the reason-

ableness of solutions to systems of linear equations.

(d) QUADRATIC AND OTHER NONLINEAR FUNCTIONS:KNOWLEDGE AND SKILLS AND PERFORMANCE DESCRIPTIONS.

(1) The student understands that the graphs of quadratic functions are affected by

the parameters of the function and can interpret and describe the effects of

changes in the parameters of quadratic functions.


(A) The student determines the domain and range values for which qua-dratic functions make sense for given situations.

(B) The student investigates, describes, and predicts the effects of

changes in a on the graph of y = ax 2.

xii

7/28/2019 algteks


(C) The student investigates, describes, and predicts the effects of

changes in c on the graph of y = x 2 + c .

(D) For problem situations, the student analyzes graphs of quadratic

functions and draws conclusions.

(2) The student understands there is more than one way to solve a quadratic equa-

tion and solves them using appropriate methods.


(A) The student solves quadratic equations using concrete models, ta-

bles, graphs, and algebraic methods.

(B) The student relates the solutions of quadratic equations to the roots

of their functions.

(3) The student understands there are situations modeled by functions that are nei-ther linear nor quadratic and models the situations.


(A) The student uses patterns to generate the laws of exponents and ap-

plies them in problem-solving situations.

(B) The student analyzes data and represents situations involving in-

verse variation using concrete models, tables, graphs, or algebraic

methods.

(C) The student analyzes data and represents situations involving expo-

nential growth and decay using concrete models, tables, graphs, oralgebraic methods.

Source:The provisions of this §111.32 adopted to be effective

September 1, 1998, 21 TexReg 7371.

xiii

7/28/2019 algteks


7/28/2019 algteks


Algebra TEKS

Assessment

Supplement

7/28/2019 algteks


b 1 Foundations for Functions

“The student understands that a functionrepresents a dependence of one quantity onanother and can be described in a variety of ways.”

With a complete understanding of this Knowledge and Skillsstatement, students should be able to perform the followingassessment task.

The time it takes a student to walk home from school is related tothe distance between home and school.

1. Identify which quantity is independent and which quantity is dependent.

2. Sketch a reasonable graph that describes this situation.

3. If Jackie walks at a rate of 3 miles per hour, complete the table shown.

Distance in Miles Time in Minutes

3 60

5 ?

? 120

7 ?

? ?

4. Graph the data shown in the table.

5. Write an equation to represent the relationship.

6. If Jackie increases her rate by 1 mph, how far was she from home if it took herthree hours to walk that distance?

2 Algebra TEKS Assessment Supplement

task

7/28/2019 algteks


end of course exam questions

When students can perform a task like this, they can

answer EOC questions like...

At the time this document was published, there were not any released items thatwould apply.

Foundations for Functions 3

7/28/2019 algteks


performance questions

Performance Descriptions and the type of assessment

items students should be able to perform

(b)(1)(A) The student describes independent and dependent quantities in functionalrelationships.

1. For a given job, the number of hours worked and the amount of moneyearned are related. Identify which quantity is independent and which isdependent. Defend your answer. Note: either quantity could be dependentdepending on the student’s response.

(b)(1)(B) The student gathers and records data, or uses data sets, to determine func-tional (systematic) relationships between quantities.

1. Using a metric tape, measure the diameter and circumference of at least 5different circles. Record the data in a table and describe the functionalrelationship between the 2 quantities.

(b)(1)(C) The student describes functional relationships for given problem situationsand writes equations or inequalities to answer questions arising from the sit-uations.

1. Membership in a CD club is $5.00 and each CD costs $10.95. Alex hassaved $85.00. Write an inequality that he could use to find the number of CDs he can purchase and not exceed his savings.


7/28/2019 algteks


(b)(1)(D) The student represents relationships among quantities using concrete mod-els, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

1. Write a story or describe a situation that the graph below could describe.Label the axes to fit your story.

(b)(1)(E) The student interprets and makes inferences from functional relationships.

1. Three students drew the graphs below to represent the relationshipbetween the number of 32 cent stamps purchased and the total cost.

Which graph is correct and why?

stamps stamps stamps

c o s t

c o s t

c o s t


7/28/2019 algteks



“The student uses the properties andattributes of functions.

”


The class is assigned the task of rolling marbles down a ramp from a

height of 10 cm, 15 cm, 30 cm, and 50 cm. The marbles must bereleased from the edge of the ramp. Students measure the distancethe marble rolls once it leaves the end of the ramp.

1. Make a table of values for the distance the marble travels, in cm, whenreleased from the various heights. Graph the data.

2. Find the domain and range for this situation.

3. Predict the distance the marble will travel if released from a height of 8 cm,12 cm, and 20 cm.

4. Make a table of the values from the time the marble is released until it comes

to a stop when releasing the marble from various heights. Graph the data.5. Find the domain and range for this situation.

6. Predict the time it will take the marble to come to a stop if released from aheight of 8 cm, 12 cm, and 20 cm.

7. Explain what you think will happen if ping pong balls are used instead of marbles.

8. What do you think will happen if marbles are rolled down a ramp to a carpet-ed surface? a tiled floor? dirt?


task

7/28/2019 algteks





1. What is the range of the function

f (x ) = x 2 – 3

when the domain is {–5, –3, – 1}?

A {7, 3, –1}

B {–28, –12, –4}

C {–13, –9, –5}

D {22, 6, –2}E {–7, –3, 1}

2. The graph shows the relationship between the number of boxes of candy sold and the amount of profit made.

How many boxes of candy must be sold to yield a $250 profit?

F 50

G 100

H 125

J 175

K 250

0 20 40 60 80 100

$10

30

50

70

90

profit

no. of boxes


7/28/2019 algteks


performance descriptions



(b)(2)(A) The student identifies and sketches the general forms of linear (y = x ) andquadratic (y = x 2) parent functions.

1. Which of the following is a graph of y = x 2

A B

C D

(b)(2)(B) For a variety of situations, the student identifies the mathematical domains andranges and determines reasonable domain and range values for given situations.

1. This graph represents a diver’s distance from the surface of the water at agiven time.

State the domain and range of this graph.

20 40 6080100

20

4060

80

100

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-2-1

1

2

3

4

-4 -3 -2 -1 1 2 3 4

-4

-3

-1

1

2

3

4

-2


7/28/2019 algteks


(b)(2)(C) The student interprets situations in terms of given graphs or creates situa-tions that fit given graphs.

1. Which of the situations below would fit this graph?

Situation 1: In Alaska the temperature was 4 degrees below zero (0) andwas increasing at a rate of 3 degrees per hour.

Situation 2: Sue was putting books on the shelf where 3 books were

already stacked. She put four more books on the shelf every five minutes.Situation 3: Jim has 4 sets of baseball cards. He plans to add 3 new setsevery week.

(b)(2)(D) In solving problems, the student collects and organizes data, makes and interpretsscatterplots, and models, predicts, and makes decisions and critical judgements.

You have been given 5 different squares. Measure the length of the diagonaland the length of a side in metric (cm) and record the data in the table below.

Square Length of Length of

Side

Diagonal

A B C D E

1. Plot the data. Draw a reasonable line of best fit for this data.

2. Enter the data in a graphing calculator and find the linear regression equation.

3. If you have a square with a diagonal of 150 cm, predict the length of a side.

4. Predict the length of the diagonal of a square if the length of a side is19 cm.

5. What does the slope of the line represent in this situation?

-3 -2 -1 1

-6

-4

-2

2

4

6

8


7/28/2019 algteks


b3 Foundations for Functions

“The student understands how algebra canbe used to express generalizations andrecognizes and uses the power of symbols torepresent situations.”


The following chart was created to describe how toothpicks canbe used to build a row of squares. Using the pictures in the visualcolumn, complete the chart.

# of Total #

Squares Visual Process Written Description of Toothpicks

1 4 it takes 4 toothpicks 4

to make 1 square 2 4 + 3 it takes 7 toothpicks 7

to make 2 squares

3 ? ? ?

... ? ? ?

... ? ? ?

12 ? ? ?

n ? ? ?


task

7/28/2019 algteks





1. Mark earns $4.50 per hour. He worked 3 hours more this weekthan last week. If h is the number of hours he worked last week,which equation shows the amount, t , that Mark earned for bothweeks?

F t = 2(4.50)(h + 3)

G t = 4.50h + 3

H t = 4.50(2h + 3)

J t = 4.50(h + 3)

K

2. Everett works at the lake renting boats to visitors. Last weekend herented 4 more sailboats than rowboats. He rented 10 boats in all.Which equation could be used to find the number of rowboats, r ,he rented last weekend?

A r + 4 = 10

B 4r = 10

C 4(r + 1) = 10

D r – 4 = 10

E r + (r + 4) = 10

4.50(h ϩ 3)t ϭ ᎏᎏᎏᎏᎏ᎑

2


7/28/2019 algteks





(b)(3)(A) The student uses symbols to represent unknowns and variables.

John inherits his father’s baseball card collection containing 100 cards andjoins a card collection club that sends him 5 new cards each month.

1. Write an expression telling how many cards he will have in his collectionafter m months.

2. Write an equation to find out in how many months John will have 195

cards.


7/28/2019 algteks


(b)(3)(B) Given situations, the student looks for patterns and represents generaliza-tions algebraically.

1. A florist designs flower arrangements using roses and carnations. A smallarrangement uses 1 rose surrounded by 8 carnations. The mediumarrangement uses 2 roses surrounded by 10 carnations. The large arrange-

ment uses 3 roses surrounded by 12 carnations. If the pattern continues,complete the table below.

Arrangement Roses Carnations

small 1 8

medium 2 10

large 3 12

extra large 4 ?

jumbo ? 16

... super size 15 ?

Texas size 20 ?


7/28/2019 algteks



“The student understands the importanceof the skills required to manipulatesymbols in order to solve problems and usesthe necessary algebraic skills required tosimplify algebraic expressions and solveequations and inequalities in problemsituations.”


A rectangle has a length that is 6 m longer than the width.

1. Write an equation that represents the perimeter in terms of w . Write anequation that represents the area in terms of w .

2. If the width is 50 m, what is the area?

3. For what value(s) of w will the area be equal to 520 m2?

4. For what value(s) of w will the perimeter be less than 60 m?

5. For what value of w is the area equal to the perimeter?


w + 6

w

task

7/28/2019 algteks





1. Nick’s rectangular bedroom has a length of (2x – 5) and a width of (x + 1). Which equation describes the area, A, of Nick’s bedroomin terms of x ?

F A = 3x – 4

G A = 6x – 8

H A = 2x 2 – 3x – 5

J A = 2x 2 – 5x + 1

K A = 2x 2 –7x – 5

2. The sides of a triangle have lengths 2x – 1, x + 3, and 3x – 4.Which of the following describes the perimeter, P , of the triangle interms of x ?

A P = 5x – 8

B P = 6x + 9

C P = 6x 3 – 12

D P = 6x – 2

E P = 5x – 1


7/28/2019 algteks


3. The Meadowbrook High School band rented a bus for a trip to afootball game. The bus company charged $475, plus $0.45 permile over 200 miles. If the bus trip cost $529, how many mileswas the trip?

4. What is the solution to the equation3(2x – 1) – 4x = –5

5. Stan is carrying a load of 50 boxes of books in his truck. Some of the boxes weigh 20 pounds each, and the rest of the boxes weigh10 pounds each. If all the boxes weigh a total of 900 pounds, howmany 20-pound boxes are in Stan’s load?


7/28/2019 algteks





(b)(4)(A) The student finds specific function values, simplifies polynomial expressions,transforms and solves equations, and factors as necessary in problemsituations.

1. The number of degrees for the sum, s , of the interior angles of a polygonwith n sides is s = 180(n – 2). How many sides would a polygon have if the sum of the angles is 1800 degrees?

(b)(4)(B) The student uses the commutative, associative, and distributive properties tosimplify algebraic expressions.

1. In mathematics class the teacher gave the following problem.

Find the area of the shaded region.

Sally’s answer was (n + 2)2 – n 2 and John’s answer was 2(n + 2) + 2n .Show that both answers are correct.


n + 2 n

n

n + 2

7/28/2019 algteks


c 1 Linear Functions

“The student understands that linearfunctions can be represented in differentways and translates among their variousrepresentations.”


One of the following situations represents a linear function.

1. Situation 1: “Refrigerator Jones,” the 346 pound fullback for the Texas Oilers,was told to lose 70 pounds. He was successful at this. However, during the 5weeks of his summer vacation, he gained weight at the rate of 3

᎑4of a pound

every 2 days. Which table represents situation 1?

Vacation Day Weight Vacation Day Weight

0 276 0 346

2 276.75 2 346.75

4

277.5 4

347.5

6 278.25 6 348.25

2. Situation 2: A biology class is studying fruit flies for four days. They start with10 fruit flies. Fruit fly populations doubled every 3 hours. Which table repre-sents situation 2?

Hours Flies Hours Flies

0 10 0 10

3 20 3 20

6 40 6 30

9 80 9 40

3. Which situation produced a linear function? Graph the function.

4. What is a reasonable domain and range for the linear function?


task

7/28/2019 algteks





1. What is the range of the function

f (x ) = x 2 – 3

when the domain is {–5, –3, –1}?

A {7, 3, –1}

B {–28, –12, –4}

C {–13, –9, –5}

D {22, 6, –2}E {–7, –3, 1}

2. When Pedro arrived at his cousin’s home in North Dakota, therewere 5 inches of snow on the ground. The next day snow startedfalling again at a rate of 2 inches per hour. The graph below showsthe amount of snow on the ground.

The equation is

s = 2h + 5

where s is the total amount of snow on the ground and h is thenumber of hours. Which graph on the following page best repre-sents the total amount of snow on the ground if it had snowed at arate of 3 inches per hour?

0 1 2 3 4 5 6

1

3

5

7

9

11

snow on ground

numberof

inches

number of hours

Linear Functions 19

7/28/2019 algteks


A B

C D

E

0 1 2 3 4 5 6

2

4

6

8

10

12

0 1 2 3 4 5 6

1

3

5

7

9

11

0 1 2 3 4 5 6

1

3

5

7

9

11

0 1 2 3 4 5 6

1

3

5

7

9

11

0 1 2 3 4 5 6

1

3

5

7

9

11


7/28/2019 algteks





(c)(1)(A) The student determines whether or not given situations can be representedby linear functions.

1. Juanita put $2.00 each week into an account for graduation expenses. Herfather occasionally adds $2.00. How could you change this situation so itcould be represented by a linear function?

(c)(1)(B) The student determines the domain and range values for which linear func-tions make sense for given situations.

1. John borrowed $40 and is paying it back at a rate of $4 per week. Hemakes the following table and uses the equation m = 4w , where m is theamount of money and w is the number of weeks.

Time in weeks Money owed

10 40

9 36

8 327 28

6 24

What is the domain in this situation? What is the range?

(c)(1)(C) The student translates among and uses algebraic, tabular, graphical, or verbaldescriptions of linear functions.

1. Given the function y = 2x + 3, describe a situation that could be repre-sented by the function.

Linear Functions 21

7/28/2019 algteks



“The student understands the meaning of the slope and intercepts of linear functionsand interprets and describes the effects of changes in parameters of linear functionsin real-world and mathematicalsituations.”


An aircraft begins its descent from an altitude of 1000 ft at a rate of 120 ft/min.

1. Write an equation that represents the altitude of the aircraft at time (t ).

2. Sketch the graph that represents this problem situation.

3. What is the slope of this line and what does it represent in this situation?

4. Find the x -intercept and explain what it means in the problem situation.

5. Sketch a graph of the glide path of a Cesna aircraft with an initial altitude of 3000 ft and a rate of descent of 60 ft/min.

6. What is the slope of this line and what does it represent in this situation?

7. After how many minutes will the first aircraft touch down on the ground?

8. If a DC-10 landed in 20 minutes and its rate of descent was 90 ft/min, atwhat altitude did the aircraft begin its descent?

9. If the equation A = 40t + 3600 is given as the model for the altitude of anMD-80 at time (t ), describe what is occurring.


task

7/28/2019 algteks





1. Which point lies on the line having as its equation 2x + y = 8?

F Point K

G Point L

H Point M

J Point N

K Point P

2. The graph of the function y = 2᎑3x – 1 is shown below. If the line is shifted

3 units up, which of the following would describe the new line?

F y = 5᎑3x – 1

G y = 2᎑3x + 2

H y = 2᎑3x (x + 2)

J y = 2᎑3x + 3

K y = 2᎑3x (x + 3)

-4 -3 -2 -1 1 2 3

-4

-3

-2

-1

1

2

3

4

-8 -6 -4 2 4 6 8

-8

-6

-4

-2

2

4

6

8

Linear Functions 23

7/28/2019 algteks


3. Which equation describes a line parallel to the graph of y = –2x + 3?

A y = – 1᎑2x – 4

B y = –2x – 1

C y = 2x + 9D y = 1

᎑2x + 6

E y = 2x – 3

4. What equation best describes the graph below?

F y = 1᎑2x + 1

G y = –2x + 2

H y = – 1᎑2x + 2

J y = – 1᎑2x + 1

K y = 2x + 1


7/28/2019 algteks


5. Which graph below best represents the equation of a line with aslope of –2 and a y -intercept of 7?

A B

C D

E

Linear Functions 25

7/28/2019 algteks





(c)(2)(A) The student develops the concept of slope as rate of change and determinesslopes from graphs, tables, and algebraic representations.

1. Stephen F. Austin High School is going to charter a bus for a school trip. Abus company provided the following table of fees.

Number of Students Total Cost

1 $216

2 $222

3 $2284 $234

From the table determine the rate of change for the graph that would rep-resent the data.

(c)(2)(B) The student interprets the meaning of slope and intercepts in situations using

data, symbolic representations, or graphs.

1. The following is the graph of y = 3x + 2. Describe a situation this portionof the graph could model. Explain what is represented by the slope and y -intercept in this situation.

-4 -2 2 4 6 8

-4

-2

2

4

6

8


7/28/2019 algteks


(c)(2)(C) The student investigates, describes, and predicts the effects of changes in m and b on the graph of y = mx + b .

1. A scuba diver starts ascending froma depth of 100 ft at a rate of 60 ftper minute. The graph represents

the diver’s depth at particular timesin seconds. How would the graphchange if the diver began his or herascent from a depth of 60 ft? Howwould the graph change if the diverbegan his or her ascent from 100 ftat a rate of 50 ft per minute?

(c)(2)(D) The student graphs and writes equations of lines given characteristics such astwo points, a point and a slope, or a slope and y-intercept.

1. Find the equation of a line through the points (–2, 3) and (6, –5).

(c)(2)(E) The student determines the intercepts of linear functions from graphs, tables,and algebraic representations.

1. Where does the graph of y = –2x + 5 cross the y -axis? Where does it crossthe x -axis? How did you find the answers?

(c)(2)(F) The student interprets and predicts the effects of changing slope and y-inter-cept in applied situations.

1. The graph at the left shows theamount a person earns at a rateof $3 per hour. The equation isa = 3h where a is the amount indollars and h is the number of hours worked. How would araise to $6 an hour change thegraph?

(c)(2)(G) The student relates direct variation to linear functions and solves problemsinvolving proportional change.

1. In the function y = 4x , how much does y change when x isincreased by 4?

hours worked

1 3 5 7 9 11 13 15

1

3

5

7

9

11

13

15

amountearned

20 40 60 80 100

–100

–90

–80

–70

–60

–50

–40

–30–20

–10

10

depth

me

Linear Functions 27

7/28/2019 algteks



“The student formulates equations andinequalities based on linear functions, usesa variety of methods to solve them, andanalyzes the solutions in terms of thesituation.”


The 38th Annual State Fair starts on October 1, 1997, and lasts 4weeks. Bill, who is 6 years old, wants to ride the Texas Cycloneroller coaster. Bill is presently 3 feet, 4 inches tall. Safety rules statethat the minimum height for riding the Cyclone is 48 inches. Billgrows at a rate of 1–

4inch per month.

1. Use a table to generate a function representing the situation.

Time in Bill’s Height

Months Process (inches)

0 40

1 2 3 4 41

... 8 42

... N

2. What equation can be used to find out when Bill will reach the minimumheight for the ride?

3. What is the solution to the equation? Explain how to solve it.

4. In how many months will Bill be 44.75 inches tall?

5. If the pattern continues, in what year will Bill be able to ride the TexasCyclone?

6. Is this a reasonable model for Bill’s height? Why or why not?


task

7/28/2019 algteks



When students can perform a task like this, they

can answer EOC questions like...

1. A student committee must decide between a band that costs $300plus 40% of the ticket sales and a disc jockey that costs $450. Thecommittee plans to charge $3 per ticket. Which inequality can beused to determine the number of tickets, t , that must be sold inorder for the band to be the better value?

A 0.40(300) + 3t < 450

B 300 + 0.40(3)t < 450

C 0.40(300 + 3t ) < 450

D 300 + 0.40(3)t > 450

E 0.40(300 + 3t ) > 450

2. Mark earns $4.50 per hour. He worked 3 hours more this weekthan last week. If h is the number of hours he worked last week,which equation shows the amount, t , that Mark earned for bothweeks?

F t ϭ 2(4.50) (h + 3)

G t ϭ 4.50h + 3

H t ϭ 4.50(2h + 3)

J t ϭ 4.50(h + 3)

K4.50(h ϩ 3)

t ϭ ᎏᎏᎏᎏᎏ᎑2

Linear Functions 29

7/28/2019 algteks


3. The Meadowbrook High School band rented a bus for a trip to afootball game. The bus company charged $475, plus $0.45 permile over 200 miles. If the bus trip cost $529, how many mileswas the trip?

4. Mark was buying a stereo that was on sale for 1᎑4 off the original

price, x . Which equation below could be used to find the amount,y , that Mark would pay, not including tax?

A y = x –1

᎑4

B y = x – 1᎑4x

C y = 1᎑4x

D y = x – 3᎑4x

E y = x – 4

5. The math club sold 64 large chocolate chip cookies for $0.75 eachand has 80 cookies left. At what sale price would the remainingcookies be sold to have an overall average price of $0.50 percookie?

F $0.25

G $0.30

H $0.38

J $0.63

K $0.90


7/28/2019 algteks





(c)(3)(A) The student analyzes situations involving linear functions and formulateslinear equations or inequalities to solve problems.

1. The drama club is going to have some posters printed to announce theiropening Christmas special. The print shop charges $3 per poster plus a$10 fee for a master copy. The drama club budget allows them to spendno more than $200. Write an inequality that could be used to solve thisproblem.

(c)(3)(B) The student investigates methods for solving linear equations and inequalitiesusing concrete models, graphs, and the properties of equality, selects amethod, and solves the equations and inequalities.

1. Given the equation, 3(x + 5) + 3(x + 5) + 4(x + 5) = 100, solve this equa-tion and show your work.

(c)(3)(C) For given contexts, the student interprets and determines the reasonablenessof solutions to linear equations and inequalities.

1. Carlos has $100 and decides to spend $25 each week for entertainmentpurposes. His 8-year-old sister, Margarita, believes he will run out of money in 3 weeks. State whether she is correct or not, and explain why.

Linear Functions 31

7/28/2019 algteks



“The student formulates systems of linearequations from problem situations, uses avariety of methods to solve them, andanalyzes the solutions in terms of thesituation.”


A set of twin boys were born, with Troy weighing 6 lbs and Tyroneweighing 3.5 lbs. Troy’s weight increases 1.2 lbs per month, andTyrone’s weight increases 1.4 lbs per month.

1. If this rate of weight gain continues, in how many months will the two boysweigh the same?

2. If this rate of weight gain continues, how much would each boy weighat age 16?

3. Is this a reasonable model for the weights of these boys at the age of 16? Whyor why not?


task

7/28/2019 algteks





1. Marie has 24 coins in quarters and dimes. The total value is $4.95.Which system of equations below will determine the number of quarters, q , and the number of dimes, d , she has?

F d + q = 4.950.10d + 0.25q = 24

G d + q = 240.25d + 0.10q = 4.95

H d + q = 24

0.35dq = 4.95

J d + q = 24d + q = 4.95

K d + q = 240.10d + 0.25q = 4.95

2. Harry bought 9 movie tickets for a total of $45. Adult tickets cost$6 each and child tickets cost $4.50 each. How many adult ticketsdid he buy?

F 2

G 3

H 4

J 5

K 6

3. Bill bought some neon fish at $2 each and some angelfish at $3each for his new aquarium. If Bill bought a total of 20 fish andspent a total of $45, how many angelfish did he buy?

Linear Functions 33

7/28/2019 algteks





(c)(4)(A) The student analyzes situations and formulates systems of linear equations tosolve problems.

1. Two (2) CDs and 5 tapes cost $68.65. Four (4) CDs and 8 tapes cost$119.40. Write a system of equations you could use to find the costof 1 CD.

(c)(4)(B) The student solves systems of linear equations using concrete models,graphs, tables, and algebraic methods.

1. The tables below describe a linear system. Solve the system.

x y

0 –30

2 –26

4 –22

x y

–2 5

0 3

2 1


7/28/2019 algteks


(c)(4)(C) For given contexts, the student interprets and determines the reasonablenessof solutions to systems of linear equations.

1. There are two 500-gallon water tanks. One is full and is to be emptied ata rate of 2.5 gallons per minute. The other is empty and is to be filled at arate of 5 gallons per minute. The valves on both tanks are opened at the

same time. The graph shows this situation. What does the point of inter-section, a , mean in this situation? What happens if the valves are left openfor 3 hours?

0 100 200

200

400

600

800

1000

gallons

time in minutes

gallons emptied

gallons filled

Linear Functions 35

7/28/2019 algteks


d 1 Quadratic & Other Nonlinear Functions

“The student understands that the graphs of

quadratic functions are affected by the

parameters of the function and can interpret

and describe the effects of changes in the

parameters of quadratic functions.”


The student council wants to build two flower gardens for its com-munity project. They want to build one at Avalon Retirement Homeand another at Country Manor Retirement Home. The garden atAvalon has a length that is 4 times its width, and the garden atCountry Manor has 3 more square feet than the area of the gardenat Avalon.

1. Write the function representing the area of Avalon’s garden and graph it.

2. Write the function representing the area of Country Manor’s garden and graphit on the same axis.

3. How are the graphs and functions for the areas of the two gardens alike? Howare they different?

4. If the student council has at most 120 feet of landscaping timbers for the fenc-ing for the garden at Avalon, what would be a reasonable domain and rangefor the area?

5. If the graph of the Avalon garden is shifted up 4 units, what conclusion can bedrawn about its area?


task

7/28/2019 algteks





At the time this document was published, there were not any released itemsthat would apply.

Quadratic and Other Nonlinear Functions 37

7/28/2019 algteks





(d)(1)(A) The student determines the domain and range values for which quadraticfunctions make sense for given situations.

1. The path of a ball that is thrown straight up in the air is modeled by thefunction H = 75t – 16t 2. H is the height in feet and t is the time in sec-onds. What is a reasonable domain and range for this situation?

(d)(1)(B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax 2.

1. Given the graphs of y = 2x 2, y = 5x 2, y = 8x 2, y = –2x 2, y = –5x 2, andy = –8x 2, explain how the difference in the functions relates to the differ-ence in graphs. How are the graphs alike and why? How are the graphsdifferent and why?


7/28/2019 algteks


(d)(1)(C) The student investigates, describes, and predicts the effects of changes in c on the graph of y = x 2 + c .

1. The graph of y = 7x 2 is shifted up 10 units. Write the equation of the newgraph.

(d)(1)(D) For problem situations, the student analyzes graphs of quadratic functionsand draws conclusions.

1. A ball is dropped from the top of a building. The graph below gives thedistance the ball is above the ground at time, t . The general function forthe distance, h , is h = –16t 2 + c , where c is the height of the building.About how high is the ball 2 seconds after it is dropped?

0 1 2 3 4 5

100

200

300

heightabove

ground

time in seconds


7/28/2019 algteks


d2 Quadratic & Other Nonlinear Functions

“The student understands there is morethan one way to solve a quadratic equationand solves them using appropriatemethods.”


The following table was generated using a

function of the form y = ax 2

+ bx + c .

1. Find the value(s) of x for whichax 2 + bx + c = 0.

2. Find the value(s) of x for whichax 2 + bx + c = 3.

3. Below is the graph of another function of the form y = ax 2 + bx + c .

Find the roots of the equation ax 2 + bx + c = 0.

4. Below is the graph of y = x 2 – 8x + 15.

Find x if x 2 – 8x + 15 = 3.

2 4 6 8

-5

5

10

15

0

–2 0 2 4 6 8 10

10

0

–10

–20

–30

x y

–1 150 8

1 3

2 0

3 –1

4 0

5 3


task

7/28/2019 algteks





1. The area of a rectangular flag is 15 square feet. The length of theflag is 2 feet longer than the width. What are the dimensions of theflag?

F 2.5 ft by 6 ft

G 3 ft by 5 ft

H 2 ft by 7.5 ft

J 5 ft by 7 ft

K 7 ft by 8 ft

2. The area of the front of a cabinet is 18 square feet. The width is3 feet longer than the height. What are the dimensions of thecabinet?

A 7.5 ft by 4.5 ft

B 15 ft by 3 ft

C 6 ft by 3 ft

D 9 ft by 6 ftE 7 ft by 4 ft

3. The equation that describes the path of a rocket after it is shot intothe air is

h = 48t – 6t 2

where h is the height, in feet, above ground level after t seconds.After how many seconds will the rocket be at a height of 90 feet?

A t = 15

B t = 3 and t = 5

C t = 8 and t = 15

D t = 18 and t = 30

E t = 8


7/28/2019 algteks





(d)(2)(A) The student solves quadratic equations using concrete models,tables, graphs, and algebraic methods.

1. Below is a graph and a partial table for the function of y = 2x 2 + 2x – 12.Solve the equation 2x 2 + 2x –12 = 0.

x y

–2 –8

–1 –12

0 –121 –8

2 0

-4 -2 2 4

-15

-10

-5

5

10

15


7/28/2019 algteks


(d)(2)(B) The student relates the solutions of quadratic equations to the roots of theirfunctions.

1. The following is a graph of a quadratic function. What are its roots?

-6 -4 -2 2 4 6

-6

-4

-2

2

4

6


7/28/2019 algteks


d 3 Quadratic & Other Nonlinear Functions

“The student understands there aresituations modeled by functions that areneither linear nor quadratic and modelsthe situations.”


Jimmy has just received his driver’s license. He went to Slick Sam’sUsed Cars and was offered the following payment plan for a$10,000 truck.

Payment Plan: Pay $0.01 on the first day, $0.02 on the second day, $0.04 onthe third day, $0.08 on the fourth day, and so on for 21 days.

1. Complete the following table

Day Payment

1 0.01

2 0.023 0.04

4 0.08

5 ?

6 ?

7 ?

8 ?

9 ?

10 ?

2. Make a scatter plot for this plan. Does this scatter plot represent a linear func-

tion, quadratic function, or neither? Explain your thinking.

3. How much would Jimmy’s payment be on the 21st day?


task

7/28/2019 algteks





1. An animal population that doubles every 6 months can bedescribed by the equation

p = n и 22t

where p is the population after t years and n is the original numberof animals. If 2 of these animals were introduced into an area,what would be the estimated population after 3 years?

F 24

G 64H 128

J 512

K 4096

2. Carla earns $6.40 per hour. She gets a 5% raise each year. The

amount she will earn per hour, x , is given by the formulax = w (1 + r )y

where w is her current wage per hour, r is the rate of increase, andy is the number of years. To the nearest cent, how much will sheearn per hour, x , after 2 years on the same job?

A $6.50

B $7.06

C $8.50

D $13.44

E $14.40


7/28/2019 algteks





(d)(3)(A) The student uses patterns to generate the laws of exponents and appliesthem in problem-solving situations.

Find the missing exponent.

1. x 2 и x 5 и x 7 и x n = x 18

2.

3. (b

4

)

n

= b

24

4. 3n = 1

(d)(3)(B) The student analyzes data and represents situations involving inverse varia-tion using concrete models, tables, graphs, or algebraic methods.

The table below shows various rates and the respective times it takes amotor bike to cover a distance of 40 miles.

Rate (mph) Time (hours)

4 10

5 8

10 4

20 2

40 1

1. How fast in miles per hour does the motorbike have to travel to cover thisdistance in 30 minutes?

2. How long would it take the motorbike to travel the same distance travel-ing at the rate of 60 mph?

z 5ᎏ᎑᎑ ϭ z 2z n


7/28/2019 algteks


(d)(3)(C) The student analyzes data and represents situations involving exponentialgrowth and decay using concrete models, tables, graphs, or algebraicmethods.

1. The number of lily pads in a pond triples every year. If there are 5 lilypads in the pond this year, how many lily pads will there be next year? in

9 more years?

Documents

algteks