algebra 1 - volume 1_2015_ student_

TIMOTHY D. KANOLD
EDWARD B. BURGER
JULI K. DIXON
MATTHEW R. LARSON
STEVEN J. LEINWAND
.
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A
Possession of this publication in print format does not entitle users to convert this publication, or any portion of it, into electronic format.
If you have received these materials as examination copies free of charge, Houghton Mifflin Harcourt Publishing Company retains title to the materials and they may not be resold. Resale of examination copies is strictly prohibited.
Copyright © 2015 by Houghton Mifflin Harcourt Publishing Company
All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage and retrieval system, without the prior written permission of the copyright owner unless such copying is expressly permitted by federal copyright law. Requests for permission to make copies of any part of the work should be addressed to Houghton Mifflin Harcourt Publishing Company, Attn: Contracts, Copyrights, and Licensing, 9400 Southpark Center Loop, Orlando, Florida 32819-8647.
Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
This product is not sponsored or endorsed by the Common Core State Standards Initiative of the National Governors Association Center for Best Practices and the Council of Chief State School Officers.
Printed in the U.S.A.
ISBN 978-0-544-36817-0
1 2 3 4 5 6 7 8 9 10 XXXX 22 21 20 19 18 17 16 15 14
4500000000 B C D E F G
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-B;CA-B
Authors Timothy D. Kanold, Ph.D., is an award-winning international educator, author, and consultant. He is a former superintendent and director of mathematics and science at Adlai E. Stevenson High School District 125 in Lincolnshire, Illinois. He is a past president of the National Council of Supervisors of Mathematics (NCSM) and the Council for the Presidential Awardees of Mathematics (CPAM). He has served on several writing and leadership commissions for NCTM during the past decade. He presents motivational professional development seminars with a focus on developing professional learning communities (PLC’s) to improve the teaching, assessing, and learning of students. He has recently authored nationally recognized articles, books, and textbooks for mathematics education and school leadership, including What Every Principal Needs to Know about the Teaching and Learning of Mathematics.
Edward B. Burger, Ph.D., is the President of Southwestern University, a former Francis Christopher Oakley Third Century Professor of Mathematics at Williams College, and a former vice provost at Baylor University. He has authored or coauthored more than sixty- five articles, books, and video series; delivered over five hundred addresses and workshops throughout the world; and made more than fifty radio and television appearances. He is a Fellow of the American Mathematical Society as well as having earned many national honors, including the Robert Foster Cherry Award for Great Teaching in 2010. In 2012, Microsoft Education named him a “Global Hero in Education.”
Juli K. Dixon, Ph.D., is a Professor of Mathematics Education at the University of Central Florida. She has taught mathematics in urban schools at the elementary, middle, secondary, and post-secondary levels. She is an active researcher and speaker with numerous publications and conference presentations. Key areas of focus are deepening teachers’ content knowledge and communicating and justifying mathematical ideas. She is a past chair of the NCTM Student Explorations in Mathematics Editorial Panel and member of the Board of Directors for the Association of Mathematics Teacher Educators.
Matthew R. Larson, Ph.D., is the K-12 mathematics curriculum specialist for the Lincoln Public Schools and served on the Board of Directors for the National Council of Teachers of Mathematics from 2010 to 2013. He is a past chair of NCTM’s Research Committee and was a member of NCTM’s Task Force on Linking Research and Practice. He is the author of several books on implementing the Common Core Standards for Mathematics. He has taught mathematics at the secondary and college levels and held an appointment as an honorary visiting associate professor at Teachers College, Columbia University.
Steven J. Leinwand is a Principal Research Analyst at the American Institutes for Research (AIR) in Washington, D.C., and has over 30 years in leadership positions in mathematics education. He is past president of the National Council of Supervisors of Mathematics and served on the NCTM Board of Directors. He is the author of numerous articles, books, and textbooks and has made countless presentations with topics including student achievement, reasoning, effective assessment, and successful implementation of standards.
Robert Kaplinsky Teacher Specialist, Mathematics Downey Unified School District Downey, California
Mindy Eden Richwoods High School Peoria School District Peoria, IL
Dustin Johnson Badger High School Math Teacher Department Chair Lake Geneva-Genoa City Union High School District Lake Geneva, WI
Ashley D. McSwain Murray High School Murray City School District Salt Lake City, UT
Rebecca Quinn Doherty Memorial High School Worcester Public Schools District Worcester, MA
Ted Ryan Madison LaFollette High School Madison Metropolitan School District Madison, WI
Tony Scoles Fort Zumwalt School District O’Fallon, MO
Cynthia L. Smith Higley Unified School District Gilbert, AZ
Phillip E. Spellane Doherty Memorial High School Worcester Public Schools District Worcester, MA
Mona Toncheff Math Content Specialist Phoenix Union High School District Phoenix, AZ
Performance Task Consultant
Michael R. Heithaus Executive Director, School of Environment, Arts, and Society Professor, Department of Biological Sciences Florida International University North Miami, Florida
STEM Consultants Science, Technology, Engineering, and Mathematics
UNIT
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Are You Ready? . . . . . . . . . . . . . . . 4
1.1 Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Modeling Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Reporting with Precision and Accuracy . . . . . . . . . . . . . . . . . . . . . 27
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Are You Ready? . . . . . . . . . . . . . . 44
2.1 Modeling with Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2 Creating and Solving Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 Solving for a Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4 Creating and Solving Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.5 Creating and Solving Compound Inequalities . . . . . . . . . . . . . . . 81
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
MODULE 1
MODULE 2
UNIT
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Are You Ready? . . . . . . . . . . . . .104
3.1 Graphing Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.2 Understanding Relations and Functions . . . . . . . . . . . . . . . . . . . 115 3.3 Modeling with Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.4 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Module Performance Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Patterns and Sequences
Real-World Video . . . . . . . . . . . 153
4.1 Identifying and Graphing Sequences . . . . . . . . . . . . . . . . . . . . . . 155 4.2 Constructing Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . . 165 4.3 Modeling with Arithmetic Sequences . . . . . . . . . . . . . . . . . . . . . . 175
MODULE 3
MODULE 4
Volume 1
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3 Linear Functions
Real-World Video . . . . . . . . . . . 197
Are You Ready? . . . . . . . . . . . . .198
5.1 Understanding Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.2 Using Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 5.3 Interpreting Rate of Change and Slope . . . . . . . . . . . . . . . . . . . . 221
Are You Ready? . . . . . . . . . . . . .238
6.1 Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 6.2 Point-Slope Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 6.3 Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 6.4 Transforming Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 6.5 Comparing Properties of Linear Functions . . . . . . . . . . . . . . . . . 281
MODULE 5
MODULE 6
Volume 1
Real-World Video . . . . . . . . . . . 299
Are You Ready? . . . . . . . . . . . . .300
7.1 Modeling Linear Relationships. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 7.2 Using Functions to Solve One-Variable Equations . . . . . . . . . . 309 7.3 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . 323
MODULE 7
UNIT
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
Multi-Variable Categorical Data
Real-World Video . . . . . . . . . . . 345
One-Variable Data Distributions
Real-World Video . . . . . . . . . . . 375
Are You Ready? . . . . . . . . . . . . .376
9.1 Measures of Center and Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 9.2 Data Distributions and Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 9.3 Histograms and Box Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 9.4 Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
MODULE 8
MODULE 9
© H
Are You Ready? . . . . . . . . . . . . .434
10.1 Scatter Plots and Trend Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 10.2 Fitting a Linear Model to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
MODULE 10
© H
Real-World Video . . . . . . . . . . . 477
Are You Ready? . . . . . . . . . . . . .478
11.1 Solving Linear Systems by Graphing . . . . . . . . . . . . . . . . . . . . . . 479 11.2 Solving Linear Systems by Substitution . . . . . . . . . . . . . . . . . . . 491 11.3 Solving Linear Systems by Adding or Subtracting . . . . . . . . . 503 11.4 Solving Linear Systems by Multiplying First . . . . . . . . . . . . . . . 515
Are You Ready? . . . . . . . . . . . . .532
12.1 Creating Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . 533 12.2 Graphing Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . 547 12.3 Modeling with Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
MODULE 11
MODULE 12
© H
13.1 Understanding Piecewise-Defined Functions . . . . . . . . . . . . . 577 13.2 Absolute Value Functions and Transformations . . . . . . . . . . . 589 13.3 Solving Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . 601 13.4 Solving Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . 611
MODULE 13
© H
Module Performance Task. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662
Real-World Video . . . . . . . . . . . 665
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634
MODULE 14
MODULE 15
© H
Are You Ready? . . . . . . . . . . . . .738
16.1 Using Graphs and Properties to Solve Equations with Exponents . . . . . . . . . . . . . . . . . . . . . . . . . 739
16.2 Modeling Exponential Growth and Decay. . . . . . . . . . . . . . . . . 751 16.3 Using Exponential Regression Models . . . . . . . . . . . . . . . . . . . . 765 16.4 Comparing Linear and Exponential Models . . . . . . . . . . . . . . . 779
MODULE 16
© H
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802
MODULE 17
© H
18.2 Multiplying Polynomial Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
18.3 Special Products of Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867
MODULE 18
© H
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
Graphing Quadratic Functions
Real-World Video . . . . . . . . . . . 887
Real-World Video . . . . . . . . . . . 935
Are You Ready? . . . . . . . . . . . . .936
20.1 Connecting Intercepts and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . 937 20.2 Connecting Intercepts and Linear Factors . . . . . . . . . . . . . . . . . 951 20.3 Applying the Zero Product Property to Solve Equations . . . 961
UNIT 8 Volume 2
MODULE 19
MODULE 20
© H
Real-World Video . . . . . . . . . . . 983
Are You Ready? . . . . . . . . . . . . .984
21.1 Solving Equations by Factoring x 2 + bx + c . . . . . . . . . . . . . . . 985 21.2 Solving Equations by Factoring a x 2 + bx + c . . . . . . . . . . . . . . 997 21.3 Using Special Factors to Solve Equations. . . . . . . . . . . . . . . . .1009
Study Guide Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1027
Module Performance Task. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1028
Assessment Readiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1030
Real-World Video . . . . . . . . . . 1031
Are You Ready? . . . . . . . . . . . 1032
22.1 Solving Equations by Taking Square Roots . . . . . . . . . . . . . . .1033 22.2 Solving Equations by Completing the Square . . . . . . . . . . . .1045 22.3 Using the Quadratic Formula to Solve Equations . . . . . . . . .1059 22.4 Choosing a Method for Solving Quadratic Equations . . . . .1073 22.5 Solving Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1089
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 982
MODULE
MODULE
21
22
© H
Real-World Video . . . . . . . . . . 1105
23.1 Modeling with Quadratic Functions . . . . . . . . . . . . . . . . . . . . .1107 23.2 Comparing Linear, Exponential, and Quadratic Models . . .1123
MODULE 23
© H
24.1 Graphing Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . .1155 24.2 Understanding Inverse Functions. . . . . . . . . . . . . . . . . . . . . . . .1167 24.3 Graphing Square Root Functions . . . . . . . . . . . . . . . . . . . . . . . .1179 24.4 Graphing Cube Root Functions . . . . . . . . . . . . . . . . . . . . . . . . . .1191
UNIT
Reading Start-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1152
10 MODULE 24
© H
xxi
Explore
Lesson 19.2 Precision and Accuracy
How do you use significant digits when reporting the results of calculations involving measurement?
Preview
Lesson Performance Task The sun is an excellent source of electrical energy. Suppose a company owns a field of solar panels. How much electricity is produced by the field? The answer depends on the amount of power the field yields per square foot, as well as the size of the field.
We’ll look at this challenge during the lesson using significant digits!
Engage Essential Question
Resource Locker
Explore Comparing Precision of Measurements. Numbers are values without units. They can be used to compute or to describe measurements. Quantities are real- word values that represent specific amounts. For instance, 15 is a number, but 15 grams is a quantity.
Precision is the level of detail of a measurement, determined by the smallest unit or fraction of a unit that can be reasonably measured.
Accuracy is the closeness of a given measurement or value to the actual measurement or value. Suppose you know the actual measure of a quantity, and someone else measures it. You can find the accuracy of the measurement by finding the absolute value of the difference of the two.
Complete the table to choose the more precise measurement.
Measurement 1 Measurement 2 Smaller Unit More Precise Measurement
4 g 4.3 g
5.71 oz 5.7 oz
4.2 m 422 cm
7 ft 2 in. 7.2 in.
Eric is a lab technician. Every week, he needs to test the scales in the lab to make sure that they are accurate. He uses a standard mass that is exactly 8.000 grams and gets the following results.
Scale Mass
The measurement for Scale is the most precise
because it measures to the nearest , which is smaller than the smallest unit measured on the other two scales.
Scale 1 Scale 2 Scale 3
Module 1 27 Lesson 3
1 . 3 Reporting with Precision and Accuracy
Essential Question: How do you use significant digits when reporting the results of calculations involving measurement?
A1_MNLESE368170_U1M01L3 27 3/19/14 3:02 PM
Comparing Precision of Measurements
12.03 g
12.029 g
11.98 g
Eric is a technician in a pharmaceutical lab. Every week, he needs to test the scales in the lab to make sure that they are. He uses a that is exactly 12.000 g and gets the following results:
Definition of Precision: The level of detail of a, determined by the smallest unit or fraction of a unit that can be reasonably measured.
Definition of Accuracy: The closeness of a given of value to the actual measurement or value.
Given two measurements, is it possible that the more precise one is not the
most accurate? Why?
Which scale is the most accurate?
Reflect
Explore Concept 1
Lesson Performance Task The sun is an excellent source of electrical energy. Suppose a company owns a field of solar panels. How much electricity is produced by the field? The an swer depends on the amount of power the field yields per square foot, as well as the size of the field.
© H
Explore Explore
Numbers are values without units. They can be used to compute or to describe measurements. Quantities are real- word values that represent specific amounts. For instance, 15 is a number, but 15 grams is a quantity.
AccuracyAccuracy is the closeness of a given measurement or value to the actual measurement or value. Suppose you know the Accuracy is the closeness of a given measurement or value to the actual measurement or value. Suppose you know the Accuracy actual measure of a quantity, and someone else measures it. You can find the accuracy of the measurement by finding the absolute value of the difference of the two.
Measurement 1
4 g
5.71 oz
4.2 m
7 ft 2 in.
Eric is a lab technician. Every week, he needs to test the scales in the lab to make sure that they are accurate. He uses a standard mass that is exactly 8.000 grams and gets the following results.
Comparing Precision of Measurements
12.03 g
12.029 g
11.98 g
Eric is a technician in a pharmaceutical lab. Every week, he needs to test the scales in the lab to make sure that they are. He uses a that is exactly 12.000 g and gets the following results:
Definition of Precision: The level of detail of a, determined by the smallest unit or fraction of a unit that can be reasonably measured.
Definition of Accuracy: The closeness of a given of value to the actual measurement or value.
Which measuring tool is the most precise?
Which scale is the most accurate?
ReflectReflectReflect
Explore Concept 1
p any
Succeeding with HMH Algebra 1 HMH Algebra 1 is built on the 5E instructional model--Engage, Explore, Explain, Elaborate, Evaluate--to develop strong conceptual understanding and mastery of key mathematics standards.
Preview the Lesson Performance Task in the Interactive Student Edition.
Explore and interact with new concepts to develop a deeper understanding of mathematics in your book and the Interactive Student Edition.
Scan the QR code to access engaging videos, activities, and more in the Resource Locker for each lesson.
ENGAGE
EXPLORE
xxii
Explore
Lesson 19.2 Precision and Accuracy
How do you use significant digits when reporting the results of calculations involving measurement?
Preview
Lesson Performance Task The sun is an excellent source of electrical energy. Suppose a company owns a field of solar panels. How much electricity is produced by the field? The answer depends on the amount of power the field yields per square foot, as well as the size of the field.
We’ll look at this challenge during the lesson using significant digits!
Engage Essential Question
Send to NotebookFormula Send to NotebookFormula
Given two measurements, is it possible that the more precise measurement may not be the more accurate?
What is the relationship between the precision used in the length and width of the rectangle and the precision of the resulting area measurement?
How are the significant digits related to the calculations using measurements?
Essential Question Check-In
How many significant digits are needed when performing operations with measurements?
Your Turn Evaluate
Elaborate
© H
Your Turn
10. The calculator screen shows the graph of ƒ(x) = 4x2 - 8x -5. Explain how the graph supports the solution in Part B.
Elaborate
11. When solving a quadratic equation in the form x 2 + bx + c = 0 by completing the square, what is the purpose of moving the constant to the other side of the equation?
12. Complete the table by answering each question yes or no for each equation. Then describe how you would decide what method to use when solving a quadratic equation.
Equation Can be solved by factoring?
Can be solved by completing the square?
x 2 + wx - 4 = 0
x 2 + 3x - 5 = 0 x 2 + 3x - 5 = 0
13. Describe any differences in the steps you take when solving the equations x 2 - 4x - 4 = 0 and 2x 2 - 4x - 4 = 0 by completing the square.
1
Reflect
7.5 8.5
My answer
19.45 19.55
Find the range of values for the actual length and width of the rectangle.
Minimum Area = Minimum width × Minimum length
= 7.5 cm × 19.45 cm
As you have seen, measurements are given to a certain precision. Therefore, the value reported does not necessarily represent the actual value of the measurement. For example, a measurement of 5 centimeters, which is given to the nearest whole unit, can actually range from 0.5 units below the reported value, 4.5 centimeters, up to, but not including, 0.5 units above it, 5.5 centimeters. The actual length, l, is within a range of possible values: centimeters. Similarly, a length given to the nearest tenth can actually range from 0.05 units below the reported value up to, but not including, 0.05 units above it. So a length reported as 4.5 cm could actually be as low as 4.45 cm or as high as nearly 4.55 cm.
In the above exercise, the location of the uncertainty in the linear measurements results in different amounts of uncertainty in the calculated measurement. Explain how to fix this problem.
Reflect
Calculate the minimum and maximum possible areas. Round your answer to the nearest square centimeters.
The width and length of a rectangle are 8 cm and 19.5 cm, respectively.
Minimum width = cm and maximum width < cm
Minimum length = cm and maximum length < cm
Determining Precision Explain Concept 2
Explore Your Turn
Elaborate
Solve the quadratic equation by factoring.
7x + 44x = 7x − 10
Question 3 of 17 Video Tutor
Check
? !
p any
Explain 1 Completing the Square with Expressions Finding the value of c needed to make an expression such as x 2 + 6x + c into a perfect square trinomial is called completing the square.
Using algebra tiles, half of the x-tiles are placed along the right and bottom sides of the x 2 -tile. The number of 1-tiles added is the square of the number of x-tiles on either side of the x 2 -tile.
To complete the square for the expression x 2 + bx + c, replace c with ( b _ 2 ) 2 . The perfect square trinomial is x 2 + bx + ( b _ 2 ) 2 and factors as ( x + b _ 2 ) 2 .
Example 1 Complete the square to form a perfect trinomial. Then factor the trinomial.
x 2 + 12 x + c
Identify b. b = 12
Find c. c = ( b _ 2 ) 2 = ( 12 __ 2 ) 2 = 36
Write the trinomial. x 2 + 12x + 36
Factor the trinomial. x 2 + 12x + 36 = (x + 6) 2
z 2 - 26 z + c
Identify b. b =
=
Write the trinomial. z 2 + z + ( ) 2 Factor the trinomial. z 2 + z + =
Reflect
Complete the square to form a perfect square trinomial. Then factor the trinomial.
3. a 2 + 18a + 4. p 2 - 5p +
Your Turn
5. In Part A, b is positive and in Part B, b is negative. Does this affect the sign of c? Why or why not?
© H
y
Learn concepts with step-by- step interactive examples. Every example is also supported by a Math On the Spot video tutorial.
Check your understanding of new concepts and skills with Your Turn exercises in your book or online with Personal Math Trainer.
Show your understanding and reasoning with Reflect and Elaborate questions.
EXPLAIN
ELABORATE
xxiii
Evaluate
Solve the quadratic equation by factoring.
7x + 44x = 7x − 10
Question 3 of 17 Video Tutor
Check
? !
Evaluate: Homework and Practice
1. The diagram represents the expression x 2 + 4x + c with the constant term missing. Complete the square by filling in the bottom right corner with 1-tiles, and write the expression as a trinomial and in factored form.
Complete the square to form a perfect square trinomial. Then factor the trinomial.
2. m 2 + 10m + 3. g 2 - 20g +
4. y 2 + 2y + 5. w 2 - 11w +
Solve the equation by completing the square.
6. s 2 + 15s = -56 7. r 2 - 4r = 165
8. y 2 + 19y + 78 = 0 9. x 2 - 19x + 84 = 0
© H
Lesson Performance Task
The quarterback of a football team is practicing throwing a 50-yard pass to a wide receiver. The quarterback can throw a pass with an initial vertical velocity of 40 feet per second and an initial height of 6 feet. He wants to throw the ball so it lands in the wide receiver’s hands at a height of 6 feet at exactly the right time.
The wide receiver can run 40 yards in 4.4 seconds and begins running at top speed when the quarterback hikes the ball. How long should the quarterback wait between hiking the ball and throwing it?
© H
Use the quadratic formula.
Since b 2 - 4ac = 0, the equation has one real solution.
Find b 2 _ 4a
(completar el cuadrado)
(fórmula cuadrática)
square root (raíz cuadrada)Equations in the form a (x + b) 2 = c
can be solved by taking square roots. Take the square root of both sides.
Solve both cases.
Using Square Roots to Solve Quadratic Equations 22
Essential Question: How can you use quadratic equations to solve real-world problems?
KEY EXAMPLE (Lesson 22.1)
Solve (x - 8) 2 = 49 by taking the square root.
(x - 8) 2 = 49
x = 15 and x = 1
Solve 9 x 2 - 6x = 20 by completing the square.
(-6) 2
_ 4 (9)
(3x - 1) 2 = 21
21 + 1 _ 3
Solve 8 x 2 - 8x + 2 = 0 using the quadratic equation.
a = 8, b = -8, c = 2
x = -b ± √ __
2 (8)
x = 8 ± √ _
0 _ 16
Module 22 1101 Study Guide Review
A1_MNLESE368187_U9M22MC.indd 1101 29/03/14 10:10 PM
© H
p any
Practice and apply skills and concepts with Evaluate exercises and a Lesson Performance Task in your book with plenty of workspace, or complete these exercises online with Personal Math Trainer.
Review what you have learned and prepare for high-stakes tests with a variety of resources, including Study Guide Reviews, Performance Tasks, and Assessment Readiness test preparation.
EVALUATE
xxiv
© H
MATH IN CAREERS
Personal Trainer A personal trainer works with clients to help them achieve their personal fitness goals. A personal trainer needs math to calculate a client’s heart rate, body fat percentage, lean muscle mass, and calorie requirements. Personal trainers are often self-employed, so they need to understand the mathematics of managing a business.
If you are interested in a career as a personal trainer, you should study these mathematical subjects:
• Algebra • Business Math
Research other careers that require knowledge of the mathematics of business management. Check out the career activity at the end of the unit to find out how personal trainers use math.
Quantities and Modeling
1Unit 1
Reviewing Expressions
3x + 43 x + 4
(función de valor absoluto)
algebraic expression (expresión algebraica)
(ecuación literal) precision (precisión) proportion (proporción) ratio (razón) scale model (modelo a escala)
Visualize Vocabulary Use the words to complete the graphic. Put one term in each section of the square.
Understand Vocabulary To become familiar with some of the vocabulary terms in this unit, consider the following. You may refer to the module, the glossary, or a dictionary.
1. The level of detail of a measurement, determined by the unit of measure,
is . 2. A statement that two ratios are equal is called a .
3. A is a comparison of two quantities by division.
Active Reading
Two-Panel Flip Book Before beginning the lessons, create a two-panel flip chart to help you compare concepts in this unit. Label the flaps “Creating and Solving Equations” and “Creating and Solving Inequalities.” As you study each lesson, write important ideas under the appropriate flap. Include any examples that will help you remember the concepts later when you look back at your notes.
2Unit 1
MODULE ©
ub lis
hi n
g C
om p
an y
• I m
ag e
C re
d it
s: P
h ot
o d
is c/
G et
ty Im
ag es
REAL WORLD VIDEO In order to function properly and safely, electronics must be manufactured to a high degree of precision. Material tolerances and component alignment must be precisely matched in order to not interfere with each other.
1 Essential Question: How do you use quantitative reasoning to solve real-world problems?
Quantitative Reasoning
MODULE PERFORMANCE TASK PREVIEW
What an Impossible Score! Darts is a game of skill in which small pointed darts are thrown at a circular target mounted on a wall. The target is divided into regions with different point values, and scoring depends on which segment the dart hits. Is there some score that is impossible to achieve no matter how many darts are thrown in a simple game of darts? Keep your eye on the target and let’s figure it out!
LESSON 1.1
Solving Equations
LESSON 1.2
Modeling Quantities
LESSON 1.3
Module 1 3
YOUAre Ready? ©
H ou
• Online Homework • Hints and Help • Extra Practice
Write a ratio using the given dimensions comparing the model length to the actual length.
Simplify.
Isolate the variable by adding 11 to both sides of the equation.
Complete these exercises to review skills you will need for this module.
One-Step Equations Example 1 Solve.
y - 11 = 7 y - 11 + 11 = 7 + 11
y = 18
2. 9 + a = 23
3. n _ 3 = 16
Scale Factor and Scale Drawings Example 2 Length of car: 132 in. Length of model of car: 11 in.
model length
____________ actual length
= 11 _ 132
= 1 ___ 12
Identify the scale factor.
4. Length of room: 144 in. Length of room on scale drawing: 18 in.
5. Wingspan of airplane: 90 ft Wingspan of model of airplane: 6 ft
Significant Digits Example 3 Determine the number of significant digits in 37.05.
The significant digits in 37.05 are 3, 7, 0, and 5.
37.05 has 4 significant digits.
Determine the number of significant digits.
6. 0.0028 7. 970.0
8. 50,000 9. 4000.01
Significant digits are nonzero digits, zeros at the end of a number and to the right of a decimal point, and zeros between significant digits.
Module 1 4
© H
Explore Solving Equations by Guess-and-Check or by Working Backward
An equation is a mathematical sentence that uses the equal sign = to show two expressions are equivalent. The expressions can be numbers, variables, constants, or combinations thereof.
There are many ways to solve an equation. One way is by using a method called guess-and-check. A guess- and-check method involves guessing a value for the variable in an equation and checking to see if it is the solution by substituting the value in the equation. If the resulting equation is a true statement, then the value you guessed is the solution of the equation. If the equation is not a true statement, then you adjust the value of your guess and try again, continuing until you find the solution.
Another way to solve an equation is by working backward. In this method, you begin at the end and work backward toward the beginning.
Solve the equation x - 6 = 4 using both methods.
Use the guess-and-check method to find the solution of the equation x - 6 = 4.
A Guess 11 for x.
x - 6 = 4
Guess 10 for x.
Is 10 the solution
of x - 6 = 4?
C Use the working backward method to find the solution of the equation x - 6 = 4.
4 + 6 = Is this the value of x before taking away 6?
- 6 = 4
Reflect
?
? ?
?
?
Module 1 5 Lesson 1
1.1 Solving Equations Essential Question: How do you solve an equation in one variable?
DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B
© H
p any
Explain 1 Solving One-Variable Two-Step Equations A solution of an equation is a value for the variable that makes the equation true. To determine the solution of an equation, you will use the Properties of Equality.
Properties of Equality
Words Numbers Algebra
Addition Property of Equality You can add the same number to both sides of an equation, and the statement will still be true.
3 = 3 3 + 2 = 3 + 2
5 = 5
a = b a + c = b + c
Subtraction Property of Equality You can subtract the same number from both sides of an equation, and the statement will still be true.
7 = 7 7 - 5 = 7 - 5
2 = 2
a = b a - c = b - c
Multiplication Property of Equality You can multiply both sides of an equation by the same number, and the statement will still be true.
3 = 3 3 ⋅ 4 = 3 ⋅ 4
12 = 12
a = b a ⋅ c = b ⋅ c
Division Property of Equality You can divide both sides of an equation by the same nonzero number, and the statement will still be true.
15 = 15 15_ 3
c ,
Example 1 Solve the equation by using Properties of Equality.
A 3x - 2 = 6
Use the Addition Property of Equality. 3x - 2 + 2 = 6 + 2
Combine like terms. 3x = 8
Now use the Division Property of Equality. 3x _ 3 = 8 _ 3
Simplify. x = 8 _ 3
B 1 _ 2
z + 4 = 10
Use the Subtraction Property of Equality. 1 _ 2 z + 4 - = 10 -
Combine like terms. 1 _ 2 z =
Now use the Multiplication Property of Equality to multiply each side by 2.
2 ⋅ 1
Module 1 6 Lesson 1
© H
es
Reflect
2. Discussion What is the goal when solving a one-variable equation?
Your Turn
3. 5x - 10 = 20 4. 1 _ 3 x + 9 = 21
Explain 2 Solving Equations to Define a Unit One useful application of algebra is to use an equation to determine what a unit of measure represents. For instance, if a person uses the unit of time “score” in a speech and there is enough information given, you can solve an equation to find the quantity that a “score” represents.
Example 2 Solve an equation to determine the unknown quantity.
A In 1963, Dr. Martin Luther King, Jr., began his famous “I have a dream” speech with the words “Five score years ago, a great American, in whose symbolic shadow we stand, signed the Emancipation Proclamation.” The proclamation was signed by President Abraham Lincoln in 1863. But how long is a score? We can use algebra to find the answer.
Let s represent the quantity (in years) represented by a score.
s = number of years in a score
Calculate the quantity in years after President Lincoln signed the Emancipation Proclamation.
1963 - 1863 = 100
Dr. Martin Luther King, Jr. used “five score” to describe this length of time. Write the equation that shows this relationship.
5s = 100
Use the Division Property of Equality to solve the equation. 5s _ 5 = 100 _ 5
s = 20
A score equals 20 years.
Module 1 7 Lesson 1
© H
p any
B An airplane descends in altitude from 20,000 feet to 10,000 feet. A gauge at Radar Traffic Control reads that the airplane’s altitude drops 1.8939 miles. How many feet are in a mile?
Let m represent the quantity (in feet) represented by a mile.
m = number of feet in a mile
Calculate the quantity in feet of the descent. 20,000 - =
A gauge described this quantity as 1.8939 miles. Write the equation that shows this relationship.
1.8939m =
Use the Division Property of Equality to solve the equation. 1.8939m _ = _
Round to the nearest foot. m ≈
There are 5280 feet in a mile.
Your Turn
Solve an equation to determine the unknown quantity.
5. An ostrich that is 108 inches tall is 20 inches taller than 4 times the height of a kiwi. What is the height of a kiwi in inches?
6. An emu that measures 60 inches in height is 70 inches less than 5 times the height of a kakapo. What is the height of a kakapo in inches?
Elaborate
7. How do you know which operation to perform first when solving an equation?
8. How can you create an equivalent equation by using the Properties of Equality?
9. When a problem involves more than one unit for a characteristic (such as length), how can you tell which unit is more appropriate to report the answer in?
10. Essential Question Check-In Describe each step in a solution process for solving an equation in one variable.
Module 1 8 Lesson 1
© H
Use the guess-and-check method to find the solution of the equation. Show your work.
1. 2x + 5 = 19
Use the working backward method to find the solution of the equation. Show your work. 2. 4y - 1 = 7
Solve each equation using the Properties of Equality. Check your solutions.
3. 4a + 3 = 11 4. 8 = 3r - 1
5. 42 = -2d + 6 6. 3x + 0.3 = 3.3
7. 15y + 31 = 61 8. 9 - c = -13
Module 1 9 Lesson 1
© H
p any
9. x _ 6 + 4 = 15 10. 1 _ 3 y + 1 _ 4 = 5 _ 12
11. 2 _ 7 m - 1 _ 7 = 3 _ 14 12. 15 = a _ 3 - 2
13. 4 - m _ 2 = 10 14. x _ 8 - 1 _ 2 = 6
Justify each step.
x _ 3 = 18
© H
y
18. In 2003, the population of Zimbabwe was about 12.6 million people, which is 1 million more than 4 times the population in 1950. Write and solve an equation to find the approximate population p of Zimbabwe in 1950.
19. Julio is paid 1.4 times his normal hourly rate for each hour he works over 30 hours in a week. Last week he worked 35 hours and earned $436.60. Write and solve an equation to find Julio’s normal hourly rate, r. Explain how you know that your answer is reasonable.
20. The average weight of the top 5 fish caught at a fishing tournament was 12.3 pounds. Some of the weights of the fish are shown in the table.
Top 5 Fish
© H
is
21. Paul bought a student discount card for the bus. The card allows him to buy daily bus passes for $1.50. After one month, Paul bought 15 passes and spent a total of $29.50. How much did he spend on the student discount card?
22. Jennifer is saving money to buy a bike. The bike costs $245. She has $125 saved, and each week she adds $15 to her savings. How long will it take her to save enough money to buy the bike?
23. Astronomy The radius of Earth is 6378.1 km, which is 2981.1 km greater than the radius of Mars. Find the radius of Mars.
24. Maggie’s brother is 3 years younger than twice her age. The sum of their ages is 24. How old is Maggie?
© H
H.O.T. Focus on Higher Order Thinking
25. Analyze Relationships One angle of a triangle measures 120°. The other two angles are congruent. Write and solve an equation to find the measure of the congruent angles.
26. Explain the Error Find the error in the solution, and then solve correctly.
9x + 18 + 3x = 1
x = - 20 _ 9
27. Check for Reasonableness Marietta was given a raise of $0.75 per hour, which gave her a new wage of $12.25 per hour. Write and solve an equation to determine Marietta’s hourly wage before her raise. Show that your answer is reasonable.
© H
The formula p = 8n - 30 gives the profit p when a number of items n are each sold at $8 and expenses totaling $30 are subtracted.
a. If the profit is $170.00, how many items were bought?
b. If the same number of items were bought but the expenses changed to $40, would the profit increase or decrease, and by how much? Explain.
© H
Resource Locker
Explore Using Ratios and Proportions to Solve Problems Ratios and proportions are very useful when solving real-world problems. A ratio is a comparison of two numbers by division. An equation that states that two ratios are equal is called a proportion.
A totem pole that is 90 feet tall casts a shadow that is 45 feet long. At the same time, a 6-foot-tall man casts a shadow that is x feet long.
The man and the totem pole are both perpendicular to the ground, so they form right angles with the ground. The sun shines at the same angle on both, so similar triangles are formed.
Write a ratio of the man’s height to the totem pole’s height. ______
Write a ratio of the man’s shadow to the totem pole’s shadow. ______
Write a proportion.
man’s height __
pole’s shadow ______ = ______
Solve the proportion by both sides by 45.
Solve the proportion to find the length of the man’s shadow in feet. x =
Reflect
1. Discussion What is another ratio that could be written for this problem? Use it to write and solve a different proportion to find the length of the man’s shadow in feet.
2. Discussion Explain why your new proportion and solution are valid.
x ft
6 ft
45 ft
90 ft
1 . 2 Modeling Quantities Essential Question: How can you use rates, ratios, and proportions to solve real-world
problems?
© H
→ →
Explain 1 Using Scale Drawings and Models to Solve Problems A scale is the ratio of any length in a scale drawing or scale model to the corresponding actual length. A drawing that uses a scale to represent an object as smaller or larger than the original object is a scale drawing. A three-dimensional model that uses a scale to represent an object as smaller or larger than the actual object is called a scale model.
Example 1 Use the map to answer the following questions.
The actual distance from Chicago to Evanston is 11.25 mi. What is the distance on the map?
Write the scale as a fraction. map
_ actual
1 in. _ 18 mi
Let d be the distance on the map. 1 _ 18 = d _ 11.25
Multiply both sides by 11.25. 11.25 _ 18 = d
0.625 = d
The distance on the map is about 0.625 in.
The actual distance between North Chicago and Waukegan is 4 mi. What is this distance on the map? Round to the nearest tenth.
Write the scale as a fraction. Let d be the distance on the map. Multiply both sides by 4.
map
_ actual
_ _ = _ _ = d
≈ d
Your Turn
→ →
© H
Explain 2 Using Dimensional Analysis to Convert Measurements
Dimensional analysis is a method of manipulating unit measures algebraically to determine the proper units for a quantity computed algebraically. The comparison of two quantities with different units is called a rate. The ratio of two equal quantities, each measured in different units, is called a conversion factor.
Example 2 Use dimensional analysis to convert the measurements.
A large adult male human has about 12 pints of blood. Use dimensional analysis to convert this quantity to gallons.
Step 1 Convert pints to quarts.
Multiply by a conversion factor whose first quantity is quarts and whose second quantity is pints.
12 pt ⋅ 1 qt
_ 2 pt = 6 qt
Step 2 Convert quarts to gallons.
Multiply by a conversion factor whose first quantity is gallons and whose second quantity is quarts.
6 qt ⋅ 1 gal
_ 4 qt = 6 _ 4 gal = 1 1 _ 2 gal
A large adult male human has about 1 1 _ 2 gallons of blood.
The length of a building is 720 in. Use dimensional analysis to convert this quantity to yards.
Step 1 Convert inches to feet.
Multiply by a conversion factor whose first quantity is feet and whose second quantity is inches.
720 in. ⋅ ft _
Step 2 Convert feet to yards.
Multiply by a conversion factor whose first quantity is yards and whose second quantity is feet.
ft ⋅ yd _
Your Turn
Use dimensional analysis to convert the measurements. Round answers to the nearest tenth.
4. 7500 seconds ≈ hours 5. 3 feet ≈ meters
6. 4 inches ≈ yards
© H
Explain 3 Using Dimensional Analysis to Convert and Compare Rates
Use dimensional analysis to determine which rate is greater.
Example 3 During a cycling event for charity, Amanda traveled 105 kilometers in 4.2 hours and Brenda traveled at a rate of 0.2 mile per minute. Which girl traveled at a greater rate? Use 1 mi = 1.61 km.
Convert Amanda’s rate to the same units as Brenda’s rate. Set up conversion factors so that both kilometers and hours cancel.
x miles _ minute ≈ 105 km _ 4.2 h
⋅ 1 mi _ 1.61 km
⋅ 1 h _ 60 min
≈ 0.2588 mi/min
Amanda traveled approximately 0.26 mi/min.
Amanda traveled faster than Brenda.
A box of books has a mass of 4.10 kilograms for every meter of its height. A box of magazines has a mass of 3 pounds for every foot of its height. Which box has a greater mass per unit of height? Use 1 lb = 0.45 kg and 1 m = 3.28 ft. Round your answer to the nearest tenth.
Convert the mass of the box of books to the same units as the mass of the box of magazines. Set up conversion factors so that both kilograms and pounds cancel.
x lb _ ft
kg ⋅ m _
ft ≈ lb __
ft ≈ lb/ft
The box of has a greater mass per unit of height.
Reflect
7. Why is it important to convert rates to the same units before comparing them?
Your Turn
8. Alan’s go-kart travels 1750 feet per minute, and Barry’s go-kart travels 21 miles per hour. Whose go-kart travels faster? Round your answer to the nearest tenth.
© H
y
Explain 4 Graphing a Proportional Relationship To graph a proportional relationship, first find the unit rate, then create scales on the x- and y-axes and graph points.
Example 4 Simon sold candles to raise money for the school dance. He raised a total of $25.00 for selling 10 candles. Find the unit rate (amount earned per candle). Then graph the relationship.
Find the unit rate. Amount earned __ Candles sold
: 25 _ 10 = x _ 1
Using this information, create scales on the x- and y-axes.
The x-axis will represent the candles sold, since this is the independent variable.
The y-axis will represent the amount earned, since this is the dependent variable.
The origin represents what happens when Simon sells 0 candles. The school gets $0.
Simon sold a total of 10 candles, so the x-axis will need to go from 0 to 10.
Since the school gets a total of $25 from Simon, the y-axis will need to go from 0 to 25.
Plot points on the graph to represent the amount of money the school earns for the different numbers of candles sold.
A local store sells 8 corn muffins for a total of $6.00. Find the unit rate. Then graph the points.
Find the unit rate. Amount earned __ Muffins sold
: _____ = x _ 1
Using this information, create scales on the x- and y-axes.
The x-axis will represent the , since this is the independent variable.
The y-axis will represent the , since this is the dependent variable.
The origin in this graph represents what happens when .
The x-axis will need to go from to .
The y-axis will need to go from to .
Plot points on the graph to represent the earnings from the different numbers of muffins sold.
Reflect
9. In Example 4A, Simon raised a total of $25.00 for selling 10 candles. If Simon raised $30.00 for selling 10 candles, would the unit rate be higher or lower? Explain.
Simon’s Earnings
y
x
10 5
20 15
© H
p any
Your Turn
Find the unit rate, create scales on the x- and y-axes, and then graph the function.
10. Alex drove 135 miles in 3 hours at a constant speed.
11. Max wrote 10 pages of his lab report in 4 hours.
Elaborate
12. Give three examples of proportions. How do you know they are proportions? Then give three nonexamples of proportions. How do you know they are not proportions?
13. If a scale is represented by a ratio less than 1, what do we know about the actual object? If a scale is represented by a ratio greater than 1, what do we know about the actual object?
14. How is dimensional analysis useful in calculations that involve measurements?
15. Essential Question Check In How is finding the unit rate helpful before graphing a proportional relationship?
x
y
x
y
© H
1. Represent Real-World Problems A building casts a shadow 48 feet long. At the same time, a 40-foot-tall flagpole casts a shadow 9.6 feet long. What is the height of the building?
Use the table to answer questions 2–4. Select the best answer. Assume the shadow lengths were measured at the same time of day.
2. The flagpole casts an 8-foot shadow, as shown in the table. At the same time, the oak tree casts a 12-foot shadow. How tall is the oak tree?
3. How tall is the goal post? 4. What is the length of the fence’s shadow?
5. Decorating A particular shade of paint is made by mixing 5 parts red paint with 7 parts blue paint. To make this shade, Shannon mixed 12 quarts of blue paint with 8 quarts of red paint. Did Shannon mix the correct shade? Explain.
6. Geography The scale on a map of Virginia shows that 1 inch represents 30 miles. The actual distance from Richmond, VA, to Washington, D.C., is 110 miles. On the map, how many inches are between the two cities?
7. Sam is building a model of an antique car. The scale of his model to the actual car is 1:10. His model is 18 1 __ 2 inches long. How long is the actual car?
Object Length of Shadow (ft)
Height (ft)
© H
es
8. Archaeology Stonehenge II in Hunt, Texas, is a scale model of the ancient construction in Wiltshire, England. The scale of the model to the original is 3 to 5. The Altar Stone of the original construction is 4.9 meters tall. Write and solve a proportion to find the height of the model of the Altar Stone.
For 9–11, tell whether each scale reduces, enlarges, or preserves the size of an actual object.
9. 1 m to 25 cm 10. 8 in. to 1 ft 11. 12 in. to 1 ft
12. Analyze Relationships When a measurement in inches is converted to centimeters, will the number of centimeters be greater or less than the number of inches? Explain.
Use dimensional analysis to convert the measurements.
13. Convert 8 milliliters to fluid ounces. Use 1 mL ≈ 0.034 fl oz.
14. Convert 12 kilograms to pounds. Use 1 kg ≈ 2.2 lb.
15. Convert 950 US dollars to British pound sterling. Use 1 US dollar = 0.62 British pound sterling.
16. The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. Convert this speed to inches per minute.
17. Tortoise A walks 52.0 feet per hour and tortoise B walks 12 inches per minute. Which tortoise travels faster? Explain.
18. The pitcher for the Robins throws a baseball at 90.0 miles per hour. The pitcher on the Bluebirds throws a baseball 121 feet per second. Which pitcher throws a baseball faster? Explain.
© H
y
19. For a science experiment Marcia dissolved 1.0 kilogram of salt in 3.0 liters of water. For a different experiment, Bobby dissolved 2.0 pounds of salt in 7.0 pints of water. Which person made a more concentrated salt solution? Explain. Use 1 L = 2.11 pints. Round your answer to the nearest hundredth.
20. Will a stand that can hold up to 40 pounds support a 21-kilogram television? Explain. Use 2.2 lb = 1 kg.
Find the unit rate, create scales on the x- and y-axes, and then graph the function.
21. Brianna bought a total of 8 notebooks and got 16 free pens.
22. Mason sold 10 wristbands and made a total of 5 dollars.
x
y
x
y
© H
p any
23. Match each graph to the data it goes with. Explain your reasoning.
A. B.
C. D.
x
y
3
6
9
12
15
Hours
4
8
12
16
20
Hours
© H
24. Multi-Step A can of tuna has a shape similar to the shape of a large water tank. The can of tuna has a diameter of 3 inches and a height of 2 inches. The water tank has a diameter of 6 yards. What is the height of the water tank in both inches and yards?
25. Represent Real-World Problems Write a real-world scenario in which 12 fluid ounces would need to be converted into liters. Then make the conversion. Use 1 fl oz = 0.0296 L. Round your answer to the nearest tenth.
26. Find the Error The graph shown was given to represent this problem. Find the error(s) in the graph and then create a correct graph to represent the problem. Jamie took an 8-week keyboarding class. At the end of each week, she took a test to find the number of words she could type per minute and found out she improved the same amount each week. Before Jamie started the class, she could type 25 words per minute, and by the end of week 8, she could type 65 words per minute.
2 in
© H
The Wright Flyer was the first successful powered aircraft. A model was made to display in a museum with the length of 35 cm and a wingspan of about 66.9 cm. The length of the actual plane was 21 ft 1 in., and the height was 2.74 m. Compare the length, height, and wingspan of the model to the actual plane and explain why any errors may occur. (Round any calculations to the nearest whole number.)
© H
Resource Locker
Explore Comparing Precision of Measurements Numbers are values without units. They can be used to compute or to describe measurements. Quantities are real- word values that represent specific amounts. For instance, 15 is a number, but 15 grams is a quantity.
Accuracy is the closeness of a given measurement or value to the actual measurement or value. Suppose you know the actual measure of a quantity, and someone else measures it. You can find the accuracy of the measurement by finding the absolute value of the difference of the two.
Measurement 1 Measurement 2 Smaller Unit More Precise Measurement
4 g 4.3 g
5.71 oz 5.7 oz
4.2 m 422 cm
7 ft 2 in. 7.2 in.
B Eric is a lab technician. Every week, he needs to test the scales in the lab to make sure that they are accurate. He uses a standard mass that is exactly 8.000 grams and gets the following results.
Scale Mass
The measurement for Scale is the most precise
because it measures to the nearest , which is smaller than the smallest unit measured on the other two scales.
Scale 1 Scale 2 Scale 3
© H
p any
Find the accuracy of each of the measurements in Step B.
Scale 1: Accuracy = 8.000 - =
Complete each statement: the measurement for Scale , which is grams,
is the most accurate because .
Reflect
1. Discussion Given two measurements of the same quantity, is it possible that the more precise measurement is not the more accurate? Why do you think that is so?
Explain 1 Determining Precision of Calculated Measurements As you have seen, measurements are reported to a certain precision. The reported value does not necessarily represent the actual value of the measurement. When you measure to the nearest unit, the actual length can be 0.5 unit less than the measured length or less than 0.5 unit greater than the measured length. So, a length reported as 4.5 centimeters could actually be anywhere between 4.45 centimeters and 4.55 centimeters, but not including 4.55 centimeters. It cannot include 4.55 centimeters because 4.55 centimeters reported to the nearest tenth would round up to 4.6 centimeters.
Example 1 Calculate the minimum and maximum possible areas. Round your answers to the nearest square centimeter.
The length and width of a book cover are 28.3 centimeters and 21 centimeters, respectively.
Find the range of values for the actual length and width of the book cover.
Minimum length = (28.3 - 0.05) cm and maximum length = (28.3 + 0.05) cm, so 28.25 cm ≤ length < 28.35 cm.
Minimum width = (21 - 0.5) cm and maximum width = (21 + 0.5) cm, so 20.5 cm ≤ width < 21.5 cm.
Find the minimum and maximum areas.
Minimum area = minimum length ⋅ minimum width
= 28.25 cm ⋅ 20.5 cm ≈ 579 cm 2
Maximum area = maximum length ⋅ maximum width
= 28.35 cm ⋅ 21.5 cm ≈ 610 cm 2
So 579 cm 2 ≤ area < 610 cm 2 .
© H
y
B The length and width of a rectangle are 15.5 centimeters and 10 centimeters, respectively.
Minimum length = (15.5 - ) cm and maximum length = (15.5 + ) cm,
so ≤ length < .
so ≤ width < .
= cm ⋅ cm ≈ cm 2
= cm ⋅ cm ≈ cm 2
Reflect
2. How do the ranges of the lengths and widths of the books compare to the range of the areas? What does that mean in terms of the uncertainty of the dimensions?
Your Turn
Calculate the minimum and maximum possible areas. Round your answers to the nearest whole square unit.
3. Sara wants to paint a wall. The length and width of the wall are 2 meters and 1.4 meters, respectively.
4. A rectangular garden plot measures 15 feet by 22.7 feet.
© H
p any
Explain 2 Identifying Significant Digits Significant digits are the digits in measurements that carry meaning about the precision of the measurement.
Identifying Significant Digits
All nonzero digits are significant. 55.98 has 4 significant digits.
115 has 3 significant digits.
Zeros between two other significant digits are significant.
Zeros at the end of a number to the right of a decimal point are significant.
Zeros to the left of the first nonzero digit in a decimal are not significant.
Zeros at the end of a number without a decimal point are assumed to be not significant.
60,600 has 3 significant digits.
77,000,000 has 2 significant digits.
Example 2 Determine the number of significant digits in a given measurement.
6040.0050 m
Significant Digits Rule Digits Count
Nonzero digits: 6 0 4 0 . 0 0 5 0 3
Zeros between two significant digits: 6 0 4 0 . 0 0 5 0 4
End zeros to the right of a decimal: 6 0 4 0 . 0 0 5 0 1
Total 8
B 710.080 cm
Nonzero digits: 7 1 0 . 0 8 0
Zeros between two significant digits: 7 1 0 . 0 8 0
End zeros to the right of a decimal: 7 1 0 . 0 8 0
Total
710.080 cm has significant digit(s).
© H
is
Reflect
5. Critique Reasoning A student claimed that 0.045 and 0.0045 m have the same number of significant digits. Do you agree or disagree?
Your Turn
Determine the number of significant digits in each measurement.
6. 0.052 kg 7. 10,000 ft 8. 10.000 ft
Explain 3 Using Significant Digits in Calculated Measurements When performing calculations with measurements of different precision, the number of significant digits in the solution may differ from the number of significant digits in the original measurements. Use the rules from the following table to determine how many significant digits to include in the result of a calculation.
Rules for Significant Digits in Calculated Measurements
Operation Rule
Addition or Subtraction The sum or difference must be rounded to the same place value as last significant digit of the least precise measurement.
Multiplication or Division The product or quotient must have no more significant digits than the least precise measurement.
Example 3 Find the perimeter and area of the given object. Make sure your answers have the correct number of significant digits.
A rectangular swimming pool measures 22.3 feet by 75 feet.
Find the perimeter of the swimming pool using the correct number of significant digits.
Perimeter = sum of side lengths
= 22.3 ft + 75 ft + 22.3 ft + 75 ft
= 194.6 ft
The least precise measurement is 75 feet. Its last significant digit is in the ones place. So round the sum to the ones place. The perimeter is 195 ft.
Find the area of the swimming pool using the correct number of significant digits.
Area = length ⋅ width
= 22.3 ft ⋅ 75 ft = 1672.5 ft 2
The least precise measurement, 75 feet, has two significant digits, so round the product to a number with two significant digits. The area is 1700 ft 2 .
© H
p any
B A rectangular garden plot measures 21 feet by 25.2 feet.
Find the perimeter of the garden using the correct number of significant digits.
Perimeter = sum of side lengths
= + + + =
The least precise measurement is . Its last significant digit is in the ones place. So round the sum
to the place. The perimeter is .
Find the area of the garden using the correct number of significant digits.
Area = length ⋅ width
= ⋅ =
The least precise measurement, has significant digit(s), so round to a number with
significant digit(s). The area is .
Reflect
9. In the example, why did the area of the garden and the swimming pool each have two significant digits?
10. Is it possible for the perimeter of a rectangular garden to have more significant digits than its length or width does?
Your Turn
Find the perimeter and area of the given object. Make sure your answers have the correct number of significant digits.
11. A children’s sandbox measures 7.6 feet by 8.25 feet.
12. A rectangular door measures 91 centimeters by 203.2 centimeters.
© H
Explain 4 Using Significant Digits in Estimation Real-world situations often involve estimation. Significant digits play an important role in making reasonable estimates.
A city is planning a classic car show. A section of road 820 feet long will be closed to provide a space to display the cars in a row. In past shows, the longest car was 18.36 feet long and the shortest car was 15.1 feet long. Based on that information, about how many cars can be displayed in this year’s show?
Analyze Information
• Available space:
Formulate a Plan
The word about indicates that your answer will be a(n) .
Available Space = Number of Cars ⋅
Find the number of longest cars and the number of shortest cars, and then use the average.
Solve
Longest:
Shortest:
The number of cars must be rounded to significant digits.
So, the club can estimate that a minimum of cars and a maximum
of cars can be displayed, and on average, cars can be displayed.
Justify and Evaluate
Because the cars will probably have many different lengths, a reasonable estimate is a
value between .
© H
Reflect
13. In the example, why wouldn’t it be wise to use the length of a shorter car?
14. Critical Thinking How else might the number of cars be estimated? Would you expect the estimate to be the same? Explain.
Your Turn
Estimate the quantity needed in the following situations. Use the correct number of significant digits.
15. Claire and Juan are decorating a rectangular wall of 433 square feet with two types of rectangular pieces of fabric. One type has an area of 9.4 square feet and the other has an area of 17.2 square feet. About how many decorative pieces can Claire and Juan fit in the given area?
16. An artist is making a mosaic and has pieces of smooth glass ranging in area from 0.25 square inch to 3.75 square inches. Suppose the mosaic is 34.1 inches wide and 50.0 inches long. About how many pieces of glass will the artist need?
Elaborate
17. Given two measurements, is it possible that the more accurate measurement is not the more precise? Justify your answer.
18. What is the relationship between the range of possible error in the measurements used in a calculation and the range of possible error in the calculated measurement?
19. Essential Question Check-In How do you use significant digits to determine how to report a sum or product of two measurements?
© H
y
1. Choose the more precise measurement from the pair 54.1 cm and 54.16 cm. Justify your answer.
Choose the more precise measurement in each pair.
2. 1 ft; 12 in. 3. 5 kg; 5212 g 4. 7 m; 7.7 m 5. 123 cm; 1291 mm
6. True or False? A scale that measures the mass of an object in grams to two decimal places is more precise than a scale that measures the mass of an object in milligrams to two decimal places. Justify your answer.
7. Every week, a technician in a lab needs to test the scales in the lab to make sure that they are accurate. She uses a standard mass that is exactly 4 g and gets the following results.
a. Which scale gives the most precise measurement?
b. Which scale gives the most accurate measurement?
8. A manufacturing company uses three measuring tools to measure lengths. The tools are tested using a standard unit exactly 7 cm long. The results are as follows.
a. Which tool gives the most precise measurement?
b. Which tool gives the most accurate measurement?
• Online Homework • Hints and Help • Extra Practice
© H
Given the following measurements, calculate the minimum and maximum possible areas of each object. Round your answer to the nearest square whole square unit.
9. The length and width of a book cover are 22.2 centimeters and 12 centimeters, respectively.
10. The length and width of a rectangle are 19.5 centimeters and 14 centimeters, respectively.
11. Chris is painting a wall with a length of 3 meters and a width of 1.6 meters.
12. A rectangular garden measures 15 feet by 24.1 feet.
Show the steps to determine the number of significant digits in the measurement.
13. 123.040 m
14. 0.00609 cm
© H
ck
Find the perimeter and area of each garden. Report your answers with the correct number of significant digits.
20. Samantha is putting a layer of topsoil on a garden plot. She measures the plot and finds that the dimensions of the plot are 5 meters by 21 meters. Samantha has a bag of topsoil that covers an area of 106 square meters. Should she buy another bag of topsoil to ensure that she can cover her entire plot? Explain.
21. Tom wants to tile the floor in his kitchen, which has an area of 320 square feet. In the store, the smallest tile he likes has an area of 1.1 square feet and the largest tile he likes has an area of 1.815 square feet. About how many tiles can be fitted in the given area?
© H
22. Communicate Mathematical Ideas Consider the calculation 5.6 mi ÷ 9s = 0.62222 mi/s. Why is it important to use significant digits to round the answer?
23. Find the Error A student found that the dimensions of a rectangle were 1.20 centimeters and 1.40 centimeters. He was asked to report the area using the correct number of significant digits. He reported the area as 1.7 cm 2 . Explain the error the student made.
24. Make a Conjecture Given two values with the same number of decimal places and significant digits, is it possible for an operation performed with the two values to have a different number of decimal places or significant digits than the original values?
The sun is an excellent source of electrical energy. A field
of solar panels yields 16.22 Watts per square feet. Determine
the amount of electricity produced by a field of solar panels
that is 305 feet by 620 feet.
© H
1 STUDY GUIDE REVIEW MODULE
Essential Question: How do you use quantitative reasoning to solve real-world problems?
Two fortnights have passed from January 3rd to January 31st. How many days long is a fortnight?
31 - 3 = 28
The scale on a map is 1 in: 8 mi. The distance from Cedar Park, TX to Austin, TX on the map is 2.5 in. How long is the actual distance?
actual _ map → 8 mi _ 1 in.
8 _ 1 = d _ 2.5
20 = d The actual distance is 20 mi.
Find the sum and product of the following measurements using the correct number of significant digits: 15 ft and 9.25 ft.
15 ft + 9.25 ft = 24.25 ft
24 ft
140 ft 2
(factor de conversión) dimensional analysis
(análisis dimensional) equation (ecuación)
(solución de una ecuación)
Round to the place value of the last significant digit of the least precise measurement.
The last significant digit of 15 is in the ones place.
Round so it has the number of significant digits of the least precise measurement.
Write the scale as a fraction.
L

algebra 1 - volume 1_2015_ student_

Documents