1 Common Core Student Edition SpringBoard Mathematics Course 2
2 About the College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the College Board was created to expand access to higher education. Today, the membership association is made up of more than 5,900 of the nation s leading educational institutions and is dedicated to promoting excellence and equity in education. Each year, the College Board helps more than seven million students prepare for a successful transition to college through programs and services in college readiness and college success including the SAT and the Advanced Placement Program. The organization also serves the education community through research and advocacy on behalf of students, educators, and schools. For further information, visit ISBN: X ISBN: Copyright 2014 by the College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, AP Vertical Teams, CollegeEd, Pre-AP, SpringBoard, connect to college success, SAT, and the Acorn logo are registered trademarks of the College Board. College Board Standards for College Success, connect to college success, English Textual Power, and SpringBoard are trademarks owned by College Board. PSAT/NMSQT is a registered trademark of the College Board and the National Merit Scholarship Corporation. Microsoft and PowerPoint are registered trademarks of Microsoft Corporation. All other products and services may be trademarks of their respective owners Printed in the United States of America
3 Acknowledgments The College Board gratefully acknowledges the outstanding work of the classroom teachers and writers who have been integral to the development of this revised program. The end product is testimony to their expertise, understanding of student learning needs, and dedication to rigorous but accessible mathematics instruction. Michael Allwood Brunswick School Greenwich, Connecticut Floyd Bullard North Carolina School of Science and Mathematics Durham, North Carolina Marcia Chumas East Mecklenburg High School Charlotte, North Carolina Kathy Fritz Plano Independent School District Plano, Texas Shawn Harris Ronan Middle School Ronan, Montana Marie Humphrey East Mecklenburg High School Charlotte, North Carolina Brian Kotz Montgomery College Monrovia, Maryland Chris Olsen Prairie Lutheran School Cedar Rapids, Iowa Dr. Roxy Peck California Polytechnic Institute San Luis Obispo, California Katie Sheets Harrisburg School Harrisburg, South Dakota Andrea Sukow Mathematics Consultant Nashville, Tennessee Stephanie Tate Hillsborough School District Tampa, Florida Product Development Betty Barnett Executive Director, SpringBoard Robert Sheffield Sr. Director, SpringBoard Implementation Allen Dimacali Editorial Director, Mathematics SpringBoard John Nelson Editor, SpringBoard Judy Windle Sr. Mathematics Instructional Specialist SpringBoard Alex Chavarry Sr. Director, SpringBoard Strategic Accounts Acknowledgments iii
4 Acknowledgments continued Research and Planning Advisors We also wish to thank the members of our SpringBoard Advisory Council and the many educators who gave generously of their time and their ideas as we conducted research for both the print and online programs. Your suggestions and reactions to ideas helped immeasurably as we planned the revisions. We gratefully acknowledge the teachers and administrators in the following districts. Vrain Valley School District Longmont, Colorado Scottsdale Public Schools Phoenix, Arizona Seminole County Public Schools Sanford, Florida Southwest ISD San Antonio, Texas Spokane Public Schools Spokane, Washington Volusia County Schools DeLand, Florida Grand Prairie ISD Grand Prairie, Texas Peninsula School District Gig Harbor, Washington iv SpringBoard Mathematics with Meaning Level 2
5 Contents To the Student Instructional Units x UNIT 1 Activity 1 NUMBER SYSTEMS Unit 1 Overview 1 Getting Ready 2 Operations on Positive Rational Numbers Paper Clips, Airplanes, and Spiders 3 Lesson 1-1 Adding and Subtracting Decimals 3 Lesson 1-2 Multiplying and Dividing Decimals 5 Lesson 1-3 Operations with Fractions 7 Lesson 1-4 Converting Rational Numbers to Decimals 11 Activity 1 Practice 13 Activity 2 Addition and Subtraction of Integers Elevation Ups and Downs 15 Lesson 2-1 Adding Integers 15 Lesson 2-2 Subtracting Integers 19 Activity 2 Practice 22 Embedded Assessment 1 Positive Rational Numbers and Adding and Subtracting Integers Off to the Races 23 Activity 3 Multiplication and Division of Integers What s the Sign? 25 Lesson 3-1 Multiplying Integers 25 Lesson 3-2 Dividing Integers 29 Activity 3 Practice 31 Activity 4 Operations on Rational Numbers Let s Be Rational! 33 Lesson 4-1 Sets of Rational Numbers 33 Lesson 4-2 Adding Rational Numbers 36 Lesson 4-3 Subtracting Rational Numbers 39 Lesson 4-4 Multiplying and Dividing Rational Numbers 41 Activity 4 Practice 44 Embedded Assessment 2 Rational Number Operations and Multiplying and Dividing Integers Top to Bottom 47 UNIT 2 EXPRESSIONS AND EQUATIONS Unit 2 Overview 49 Getting Ready 50 Activity 5 Properties of Operations What s In a Name? 51 Lesson 5-1 Applying Properties of Operations 51 Lesson 5-2 Applying Properties to Factor and Expand 54 Activity 5 Practice 58 Contents v
6Have I Got a Job for You! 123 Lesson 12-1 Percent Increase and Decrease 123 Lesson 12-2 Markups and Discounts 125 Lesson 12-3 Interest 127 Lesson 12-4 Percent Error 129 Activity 12 Practice 131 vi SpringBoard Mathematics with Meaning Level 2
8 Contents continued UNIT 5 PROBABILITY Unit 5 Overview 225 Getting Ready 226 Activity 20 Exploring Probability Spinner Games 227 Lesson 20-1 Making Predictions 227 Lesson 20-2 Investigating Chance Processes 231 Lesson 20-3 Estimating Probabilities 239 Lesson 20-4 Making Decisions 243 Activity 20 Practice 249 Activity 21 Probability Probability Two Ways 251 Lesson 21-1 Equally Likely Outcomes 251 Lesson 21-2 Theoretical Probability 259 Lesson 21-3 Comparing Probabilities 265 Activity 21 Practice 268 Embedded Assessment 1 Finding Probabilities Spinning Spinners and Random Picks 272 Activity 22 Games and Probability Rock, Paper, Scissors and Other Games 275 Lesson 22-1 Rock, Paper, Scissors 275 Lesson 22-2 More Rock, Paper, Scissors 279 Lesson 22-3 Boxes and Drawers 285 Lesson 22-4 More Boxes and Drawers 291 Activity 22 Practice 295 Activity 23 Probability Estimating Probabilities Using Simulation 297 Lesson 23-1 What is Simulation? 297 Lesson 23-2 Using Random Numbers to Simulate Events 302 Lesson 23-3 Simulating a Compound Event 306 Lesson 23-4 Finding Probabilities Using Simulation 310 Activity 23 Practice 316 Embedded Assessment 2 Probability and Simulation Flipping Coins and Random Choices 320 UNIT 6 STATISTICS Unit 6 Overview 323 Getting Ready 324 Activity 24 Statistics Summer Reading Club 325 Lesson 24-1 Population and Census 325 Lesson 24-2 Sampling from a Population 328 Activity 24 Practice 336 viii SpringBoard Mathematics with Meaning Level 2
9 Activity 25 Exploring Sampling Variability Sample Speak 341 Lesson 25-1 Sample Statistic and Sampling Variability 341 Lesson 25-2 Predictions and Conclusions 346 Activity 25 Practice 353 Embedded Assessment 1 Random Sampling and Sampling Variability School Populations 357 Activity 26 Comparative Statistics Seventh-Grade Students 361 Lesson 26-1 Two Sample Means 361 Lesson 26-2 Difference in Terms of MAD 371 Lesson 26-3 Calculating MAD for a Sample 380 Activity 26 Practice 387 Embedded Assessment 2 Comparing Populations One Mean Arm Span 391 UNIT 7 PERSONAL FINANCIAL LITERACY Unit 7 Overview 393 Getting Ready 394 Activity 27 Budgeting and Money Management How Much Is Too Much? 395 Lesson 27-1 Understanding Earnings and Budgets 395 Lesson 27-2 Financial Planning 400 Activity 27 Practice 404 RESOURCES 405 Formulas 406 Learning Strategies 410 Glossary 413 Academic Vocabulary Graphic Organizers 423 Contents ix
10 To the Student Welcome to the SpringBoard program. We hope you will discover how SpringBoard can help you achieve high academic standards, reach your learning goals, and prepare for success in future mathematics studies. The program has been created with you in mind: the content you need to learn, the tools to help you learn, and the critical thinking skills that help you build confidence in your own knowledge of mathematics. The College Board publishes the SpringBoard program. It also publishes the PSAT/NMSQT, the SAT, and the Advanced Placement exams all exams that you are likely to encounter in your student years. Preparing you to perform well on those exams and to develop the mathematics skills needed for high school success is the primary purpose of this program. Standards-Based Mathematics Learning Knowledge of mathematics helps prepare you for future success in college, in work, and in your personal life. We all encounter some form of mathematics daily, from calculating the cost of groceries to determining the cost of materials and labor needed to build a new road. The SpringBoard program is based on learning standards that identify the mathematics skills and knowledge that you should master to succeed in high school and in future college-level work. In this course, the standards follow these broad areas of mathematics knowledge: Mathematical practices Number and operations Expressions, equations, and relationships Ratio and proportionality Geometry Statistics and probability Mathematical practice standards guide your study of mathematics. They are actions you take to help you understand mathematical concepts rather than just mathematical procedures. For example, the mathematical practice standards suggest the following: Make sense of and connect mathematics concepts to everyday life and situations around you. Model with mathematics to solve problems, justify solutions and their reasonableness, and communicate mathematical ideas. Use appropriate tools, such as number lines, protractors, technology, or paper and pencil, strategically to help you solve problems. Communicate abstract and quantitative reasoning both orally and in writing through arguments and critiques. Analyze mathematical relationships through structure and repeated reasoning to connect ideas. Attend to precision in both written and oral communication of your mathematical ideas. In the middle school years, your study of mathematics begins with a basic understanding of fractions and the operations performed with them. Your study continues with the development of a deep understanding of the rational numbers, their different representations, and the connections between these numbers and other number systems and operations. You will need a broad x SpringBoard Mathematics with Meaning Level 2
11 understanding of addition, subtraction, and multiplication with rational numbers, along with computational fluency with whole-number operations. As you continue your studies, you will examine ratios and rates, which will allow you to make comparisons between numbers. Ratios and rates represent proportionality. Understanding the concepts of proportionality and linear equations are critical to future success in your study of algebra and the rest of the high school mathematics curriculum. See pages xiii xvi for a complete list of the Common Core State Standards for Mathematics for this course. Strategies for Learning Mathematics Some tools to help you learn are built into every activity. At the beginning of each activity, you will see suggested learning strategies. Each of these strategies is explained in full in the Resources section of your book. As you learn to use each strategy, you ll have the opportunity to decide which strategies work best for you. Suggested learning strategies include: Reading strategies, which help you learn to look at problem descriptions in different ways, from marking the text to highlight key information to turning problem information into questions that help you break the problem down into its separate parts. Writing strategies, which help you focus on your purpose for writing and what you re writing about. Problem-solving strategies, which give you multiple ways to approach the problem, from learning to identify the tasks within a problem to looking for patterns or working backward to see how the problem is set up. Collaborative strategies, which you ll use with your classmates to explore concepts and problems in group discussions and working with partners. Building Mathematics Knowledge and Skills Whether it is mathematics or sports or cooking, one way we learn something really well is by practice and repetition. To help you learn mathematics, the SpringBoard program is built around problem solving, reasoning and justification, communication, connections between concepts and ideas, and visual representation of mathematical concepts. Problem Solving Many of the problems in this book are based on real-life situations that require you to analyze the situation and the information in the problem, make decisions, determine the strategies you ll use to solve the problem, and justify your solution. Having a real-world focus helps you see how mathematics is used in everyday life. Reasoning and Justification One part of learning mathematics, or any subject, is learning not only how to solve problems but also why you solved them the way you did. You will have many opportunities to predict possible solutions and then to verify solutions. You will be asked to explain the reasoning behind how you solved the problem, the mathematics concepts involved, and why your approach was appropriate for solving the problem. To the Student xi
12 To the Student continued Communication When learning a language, saying words out loud helps you learn to pronounce the words and to remember them. Communicating about mathematics, orally and in writing, with your classmates and teachers helps you organize your learning and explain mathematics concepts and problemsolving strategies more precisely. Sharing your ideas and thoughts allows you and your classmates to build on each other s ideas and expand your own understanding. Mathematics Connections As you study mathematics, you will learn many different concepts and ways of solving problems. Reading the problem descriptions will take you into the real-life applications of mathematics. As you develop your mathematics knowledge, you will see the many connections between mathematics concepts and between mathematics and your own life. Representations Artists create representations through drawings and paintings. In mathematics, representations can take many forms, such as numeric, verbal, graphic, or symbolic. In this course, you are encouraged to use representations to organize problem information, present possible solutions, and communicate your reasoning. Creating representations is a tool you can use to gain understanding of concepts and communicate that understanding to others. We hope you enjoy your study of mathematics using the SpringBoard program. We, the writers, are all classroom teachers, and we created this program because we love mathematics. We wanted to inspire you to learn mathematics and build confidence that you can be successful in your math studies and in using mathematics in daily life. xii SpringBoard Mathematics with Meaning Level 2
13 Common Core State Standards for Mathematics Grade 7 Standards for Mathematical Practice MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.4 Model with mathematics. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.7 Look for and make use of structure. MP.8 Look for and express regularity in repeated reasoning. 7.RP Ratios and Proportional Relationships 7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1 2 mile in each hour, compute the unit rate as the complex fraction 214 miles per hour, equivalently 2 miles per hour. 7.RP.A.2 Recognize and represent proportional relationships between quantities. 7.RP.A.2a Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 7.RP.A.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 7.RP.A.2c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. 7.RP.A.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 7.NS The Number System 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.NS.A.1a Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 7.NS.A.1b Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show Common Core State Standards for Mathematics xiii
14 that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. 7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. 7.NS.A.1d Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.A.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1) ( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then p = ( p) p =. Interpret quotients of rational q q ( q) numbers by describing real-world contexts. 7.NS.A.2c Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.A.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers. 7.EE Expressions and Equations 7.EE.A.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a a = 1.05a means that increase by 5% is the same as multiply by EE.B.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 10 1 of her salary an hour, or $2.50, for a new salary of $ If you want to place a towel bar inches long in the center of a door that is 27 1 inches wide, you will need to place the bar about 9 inches from each edge; this 2 estimate can be used as a check on the exact computation. xiv SpringBoard Mathematics with Meaning Level 2
15 7.EE.B.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 7.EE.B.4a Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? 7.EE.B.4b Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. 7.G Geometry 7.G.A.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.G.A.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. 7.G.A.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7.G.B.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. 7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. 7.SP Statistics and Probability 7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. Common Core State Standards for Mathematics xv
16 7.SP.B.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. 7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. 7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. 7.SP.C.7a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. 7.SP.C.7b Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? 7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 7.SP.C.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 7.SP.C.8b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ), identify the outcomes in the sample space which compose the event. 7.SP.C.8c Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? xvi SpringBoard Mathematics with Meaning Level 2
SPRINGBOARD MATHEMATICS COURSE 2 ANSWER KEY
Can you get a degree online?
A college education doesn't have to be inconvenient. Our online college degree programs let you work towards your academic goals without dropping your family or professional obligations. You can get an associate, bachelor's, master's or doctoral degree online.
Are online courses worth it?
Cost is another benefit, as most online courses are much cheaper than a traditional classroom program. Tuition is usually lower and there are practically no travel costs involved. That said, online education is only worth your time if you are earning accredited online degrees from accredited colleges.
Is financial aid available?
Just as financial aid is available for students who attend traditional schools, online students are eligible for the same – provided that the school they attend is accredited. Federal financial aid, aid on the state level, scholarships and grants are all available for those who seek them out. Here’s what students need to know about financial aid for online schools.
How do I prepare for an online class?
You need a reliable internet connection to participate in online courses. Many programs will tell you the requirements you need to succeed in their courses, but make sure to consider if other people in your household will use the internet at the same time. Online classes often require streaming videos or uploading content, so make sure you have the necessary speed and signal reliability to participate without interruption.
Middle School Math
Discover what middle school students will learn when they take SpringBoard Math courses.
Through exposure to rigorous math content, students begin developing the critical thinking skills needed to analyze, solve, and explain complex problems.
Why It Helps
Right from the start—as early as grade 6—students begin applying mathematical thinking to real-world situations, gaining skills that will later help them succeed in algebra, geometry, and beyond.
How It Works
Each unit contains:
- Suggested learning strategies.
- Reading Math and Writing Math callouts.
- Discussion Group Tips to encourage collaboration.
- “Mini-lessons” that review prior concepts.
- Additional practice that can be used as homework.
- Formative and summative assessments.
- Differentiated instruction—so both struggling and advanced students always have the support they need.
Grade 6 students learn to:
- Model functions in numbers, equations, tables, and graphs.
- Communicate math verbally, with the ability to justify answers and clearly label charts and graphs.
- Read and represent data in a variety of forms.
- Use multiple representations to communicate math concepts.
Grade 7 students learn to:
- Acquire an understanding of functions in the context of algebra and graphs.
- Write, solve, and graph linear equations, recognize, and verbalize patterns, and model slope as a rate of change.
- Communicate problem-solving methods and interpret results clearly.
- Investigate concepts presented visually and verbally.
Grade 8 students learn to:
- Write algebraic models from a variety of physical, numeric, and verbal descriptions.
- Solve equations using a variety of methods.
- Justify answers using precise mathematical language.
- Relate constant rate of change to verbal, physical, and algebraic models.
- Use technology to solve problems.
- Reinforce and extend the vocabulary of probability and statistics.
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She stood out in everything, in her posture, in her gaze, she simply radiated it and would be beautiful if she hadn't been so drunk. And she drank, apparently, not the first day. Yesterday or the day before yesterday, the neat clothes were changed, the hair was frayed and demanded a quick shower, as.