Little Rock School District June 2,1999 The main thing is to keep the main thing the main THING! Dear Colleagues: What a year! I hope all of you have the sense of accomplishment that 1 do! The positive opportimities for the school district continue to develop. We have had great individual and collective accomplishments, but the most exciting aspect of the year has been that the instructional indicators are pointing to very positive future results. A parent and a teacher asked me how that could be true. Hadnt I seen the newspaper story showing results of the 4* grade literacy and math exam? Yes, but when we looked at those we noticed the schools that were pilots of the ELLA program K-3 and math, all had strong gains. Staff development has not been universally accepted or appreciated, primarily because of the many failed promises of programs and quick fix elixirs of the past! But this time this very basic and focused approach appears to be just what the doctor ordered. Most educational reformers who have produced results over time have agreed it is a matter of focusgetting everyone on the same page with a standard or basic program. All the efforts of parents, students and teachers, regardless of the specialty, are focused to produce progress on a limited set of goals. ELLA training and the focused math training will continue to produce results on basic reading and math achievement goals. And let us not forget district graduates really had no peers when it came to top student awards in a number of state and national academic categories. Student activities have been a very pleasant surprise. The numbers of students participating grew impressively, and I know this helped to produce an improved year in student behavior. Many of you remember when state winners were most often decided within the schools of Little Rock, and Im talking about a variety of co-curricular activities. The district again has asserted itself with impressive performances in music, art, drama, speech, debate and athletics. With the implementation of the middle schools and neighborhood schools our students and programs will only become stronger. Without question, the academic reasoning for middle schools and 9-12 high schools will provide academic strength, and it will have a positive impact with increased student participation in career/technical programs and co-curricular activities. (continued) 810 West Markham Street Little Rock, Arkansas 72201  (501)324-2000We have mentioned before that we are moving toward stronger alternative education programs addressing behavior and academics. I am convinced we can provide many of our young people a more appropriate education in an alternative setting. One size doesnt fit all, and we must realize that these are times when public education is under attack from a variety of fronts. On one hand, we must be proactive in education for all of our children, but at the same time we must have appropriate alternatives for children who are not successful in the regular classroom. Being well intentioned is very noble, but the classroom must be a place of learning
we must have the appropriate learning climate to be successful. Improved instruction is essential, but alternative learning classrooms must also be a viable option. Your efforts have been commendable as you have participated in the planned restructuring of the Little Rock School District. Next year is the year for implementation and celebration. We will succeed in making the transition, and the change will provide us with the momentum to move forward as an exemplary educational institution for all children! In the 2000-2001 school year we will succeed in being declared unitary as a system which provides and will continue to provide a high quality education for all Little Rock children. This not a dream, but a vision of reality. Have a pleasant summer, and to those who are winding up their educational career this year, we offer our best wishes and heartfelt thanks from the whole community. To those who are moving, we say thank you for your assistance and effort during the year. It has been a great year! Thanks again. Sincerely, Leslie V. Gamine Superintendent of SchoolsLittle Rock School District Middle School Parents Mathematics Packet Purpose: The purpose of the packet is to inform middle school parents about the Districts middle school mathematics program and to provide resources which will help parents work with their children. Contents: DECEIVED 1. Connected Mathematics Program - Whats It All About? DEC 6 2000 2. 3. 4. Connected Mathematics Curriculum - Grade level modules and topics Comparison of Connected Mathematics to Past Mathematics Programs Why Connected Mathematics?  Math Performance of the Past  Desired Math Performance for the Future  Why Connected Mathematics Can Improve Mathematics Achievement 5. Assessments for Middle School Mathematics 6. What effect will Connected Mathematics have on SAT-9, ACT, and SAT scores? 7. Will Connected Mathematics prepare my child for Algebra 1 and higher level mathematics? 8, 9. How will the District know whether Connected Mathematics is working? Middle School Mathematics Lead Teacher, Middle School Mathematics Consultant 10. Connected Mathematics Module Order Form Resources Available for Parents (These are available at your Middle School.)  The Arkansas Mathematics Framework (This is Required Mathematics for All Students)  Correlation of Connected Mathematics to Arkansas Mathematics Framework  petting to Know Connected Mathematics - A Guide to the Connected Mathematics Curriculum  Abbreviated Connected Mathematics Teachers Editions  Computation Drill for Homework (These drill sheets should be brought home by students on a weekly basis for homework. A Complete Set can be checked out from your Middle School.) 1Little Rock School District Connected Mathematics Program Whats It All About? This document has been produced to provide information to parents regarding the mathematics curriculum currently being taught, tested and reviewed in all eight middle schools of the Little Rock School District. Q: What is Connected Mathematics? A: Connected Mathematics (CMP) is a middle school mathematics curriculum. Designed for grades 6, 7, and 8, it is a problem-based curriculum connecting different areas within mathematics, mathematics to other subject areas, and mathematics to applications in the world outside school. Q: Why are we making changes in the middle school curriculum? A: The curriculum in CMP offers Little Rock a much more rigorous middle school curriculum than we have had in the past. Little Rock needed a more rigorous curriculum at the middle school in order for our students to be successful in Algebra I and other higher level mathematics courses. By law, the Arkansas Mathematics Frameworks outlines the mathematics standards that all students are expected to achieve. Students success in reaching the middle grades (5-8) standards is measured by an exam called the 8^ grade Benchmark exam. 89% of the students taking the Arkansas Benchmark exam given in the 8*' grade scored below basic. Our past middle school mathematics curriculum has not always provided good preparation for our students. The Third International Math and Science study shows that only the top 5% of American students can perform as well as the top 25% of students internationally. Q: Does CMP emphasize the basic skills? A
Basic skills are a vital part of CMP. In addition to the basic skills practice embedded in CMP, Little Rock teachers will reinforce basic skills through homework assignments and mini-units as time permits. Calculators are used as a valuable teaching tool in CMP as well as in the mathematics programs at all Little Rock schools, but they are not used as a substitute for students learning the basics. Q: How has Connected Mathematics been implemented in Little Rock? A: Little Rock is entering its second year of implementation of the Connected Mathematics curriculum in 2000-01. Year one implemented 6^ grade
year two added 7*^ and 8** grade. Q: Is there any research about Connected Math? A: Yes. The National Science Foundation funded several curriculum projects to design and implement curriculum that was outlined in the Curriculum and Evaluation Standards developed by the National Council of Teachers of Mathematics in the late 198Os. The Connected Math Project was developed at Michigan State University with a National Science Foundation grant. Units were developed and extensively piloted in 19 states over several years with the final version published in 1996. CMP has more research behind it than the programs we have used in the past. It is the only program that has been developed from the ground up rather than taking an existing program and modifying it by adding suggestions for manipulative, alternative assessment, group work, etc. Additional information is available on the Web at www.mth.msu.edu/CMP. 2Q. What is the result on student achievement from implementing Connected Mathematics? A
The results will be monitored closely in Little Rock. At the end of each module Ct I InOnto v-i - + > 11 L. ________...t t* ... __ ..  students achievement will be measured according to the Districts benchmarks for mathematics. Stanford 9 scores for the grade and ACTAAP scores for 6* and 8 grades will be closely monitored. In addition, individual student growth will be measured at ah grade levels by the use of the levels tests (ALTs) given in the fall and spring of each year. Schools that have been using CMP for several years (Travers City, Michigan, and Bloomfield Hills, Michigan, for example) state their students perform I,  J u ....  cAdiiipie
biaie ineir siuaenis perrorm very well in high school. Many of them have such strong mathematical backgrounds that they are able to skip Algebra I and move directly into geometry. Q: How is Connected Mathematics different from the math taught in the traditional math class? A: Content presented in the CMP is very similar to a traditional program. CMP students are required by state law to cover the same material as students in other programs. Therefore, CMP students will be well prepared to enter Algebra I. What is different is the way the content is delivered. WHAT students learn is shaped by HOW they learn Students work individually, in pairs and in groups of four. This strategy allows students to experience different points of view and offers opportunities for students to share their opinions and strategies about mathematical processes. Students are assessed individually with homework grades, quizzes, tests, and projects. Q: Are there Connected Mathematics textbooks? A
Yes. All students have their own textbooks. However, the textbook is published in separate booklets. The booklets are three-hole punched and can be carried in students three-ring binders. Parent handbooks for each unit will be available for checkout from the school libraries to aid parents in helping their children at home. Q
Do teachers receive training in order to teach Connected Mathematics? A: Teachers received two days of training for each booklet. Since CMP is so heavily correlated to the Arkansas State Benchmark exam given in the 8^ grade, and this year in the 6 grade, our teachers were already very well prepared in terms of curriculum The only difference is the methodology of delivery. Q: is this anything like the old new Math? A: No. The content of CMP is based on traditional mathematics concepts. The difference is that instead of working on computation skills alone, the students learn mathematics in the context of actual situations, such as those they will have to face in the work place. Q: Where can I learn more about Connected Math? A, You can get information from the Connected Mathematics web site. The address for this web site is www.mth.msu.edu/CMP/CMP.html. Please feel free to talk to your child s teacher or school principal about the mathematics program if you have additional questions, or call Dennis Glasgow, Director of Math and Science, or Docia Jones middle-school math lead teacher, at 324-0520. 3LITTLE ROCK SCHOOL DISTRICT CMP CURRICULUM for Grades 6, 7, & 8 6*" Grade 7'" Grade 8" Grade Prime Time Number theory
primes
composites, factors and multiples Bits and Pieces II Using rational numbers: computation Data About Us Data investigation
formulating questions, gathering data, organizing and analyzing data, making decisions based on data Accentuate the Negative Understanding and using integers Moving Straight Ahead Linear relationships expressed in words, tables, graphs, and symbols Bits and Pieces I Understanding rational numbers: fractions, decimals, and percents Shapes and Designs Reasoning about shapes
shape properties
angle measure and Variables and Patterns Introducing Algebra
variables, tables, graphs, and symbols as representations Filling and Wrapping 3-D measurement For Pre-AP add Moving Straight Ahead Linear relationships expressed in words, tables, graphs and symbols Covering and Surrounding Measurement: area and perimeter Thinking with Mathematical Models Introduction to functions and modeling Growing, Growing, Growing... Exponential growth Looking for Pythagoras Pythagorean Theorem, slope, area and irrational numbers For Pre-AP add Frogs, Fleas, and Painted Cubes Quadratic growth I How Likely Is It? Probability For Pre-AP add Bits and Pieces II Using rational numbers: computation 4Comparison of the Little Rock School Districts Current Mathematics Program to Past Mathematics Program Previous Program Current Program Mathematical content is the same in both programs. Content is presented in a spiral effect through 6*, 7'", and s grade. The order in which the concepts are introduced differs from the previous program
concepts are presented in thematic modules. Basic math skills are emphasized. Isolated skills are taught with little problem solving application. Integrated skills are taught in order to solve real- world problems and promote critical thinking. Formulas and application of formulas are emphasized. Formulas are taught first and little application or extension activities are provided. Application problems are presented first and students are asked to derive formulas from logical reasoning of patterns. Calculator use is integrated into instruction when appropriate. Most students lack the confidence to solve problems without a calculator. CMP students use calculators as tools to be used at appropriate time to solve problems. Teacher editions provide examples and answers to problems. The curriculum expects teachers to see the same outcome from each student
some traditional books provide outlines. Possible student scenarios are given
provides Daily progressive outlines are provided in order to reach targeted objectives. Cooperative learning is up to the individual teacher. Lessons cater to the individual learner. A wealth of opportunities to incorporate cooperative learning is provided. 5 IPrevious Program I Current Program Student textbooks are provided. Traditional books provide examples and practice in order for students to achieve mastery through concentrated repetition. CMP books provide problems that encourage variety of solution strategies with teachers providing additional computation practice. a Homework problems are assigned for practice daily. 20-50 problems concentrating on one objective are provided
most problems are repetitive practice. 3-8 problems are assigned over the objective, while incorporating other objectives and asking students to explain answers. Students are taught problem-solving strategies. Problem-solving is isolated in one unit of the text. Problem-solving is taught throughout the curriculum. Teacher directed learning is provided in both programs^ Students are passive learners. Teachers guide students to be active learners. Assessment materials are provided for the teacher. All assessments follow a similar format in which students are asked to recall facts and formulas. A variety of assessments are provided, allowing students to express their knowledge through a wide spectrum of higher order thinking skills. Students are expected to show how they arrived at an answer. Work shown is typically mathematical and very little verbal explanation is given. Students are expected to be able to verbally explain how they worked their problem and why they chose a particular strategy. 6Why Connected Mathematic^ The Little Rock School Districts Secondary Mathematics Textbook Selection Committee recommended Connected Mathematics to the Superintendent for adoption in the middle grades (6-8) for two primary reasons: 1. The traditional mathematics program was not producing desired achievement results. 2. Much evidence convinced the committee that Connected Mathematics would improve the academic performance of students in the middle grades. Mathematics Achievement of Past Years  Benchmark Examination for eighth grade mathematics - Students took this examination required for state accountability purposes for the first time in 1998-99. (The results for 1999-00 are not available yet.) Results were reported as the percentage of students who were either advanced, proficient, basic, or below basic. The state considered students who scored in the proficient or advanced range as being at grade level or above and those who scored in the basic or below basic as being below grade level. Results from the 1519 eighth grade students who took the exam were: Scoring Range Advanced Proficient Basic_______ Below Basic 1998-99 2% 24% 6^ Only 11 % of the students were at or above grade level on the examination. 31 % of white students were at or above grade level and 3% of black students were at or above grade level.  Stanford Achievement Test, Edition taken by seventh grade students  This national test is taken by all seventh grade students in the area of mathematics. The students who took the test in 1997-98 through 1999-2000 were compared to all students nationwide and grouped according to the number who were scored in the 0 - 25' percentile range (first quartile), the 26 - 50 percentile range (second quartile), the 51 - 75 percentile range (third quartile), and the 76 - 99 percentile range (fourth quartile). Those students in the third and fourth quartiles scored above the national average, and those students in the first and second quartiles scored below the national average. Results from the test were: 7Grade 7 Quartile 97-98 98-99 99-00 Fourth Third Second First 17% 18% 21% 44% 16% 19% 26% 40% 18% 18% 22% 41% Results sho\A/ that the number of students above the national average in mathematics in grade 7 was only about 36%. The remaining students, about 64%, were below the national average in mathematics. Desired Mathematics Performance for the Future The Arkansas Comprehensive Testing, Assessment, and Accountability Program (ACTAAP) has established the following performance goals:  Tier 1 - 100% of a school's students shall perform at or above the proficient level in mathematics on the eighth grade Benchmark Examination in Mathematics.  Tier 2 - (If the 100% goal is not met, a trend or improvement goal must be met.) The percent of students performing at or above the proficient level in mathematics on the eighth grade Benchmark Examination in Mathematics will meet or exceed a 10% growth each year. The National Science Foundation Cooperative Agreement contains the following goals:  The Little Rock School District agrees to enact policies and practices that will contribute to an annual increase of 5% for Years 3-5 in the test scores of underrepresented minority students in mathematics as measured by the following tests: Mathematics Test Measures Achievement Level Tests___________ Stanford Achievement Test______________ Arkansas Benchmark Examination________ American College Testing (ACT) EXPLORE exam Grade(s) Implemented 3-11 5,7,10 4,6,8 8 8Why Connected Mathematics Can Improve Mathematics Achievement State pilot school district data and national reports indicate that Connected Mathematics improves achievement in mathematics. In Arkansas five middle schools representing five different school districts piloted Connected Mathematics for three years before students took the eighth grade Benchmark Examination in mathematics. The percentage of students in each of the four achievement categories for the pilot school districts, the state, and the Little Rock School District are presented in the chart below: LR Schools State Schools Below Basic Basic Proficient Advanced 65% 24% 9% 2% 47% 40% 11% 2% CMP Pilots Schools 28% 49% 19% 4% The greater the number of students in the below basic category, the farther behind the schools are in meeting the achievement goals set by the state. Districts piloting Connected Mathematics were much ahead of the state and the Little Rock School District in mathematics achievement on the eighth grade Benchmark Exam. The Portland, Oregon schools have implemented Connected Mathematics in its middle schools. A recent article in the Portland newspaper highlighted the increase in math scores on the state math test. The headline read City schools make big gains on state tests. The districts math coordinator predicted Portlands math scores would rise the first year the new math approach was instituted. Results proved him to be correct. The districts new math programs, Investigations for elementary school and Connected Mathtor middle school, emphasize problem-solving instead of drills and call on students to discover math concepts for themselves rather than memorize formulas. Connected Mathematics was designated as an Exemplary Program by a United States Department of Education Expert Panel. The panel found strong evidence for the impact of the program on gains in students understanding of mathematics, mathematical reasoning, and problem solving. The strength of the evidence was based on use of multiple measures, including tests aligned with the National Standards in Mathematics, and several types of comparisons. The American Association for the Advancement of Science rated Connected Mathematics as the number one middle school math textbook. AAAS evaluated twelve middle school math textbooks using a set of instructional criteria. 9ASSESSMENTS FOR MIDDLE SCHOOL MATHEMATICS The following measures are administered annually at targeted grade levels: Name of Assessment SAT^9 ~ Stanford Achievement Test 9*^ Edition Type National Norm Referenced Grade Levels Grades 5, 7,10 EXPLORE American College Test ACTAAP State Benchmark Examination National Criterion Referenced___________ State Criterion Referenced Grade 8 Grade 8 ALT Achievement Level Test EMT End of Math Module Tests ACTAAP State End of Course Exams Local Criterion Referenced with National Comparison Group ___________ Local Criterion Referenced State Criterion Referenced 10 Grades 6-8 Grades 6-8 Algebra, geometryWhat Effect Will Connected Mathematics Have on SA T-9, ACT, and SA T Scores? SAT-9 A study was done which compared the performance of eight pilot Connected Mathematics middle schools to the state average on the Stanford Achievement Test in mathematics given to fifth grade students in 1995 and to the same students again as seventh graders in 1997. At the time the Stanford-9 was given the pilot school students had only had one year of the Connected Mathematics curriculum. The results were: CMP schools State schools Stanford-9 1995 NCE 44.58 43.00 Stanford-9 1997 NCE 49.92 47.00 Difference +5.34* +4.00 This increase is statistically Significant ACT. SAT Although students who have taken Connected Mathematics have not moved into the high schools yet, we are confident that students will do well as well as or better than before on the mathematics part of ACT and SAT. These assessments, like many others, have responded to the NCTM Standards in Mathematics in developing assessment questions. The tests require students to use their reasoning skills to solve practical problems in mathematics which is a strength of Connected Mathematics. 11Will Connected Mathematics Prepare My Child for Algebra 1 and Higher Level Mathematics? We think that Connected Mathematics will superbly prepare students for Algebra 1 and above. Many students will be ready to take Algebra 1 in the eighth grade after two years of Connected Mathematics. Others may continue in Connected Mathematics in the eighth grade and either take Algebra 1 in its traditional location in the ninth grade or take the Algebra 1 End of Course Exam after eighth grade Connected Mathematics. \Ne feel that many students will have all their Algebra 1 skills mastered after eighth grade Connected Mathematics. Following is an excerpt from Getting to Know Connected Mathematics that addresses the algebra in CMP. are Many people think of algebra as a course at high school, towards which all prior mathematics has been heading. Students enter the traditional course somewhat in awe of the lofty position they have reached, and are often discouraged to find that the mathematical ideas and skills they have previously developed do not seem related to success in a traditional Algebra 1. The popular impression in the community at large is that algebra is the ability to manipulate symbols, usually following instructions to simplify and expression, or solve an eguation. Traditionally students memorize rules, focusing on specific strategies for specific problems. The symbols and rules often__ meaningless to students, who try to survive by memorizing and, thus, only retain the ideas for a short time. There is little evidence that students develop algebraic reasoning, including symbolic reasoning, from this kind of experience. In fact, the development of algebraic ideas can and should take place over a long period of time, prior to attempts to deal solely with abstract symbols, and well before the first year of high school. The philosophy of how students learn mathematicsthat is, students develop algebraic reasoning and understanding while pursuing solutions to interesting problems. Having students approach algebraic ideas through investigations and problems allows students to search for patterns and relationships in data, and to find ways to express these patterns, first in words and then in symbols. The algebraic units in CMP develop three important patterns, which model many real situations. By focusing on the patterns underlying the problem, asking students to describe similarities and differences across patterns, and challenging students to predict answers, the mathematics of functions and relationships is developed. For example, in the Walkathon Problem, in the unit entitled Moving Straight Ahead, students investigate the distance each person walks. 2.3 Walkathon Your class decides to raise money for a charity by competing in t he citys Walkathon. Each person in the class finds sponsors who each pledge to pay a certain amount of money for each kilometer that the person walks. The money goes to a charity. The person who raises the most money receives a new pair of Rollerblades. Sponsors 12 often ask for a suggestion of how much they should pay per kilometer. Others will follow the example of the first person who fills out the form. Your class would like to agree on how much money to ask the sponsors for. Jane argues that one dollar per kilometer would be appropriate. Bill argues for two dollars a kilometer because it would bring in more money. Amy suggested that if the price was too high not as many people would be sponsorsso she suggested that each sponsor pay a $5 donation plus an extra 50 cents per kilometer. For each price suggestion made by Bill, Jane, and Amy:  Make a table showing the total amount of money a sponsor pays for distances between 1 and 10 miles. This is called a pay plan.  Sketch a graph of the amount of money a sponsor would pay under each pay plan. Display all the graphs on the same set of axes.  Write an equation for each of the pay plans which shows how the amount of money a sponsor owes can be calculated for the total distance that a student walked. 1. 2. 3. 4. As the amount of money charged per mile is increased, what effect does this have on the table? On the equation? On the graph? A student walked 8 miles in the Walkathon. How much would her sponsors pay under each of the pay plans? Explain how you arrived at your answers. One of the sponsors paid $10 after the race. How many miles would that sponsors student have walked under each of the three pay plans? Explain how you arrived at your answers. How is Amys fixed $5 cost represented in the table? In the equation? In the graph. In this situation the focus quickly centers on the relationship between the variables. If the rate at which a person walks is fixed, then distance depends upon (or is a function of the) time
that is, distance = rate x time. As students reason about this situation they are encouraged to use multiple representations, which help them to show their reasoning and also to develop understanding of the important linear relationship underlying this situation. As the time changes by one unit, the distance changes by a constant amount. This is the key feature of recognizing this constant rate of change in various representations, students develop a deep understanding of the basic linear pattern. The rate at which a person walks is the constant rate of change (or the slope of a straight line). This pattern shows up as a straight line in a graph, as an equation of the form, y = ax, and as increments of change in a table. The rate is indicated by the steepness of the line, the coefficient, a, of x in the equation, and the constant increment of change in the table. A variation of the problem has two brothers involved in a walking rave with the younger brother getting a head start. This leads to the equation, y = ax + b
where a is the rate at which the younger brother walks and b represents the amount of head start. Once students have a beginning understanding of linear patterns, they then need help in recognizing this pattern in many different disguises. For example, the cost, C, of buying N compact discs at $15 each is C + 15N. Having recognized that this is the 13 same pattern as that in the Walkathon problem, students are quickly able to reason about the relationship between the variables. Using problem situations and multiple representations to represent these situations gives rise to solving equations. For example, students can use tables or graphs or they can reason about the problem. Once students have a good sense of what it means to solve and how these solutions relate to the variables and the problem, then the techniques of solving a linear equation symbolically are developed. CMP Algebra Goals - by the End of Eighth Grade in CMP Most Students Should be Able to:  Recognize situations in which important problems and decisions involve relations among Quantitative variablesone variable changing overtime or several variables changing in response to each other.  Use numerical tables, graphs, symbolic expressions, and verbal descriptions to describe and predict the patterns of change in variables.  Recognize (in various representational forms) the patterns of change associated with linear, exponential, and quadratic functions.  Use numeric, graphic, and symbolic strategies to solve common problems involving linear, exponential, and quadratic functions. Inevitably parents and teachers will wonder how the CMP algebra strand relates to and can be compared to a traditional school algebra curriculum. The only helpful comparison to make is how well students understand algebraic ideas, and how well they employ algebraic reasoning in solving problems. Since the focus of CMP is on developing understanding, rather that on memorizing rules and processes to apply in response to instructions such as simplify or solve," a simple checklist of concepts and skills that appear in CMP and in traditional programs is not particularly relevant. The question of most interest to parents and teachers should be,  How well will the next mathematics course my student takes support the strong understanding of algebraic ideas developed in CMP7" If the next course is a traditional Algebra 1, CMP students will find themselves spending large amounts of time studying processes and rules for which they see no application. The motivation for learning these rules may shift from the focus of making sense to simply memorizing to survive. Certainly, CMP students will find that many of the big ideas in Algebra 1, such as representing linear functions, solving equations, and finding equivalent forms of an expression, are familiar, and perhaps a repetition of what they already know. They will also wonder why the problems are restricted mainly to linear patterns, whereas in CMP they had also learned about other important patterns. The algebra strand in CMP will certainly prepare students for successful, if somewhat repetitive year in a traditional Algebra 1. It is to be hoped, however, that students will be fortunate enough to find themselves able to take courses that continue to challenge them to develop algebraic reasoning. High school teachers of these students need to be aware that these students have begun to develop an understanding of functions that is far beyond a traditional Algebra 1 approach. 14How Will the District Know Whether Connected Mathematics is Working? A number of avenues will be used to know whether Connected Mathematics is working. First, after each module, an End of Module Test will be given to every student. The End of Module Tests will assess the mathematics content and skills that were targeted for that module. We will be looking to see if the majority of students achieved mastery (75% of the items were answered correctly) on each test. The End of Module Tests were developed locally by teachers and specialists to match the Connected Mathematics curriculum. Secondly, students will take fall and spring Achievement Levels Tests (ALT) in mathematics. This test is based on the Little Rock School District and state standards for mathematics. The achievement on this math test will tell us whether our students are learning what we expected them to learn and will also let us see how our students are achieving compared to a large comparison group of students nationally who take the same test. We will look closely at the ALT scores to see how our students perform compared to the national group. We expect our students to make a full years growth in mathematics each year and to perform at or near the same level as the national group. Based on the first administration of the ALT, our students performed well, only slightly behind the national comparison group. We will be looking forthat gap to close during the next few administrations. Thirdly, and most importantly, students in the eighth grade will take the high stakes Sate Benchmark Examination in mathematics. We expect good growth in the percentage of students who move from one achievement level to the next higher level (from basic to proficient for example). This growth should start appearing in the Benchmark scores from the spring 2001 administration of the exam. This will be the first administration where the students have been enrolled in a full year of Connected Mathematics before they take the exam. At the same time that the measures mentioned above are being watched, we want to make sure that our nationally normed tests such as the SAT-9 (7*^ grade) and the EXPLORE (8 grade) hold steady or increase. While our main objective is for students to achieve on the critically important State Benchmark Examination, we want to make sure that our students are improving in relation to other students around the country. 15MIDDLE SCHOOL MATHEMATICS LEAD TEACHER/CONSULTANT The District has employed a lead teacher for middle school mathematics. The name, job goal, and performance responsibilities of the lead teacher are given below. Please call the middle school math lead teacher if you have questions or concerns. The District has also retained the services of a part-time mathematics consultant to work through the schools to help parents understand the middle school mathematics program and to support their children in learning mathematics. NAME/PHONE Docia Jones, Middle School Lead Teacher for Mathematics 324-0520, ext 867 JOB GOAL
Facilitate the change from a traditional mathematics curriculum to a standards-based curriculum for the purpose of increasing both the participation of students in challenging mathematics programs and the success of students in achieving District Curriculum Standards and Benchmarks in mathematics. PERFORMANCE RESPONSIBILITIES:  Provide training for mathematics teachers on District adopted standards-based math curricula [Connected Mathematics for example) and related content and teaching strategies.  Provide weekly classroom support for middle school mathematics teachers who are implementing Connected Mathematics. This support will include encouragement, extra hands, demonstration teaching, team teaching, material resources, trouble shooting, and other technical support deemed necessary.  Provide follow-up training for teachers to address observed/identified needs.  Assess the implementation level of the teachers/schools in the middle schools. Inform principals and the Division of Curriculum and Instruction of implementation problems/concerns.  Assist with the development and implementation of end of module assessments to evaluate the effectiveness of the Connected Mathematics in achieving desired student results. Inform principals and teachers of findings.  Communicate to parents and the community about the changes that are taking place in mathematics. Judy Trowell, Part-time Consultant (Contact Judy through Docia Jones) Assists the Lead Teacher and principals in communicating with parent groups about Connected Mathematics. Establishes and leads dialogue with parents about questions and concerns about the philosophy, mathematical content, and structure of Connected Mathematics. 16Name: School name: Street
City, State, Zip: ' Phone (inc. area code):_ Visa or Master Card Exp Date
Connected Mathematics Order Form Card Holders Street #: Zip Code
If ordering by mail send check or money order to: Dollar BUI Copying Attn. CMP books 611 Church Street Ann Arbor, MI, 48104 Phone (734) 665-9200 Fax (734) 930-2800 P.O. # (for schools only, include a copy with your order):. 6 grade-(pink) Quantity Price Poes not include tax or dipping Approx. Weight Total Price before ox and shipping Prime Time Data About Us Bits and Pieces H  Shapes and Designs I.  How Likely Is It?______________ Covering and Surrounthng Ruins of Montarek Bits and Pieces I 7" grade-{blue) Comparing and Scaling Stretching and Shrinking Variables and Pattenu Accentuate the Negative Moving Straight Ahead Data Around Us Riling and Wrapping______________ What do you Expect 8* grade -(green) Looking for Pythagesas Thinking with Matbemadcal Models Ftogs, Reas, and Painted Cubes Growing, Growing, Growing Clever Counting Samples and Populations Kaleidoscopes. Hubcaps, and Mirrors Say II with Symbols For Teachers Only 6 Grade Curriculum Guide 7* Grade Curriculum Guide 8* Grade Curriculum Guide For Parents Only K-S** Elementary Math Handbook Please allow up to two weeks for shipping $437 $4.95 $4.95 $4.95 J3J0 J4.70 $4.20 $5.06 S4.S4 $5J0 S4.40 $5.61 $5.20 $3.80 $4.60 $638 $539 $4.51 $6.27 $4.73 $3.90 $4.90 $5.20 $5.94 $4435 $52.05 $41.01 $4.95 I pound Ml pound pound Ml pound pound pound pound pound pound pound pound Vi pound pound Vi pound impound 1 pound pound Vi pound pound pound Mi pound pound pound pound 5 pounds 6 Vi pounds 5 pounds pound Sub-Total Sales Tax (DBC will fill in price) Shipping & Handling GRAND TOTAL i"?*' * & *^)i>FlUl'<'v ".' 'f- XtJ 'CT'errSir- ' n- rr Connected *^ Mathematics TM 9 ffi Representing Relationships Teachers Edition Glenda Lappan James T. Fey William M. Fitzgerald Susan N. Friel Elizabe Difanis Phillips Developed at Michigan State University DALE SEYMOUR PUBUCATIONS*Connected Mathematics was developed at Michigan Sute University wi financial support from the Michigan State University Office of the Provost, Computing and Technology, and the College of Natural Science. This material is based upon work supported by the National Science Foundation under Grant No. MDR 9150217. s This project was supported, in part. b by the ? National Science Foundation Opinions expressed are tnose of the authors and net necessarily those of the Foundation The Mirhigan State Univeisity authors and administration have ^eed that all MSU royalties arising from tins publication will be devoted to purposes supponed by the Department of Mathematics and the MSU Mathematics Education Enrichment Fund. This book is published by Dale Seymour Publications
an imprint of Addison Wesley Longman, Inc. Managing Editor: Catherine Anderson Project Editor: Stacey Miceli Book Editor: .Mali Apple ESL Consultant: Nancy Sokol Green Produaion/Manufacturing Director: Janet Yearian ProductionIManufacturing Coordinator: Claire Flaherty Design Manager
John F. Kelly Photo Editor: Roberta Spieckerman Design: Don Taka Composition: London Road Design, Palo Alto, CA Illustrations: Pauline Phung, Margaret Copeland, Ray Godfrey Cover: Ray Godfrey Photo Acknowlaigements: 5  Ron SanfordTony Stone Images
23  Owen Franken/Stock, Boston
27  Frank Siteman/Stock, Boston
41  Joseph Schuyler,'Stock, Boston
45  Lionel Delevingne'Stock, Boston
47  Bert SagaraTony Stone Images
54 (Big Ben)  George HunterZTony Stone Im^es
54 (Eiffel Tower) John Lawrence/ Tony Stone Images Copyright  1998 by Michigan State University, Glenda Lappan, James T. Fey, William M. Fit^erald, Susan N. Friel, and Elizabeth D. Phillips. All rights reserved. Limited reproduction permission: The publisher grants permission to individual teachers who have purchased this book to reproduce the bbckline masters as needed for use with their own students. Reproduction for an entire school or school district or for commercial use is prohibited. ALE SEYMOUR PUBLICATIONS Tha Boek a Pnnted on RxcycM Paper Order number 21474 ISBN 1-57232-179-2 123456789 lO-ML-01 00 99 98 97 Contents Overview The Mathematics in Thinking with Mathematical Models Technology Mathematical and Problem-Solving Goals Summary of Investigations Connections to Other Units Materials Resources Pacing Chart Thinking with Mathematical Models Vocabulary Assessment Summary la lb Ic If Ig lb li If Ij Ik Ik Introducing Your Students to Thinking wiSi Mathematical Models 2 Mathematical Highlights Using This Teachers Edition Investigation 1: Linear Models Student Pages Teaching the Investigation Investigation 2: Nonlinear Models Student Pages Teaching the Investigation Investigation 3: More Nonlinear Models Student Pages Teaching the Investigation Investigation 4: A World of Patterns Student Pages Teaching the Investigation Assessment Resources Blackline Masters Additional Practice Descriptive Glossary Index 4 4a 4b 5 25a 25p 26 36a 36) 37 46a 46) 59a 61 91 115 136 137Overview The focus of this unit is mathematical models: what they are, how they are constructed, and what they enable us to do. Just as a designer might construct a model of a building and then tinker with the components to see what relationships are affected by changing certain variables and what outcomes are pleasing and functional, a mathematician can construct an algebraic model to represent a situation. By choosing appropriate variables, collecting and graphing data, and manipulating the variables to see how relationships are affected, mathematicians can predict outcomes. The investigations in this unit introduce students to several kinds of algebraic relationships including hnear relationships, inverse relationships, and exponential relationshipsused to model real-life situations. Students also explore other interesting relationships with graphing calculators. Linear relationships are characterized by a constant rate of change in one variable associated with a constant rate of change in the other variable. Students have probably had more experience with linear relationships that display a constant increase in one variable compared to the other (associated wi straight lines having a positive slope) than those that display a constant decrease in one variable compared to the other (associated with straight lines having a negative slope). In this unit, they will review and deepen their understanding of both kinds of linear relationships and their represenution in all three forms: tabular, graphic, and symbolic. Students will also encounter inverse relationships, in which one variable decreases as the other increases but not at a constant rate. They will also look at exponential growth and exponential decay relationships, which are obtained by repeatedly multiplying by a constant factor. Students continued study of mathematics will include further analysis of these relationships
the intention of this unit is to provide only a brief introduction to them. Graphing calculators allow students to manipulate data and to test conjectures more easily than can be done with paper-and-pencil methods. Students need to develop facility in the use of the calculator. At the same time, they need to be able to determine when using the calculator is an advantage and when making a quick estimate or mental calculation or sketching a graph by hand is more appropriate. 1a IntroductionThe Mathematics in Tbinking with Mathematical Models In this unit, students explore the advantages of using algebraic models, in the form of graphs and equations, to describe situations. A table of data is often a good starting point for deciding what type of relationship is suggested by the dam. For example, table 1 below shows values for the linear relationship represented by the equation y = 3x + 5. As X increases by increments of 1, y increases by increments of 3. Table 2 shows values for the linear relationship represented by the equation y = 20 - 2x (orj = ~2x + 20). As X increases by increments of 1, y decreases by increments of 2. Table 3 shows values for the nonlinear relationship represented by the equation y = -y (or xy = 40). As x increases by increments of 1, j decreases but not at a constant rate. Table 4 shows values for the nonlinear relationship represented by the equation y = (1.06/, which gives the amount j in a bank account after x years at an interest rate of 6% compounded annually. From the table, we can see that the doubling time for the investment is 13 years.- it takes about 13 years for $1 to increase to $2 at a 6% interest rate compounded annually. Table 1 Table 2 Table 3 Tablei X q 1 2 3 4 5 6 y 5 8 11 14 17 20 23 X 1 2 5 4 5 6 y 20 18 16 14 12 10 8 X T 2 3 4 5 6 y 40.0 20.0 13.3 10.0 8.0 6.7 X T 3 5 7 9 11 13 y 1.06 1.19 1.34 1.51 1.70 1.91 2.14 Fitting a Curve to Given Data Real situations usually generate messy data, and we cannot expect a line or a curve to fit such data exactly. What we do look for is a graph model that fits the data pattern well enough to be useful as a predictive tool. The process of curve fitting is technically complex, but students can informally understand the goals of the process by drawing a simple curve to fit plotted points. At the simplest level, for plotted data that suggest linear relationships, students can eyeball e data and, using a straightedge as a guide, try several modeling lines until they find one that seems to be a good fit. In this process, students often try to hit as many points as possible. However, this strategy sometimes results in lines that do not give very good predictions. The best rule of thumb is to look for a line that seems to catch the overall trend in the data. Then, students can -write an equation for the line. Introduction 1bTechnology Students will need access to graphing calculators for much of their work in this unit. Its best if their calculators have the capacity to display a function as a table and that an overhead display model of their calculator is available. Connected Mathematics was developed with the belief that calculators should always be available and that students should decide when to use them. The graphing calculator provides students with a useful method for finding information about a situation by examining its graph. In addition, it allows students to investigate many examples quickly, helping them to observe patterns and to make conjectures about relationships. In this unit, students will be looking at linear and nonlinear relationships. For linear relationships, finding a graph model and an associated equation model can often be accomplished by sight alone. However, graphing calculators and computer software can help facilitate the modeling process, and students often find technological approaches fascinating. The instructions below are written for the TI-80 graphing calculator. If your students use a different calculator, consult the manual for instructions on these various procedures. Entering Data To enter a list of (x, y) data pairs, press ISTATI and the screen will display something similar to that shown at the top left of the next page. Press I ENTER I to select the Edit mode. Then enter the pairs into the LI and L2 columns: enter the first number and press I ENTER I, use the arrow keys to change columns, enter the second number and press I ENTER I, and use the arrow keys to return to the LI column. gJBH CALC 2:SORTA( 3:SORTD( 4:CLRLIST LI a 1 2 3 4 5 LI (l)=0 1c introduction L2 4 5 8 10 11 12Plotting the Points To plot the dau you have entered, use the commands in the STAT PLOT menu. First display the STAT PLOT menu, which looks like the screen shown below left, by pressing i~2rid) E3. Press I ENTERl to select PLOTl, which looks like the screen shown below right. Use the arrow keys and I ENTER I to move around in the screen and to highlight the elements as shown (ON, IE , LI, L2, and D). STAT PLOTS 1: ON LL Ll L2 2:PLOTl . . . ON LL Ll L2 3:PLOTl . . . ON LL Ll L2 4'IPLOTSOFF PLOTl SSJ OFF TYPE r IM IZ~ pTh XL:IB L2 L3 L4 L5 L6 YL: Ll EB L3 L4 L5 L6 MARK:0 PLOTl . .    + Next, press IWINDOWI to display a screen similar to that shown below left. To accommodate the dau you have input, adjust the window settings by entering values and pressing [ENTER L allowing some margin beyond the data points if possible. Then, press iGRAPHl. WINDOW XMIN=0 XMAX=7 XSCL=1 YMIN=0 YMAX=16 YSCL=1      Testing Graph Models To experiment with various equations to find a good fit for modeling a particular dau pattern, enter equations into the Y= list (press to access the list) and graph the equations along with the plotted dau. Below, y = 3 + 2x has been entered into Y1 and the equation has been graphed over the entered dau by pressing IGRAPHl. Ylg3+2X X2= Y3= Y4= introduction 1d Error Messages in Calculator Tables If the table for an equation is displayed with an x value for which the equation is undefined, the table will display the word ERROR in the column for e corresponding y value. You may need to talk with your students about what this means. X  1 2 3 4 5 X=0 Y1 ERROR 40 20 13.33 10 8 Using Regression Features Graphing calculators also offer curve-fitting routines
you may or may not want to introduce these features to your class. If your students are ready and you have the time, consult the graphing calculator manual for instructions. 1e introductionMathematical and Problem-Solving Goals ^binking udtb Mathematical Models was created to help students  Develop skill in collecting dau from experiments and systematically recording that data in tables  Construct coordinate graphs to represent data  Make predictions from data tables or graph models  Use patterns in dau to find equations that model relationships between variables  Use tables, graphs, and equations to model linear and nonlinear relationships between variables  Distinguish between linear and nonlinear relationships  Identify inverse relationships and describe their characteristics  Use a graphing calculator to find and study graph models and equation models of relationships between variables  Use intuitive ideas about rates of change to sketch graphs for, and to match graphs to, given situations  Use intuitive ideas about rates of change to create stories that fit given graphs The overall goal of the Connected Mathematics curriculum is to help students to develop sound mathematical habits. Through their work in this and other algebra units, students I Pam important questions to ask themselves about any situation that can be represented and modeled mathematically, such as
Wbat are the variables? How are they changing in relation to each other? How is an increase in the independent variable related to a change in the dependent variable? Where is they value changing the most? The least? How can this change be seen in a table? Detected in a story? Observed in a graph? Read from a symbolic representation? Where does the graph cross the x-axis and the y-axis, and what is the significance of each intersection? Where does they variable reach its greatest value and its least value, and what is the significance of each location? IntroductionSummary of Investigations Investigation 1: Linear Models Students conduct an experiment to investigate the relationship between the thickness and the strength of a bridge. They observe patterns in their tables and graphs, find equation models in the form y = mx+ b to describe experimental data, and make predictions based on their models. They interpret the slope and the j-intercept in terms of the experimental situation, and they graph lines given e slope and the jy-intercept. They practice writing equations for fines given (a) the graph of the fine, (b) the slope and 7-intercept, (c) the slope and the coordinates of one point on the fine, and (d) the coordinates of two points on the fine. Investigation 2: Nonlinear Models Students continue to observe patterns in tables and graph models and to use them to seek equation models that describe given data and to make predictions. However, in the situations in this investigation, the value of one variable increases as the other decreases but not at a constant rate
such inverse relationships are modeled with nonlinear graphs. Students are not expected to acquire formal vocabulary about inverse relationships at this time but to express the relationships in their own words, focusing on the panem of change in the variables
how this pattern is reflected in shape of the graph
and, possibly, how an equation might model the relationship. Investigation 3: More Nonlinear Models Students meet another type of graph model in this investigation. The patterns in exponential growth and exponential decay relationships are characteristic of many common situations and can be recognized from their graphs. Students first analyze the growth of an investment over time. Then, they conduct another experiment, pouring water from one glass into another so that each successive glass contains half as much water as the previous glass. They look at the relationship between the number of the glass in the series and the amount of water it contains. In both problems, a quantitymoney or watergrows or decays by being repeatedly multiplied by a constant factor. Investigation 4: A World of Patterns In previous investigations, students have collected data from experiments or generated data from descriptions of familiar situations. They have made tables, graph models, and equation models to fit dau. In this investigation, they concentrate on working with and interpreting graph models, describing situations that could fit given models or creating graphs to fit a variety of real-world situations. Some of the graphs they will work with are linear, some are nonlinear, and some are combinations of linear and nonlinear graphs. Students use a graphing calculator to further study graph models and equation models and, in the process, discover that some of the unusual graphs they have constructed in previous problems can be represented by equation models. Students will come away from this unit with an understanding that graph and equation models can be used to represent many of the patterns found in their world. ig IntroductionConnections to Other Units The ideas in Thinking with Mathematical Models build on and connect to several big ideas in other Conneaed Mathematics units. Bigidea Prior'Work PutnreWork building and analyzing mathematical models generalizing formulas (Covering and Surrounding
Filling and Wrapping) using symbols to represent relationships (Variables and Patterns
Moving Straight Ahead) making tables and graphs by hand and with a graphing calculator (Data About Us
How Likely Is It?
Accentuate the Negative
Moving Straight Ahead) recognizing families of functions (Growing, Growing, Growing
Frogs, Fleas, and Painted Cubes) manipulating symbols to find alternative forms of expression and to solve linear equations (Say It with Symbols) recognizing patterns of growth and change from tables and graph models (Growing, Growing, Growing
Frogs, Fleas, and Painted Cubes) fitting a line to experimental dau writing the equation of a line from information about its slope and points on the line (Moving Straight Ahead) fitting lines and making predictions (Samples and Populations) identifying the variables of interest in a situation identifying dependent and independent variables and observing how change in the dependent variable is related to change in the independent variable (Variables and Patterns
Accentuate the Negative
Moving Straight Ahead) determining panems of growth in different kinds of equations (Growing, Growing, Growing
Frogs, Fleas, and Painted Cubes) conducting experiments to gather dau about how variables are related conducting experiments involving linear relationships (Data About Us
Variables and Patterns
Moving Straight Ahead) conducting e^eriments involving exponential and quadratic relationships (Growing, Growing Growing
Frogs, Fleas, and Painted Cubes) .. introduction 1hResources For the teacher Carlson, Ronald J., and Mary Jean Winter. Algebra Experiments H. Palo Alto, Calif: Addison- Wesley, 1993. Heid, Kathleen. Algebra in a Technological World, NCTM Addenda Series Grades 9-12. Reston, Va.: National Council of Teachers of Mathematics, 1995. Winter, Mary Jean, and Ronald J. Carlson. Algebra Experiments I. Palo Alto, Calif.: Addison- Wesley, 1993. Technology Dugdale, Sharon, and David Kibbey. Green Globs and Graphing Equations (IBM, Apple H, Macintosh). Pleasantville, N.Y: Sunburst Communications. Rosenberg, Jon. Math Connections: Algebra 1 (Macintosh). Pleasantville, N.Y: Sunburst Communications.Thinkinff with Matbcmatical Models Vocabulary The following words and concepts are used in Thinking with Mathematical Models. Concepts in the left column are ose essential for student understanding of this and future units. The Descriptive Glossary gives descriptions of many of these terms. Essential terms developed in this unit equation model fulcrum graph model inverse relationship linear relationship mathematical model relationship Terms developed in previous units coordinate grid exponent function graph linear equation panems of change slope table variable y-intercept Assessment Summary Embedded Assessment Opportunities for informal assessment of student progress are embedded throughout Thinking with Mathematical Models in the problems, the ACE questions, and the Mathematical Reflections. Suggestions for observing as students explore and discover mathematical ideas, for probing to guide their progress in developing concepts and skills, and for questioning to determine their level of understanding can be found in the Launch, Explore, and Summarize sections of all investigation problems. Some examples:  Investigation 4, Problem 4.3 Launch (page 59d) suggests a procedure for assessing what students observe about the form of algebraic equations.  Investigation 3, Problem 3-2 Explore (page 46d) suggests questions you can ask to assess students understanding of how to distinguish the independent variable from the dependent variable in a relationship, and how to determine on which axis each variable belongs.  Investigation 2, Problem 2.2 Summarize (page 36c) suggests a way you can help lead students to an understanding of inverse relationships. ACE Assignments An ACE (ApplicationsConnectionsExtensions) section appears at the end of each investigation. To help you assign ACE questions, a list of assignment choices is given in the margin next to the reduced student page for each problem. Each list indicates the ACE questions that students should be able to answer after they complete the problem. Ik IntroductionI* 
F Introducing Your Students to Thinking with Mathematical Models Introduce this uAt by telling students that Aey will be learning about mathematical modelswhat they are, how to construa Aem, and how Aey can be used. Read through e three opening questions with the class, and ask whether students have any ideas about how they might answer them. At Ais point, these problems present an opportunity for students to Aink out loud and to compare ideas. Let students brainstorm and make conjectures. Dont take the time to act on Aeir suggestions or to try to find solutions, but explain Aat Aey will have many opportunities Aroughout Ae umt to revisit questions like Aese and to find solutions to Aem. ^mg^M fMathematicJPlVlodels '^^Fallot and bis mother want llooggoo on the teeter- I 1 The pep club surveyed 500 students to find out which totter at the park. FaUoi weighs 75 pounds - and his mother weighs 150 pounds. How can they sit on the teeter-totter so  that it balances? ' ^' I of several amounts they would be willing to pay for a spirit-week T-shirt. They found that 400 students would pay 2 dollars, 325 would pay 4 dollars, 230 would pay 6 dollars, 160 would pay 8 dollars, and 100 would pay 10 dollars. How can they use this information to predict how many students would pay 5 dollars? On the day Chantal was bom, her Uncle Charlie used SlOO to open a savings account. He plans to give Chantal the money in the account on her tenth birthday. The account earns 8% interest at the end of every year. If Charlie does not deposit or withdraw any money, how much money will be in the account on Chantals tenth birthday? Tips for the Linguistically Diverse Classroom Rebus Scenario The Rebus Scenario technique is described in detail in Getting to Know Connected Mathematics. This technique involves sketching rebuses on the chalkboard that correspond to key words in the story or information that you present orally. Example: Some key words and phrases for which you may need to draw rebuses while discussing these introductory questions: pep club (stick-figure cheerleaders), T-shirt (rebus of one), dollars ($), savings account (bank), tenth birthday (cake with ten candles), teeter-totter (rebus of one). 1 Introduction People in many professions make important predictions every day. Sometimes millions of dollars, and often peoples lives, depend on the accuracy of these predictions. A company may develop a wonderful new product, but without accurate predictions about the number and location of potential customers and the price they would be willing to pay, the company could lose a great deal of money. When designing a bridge, a civil engineer must make accurate predictions about how much weight e bridge can hold and about how it will stand up to strong winds and earthquakes. People often use mathematical models to help them make such predictions. People construct mathematical models by gathering data about a situation and then finding a graph or equation that fits the data. In this unit, you will find mathematical models yH for many situations, and you will use your models to answer interesting questions like those on the previous page. Introduction 3 <P' Mathematical Highlights The Mathematical Highlights page provides information for students and for parents and other family members. It gives students a preview of the activities and problems in Thinking with Mathematical Models. As they work through the unit, students can refer back to the Mathematical Highhghts page to review what they have learned and to preview what is still to come. This page also tells students families what mathematical ideas and activities will be covered as the class works through Thinking with Mathematical Models. 4 Introduction I I i i V  -s Mathematical Highlights In this unit, you will leant about ways to model relationships between variables. Testing paper bridges to find out how thickness affects strength and then fitting a straight line to your experimental data introduces you to the idea of a linear graph model. Using what you know about slope andy-intercept, you find equations for linear graph models. Using graph models and equation models, you make predictions about values that are not in your data set. Testing paper bridges to find out how length affects strength and then fitting a curve to your experimental dau introduces you to the idea of a nonlinear graph model. Finding and graphing (distance, weight) combinations that balance a teeter-totter and (speed, time) combiiutions needed to complete a trip shows you that sometimes very different situations have similar graph models. Calculating interest on a savings account introduces you to another common nonlinear panem. Drawing graphs to model events in a story and writing stories that match the information in graphs give you practice undersunding and interpreting graphs of linear and nonlinear relationships. Using a calculator In this unit, you might use a graphing calculator to plot data and to fit lines or curves to the data points. As you work on the Connected Mathematics units, you decide whether to use a calculator to help you solve a problem. Thinking with Mathematical Models 1Using This Teacher's Edition The Investigations The teaching materials for each investigation consist of three parts: an overview, student pages with teaching outlines, and detailed notes for teaching e investigation. The overview of each investigation includes brief descriptions of the problems, the mathematical and problem-solving goals of the investigation, and a list of necessary materials. Essential information for teaching the investigation is provided in e margins around the student pages. The At a Glance overviews are brief outlines of the Launch, Explore, and Summarize phases of each problem for reference as you work with the class. To help you assign homework, a list of Assignment Choices is provided next to each problem. Where space permits, answers to problems, follow-ups, ACE questions, and Mathematical Reflections appear next to the appropriate student pages. The Teaching the Investigation section follows the student pages and is the heart of the Connected Mathematics curriculum. This section describes in detail the Launch, Explore, and Summarize phases for each problem. It includes all the information needed for teaching, along with suggestions for what you might say at key points in the teaching. Use this section to prepare lessons and as a guide for teaching the investigations. Assessment Resources The Assessment Resources section contains blackline masters and answer keys for the check-up, the quiz, the Question Bank, and the Unit Test. Blackline masters for the Notebook Checklist and the Self-Assessment are given. These instruments support student self-evaluation, an important aspect of assessment in e Connected Mathematics curriculum. Discussions of how teachers can suppon the work of special education students, and strategies they might use to record and display students work, are also included. Blackline Masters The Blackline Masters section includes masters for all labsheets and transparencies. Blackline masters of grid paper and graphing calculator grids are also provided. Additional Practice Practice pages for each investigation offer additional problems for students who need more practice with the basic concepts developed in the investigations as well as some continual review of earlier concepts. Descriptive Glossary The Descriptive Glossary provides descriptions and examples of the key concepts in Thinking with Mathematical Models. These descriptions are not intended to be formal definitions but are meant to give you an idea of how students might make sense of these imponant concepts. Introduction 4aBWMBSOWH Linear Models In this investigation, students observe panems in tables and graphs, find equation models in the form y = mx + b to describe experimental data, and make predictions based on these models. They interpret the slope and thej-intercept in terms of the situation, and they graph lines given the slope and the jy-intercept. They practice writing equations for lines given (a) the graph, (b) the slope and the >-inter-cept, (c) the slope and the coordinates of one point, and (d) the coordinates of two points. In Problem 1.1, Testing Paper Bridges, students conduct an experiment to investigate the relationship between the thickness and the strength of a bridge. Their model involves paper strips and pennies: as the number of layers increases, the number of pennies the bridge can suppon increases. The resulting graph can be convincingly linear. In Problem 1.2, Drawing Graph Models, students attempt to draw a line to fit the pattern in their graphs, and they use the pattern to make predictions. In Problem 1.3, Finding Equation Models, they are introduced to finding equation models that describe experimental data. In Problem 1.4, Setting the Right Price, they make a graph and an equation model for a set of data. In Problem 1.5, Writing Equations for Lines, they write and evaluate equations to model the information in graphs. Mathematical and Problem-Solving Goals  To collect data, record data in tables, and represent data in coordinate graphs  To fit a linear model to a graph  To make predictions from data tables and graph models  To write an equation given the graph of a line  To review the meaning of slope and y-intercept in relation to a set of data  To write the equation of a line given the slope and the y-intercept, the slope and the coordinates of a point on the line, or the coordinates of two points on the line Materials Problem For students For the teacher AU Grid paper, graphing calculators Transparencies: 1.1 to 1.5B (optional) 1.1 11" by 4" strips of paper (15 per group), small paper cups, pennies (50 per group), chalk or masking tape (optional), blank transparencies (optional) 1.2 Labsheet 1.2 (1 per student), transparent grids or blank transparencies (optional) Transparency of Labshect 1.2 (optional) 1.3 Blank transparencies (optional) Overhead graphing calculator (optional) 1 1.4 Transparent grids (optional) 4b Investigation 1 Student Pages 5-25 Teaching the Investigation 25a-25n 1.1 * Testing Paper Bridges In their previous mathematics work, students have explored relationships between variables and have found patterns to help them express those relationships in symbolic form. They have used graphs, tables, and equations to represent situations. In this experiment, they will collect dau, search for patterns in their graphs and tables, and extend those patterns to make predictions. Launch Prepare a set of paper strips in advance, folding up e edges as explained in the student edition. Consider whether you want to take the class time necessary for each group to prepare their own strips or whether to have groups use pre-prepared strips that you or student aides have made. The strips need to be folded carefully, especially when several will be placed together to make a thicker bridge. This takes time to do well. To introduce the topic, have a short discussion about the design and strength of bridges. Have you ever walked across a shaky bridge and wondered if it would hold your weight? Do you get nervous when a car, bus, or train you are in crosses a high bridge? How do you suppose bridge designers know which materials will give a bridge the strength it needs? Read with students the information about bridges in the student edition and the experiment they will conduct to model how the strength of a bridge changes as its thickness increases. With the help of one or two students, demonstrate the experiment using a single strip of paper. Talk with the class about the mechanics of the experiment. Why might it be important that the ends of the bridge always overlap the supportthe books at either endby 1 inch? This is important, because for the comparisons to be meaningful, all of the bridges must be suspended over the same distance. Where should we place the cup? (in the center of the bridge, equidistant from the ends) How should we add the pennies? (carefully, using the same technique each time) Should we reuse our paper bridges after they have failed once? (No
once a bridge has broken, the paper that composed it is structurally different from unused paper strips and would yield different results.) 25a Investigation 1Fold a new bridge, but use a sloppy technique. Hold it up for the class to see. Does it matter if the paper strips are folded the same way every time? Would it make a difference if you used a bridge that looked like this? What might a bridge like this do to your results? When the class has an idea of how to conduct the experiment, talk about the data they will collea. What does it mean for a bridge to "break"? We need to agree on a definition for this before we begin. Does it mean that the bridge just touches the tabletop? In order for the groups to be able to compare their results, they will all have to use books of the same thickness. The experiment works best if books are at least 1 inch thick, as students will be better able to tell when the bridges have broken. Have students conduct the experiment, answer the questions, and work on the follow-up question in groups of four. Explore Groups may want to share the work by having each student play a defined role. One possible division of labor is to have  one student manage the paper,  one student manage the pennies,  one student verify that the ends of the bridges overlap each book by 1 inch and that the cup is positioned at the center of the bridge, and  one student record the data. You might suggest that students mark the books with chalk or tape 1 inch from the edge to lend greater accuracy to the data gathered. Also, marking the center of the top layer of each bridge would be helpful for placing the cup. For the Teacher: Choosing Independent and Dependent Variables You may need to review with students how to tell which variable is the independent variable and which is the dependent variable. The dependent variable belongs on the y-axis, and the independent variable belongs on the x-axis. You can often determine which variable to assign to which axis by thinking about how the two variables are relatedby asking yourself if one variable depends on the other. In this experiment, the number of pennies a bridge will hold depends on the bridges thickness, so the number of pennies, or breaking weight, is the dependent variable and belongs on the y-axis. Investigation 1 25bAsk parh student to draw a graph for part B. Groups should discuss which variable should be on the horizontal axis (thickness, or number of layers) and which should be on the vertical axis (breaking weight, or number of pennies). You may want to distribute blank transparencies for some groups to record their findings for sharing during the summary. Have groups do the follow-up as soon as they finish the experiment. Summarize If some groups have put their work on transparencies, ask them to display their results. Compare your group's results and predictions to those shown here. Why are there differences in the various groups' results and predictions? The data gathered will vary slightly from group to group, and groups may have used different methods to make their predictions. Talk about the various methods used. For every layer of paper you added to the bridge, what was the approximate increase in the number of pennies it would hold? Was the increase in the number of pennies constant? If not, how did you make your predictions for other bridge thicknesses? Students will probably have used a formal or informal averaging procedure based on their intuition that the rate of increase ought to be constant If groups do not seem to have used the idea of a constant rate, ask them to explain what guided them in making their predictions. They may offer some interesting rationales. Discuss the follow-up question, in which students consider the effects of building bridges using a stronger material. 1.2  Drawing Graph Models In this problem, students fit a line to approximate the pattern of results that they graphed in the bridge-thickness experiment. The creation of a graph model is a new concept for students. They will need time to assimilate why a graph model would give reasonable predictions and how it is related to the informal averaging students did when they made predictions in Problem 1.1. Launch Introduce the graph model that the Maryland class constructed to represent their data. You may want to display Transparency 1.2A, revealing first the table of data, and then the plotted points, and then the graph model as the discussion progresses. When you investigated the relationship between the thickness of a bridge (the number of layers of paper) and its strength (the number of 25c Investigation 1pennies it would support before breaking), you organized your data in a table and a graph. This class in Maryland did the same, creating a table and then plotting their data on a coordinate grid. How does their data compare to yours? Are their data similar? How? Explain that the Maryland class thought that the dau were somewhat linear, so they drew a line that seemed to fit the dau and to show the trend. Would you say that the line the Maryland class drew fits their experimental data well? Where does the line lie in relationship to the points? Why do you think they didn't just connect the dots on their graph? You might want to join two consecutive dau points with a line and then ask: If we extended this line segment, would it come close to, or predict, the next collected data point? Explain that students are to work in the same groups in which they explored Problem 1.1. Save the follow-up for the summary. Explore Each group should discuss and decide on the placement of their graph model. If they have trouble gening started, suggest that they try the strategy that the Maryland class used (they tried to suy as close as possible to the majority of the points). As you circulate, ask groups whether they think it is necessary for the line they draw to touch any of the plotted points, and if so, why. Also, suggest that they try more than one line and then, as a group, decide which they feel fits the dau best and be prepared to explain their reasoning. Keep students focused on the idea of drawing a graph model with the intention of being able to use it to make good predictions. Verify that groups are actually using their graph model, not the collected data, to make their predictions. You may want to have two or three groups put their graph models and predictions on transparencies for use in the summary discussion. Summarize If some groups have put their work on transparencies, ask them to display their results. Did this group draw a line that seems to fit the pattern in their data? If students have not drawn a straight line but have merely connected the points, make sure this is discussed. Ask these groups to revise their work, fitting a straight line to the dau. Ask the oer groups to explain how they decided where to draw the line. Investigation 1 25dDoes the line this group drew seem to be a reasonable fit? Explain. Look at this group's predictions for breaking weights. How do these predictions compare to those your group made? Explain any differences you see. Groups dau will vary, so graph models as well as predictions will vary from group to group. When students seem to have nude sense of Problem 1.2 and the idea of fitting a line to a somewhat linear dau set, distribute a copy of Labsheet 1.2 to each student. Have groups work on the follow-up, in which they will focus on fitting a line to plotted dau. In some of the dau sets in question 2, a line will fit very well
in others, a good fit is not possible. If students try to draw linear graph models for obviously nonlinear dau, ask whether the line would allow them to make good predictions. Remind them that if a graph model gives poor predictions, it is useless. When the class has finished working on the follow-up, discuss the follow-up questions, perhaps displaying a transparency of Labsheet 1.2 for talking about question 2. When students explain their strategies for fitting graph models to the dau sets on the labsheet, they are not expected to talk in terms of minimizing error directly, yet you may hear them express ideas that show they are beginning to think along these lines. They will probably have good suggestions, such as being in the middle of all the points or hitting as many points as possible. Although these are reasonable strategies, they wont always work well. For example, drawing a line to touch as many points as possible will sometimes lead to a poor fit for the remaining points. Similarly, joining the first and last points can lead to a poor fit for the intervening points. 1.3  Finding Equation Models In this problem, it is assumed that students are familiar with the concepts of slope andy-inter- cept and that they have had experience writing the equation of a line given the slope and the y-intercept. They will review those concepts and skills and extend their knowledge using data from the bridge-thickness experiment, interpreting a given slope and y-intercept of a graph model in terms of thickness and breaking weight. They will write an equation model that corresponds to the graph and closely represents the data. Launch Talk about the Maryland students graph model, perhaps displaying Transparency 1.2A again. Where does this linear graph model cross the y-axis? (at 0) How steep is the line? How can we measure the steepness of the line? 25e Investigation 1When you learned to write equations for lines, you learned that the steepness of the line (the slope) and where the line crosses the y-axis (the y-intercept) were important information for writing an equation for the line. Does anyone remember the general form for a linear equation? If no one recalls it, write y = mx + b on the board. If the slope of the graph model produced by the Maryland class is 8.7 and the y-intercept is 0, what would the equation of the line they have drawn be? (y = 8.7x + 0,ory = 8.7x) Read through Problem 1.3 with students, and have them work in pairs on the problem and follow-up. Explore Ask each student to record the results obtained on the graphing calculator in parts A and B and the responses to pans C and D. You might ask one or two students to put their tables and graphs on transparencies for sharing in the summary. Pan A asks students to make a table using a graphing calculator. If students do not have graphing calculators that can display a function in a table, have them make ubles by hand. For the Teacher: Displaying the Table with a Graphing Calculator The table feature of the graphing calculator is a powerful tool. You may want to consult your calculator's manual to be certain how to use this feature. On the Tl-80, the |TABLE] button is accessed by pressing [2rid1 IGRAPHI. The standard setting uses increments of 1 for x, which work well for this problem. The |TblSet| key (accessed by pressing [2rid11 WINDOW I) can be used to change the increment. Students do not need to use |TblSet| in this problem unless the increment has been changed from 1. For more assistance with using the graphing calculator to make tables and graphs, see the technology section in the front of this book. Some students may question e meaning of 8.7 pennies. If so, ask what they think it means, and suggest ey raise their question during the summary for the class to discuss. Investigation 1 25fSummarize Review each part of the problem with the class. If some students have put their tables and graphs on transparencies, display their work. Focus the summary on the topic of using equation models, as well as tables and graphs, to represent relationships. Spend time discussing e meaning of the slope and the j'-intercept in terms of a graph, an equation, and experimental data. Having students be able to move confidently between a situation and the models of the situationtabular, graphical, and symbolicand to understand how changes in one aspea of a situation affect the models, are the long-term goals of smdents work in this unit. Review the follow-up, asking the questions below to help students come to a better understanding of these ideas. What does it mean if one model has a steeper slope than another model? (It means that the rate of change is greater.) If you did the experiment using paper that was twice as thick, what would happen to the results in the table? (The number of pennies needed to break the bridge would increase.) To the slope of the line? (It would increase.) To the y-intercept? (nothing) Help the class review what they have learned so far. Look again at the Maryland class's graph model. Does their graph model fit all the data exactly? Why or why not? (The graph fits most of the data quite well but none of the data exactly.) Does the equation model exactly fit the data? Why or why not? (No
the equation model is essentially the same as the graph model.) How can you decide whether a model fits a given set of data? (A good model is close to the majority of the data points, reflects the trend in the data, and is helpful for making predictions.) What is the purpose of a model? (To aid in making predictions for data points between and beyond those we have gathered.) 1.4  Setting the Right Price In this problem, students will apply the ideas developed in Problem 1.3- They will consider data collected from a survey rather than from an experiment, again finding that although the data are not precisely linear, the pattern is close enough for a line to fit well and to make good predictions possible. The line in this problem has a negative slope and a nonzero j-intercept, which students may find somewhat challenging. 25g Investigation 1Launch Tell the story of the SGA survey, and ask whether students think the survey responses are reasonable. Help them to think about the data in realistic terms. Is $2 a realistic price to pay for an embroidered baseball cap? (It's not realistic, but it is an incredible deal that many students would take.) Is $12 a reasonable price? What overall pattern do you see in the data? If you graphed these data, what variables would you put on the axes? What would the graph look like? Have students work on the problem in groups of two or three. Save the follow-up until after the summary of the problem. Explore Have students do parts A and B individually and compare and discuss their answers in their groups. Encourage students in each group to share their graphs, note any differences, and choose e best graph model from among them. The collection of sbghtlv different graph models within each group will encourt^e a richer understanding of what a model is: not an algorithm for producing exact answers, but a judiciously chosen predictor. You might ask groups to put their graphs on transparent grids for sharing in the summary. To write eguatinns for their graph models, students will have to apply the concepts of j-intercept and slope. Have groups decide together what slope and y-intercept, and thus what equation, best represents their chosen graph model. Do not be too concerned if the slope is not an exact match for what students have drawn, as it may be difficult for them to determine the slope by using the scales on the axes. If graphing calculators are available and your students seem ready, have students enter the survey dau into their calculators, graph the data, and enter the chosen equation so that the group can see how closely their equation really matches the dau. See the technology section in the front of this book for information about entering dau pairs into a graphing calculator. Summarize Give each group an opportunity to present their graphs and equations. The graphs should all look similar. The equation models may be quite different, as students may have had trouble calculating the slope. Have groups justify their choice for the slope
some will have used the grid marks to count the rise and run
some will have made a guess from the survey dau. You will want to point out any common errors in finding slope, such as ignoring or misreading the scales on the axes. Students will get more practice with this idea in the follow-up to Problem 1.5. Have groups do the follow-up after the problem has been sufficiently reviewed. Be sure to ask students about the shape of the graph representing the relationship between price and income, which is drawn for follow-up question 4. This graph should alert students to the facts that not every relationship is linear and that trying to fit a straight line to a coordinate graph isnt always reasonable. You might also discuss the meaning of the negative values for the number of buyers in the uble. Investigation 1 25h1.5  Writing Equations for Lines In this problem, students build on their previous experiences with the concepts of slope and j-intercept, matching linear graphs to given situations and writing an equation for a given line. Launch Tell the story of the lawn-mowing earnings, and discuss the graphs shown in e student edition (and reproduced on Transparency 1.5A). When the class is ready, have students work in pairs on the problem. Save the follow-up until after the summary of the problem. Explore Have students discuss the questions with their parmers but write their own answers. Some students will immediately identify the panem implied in the problem statement: Denises rate is S5 per week, implying a slope of 5. Some will work from the visual pattern in the graphs
for example, Jonahs line shows that his earnings rise 530 over 10 weeks and so has a slope of 3. Encourage students to explain to their partners how they are thinking about this. The context will help them to make sense of and to identify the slope and the y-intercept for each graph. Summarize Let students share their answers to the problem. Then, introduce the follow-up. You were able to write equations for Jonah's and Denise's earnings by making sense of the context and the graphs. You can always write an equation for a linear relationship if you know the slope, m, and the y-intercept, b. The follow-up questions will help you to review ideas you have explored in your previous work in mathematics: how to find the equation of a line if you know the slope and a point on the line, or if you know two points on the line. For example, we could have figured out the slope of Jonah's graph if all we had was two points on the line, such as (0, 20) and (10, 50). We can see that as x changes from 0 to 10, y changes from 20 to 50: for every 10 units of change in x, there is a change of 30 units in y. In other words, for every 1 unit of change in x, there is a change of 30 10 = 3 units of change in y. \Ne call 3 the slope of the line. Have students work on the follow-up with their partners. Ask them to read each question carefully. Even though these questions are review
they are abstract and students will need time to work on them. When the class is ready, discuss each question, having students explain how they found their solutions. Then, help them to summarize what they have discovered. What do you need to know to identify the equation that matches a linear graph? (the slope and the y-intercept) 25i Investigation 1Can you get this information from the graph? (You can often easily read the y-intercept from the graph. If you can identify two points on the line, you can use them to calculate the slope. Identifying the equation is more difficult when you cannot read the y-intercept from the graph.) You may want to give students more practice with the difficult cases. For example, ask each student to write four linear equations, identify the slope (some of which should be negative) and the j-intercept of each, carefully graph them on grid paper, and exchange papers with another student. Then, ask students to try to write an equation that matches each graph on their panners sheet. Investigation 1 25jBMBBaWMB Nonlinear Models In this investigation, students continue to observe patterns in tables and graph models, and to use those patterns to seek equation models at describe data and to make predictions. In the situations in this investigation, the value of one variable decreases as the other increases but not in a linear fashion. In Problem 2.1, Testing Bridge Lengths, students again experiment with paper bridges, this time varying their length. The longer the bridge, the fewer the pennies it will support. Problem 2.2, Keeping Things Balanced, and Problem 2.3, Testing Whether Driving Fast Pays, pose other situations that lead to graph models characteristic of inverse relationships. Students are not expected to acquire formal vocabulary about inverse relationships at this time but to learn to express the relationships in their own words. They will focus on the pattern of change in the variables
how this pattern is reflected in the graph
and, perhaps, how an equation can model the relationship. The symbolic model is easier to understand in familiar distance-rate-time relationships than in the bridge-length and teeter-totter problems
however, the essence of the relationship is the same, and the graph and equation models have the same form. Mathematical and Problem-Solving Goals  To express data in tables and graphs  To make predictions from tables and graph models  To distinguish between linear and nonlinear relationships  To identify inverse relationships and describe their characteristics Materials Problem For students For the teacher AU Grid paper, graphing calculators Transparencies
2.1A to 2.3 (optional) 2.1 4" strips of paper cut to lengths of 4" to 11" plus extra strips to cut to length as needed (see Problem 2.1), small paper cups, pennies (50 per group), chalk or masking upe (optional), large sheets of paper (optional) 2.2 Weights such as heavy washers (20 per group) or pennies (80 per group), blocks of wood or cardboard fulcrums (1 per group), metersticks (1 per group), large sheets of paper (optional) I I II i i I 2.3 Large sheets of paper (optional) I I Student Pages 26-56 Teaching the Investigation 36a-36i Investigation 2 25p Ire 2.1  Testing Bridge Lengths In this investigation, students will encounter a different kind of relationship
an inverse relationship. In an inverse relationship, as one variable decreases e other increases, but not in a linear fashionthat is, not by constant decreases. There is an underlying constant in inverse relationships
when the two variables are multiplied, they yield a constant product. (The inverse relationship generated by the data in this experiment also involves a translation along the x-axis. Students will not be expeaed to find an equation model for this experiment, but they will see that the panem of change in the dau is similar to that in the other inverse relationships they will investigate.) Launch Prepare a set of folded paper strips in advance. Consider whether you want to use the class time necessary for each group to do this, or whether you prefer to have groups use pre-prepared strips that you or student aides have made. Read with students about the new experiment they will conduct to model how the strength of a bridge changes as its length increases. What do you expect to happen in this experiment? We are using equipment similar to what we used before. What are the variables this time? (length and breaking weight) What will the data look like? What shape do you think this graph will be? Students will probably guess that longer bridges will not support as much and at the graph model will be linear. Talk about the reasoning behind their conjectures, but let them discover the actual shape fitom their experimenution. As in Problem 1.1, esublish with the class what it means for a bridge to break. Also discuss ways to minimive variability, such as marking the books with chalk or upe to indicate where the strips will be placed, marking the strips to indicate where the cup will be placed, and using a consistent method for adding pennies to e cup. In addition, labeling each strip with its length will help students to avoid errors in recording. Have students work in pairs on the problem and follow-up. Explore Have students do e experiment and discuss the questions with their partners, but ask each student to make a table and a graph and to write his or her own answers. You might have pairs put their tables and graphs on large sheets of paper for sharing in the summary. Some students may try to use a straight line as their graph model. If so, ask how well it fits the dau. If they dont recognize that a line isnt a good fit, leave the issue for now and be sure these students share their work with the class in the summary discussion. At that time, you and the class can challenge the reasonableness of a linear model. investigation 2 36aSummarize If students have put their results on large sheets of paper, ask them to display their work. Have several pairs explain the reasoning behind their work. Then, ask
Are there similarities in the results of the various groups? Are there differences? What might have caused those differences? As bridge length increases, what happens to the number of pennies that can be supported? (It decreases.) As bridge length decreases, what happens to the number of pennies that can be supported? (It increases.) Focus the classs attention on this inverse relationship. What shape or pattern do you see in your graph? Is it linear? (No, it is definitely a curve.) How might you be able to tell from your table that the graph model will be a curve? (The difference between two consecutive breaking weights decreases less and less as the length increases
it is not a constant change.) Students will probably have been able to fit a smooth curve to the data. Encourage them to consider the possibility of there being an equation model that would fit their graph model. For now, it is enough to help them to realize that equation models of the familiarj = mx + b form will not fit these dau. Have students share their answers to the follow-up questions. Ask that they explain how they determined the breaking weights for lengths of 12,13,14, and 15 inches. If they are making predictions based on a linear model, you will want to challenge this idea, as a linear model wont match the collected dau and therefore is not reasonable. 2.2  Keeping Things Balanced The data generated in this activity will produce a more dearly inverse relationship. The product obtained by multiplying the two variables in this balancing experimentdistance and weight is theoretically a constant. Launch Read with the class, or simply talk about, the information about teeter-totters presented in the student edition. Smdents will be able to bring their experiences with similar playground equipment to the discussion, but they may not have consciously thought about the positions that create a balance. 36b Investigation 2What will happen if a very heavy person and a very light person sit on a teeter-totter that has its balance point in the center? Students should be able to relate that, unless the two riders change their positions enough to compensate, the heavy person will be on the ground and the light person will be stuck in the air. With the class, read the directions for conducting the experiment. Make sure everyone understands the procedure. Have students work in groups of four on the problem. Save the follow-up questions until after the summary of the problem. Explore Ask students to collect data in their groups but to make their own tables and graphs and to record their own set of answers. You mi^t want to have each group record their answers on large sheets of paper for sharing later. After students have had a chance to explore what (distance, weight) combinations will balance 3 weights at 40 centimeters from the fulcrum, encourage them to find a systematic way to collect and organize their data. They might look at the distance necessary to balance 4, 5,6,7, and 8 weights. Or, they might try to find the number of weights needed to balance the 3 weights at 40 centimeters by placing weights 10,15, 20, 25, 30, 35, 45, and 50 centimeters from the fulcrum. Some distances, such as 25 centimeters, wifi require a non-whole number of weights to balance the meterstick, which may lead some students to conclude that it is not possible to balance the stick. You can address such ideas in the whole-class summary. Listen to how students decide v.-hich weights to try at each distance, and capitalize on their ideas in the summary They should make some observations that the greater the number of weights, the shoner the distance they need to be placed from the fulcrum. Some might also notice, for example, that the weight needed at 15 centimeters is twice the weight needed at 30 centimeters. Encourage students to look for patterns in their tables. Even if the data are not precisely accurate, the proportional aspect of this relationship should emerge. Summarize Make a table on the overhead or board for collecting the class dau. To balance 3 weights 40 cm from die fulcrum, you need Number of weights Distance from fulcrum to fill in a weight and its corresponding distance. If no one Ask a represenutive from each group mentions lengths of 15, 20, 30, or 40 centimeters, have students return to the experimental setup and find results for these lengths. investigation 2 36cAfter collecting several dau entries, you might suggest organizing the dau to help in the search for patterns. The dau can be organized from least to greatest, or vice versa, for either the number of weights or the distance from the fulcrum. How many weights would have to be placed 10 centimeters to the right to balance the 3 weights 40 centimeters to the left? (12 weights) How did you find that solution? fWe read it from the chart
we found it during the experiment.) How many weights would have to be placed 20 centimeters to the right to balance the 3 weights 40 centimeters to the left? (6 weights) Does it seem reasonable that as we move the weights farther from the fulcrum, we need fewer weights to balance the meterstick? Where would 24 weights have to be placed to balance the 3 weights 40 centimeters from the fulcrum? (5 centimeters from the fulcrum) How did you find your solution? Does it seem reasonable compared to the other data entries in our table? Why or why not? Propose conducting a similar experiment with a longer teeter-totter. Imagine a teeter-totter 120 centimeters long with the fulcrum at the center. How many weights would have to be placed 60 centimeters from the fulcrum on the right side to balance 3 weights placed 40 centimeters from the fulcrum on the left side? (2 weights) How can you predict this answer without conducting the experiment? Encourage students to explain their thinking. If they can predia that doubling the distance would halve the required weight, or that doubling the weight would halve the required distance, they are using proportional reasoning and understand the inverse nature of the relationship. If they can explain the examples that involve a factor of 2, ask questions that involve a factor of 3. However, if students are still struggling with these concepts and dont see the pattern in the dau set, offer more examples. Where would you place 12 weights to balance 4 weights at 30 centimeters? (10 centimeters from the fulcrum) How did you find that distance? Why is it reasonable? Suppose we use an even longer teeter-totter. How many weights would have to be placed 120 centimeters from the fulcrum to balance 4 weights placed 30 centimeters from the fulcrum? (1 weight) Why does that answer make sense? When you are ready, have groups work on the follow-up. When they are finished, discuss the follow-up questions, offering additional examples if necessary. You might set up a format such as the following, and ask students to fill in the missing information: 36d Investigation 22 weights at 2 weights at 2 weights at will balance 6 weights at will balance 8 weights at 10 cm will balance 5 weights at 20 cm weights at 30 cm will balance 6 weights at 50 cm Encourage students to describe the strategies they use to answer these questions. Some may sute a general rule in words. If so, record their words on die board, and ask if thev can find a way to express their rule in symbols. If they are ready to proceed with an equation model, continue the discussion. For example, the last question above could be bought of as 30r=6x50 so W = 10. 30 This analysis focuses on the constant product, WD = 300. To review question 2, ask students to explain how they decided which equations satisfied the situation. This may be the understanding students need to be able to generalize the panem governing the teeter-toner experiment. 2.3  Testing Whether Driving Fast Pays In this problem, students will encounter another inverse relationship, that based on the relationship distance = rate x time. Distance in this story setting is fixed at 300 miles, so as speed increases, time decreases. This relationship can be wrinen as a constant product (speed X time = 300) or as a fraction with a constant numerator (time = In this problem, the dau are exact, so students should discover the form of the equation governing the relationship if they havent done so previously. They should also recognize that this graph has the same shape as those in Problems 2.1 and 2.2. Launch Tell the story of the bike tour and the bus drivers desire to get e riders home as quickly as possible. What would a typical speed for such a bus trip be? (say, 55 mph) How long would this journey take at that speed? (about 5^ hours) What would the time be if we went slower- (about hours) lay, 45 miles per hour? Do you think that, in general, driving fast saves time? Have students work on the problem and follow-up in groups of two or three. Investigation 2 36eExplore Students should recognize the shape of the graph that they produce. Though they explicitly use a maAematical relationship (distance = rate x time, or time = to generate the dau, some students may not realize that this relationship implies a constant product. As you circulate, ask students to explain how they calculated their answers. If students struggle wi writing an equation in pan C, ask how their tables of dau and their graphs are similar to those from Problems 2.1 and 2.2. Suggest revisiting question 2 in Problem 2.2 Follow-Up
looking at the equations that worked for that dau set may help students to think about how to write an equation for this problem. The answer to part D will surprise many students, as it is counterintuitive that the amount of time saved by increasing the speed is not a constant. In the follow-up, you may need to help students understand how minutes relate to fractions of hours
shortening the trip time from hours to 6j hours cuts | hour x 60 minutes/hour = 50 minutes off the trip. Summarize Talk with the class about the tables and the graphs they produced. How does this table and this graph compare to those you made in Problems 2.1 and 2.2? (It has the same basic shape and same general pattern: faster speeds go with shorter times, more weight goes with shorter lengths and shorter distances.) Ask different groups to write their equations from pan C on the board until all the equations the class found are displayed. Then, ask: Are their any equations that you disagree with because you don't believe they represent the situation? Why don't you think those equations work? Find a way to prove they don't work. Continue this discussion until all incorrect equations have been identified and eliminated and all correct equations have been verified. Of the equations that remain, explain how you developed them. What did you think about? How did you tackle this task? In the discussion of the follow-up, ask students to share their explanations of how they can tell from the graph that increases in speed result in less and less time savings. This understanding is the essence of making sense of this graph model. Follow-up question 3 asks students to revisit eir data from Problem 2.2 with this new mathe- matical relationship in mind. If they are having trouble with this question, you might ask In this problem, speed x time = 300. In the teeter-totter experiment, does weight x distance = 300? (No, the constant for this problem is not 300.) 36f Investigation 2Can you now find an equation model that is a good fit for the data in Problem 2.2? For the Teacher: Identifying a Graph Model for Problem 2.1 The data gathered in the bridge-length experiment also conform to an inverse relationship. However, as this is truly experimental data, results will probably vary widely from group to group. Finding an equation to fit the data would likely be quite difficult for students. For example, the graph model given in the answers to Problem 2.2 might be approximated by the equation model y = x-3.2' Bridge-Length Experiment 60 "S 50 c 1,40 SO s cn re oi co 20 10-------- 4 6 Length (indies) 10 12 V 0 0 2 + 8 Students should now recognize the idea tha similar graphs can signal similar underlying patterns and similar equation models. At this point, though, it is more important that they grasp the inverse nature of the pattern than that they become fluent with the symbols. Investigation 2 36gHMH^non More Nonlinear Models In this investigation, students meet another type of graph model. The patterns in exponential growth and exponential decay relationships are characteristic of many common situations and can be recogniaed from their graphs. The questions in this investigation emphasize making comparisons with linear graphs and noticing how the patterns of exponential growth and decay are different from ose of linear growth and decline. The ACE questions bring in all three models encountered thus farlinear, inverse, and exponential and continue to ask students to think about fitting a graph model to the data to make a trend or pattern more apparent and useful. In Problem 3.1, Earning Interest, students use their knowledge of percent to analyze the growth of an investment over time. They will be surprised by the discovery that the amount grows by greater and greater increments. In Problem 3-2, Pouring Water, the class conducts another experiment, pouring water from one glass into another so that each successive glass has half as much water as the one before. Then, students look at the relationship between the number of the glass in the series and the amount of water it contains. In both problems, a quantitymoney or water grows or decays by being repeatedly multiplied by a constant factor (in these cases, 1.08 and 0.5). Mathematical and Problem-Solving Goals  To use knowledge about percents and fractions to generate data  To explore a new type of graph model and to compare it to those explored previously  To use a graph model to make predictions  To continue to develop the idea of using a graph to model the trend in a data set \ I Materials Problem For students For the teacher AU Grid paper, graphing calculators Transparendes
3.1 and 3.2 (optional) 3.1 Blank transparencies or large sheets of paper (optional) 3.2 8 identical clear drinking glasses, water, food coloring (optional), stick-on notes or masking tape ACE Labsheet 3ACE (1 per student) 36j Investigation 3 Student Pages 37-46 Teaching the Investigation 46a-46i WMi T W * Earning Interest In Investigations 1 and 2, students coUected dau from experiments or generated tables of dau 2X1 IXXV A -------------- - J 1_ J 1 from their own knowledge of the relationship between two variables. They aeated graph models that fit the data weU enough to enable them to make predictions. They wrote equations for some of these graph models. In this problem, students will make graph models for a relationship that is neither linear nor inverse, but exponential
a graph model with a different shape is thus appropriate. (Students are not asked to write equation models for this type of relationship.) Launch Tell the story of the bank account Uncle Charlie opened for Chantal. Make sure everyone understands what yearly interest is and how it is calculated. a bank that pays 6% interest each year. Suppose you deposited $20 in How much interest would you earn after the first full year? ($1.20) How much will you have in the bank at the end of that year? ($21.20) How did you calculate your answer? Did anyone do it another way? Why do banks pay interest? (so that people will save their money with the bank, which allows the bank to make a profit by lending the money to others) Can you take out the interest you earn each year and leave the original balance in the account? (Yes. In fact, you can usually withdraw any portion of your money from a savings account, but you don't earn interest on what you take out.) What happens when you leave the interest you have accumulated in your account? What does the bank do? (The banks pays you interest on whatever is in your account, so you earn interest on the interest) Talk about any other questions students have, and then return to the story of Chantals option. What do you think Chantal should do, take the money or leave it in the account? Would it help you to make a decision if you knew how much money was in the savings account now? Have students work in pairs on the problem and follow-up. Explore Circulate as pairs work. Watch for students who apply faulty reasoning, such as figuring 8% of SlOO and then multiplying by Chantals age without considering the impact of compounding. Some students may get confused as they work through *e compounding equations
if so, you Investigation 3 46amay want to have them look at the strategies that other pairs are using. You might also ask pairs to justify their strategies and answers to each other. Whether to work with the decimal places used by the calculator or to round to fewer decimal places will be an issue in these calculations. (Banks would, of course, use figures rounded to two decimal placesthe amount in the accountin each new calculation.) However students decide to round, within a few calculations it will become apparent that the yearly increase is growing by greater and greater increments. The follow-up will help students to structure eir thinking about this problem. As they work through these questions, ask how they are using their calculators. Since the essence of an exponential relationship is that it is recursive multiplication, there are efficient ways to use a calculator to shorten the process. Using efficient calculator algorithms can sometimes enhance students understanding of the relationship: if they are not lost in the mechanics of the calculations, they will more easily see the underlying numerical pattern. For the Teacher: Recursive Multiplication on the Calculator The calculator can make this process more efficient. For example, here is the output from a calculator that is efficiently repeating the process of taking an answer, multiplying it by 8%, and adding the interest to the answer. To accomplish this on the TI-80 graphing calculator, press 100 and I ENTER I. Then, to enter 100 100 ANS+.08ANS 108 116.64 125.9712 136.048896 the formula ANS + .08ANS, press [2n3OH-08 |2nd| O- You have created a short program that takes the answer each time and adds to it the interest, 0.08 x the answer. By repeatedly pressing [ENTER |, you generate the value in the account at the end of each successive year. Use your best judgment about whether your students will gain from using such a shortcut. They need to understand the process before they move to this kind of algorithm, but if they know what they intend to do and are simply getting caught in the complexity, this technique may be appropriate. You may want to distribute blank transparencies or large sheets of paper for pairs to record their tables and graphs for sharing during the summary. Graphs should have the appearance and characteristics of exponential models, though students may find it difficult to draw smooth curves and may be unsure how to extend the curves to make predictions beyond the dau. 46b Investigation 3Summarize Have students share their tables and answers. Some will have made errors in the multistep calculation
comparing the results in the tables should help them find and correct their errors. Once the mechanics are out of the way, have students share their graph models. Some students may need to adjust theirs, as the data definitely produce a curve. Look at your graph models. How do we know this relationship is not linear? (The graph is not a straight line, so there is not a constant rate of change.) How could you have known this from looking at the data in the table? How is this relationship like the inverse relationship you found in the teeter-totter experiment? (It is represented by a curve. The change is not the same each time.) How is it different from that inverse relationship? (In this graph, both variables are increasing.) From the curve, what can you tell about how the account is growing? (It grows slowly at first and by greater and greater amounts as time passes.) In follow-up question 3, students may have calculated e account balance at the end of 18 years. What would the graph look like if the money were kept in the bank for 4 more years? 3.2  Pouring Water In this problem, students investigate another exponential relationship. They generate dau by conducting an experiment in which they line up a series of glasses, fill the first with water, and consecutively pour half the water from one glass into the next. They make a table and a graph showing the relationship between the glass number and the amount of water in the glass. Launch Talk about the experiment with the class. With their help, line up the eight glasses on a flat surface, label them, and fill glass 1 with water (adding food coloring will enhance the effect). Have someone carefully pour half the water from glass 1 into glass 2. Point out the interesting fact that the amount of water can be divided in half fairly well visually without the need for measuring. (If we wanted to be more exact, measuring tools would help.) Continue the process, as outlined in the student edition, until half the water from glass 7 is poured into glass 8. Then pour half the water from glass 8 into another container. Investigation 3 46cLook carefully at the water level in each glass. How does it change as the glass number increases by 1? (The water level decreases by half.) What do you think would happen if we continued this process with more glasses? (It would be almost impossible to continue because the amount of water would be so little.) Because we don't have an actual measure of the water in each glass, we can't make an accurate graph. In Problem 3.2, you are given a specific amount of water and challenged to think about what would happen if you conducted this experiment starting with that amount of water. Have students work in pairs on the problem and follow-up. Explore As pairs work, check their tables. If students are finding the process easy, challenge Them to extend their table for several more glasses. As students make their graphs, you may want to ask these questions: How can you tell which variable is the dependent variable and which is the independent variable? (You have to figure out which variabie depends on which.) Which variablethe independent variable or the dependent variable will you put on the x-axis? Why? (By convention, the independent variable goes on the x-axis.) Which variable is the independent variable? How do you know? (Glass number is the independent variable. The amount of water in a glass depends on the number of the glass.) Summarize Have students share their answers and compare their graphs. In what other problem have you seen a relationship that looks like this one? Students may think the pattern in this graph most resembles the inverse relationships in the bridge-length and teeter-totter experiments, because the amount of water decreases in each succeeding glass. Or, they may think it resembles the decreasing number of buyers in the problem about selling school-spirit caps. It is, in fact, mathematically similar to the exponential pattern seen in Problem 3.1. Students may not make this connection, as the savings-account problem involves an increasing relationship. Accept whatever ideas students offer, as they do not yet have a knowledge of exponential relationships to guide them. However, do challenge their ideas by pointing out how this graph is different from those in Investigation 2 and Problem 1.4. 46d Investigation 3You might ask whether e data in the table show the same kind of numerical relationship as the dau in the teeter-toner experiment (in which distance x weight = constant). And you can ask whether the graphs and tables from Problems 3-1 and 3-2 resemble each other in anyway. Students should be able to recognize that the patterns in the tables are not the same, but they probably wont spot that both tables are generated by repeatedly multiplying by a constant factor. (On the other hand, if someone sees this, capitalize on the discovery.) Investigation 3 46eA World of Patterns In previous investigations, students have collected data from experiments or generated data from descriptions of familiar situations. They have made tables, graph models, and sometimes equation models to fit data. In this investigation, they concentrate on e graph model, describing situations that could fit given models or creating graphs to fit given situations. Some of the graphs they will work with are linear, some are nonlinear, and some have linear and nonlinear sections. Students will come away firom this unit with an understanding that graph and equation models can be used to represent many patterns found in our world. In Problem 4.1, Modeling Real-Life Events, students read a story about a bus tour. They match graphs to, or create graphs to represent, a variety of activities in which the tour participants are involved. The focus of Problem 4.2, Writing Stories to Match Graphs, is on interpreting graphs. Students write stories that might have the given graphs as a model. In Problem 4.3, Exploring Graphs, students use a graphing calculator to examine equation models. In the process, they discover that some of the unusual graphs they have made in previous problems can be represented by equation models. Mathematical and Problem-Solving Goals  To use intuitive ideas about rates of change to sketch graphs for, and to match graphs to, given situations  To use intuitive ideas about rates of change to create stories that fit given graphs  To extend understanding of graph models to include new shapes  To explore symbolic representations for several graph models Materials Problem For students For the teadier AU Grid paper, graphing calculators Transparencies: 4.1 to 4.3 (optional), overhead graphing calculator 4.2 Blank transparencies (optional) 4.3 Paper or transparent graphing calculator grids (optional
provided as a blackline master) 46j Investigation 4 Student Pages 47-59 Teaching the Investigation 59a591 SUES] * Modeling Real-Life Events In this problem, students consider several situations that involve relationships between variables, applying their previous experiences with interpreting graphs. They match graphs to stories
create graphs to fit given situations
and invent a story and make a corresponding graph, identifying the variables in the story and showing how they are related. Launch Read e story about the bus tour to the class. If there are contextual elements that students do not understand, clarify them. You may want to bring photographs to help explain a low tide or the spon of parasailing. Continue the discussion by asking these questions: What variables do you see in this story? What kinds of things change as time passes? Are there relationships between any of the variables? If so, how might you model those relationships? When students have offered some ideas, read through the problem with them. Then, have them work in groups of three or four on the problem. Save the follow-up until after the summary of the problem. Explore As groups work, encourage them to discuss the relationships they see among the variables in the story. They need to dearly discern how one variable depends on another and the associated pattern of change between the variables. In part A, groups are to locate events in t
This project was supported in part by a Digitizing Hidden Special Collections and Archives project grant from The Andrew W. Mellon Foundation and Council on Library and Information Resoources.