rethinking middle school mathematics rethinking middle school mathematics: algebraic reasoning v...

Mathematics InstituteMathematics Institutehttp://www.utdanacenter.org/ssi/projects/texteams

Algebraic Reasoning

Across the TEKS

Algebraic Reasoning

Across the TEKS

Algebraic Reasoning

Across the TEKS

Rethinking MiddleSchool MathematicsRethinking MiddleSchool Mathematics

Permission is given to any person, group, or organization to copy and distribute Texas

Teachers Empowered for Achievement in Mathematics and Science (TEXTEAMS)

materials for noncommercial educational purposes only, so long as the appropriate

credit is given. This permission is granted by The Charles A. Dana Center, a unit of the

College of Natural Sciences at The University of Texas at Austin.

Dwight D. Eisenhower Professional Development Program, Title II, Part B

Texas Education Agency

Texas Statewide Systemic Initiative in Mathematics, Science, and Technology Education

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

Completed Summer 2001

iiTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Acknowledgments

The TEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning Institute was

developed under the direction and assistance of the following:

Academic Advisor:

Paul Kennedy TexasChristian University

Writers:

Paul A. Kennedy Texas Christian University.

Pamela Weber Harris Consultant

Jane Schielack Texas A&M University.

Contributing Writers:

Eva Gates

Lori Gonzales

Mary Alice Hatchett

Noemi Lopez

Diane McGowan

Bonnie McNemar

Michelle Morvant

Louise Nutzman

Thanks go to our student editor, SWT student Sarah McKinney.

Some of the materials were adapted from Dr. Kennedy’s work in the Holt, Rinehart and Winston

secondary series, Mark Driscoll's Fostering Algebraic Thinking: A Guide for Teachers Grades 6– 10, and Texas Instruments’ publications. Related publications are listed in the reference list.

iiiTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Table of Contents

Page

About TEXTEAMS Institutes iv

Introduction v

Section Overviews vi

Movin' on Down the Line 1

Moving with Technology 17

Unit I: Reflect and Apply 31

Stretching Sequences 33

Pipe Cleaners 67

Unit II: Reflect and Apply 76

Cross-Country Cycling 78

Speed Trap 105

Unit III: Reflect and Apply 117

Making Connections 119

Grids Galore 143

Unit IV: Reflect and Apply 159

Swimming Pool 161

Cover Up 171

Unit V: Reflect and Apply 179

Ups and Downs 181

Looking Back and Looking Forward 198

Appendix A—Materials List 204

Appendix B—References 205

Appendix C—Graphing Calculator Keystrokes 207

ivTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

About TEXTEAMS Institutes

TEXTEAMS Philosophy

• Teachers at all levels benefit from extending their own mathematical knowledge and

understanding to include new content and new ways of conceptualizing the content

they already possess.

• Professional development experiences, much like the school mathematics curricu-

lum itself, should focus on few activities in great depth.

• Professional development experiences must provide opportunities for teachers to

connect and apply what they have learned to their day-to-day teaching.

Features of TEXTEAMS Institute Materials

Multiple representations (verbal, concrete, pictorial, tabular, symbolic, graphical)

Mathematical ideas will be represented in many different formats. This helps both teachers

and students understand mathematical relationships in different ways.

Integration of manipulative materials and graphing technology

The emphasis of TEXTEAMS Institutes is on mathematics, not on learning about particular

manipulative materials or calculator keystrokes. However, such tools are used in various

ways throughout the institutes.

Rich Connections within and outside mathematics

Institutes focus on using important mathematical ideas to connect various mathematical

topics and on making connections to content areas and applications outside of mathemat-

ics.

Questioning strategies

A variety of questions are developed within each activity that help elicit deep levels of

mathematical understanding and proficiency.

Hands-on approach with “get-up-and-move” activities

Institutes are designed to balance intense thinking with hands-on experiences.

Math Notes and Reflect and Apply

A feature called Math Notes includes short discussions of mathematical concepts accompa-

nying the learning activities. Similarly, the Reflect and Apply feature is designed to extend

and apply participants’ understanding of the mathematical concepts.

The Charles A. Dana Center is approved by the State Board for Educator Certification as a registered Continuing Profes-

sional Education (CPE) provider. Hours received in TEXTEAMS institutes may be applied toward the required training for

gifted and talented in the area of curriculum and instruction. Individual district/ campus acceptance of these hours for

gifted and talented certification is a local decision.

vTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Introduction

Middle school students are in a stage of intellectual transition from concrete

to formal operational reasoning. Understanding how middle school students think

enables us to help them make the transition from numerical and concrete repre-

sentations of mathematics to algebraic, abstract representations. In the institute

we follow a model of developing algebraic reasoning based on giving participants

a concrete experience from which they can explore and investigate problems and

patterns. From their explorations and investigations, participants make informal

generalizations. Then they use language to describe the informal generalizations

and connect symbols to the language, a process which produces formal (alge-

braic) generalizations. The components of the model are listed below.

A Model for Developing Algebraic Reasoning

Concrete Experience

Pattern or Problem

Informal Generalization

Language

Formal (Algebraic) Generalization

The language component is critical in that students are required to articulate

their own movement from concrete to abstract. In fact, abstractions – writing a

linear function rule for a real-world situation, for example – become concrete, a

result that enables students to then access higher levels of abstraction. More-

over, the algebraic reasoning institute highlights the algebraic knowledge and

skills called for in the TEKS for middle school mathematics: building and making

connections among concrete, verbal, numeric, graphic, and symbolic representa-

tions of relationships between quantities. Careful attention to these connections

facilitates the move from concrete to abstract, enabling students to reason alge-

braically.

In the development of the institute, we relied on the ideas contained in Mark

Driscoll’s Fostering Algebraic Thinking: A Guide for Teachers Grades 6 – 10. We

pay particular attention to the three “Algebraic Habits of Mind”: building rules to

represent functions, doing-undoing, and abstracting from computation. Partici-

pants have an opportunity to identify and describe these habits throughout the

institute.

viTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Unit Overviews

Rethinking Middle School Mathematics:

Algebraic Reasoning

Unit I Focus: To use graphs to represent relationships.

Unit II Focus: To develop a model of algebraic reasoning.

Activity Name Concept(s) Materials

Movin' on Down theLine

Use a graph to explorerelationships between thequantities time and distance.

Stop watches or watches witha second hand (one pergroup), 100' or 150'

Measuring tapes (one pergroup), Data Collection Cards,

Secret Instruction Cards, 1"grid wall charts, Peel-and-stickdots or markers

Moving with Technology Represent and interpret

relationships between thequantities time and distance

by using technology to recordthe positions of walkers.

Data collection device or

devices (one per group),Motion sensors (one per

group), Overhead graphingcalculator

Unit I: Reflect andApply

Summarize and extendconcepts from Unit I.

Worksheet/ Transparency

Stretching Sequences Recognize patterns and writea rule for each pattern.Explore the connection

between sequences andrelationships.

Color tiles, Overhead colortiles, 1" grid paper, Peel-and-stick dots, Markers

Pipe Cleaners Gather data to determine the

relationships between thebase and height of rectangleswith a fixed perimeter and

express the relationship usingwords and symbols.

Pipe cleaners (light colored,

one per participant), Markers,1" grid paper, Peel-and-stickdots

Unit II: Reflect andApply

Summarize and extendconcepts from Unit II.


viiTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Unit Overviews

Unit III Focus: To explore constant rates of change.

Unit IV Focus: To connect multiple representations.


Cross-Country Cycling Construct function rules

and graphs using unit ratesand starting points, and

solve related linearequations.

Graphing calculator, 1" grid

paper, markers, Peel-and-stick dots

Speed Trap Collect distance data usinga motion detector and find

a trend line for the data.

Aluminum foil, Maskingtape, Small battery

operated vehicles (1 pergroup), Data collection

devices with motiondetectors (1 per group)

Unit III: Reflect andApply

Summarize and extendconcepts from Unit III.


Making Connections Formalize the input/ outputmodel for function andconnect mulitple

representa-tions: tables,function rules, equations,

and graphs.

1" grid paper, Coffeestirrers or flat spaghetti,Graphing calculator,

Markers, Peel-and-stickdots

Grids Galore Explore situationsnumerically usingcombination charts and

solve for unknowns giventwo equations.

1" grid paper, Cuisenairerods, Linking cubes,Markers, Dimes, Quarters

Unit IV: Reflect andApply

Summarize and extendconcepts from Unit IV.


viiiTEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Unit Overviews

Unit V Focus: To compare constant rates of change and changing rates of

change and explore precursors to equation solving.

Unit VI Focus: To apply the algebraic reasoning model and algebraic habits

of mind.

Ups and Downs Distinguish between linearand exponential growth

and decay.

1" grid paper, Peel-and-stick dots, Markers, Color

tiles, Paper cups, M&M's,Graphing calculator,Motion detector, Ball

(Raquet balls and golf ballswork well.)

Looking Back and

Looking Forward

Connect institute materials

to your classroom.

Blank transparencies,

Overhead pens, Paperpresentation pad orcomputer with projection

device


Swimming Pool Open-ended investigationof relationships between

quantities, both linear andnon-linear.

Two colors of color tiles, 1"grid paper, Markers, Peel-

and-stick dots

Cover Up Develop the cover-upmethod for equation

solving.

Overhead projector,Graphing calculator

Unit V: Reflect andApply

Summarize and extendconcepts from Unit V.

Worksheet/Transparency

1TEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning

Movin' on Down the Line

Institute Notes

Concept: Use a graph to explore relationships between thequantities time and distance.

Overview: Participants actively collect distance data in relation totime and graph it. Through questioning and richdiscussion, participants make the connection betweenthe motions observed during the activity and therelationship shown in the graph. This activity requiresextra physical space and can be done in a hallway orgym but gives the perfect opportunity to go outdoorson a day that weather permits.

TEKS Focus: 6.13— The student uses logical reasoning to makeconjectures and verify conclusions.7.15— The student uses logical reasoning to makeconjectures and verify conclusions.8.16— The student uses logical reasoning to makeconjectures and verify conclusions.

Materials: Stop watches or watches with a second hand (one pergroup), 100’ or 150’ Measuring tapes (one per group),Data Collection Cards, Secret Instruction Cards, 1”grid wall charts, Peel-and-stick dots or markers

Procedure: 1. To prepare for this activity make a copy of the DataCollection Cards and Secret Instruction Cardssheets for each group and cut the cards apart.You may wish to laminate the cards.

2. Go over the directions on a transparency ofActivity 1 - Instruction Sheet with the whole group.Do a quick demonstration and allow participants toask questions about the procedure.

3. Divide the participants into groups of 12, distributesupplies, and have them complete Activity 1 byrecording the distance data for the five movers.

4. Once the participants return to the room, havethem complete Activity 1 by compiling their data

into the tables and graphing the data on the grids.

Also:

Grade 6

4A, 7, 8B, 10D, 12A, 13AGrade 7

2G, 4B, 7A, 11B, 14A, 15AGrade 8

4, 5A, 12B, 15A, 16AAlgebra I

b.1A, 1D, 1E, b.2C, 2D,c.1B, c.2B, 2F



5. Have participants do Activity 2 by graphing the dataon 1” grid wall charts. Label each graph with thename and number of the mover. You may wish toassign a mover (1 – 5) to each group so that thedata for each mover is graphed only once.

6. When all graphs are completed, use the Reason andCommunicate questions to discuss the informationyou can determine from the graphs.

7. Have participants work in their groups of twelve todiscuss the graphs and determine the writteninstructions that each mover was given.

8. With the whole group, use a transparency of theSecret Instruction Card sheet to compare theinstructions written by the groups to the actual secretinstructions given to each mover.

9. Have participants reverse the process by doingActivity 2. Discuss using the Reason and Communi-cate questions.

10.Have participants use the rules they have formed toanalyze the graphs in Activity 3. Discuss using theReason and Communicate questions.

Extensions: Write a story about a person or some people moving.Draw a graph to illustrate it. The graph should showthe relationship between the distances of the personor people from some fixed place at different times.Exchange graphs with another group. Write a story togo with the graph drawn by the other group. Compareyour story with the other group’s story.

Select a straight line section of one of the “Movin’ onDown the Line” graphs and determine the speed ofthe mover and the direction in which he or she wasmoving. Repeat for several different sections. Write abrief description explaining how you determined thespeed and direction.

Graph the data in both inches and feet and comparethe graphs.



Assessment: Illustrate the following situation with a graph thatshows distance from the bus stop in relation to time:

Ann was walking toward the bus stop when shesaw the bus coming. She ran as fast as she couldtoward the bus stop, but the bus left before she gotthere. She walked slowly the rest of the way to thestop and sat down to wait for the next bus.

Write a brief statement describing how you can deter-mine the following when looking at a graph of distancefrom a fixed place over time.

• How can you tell when the mover is travelingfast, slow, or standing still?

• How can you tell if the mover is traveling towardor away from the beginning of the measuringtape?

Notes:

TEXTEAMS Rethinking Middle School Mathematics: Algebraic Reasoning Activity-4

Movin' on Down the LineSecret Instruction Cards

Movin’ on Down the Line

Secret Instruction Card for Mover 1

Do not show this card to anyone.

Start at the beginning of the measuringtape. Walk forward slowly at a constantrate. When the timekeeper calls the10th second, stand still. When youhear 16, walk at a faster rate until timeruns out.




Start at the far end of the measuringtape. Walk fast toward the beginningof the measuring tape. Keep yourspeed as constant as you can. Whenthe timekeeper says 10, walk veryslowly. Continue to walk very slowlytoward the beginning of the tape.




Start at the 10 foot mark of the mea-suring tape. Walk very, very slowly atfirst. Then very gradually increaseyour speed until you are running.When you reach the end of the tape,

stop running and stand still.




Start at the 25 foot mark of themeasuring tape, facing the beginningof the measuring tape. Stand still for 6seconds. When the timekeeper callsthe 6th second, walk along the tapetoward the beginning of the tape at aconstant speed. When you hear 12,stand still. When the timekeeper says

18, walk again until time runs out.




Start at the beginning of the measuringtape. Jog along the tape at a moder-ate speed for 10 seconds, keepingyour speed as constant as you can.When the timekeeper calls the 10th

second, walk back toward the startingpoint at a slow, steady pace until timeruns out.



Data Collection Cards

Mover

1

2

3

4

5

Position at:Mover

1

2

3

4

5

Position at:

Mover

1

2

3

4

5

Position at:Mover

1

2

3

4

5

Position at:

Mover

1

2

3

4

5

Position at:Mover

1

2

3

4

5

Position at:

0 seconds 2 seconds 4 seconds 6 seconds




Movin' on Down the LineActivity 1 - Instruction Sheet

Purpose: Collect data about the relationship between the distance of a personfrom the beginning of a measuring tape and the elapsed time.

In each group identify: 1 Timekeeper with a stopwatch or a watch with a second hand5 Movers, each with a different Secret Instruction Card6 Data collectors, each with a Data Collection Card

Description of the activity:

Each data collector will be given a Data Collection Card with two times writtenon the top. Each is responsible for recording the position of the mover at eachtime on his or her card. For example, if your card has 4 and 6 seconds on it,you are responsible for recording the position of each mover when the time-keeper calls 4 seconds and 6 seconds.

Each mover will be given a Secret Instruction Card. One at a time, follow thesecret instructions on your card without showing others your card. The time-keeper will give the start signal and then call out each second for 22 seconds.(“Go,…1,…2,...”)

First, mover 1 will practice moving down the line to give the data collectors theopportunity to position themselves where the mover will be at their assignedtime. The second time Mover 1 moves down the line, each data collector willrecord the position of the BACK of the mover’s back foot at his or her assignedtime, rounding to the nearest foot.

Repeat this procedure for each mover.

Once all of the data is collected, return to the classroom.



Math Notes:

In this activity, participants have recorded movers' distances at certaintimes, organized this data into a table, and then represented the data witha graph. From the data and graphs, they made conjectures about themovers’ instructions. In the next activity, they will reverse the process.We call this habit of mathematical (algebraic) thinking doing and

undoing. One of the processes we want to help develop during theinstitute and in our students is this notion of reversibility. That is, once wedo something, often we can work backwards and gain new insight, abetter understanding of what we have done, and a concrete knowledge ofthe concepts.

Also, one goal of this activity is for participants to form ideas about howthe graph of a mover’s movement looks for different instructions. Ineffect, we are building intuition for what motion looks like graphically as arelationship between distance and time. This is the first step towarddeveloping algebraic reasoning by generalizing rules from data. We havenot moved to a symbolic form with variables and equations yet, but wehave begun to move from one representation to another. We call thishabit of mathematical (algebraic) thinking patterns to rules (Driscoll).

Reason and Communicate:

• What does the ordered pair (x,y)mean in these graphs? (time,distance)

• In which graphs did the moverstand still for some period of time?How can you tell? Graphs 1, 3, and5. They have a horizontal line,which indicates that for more thanone second the mover was at thesame distance from the beginning ofthe tape.

• In which graphs did the moverstart at the beginning of the tape?Graphs 1 and 2.

• If the mover starts at the beginningof the tape, at what point does thegraph start? (0,0)

• If the mover did not start at thebeginning of the tape, where did heor she start? In Graph 3, the moverstarted 25 feet from the beginning ofthe tape since the distance at t=0 is25 feet (300 in.). In Graph 4, themover started at the end of the tapesince the distance from the begin-ning of the tape at t=0 is equal to thelength of the tape. In Graph 5, themover started 10 feet (120 in.) fromthe beginning of the tape since thedistance at t=0 is 10 feet.

• In which graphs is the mover

moving away from the beginning ofthe tape? How can you tell? InGraphs 1 and 5, the mover movesaway from the beginning of the tapesince the distance from the begin-ning of the tape is increasing overtime. In Graph 2, the mover ismoving away from the beginning ofthe tape at first.

• In which graphs is the mover

moving toward the beginning of thetape? How can you tell? In Graphs3 and 4, the mover is moving towardthe beginning of the tape since thedistance from the beginning of thetape is decreasing over time. InGraph 2, the mover moves towardthe beginning of the tape in thesecond part.


Movin' on Down the LineActivity 1, cont.

Enter the data for each mover into the tables below. Graph the data on the corre-sponding grids below and then transfer to the 1" grids to display on the wall.Based on each graph, write a description of the instructions the mover followed.

Dis

tan

ce

Dis

tan

ce

Time

Time

Mover 2

Time Distance

Mover 1

Time Distance


Movin' on Down the LineReason and Communicate, cont:

• What does the graph look like if

the mover moves away from thebeginning of the tape? The graphrises from left to right.

• What does the graph look like if

the mover moves toward thebeginning of the tape? The graphfalls from left to right.

• In Graph 1, during which part is

the mover moving the fastest? Themover is moving fastest in the thirdsection of the walk because then thechange in his or her distance overone second is the greatest .

• In Graph 2, when is the mover

moving the slowest? The mover ismoving the slowest in the secondsection because the change in his orher distance over one second is theleast then.

• Comparing all five graphs, in

which one is the mover moving thefastest? How can you tell? Look forthe steepest segment.

• Comparing all five graphs, in

which one is the mover moving theslowest? How can you tell? Lookfor the most shallow segment.

• Which graph does not look similar

to the others? Why? Graph 5 has acurved line instead of a straight line.The mover’s rate is changing.

• Should you connect the data

points? Is the data discrete orcontinuous? Discuss that the onlyway you would have a break in thegraph is if the mover left the courseand then came back.

• The graph represents a relation-

ship between what two quantities?Elapsed time and distance from thebeginning of the measuring tape.



Dis

tan

ce

Time

Time

Mover 3

Time Distance

Mover 4

Time Distance

Dis

tan

ce


Dis

tan

ce

Time


Mover 5

Time Distance



Math Notes:

Point out that in Activity 1 participants recorded movers' distances atcertain times, organized the data into a table, and represented the data ona graph. From the data and graphs, we made conjectures about themovers’ instructions. Now in Activity 2, we have reversed the process.We are undoing by taking the verbal description of a mover's walk andrepresenting it with a graph.

Also, we used the algebraic thinking skill of patterns to rules by applyingthe rules we found in Activity 1. For example, a faster pace is representedby a steeper line.


• How are the graphs alike? How

are they different?

• Which graph has a horizontal

section? Graph 1

• How do you represent on the

graph a mover standing still? Ex-plain why. By drawing a horizontalline to show that the distance isstaying the same as the timechanges.

• Which graph slopes down from left

to right? Why does it slope thatway? Graph 2 slopes down from leftto right because the mover started atthe end and walked toward thebeginning. Thus, the distance fromthe beginning of the tape decreasedin relation to time. Also, Graphs 3and 4 had segments where themover turned around and walkedtoward the beginning, so they havesegments that slope down from leftto right.

• Which graph has the steepest

segment? Why? Graph 3, becausethe mover ran and there was agreater change in distance persecond.

• How did you represent a mover

walking at a slow pace? Why? Bydrawing a shallow line to show asmall change in distance for eachsecond of time.

• Does it matter how shallow you

draw the line? At this point, we arenot overly concerned with findingspecific rates of change. We justwant participants to draw things slowand fast relative to one another.


Movin' on Down the LineActivity 2

Some students in another class were given different Secret Instruction Cards for“Movin' on Down the Line.” Data was collected for 20 seconds. Below is a list ofwhat was on their cards. Draw a sketch of what you think the graph should looklike for each one.

1. Start at the beginning of themeasuring tape. Walk forward at asteady, slow pace for 5 seconds.Then stand still for the next 5 sec-onds. Repeat these steps until timeruns out.

2. Start at the end of the tape andwalk towards the beginning of thetape at a steady pace.

3. Start at the beginning of themeasuring tape and walk slowly for8 seconds then run for 8seconds. Then turn around andwalk slowly back towards the begin-ning of the tape.

4. Start at the 50 foot mark on themeasuring tape. Walk forwardquickly for 7 seconds. Then turnaround and walk slowly backtoward the beginning of the tape for7 seconds. Turn around again andwalk away at a medium rate fromthe beginning of the tape until timeruns out.


Answers:

1. Start about in the middle of the tape measure, walk further away for ashort while, and then stand still for the remainder of the time.

2. Start at the beginning of the tape measure and stand still for a fourthof the time. Then walk away from the beginning of the tape measure untiltime runs out.

3. This graph is not possible for this activity. At one time, the mover isthree different distances from the beginning of the tape. This implies thatthe walker was in three places at once. Note that Exercises 1 and 2 arefunctions and Exercise 3 is not a function.

Math Notes:

Remember that we are looking atthe relationship between the twoquantities time and distance fromthe beginning of the tape. Thegraphs represent these relation-ships. The graphs do not representa picture of the path taken. Themover did not walk a course thatlooks like the graph.

In Activity 2, we went from a descrip-tion to a graph. In Activity 3, wemove from a graph to a description.This idea of going back and forthbetween representations, doing and

undoing, is a major mathematicalhabit of thinking that builds algebraicreasoning. Keep looking for itthroughout the institute.

In Activity 3, we build intuition for thealgebraic concept of function.Algebra students will need todifferentiate between relations thatare functions and relations that arenot functions.


• Which graphs are possible graphs

for Secret Instruction Cards? Why?Graphs 1 and 2 show that eachmoment the mover is only at onedistance from the beginning of the

measuring tape.

• Which graph is not a possible

graph of distance from the beginningof the measuring tape in relation toelapsed time? Why? Graph 3 is notpossible because to produce thisgraph, the mover would have to bein up to three places at the samemoment in time.

• What do we mean by doing and

undoing? Doing and undoing is amathematical (algebraic) habit ofmind. It is the idea of reversibility,strengthening of mathematicalconnections by going backwards.Activity 2 reverses Activity 1. Activ-ity 3 reverses Activities 1 and 2.



Movin' on Down the LineActivity 3

Study the graphs below made by a group of students in another class. Are thesegraphs possible graphs for other Secret Instruction Cards for “Movin’ on Downthe Line”? If not, explain why. If possible, write a description of how the personwas moving.

1.

2.

3.


Moving with Technology

Institute Notes

Concept: Represent and interpret relationships between thequantities time and distance by using technology torecord the positions of walkers.

Overview: Participants use a motion detector to record thewalkers’ distance from a given point. Participants areasked to use these graphs to discuss the relationshipbetween time and distance. Finally, participants willuse their understanding of the representations of therelationships to walk different shapes of graphs.

TEKS Focus: 6.13—The student uses logical reasoning to makeconjectures and verify conclusions.7.15—The student uses logical reasoning to makeconjectures and verify conclusions.8.16—The student uses logical reasoning to makeconjectures and verify conclusions.

Materials: Data collection devices (one per group), Motion sen-sors (one per group), Overhead graphing calculator

Procedure: 1. Set up the motion detector to collect distance datain real-time on an overhead calculator unit. Askvolunteers to stand in front of the motion detectorand walk. Make sure that the overhead unit isprojecting where the walker can see the graph ashe or she walks. Have each walker start a differentdistance from the motion detector and walk towardand away from the motion detector and stand still inreal time for 15 seconds. Ask participants to notehow the changes in the walk are reflected in thegraph. Keep the comments limited at this time aswe want participants to experience walking them-selves before they hear other’s interpretations.

2. Have participants complete Activity 1 in small groupsby using patterns to develop rules about how thedirection of a walk is represented in the graph of therelationship between distance and time. Then usethe Reason and Communicate questions to discussthe activity.

Also:

Grade 6

4A, 8B, 10D, 11B, 12A,13AGrade 7

3B, 4B, 9, 13B, 14A, 15Grade 8

4, 5A, 12B, 14B, 15A, 16A



3. Have participants complete Activity 2 by using pat-terns to develop rules about how the rate of walkingis represented in the graph. Then use the Reasonand Communicate questions to discuss the activity.

4. Have participants complete Activity 3 by using pat-terns to develop rules about how changing speeds isrepresented in the graph. Then use the Reason andCommunicate questions to discuss the activity.

5. Have the participants complete Activity 4 by applyingtheir rules about the graphic representations. Thenuse the Reason and Communicate questions todiscuss the activity.

6. Discuss with participants the differences and similari-ties between “Movin’ on Down the Line” and “Movingwith Technology.”

Extensions: Use a match-the-graph activity that has students lookat a graph and then use the motion detector to make agraph to match it by walking in an appropriate way.

Assessment: Write the alphabet. Explain which letters could be agraph representing the relationship between distanceand time in a “walk”. Also, note which letters cannotrepresent the relationship between distance and timein a “walk” and why not.

Notes:


Moving with TechnologyReason and Communicate:

• What does the graph look like

when you walk away from themotion detector? Why? It risesfrom left to right because as elapsedtime increases, the distance fromthe motion detector increases.Thus, both x- and y-values increase,and the graph rises.

• What is walking away from the

motion detector similar to in “Movin’on Down the Line”? It is similar towalking away from the beginning ofthe tape measure.


when you walk toward the motiondetector? It falls from left to rightbecause as elapsed time increases,the distance from the motion detec-tor decreases. Thus, y-valuesdecrease as x-values increase, andthe graph declines.


when you stand still in front of themotion detector? It is a horizontalline because as elapsed timeincreases, the distance from themotion detector is constant.



Activity 1

1. Use a data collection device and a motion detector to record the relationshipbetween distance from the motion detector and elapsed time for each of thefollowing walking instructions.

a. Collect data for 3 seconds. Startabout 2 feet from the motion detectorand walk away from the motiondetector for 3 seconds. Sketch thegraph in the box.

b. Collect data for 3 seconds. Startabout 10 feet from the motiondetector and walk toward the motiondetector for 3 seconds. Sketch thegraph in the box.

c. Collect data for 3 seconds. Startabout 5 feet from the motion detectorand stand still for 3 seconds. Sketchthe graph in the box.



Answers:

a. The graph in Exercise 1, Part a goes up from left to right because aselapsed time increases, the walker's distance from the motion detectorincreases.

b. The graph in Exercise 1, Part b goes down from left to right becauseas elapsed time increases, the walker's distance from the motion detectordecreases.

c. The graph in Exercise 1, Part c is a horizontal line because as elapsedtime increases, the walker's distance from the motion detector remainsthe same. The distance is constant.

d. From left to right, the graph will start off as a straight horizontal line,then go down for a while, and then go up the remainder of the time. Also,see snapshot.

f. Answers will vary but may include that standing still makes a horizon-tal line, walking toward the motion detector makes the line go down fromleft to right, and walking away from the motion detector makes the line goup from left to right.

g. The graph would start at 7 on the y-axis instead of 10 but otherwiselook the same.


• How did the graph change whenyou started farther from the motiondetector? The graph started higheron the y-axis at x=0. Just as yourdistance from the motion detector isgreater so is the distance of thegraph from the x-axis.


2. Answer the following questions.

a. Why does the graph in Exercise 1, Part a go up from left to right? Use thewords time and distance in your explanation.

b. Why does the graph in Exercise 1, Part b go down from left to right? Usethe words time and distance in your explanation.

c. Why is the graph in Exercise 1, Part c a horizontal line? Use the wordstime and distance in your explanation.

d. What will the graph look like if you combine the three instructions aboveinto one walk as follows: stand still, then walk toward the motion detector,and then walk away from the motion detector?

e. Walk the directions in Part d and confirm your sketch.

f. Write a paragraph summarizing what you know about how the direction youmove is represented in the graph of distance versus time.

g. How will your graph in Exercise 1, Part b change if you start your walk 7feet away from the motion detector instead of 10? Test your conjecture.


Activity 1, cont.




when you walk slowly? It rises orfalls slowly because as elapsed timeincreases, the distance from themotion detector is only changing alittle. Thus, as x-values increase, y-values only change a little.


when you walk quickly? It rises orfalls quickly because as elapsedtime increases, the distance fromthe motion detector is changing alot. Thus, as x-values increase, y-values change a lot.

• How are these graphs similar to

the graphs in the slow and quickmoving sections of "Movin' on Downthe Line"?

Technology Note:

If the calculator window automati-cally resets each time you collectdata, you may not see the desiredcontrast between rapid, medium,and slow speeds.

For the graph of rapid walking, notethe window and sketch the graph inthe rectangle of the overheadcalculator. Then, after the mediumrate walk, change the window tomatch the rapid walk window. Addthe sketch of the medium rate walkto the sketch of the rapid walk.Repeat for the slow walk. Display all3 sketches in the same rapid walkwindow.


1. Use a data collection device and a motion detector to record the relationshipbetween distance from the motion detector and time for each of the followingwalking instructions.

a. Collect data for 3 seconds. Startabout 2 feet from the motiondetector and walk rapidly awayfrom the motion detector. Sketchthe graph in the box.

b. Collect data for 3 seconds. Startabout 2 feet from the motiondetector and walk at a medium

rate away from the motion detec-tor. Sketch the graph in the box.

c. Collect data for 3 seconds. Startabout 2 feet from the motiondetector and walk slowly awayfrom the motion detector for 3seconds. Sketch the graph inthe box.

d. Collect data for 3 seconds.Startabout 2 feet from the motiondetector and stand still for 3seconds. Sketch the graph inthe box.


Activity 2



Answers:

a. Graph 1(a) is the steepest because the distance from the motiondetector over time is increasing at the highest rate.

b. Graph 1(d) is really the most shallow because it has 0 change indistance over time. However, of the graphs whose change over time isgreater than 0, 1c is the least steep because the distance from the motiondetector over time is increasing at the lowest rate.

c. The graph starts off increasing with a medium slope, then it increaseswith a steeper slope. Then it increases with a very shallow slope, andfinally it is a horizontal segment.

e. Answers will vary but may include that the faster the walker is walking,the steeper the graph. In other words, when one walks fast, the distancechanges significantly over an increment of time. When one walks slowly,the distance changes a little over an increment of time, and therefore, thegraph is less steep.

f. The graph will start higher on the y-axis (at 5 instead of 2) but otherwiseremain the same.


2. Answer the following questions.

a. Which graph in Exercise 1 is the steepest? Why? Use the words time anddistance in your explanation.

b. Which graph in Exercise 1 is the most shallow or the least steep? Why?Use the words time and distance in your explanation.

c. What would the graph look like if you combined these four instructions intoone walk as follows: walk at a medium rate, walk rapidly, walk slowly andstand still?

d. Walk the directions in Part c above and confirm your sketch.

e. Write a paragraph summarizing what you know about how the speed youmove is represented in a graph.

f. How will your graph in Exercise 1, Part c change if you start your walk 5 feetaway from the motion detector instead of 2? Test your conjecture.


Activity 2, cont.



Discuss the difference between aconstant speed and a changingspeed. Ask questions about thegraphs:

• Where is the walker slowing?

• Where are the graphs getting

flatter?

• Where is the walker speeding up?

• Where are the graphs getting

steeper?

Math Notes:

This is a good place to count pixels on the graph on the graphing calcula-tor. For every unit of time, count the number of pixels on the graph that goup or down. On a graph of constant speed, for every unit of time, thegraph rises or falls the same amount. On a graph of changing speed, forevery unit of time, the graph rises or falls different amounts. Speeding upis acceleration. When a walker speeds up, the graph will be concave up,like a smile. Slowing down is deceleration. When a walker slows down,the graph will be concave down, like a frown.

Technology Note:

Remind participants that motion detectors have a range of detection andthat the motion sensor only collects data in that range. Participants needto accomplish all of their movement in that range. In this activitiy, partici-pants will need to speed up and slow down quickly to get the desiredresults.


Use a data collection device and a motion detector to record the relationshipbetween distance from the motion detector and time for each of the followingwalking instructions.

1. Collect data for 3 seconds. Startabout 2 feet from the motion detectorand walk away from the motiondetector slowly at first but speedingup to a run at the end of 3 seconds.Sketch the graph in the box.

2. Collect data for 3 seconds. Startbehind the motion detector. Beginrunning and when you reach thefront of the motion detector, begincollecting data. Slow gradually sothat you are barely moving at theend of 3 seconds. Sketch the graphin the box.

3. Collect data for 3 seconds. Startabout 5 feet from the motion detectorand stand still for 3 seconds. Sketchthe graph in the box.


Activity 3



• How did you use the rules you

made about the representation ofthe direction you walked to createthe graphs?

• How did you use the rules you

made about the representation ofthe speed you walked to create thegraphs?

• What generalization could you

make about impossible shapes forthe graph? Anything that shows twodifferent distances at the same timewould be impossible. Any two pointsthat belong to the same verticalsegment are impossible.

Math Notes:

Since vertical lines cannot becreated in these graphs of distanceversus time, many of these shapesare possible only if you slant thelines.

To obtain a segment with a steepslope, you need to move at a veryfast rate. We build intuition here forthe algebraic concept of slope. Avertical line has undefined slope, asteep line has a higher slope then ashallow line, and a horizontal linehas a slope of 0.

Answers, cont.:

8. This graph is not possible sinceyou cannot walk negative distances.One could walk the same shape butshifted up one unit as discussed inExercise 7.

9. Start at the motion detector andwalk 1 meter away for about oneand a half seconds, slowing as yougo. Walk back for one and a halfseconds, speeding up as you go.Repeat three times.

10. This is not possible because inone moment of time, you cannot bein more than one place.

11. Stand still in front of the motiondetector.

Answers:

1. Start away from the motion detector, walk quickly toward it for a veryshort time, and then stand still.

2. Start at the motion detector, walk quickly away from it for a short time,and then walk back toward it until you are where you started. Finally,quickly walk away from the detector.

3. Start away from the detector and walk quickly toward it. Next, walkquickly away from it, returning to your original spot. Repeat the move-ments ending in your original spot.

4. This is not possible because in one moment of time, you cannot be inmore than one place.

5. Start near the motion detector, quickly walk away, and then quicklyback.

6. Start away from the motion detector, slowly walk toward it, and thenslowly back.

7. Start 1 meter from the motion detector and walk 1 m. away for one anda half seconds, slowing as you go. Turn and walk 2 m. toward the motiondetector for 3 seconds, speeding at the beginning and slowing at the end.Keep reversing your walk until time runs out.


Design a walk that would be represented by a graph with each of the the follow-ing shapes, if you can. If the graph is not possible, explain why.

1. The letter L2. The letter N3. The letter W4. The letter O5. A steep mountain6. A shallow valley

7.

8.

9.

10. A vertical line11. A horizontal line


Activity 4


Unit I: Reflect and Apply

Sample Responses:

1a. When collecting data without technology, one must physically find allthe data and create a table and graph. When collecting data withtechnology, the calculator gives the data in a table and graph.

b. The relationship between the variables are represented numerically intables. Graphically the relationship is represented by ordered pairs.

c. The graph of a walker's path represents his or her distance from acertain point over time, not the directions in which the walker moved. Forexample, if a walker stands still, the graph of the walker's path is ahorizontal line, but he or she did not walk a horizontal line.


Unit I: Reflect and Apply1. Reflect on the activities in "Movin’ on Down the Line" and "Moving with

Technology."

a. Compare collecting motion data with technology and without technology.

b. Explain how the relationships between variables can be representednumerically and graphically.

c. Describe how the actual path of the walker differs from its graph.

2. Consider ways you might adapt these activities for your specific gradelevel.

3. Using the graph below, fill in the table.

Time

Distance from the

Beginning of Measuring

Tape

2

6

10

14

18

22

26

28

Dis

tance fro

m B

egin

nin

g o

fM

easuring T

ape (

feet)

30

25

20

15

10

5

2 6 10 14 18 22 26 28

Time (seconds)


Stretching Sequences

Institute Notes

Concept: Recognize patterns and write a rule for each pattern.Explore the connection between sequences andrelationships.

Overview: Participants will represent sequences concretely withcolor tiles and numerically in a table. Then they willinvestigate each pattern and generalize an expressionfor the nth term in the sequence. The term variable isintroduced in this context. Finally, participants willrepresent the relationship graphically.

TEKS Focus: 6.13A—The student is expected to make conjecturesfrom patterns or sets of examples and nonexamples.7.4C—The student is expected to describe the relation-ship between the terms in a sequence and their posi-tions in the sequence.8.3B— The student is expected to use an algebraicexpression to find any term in a sequence.

Materials: Color tiles, Overhead color tiles, 1” grid paper, Peel-and-stick dots, Markers

Procedure: 1. Show Table 1 on the Transparency and have partici-pants build the next three figures in the pattern withcolor tiles. Then record the next three terms in thesequence.

2. Use the Reason and Communicate questions andMath Notes to complete and discuss Table 2. Showparticipants that they have now done Activity 1,Exercise 1.

3. Assign each group a problem from Activity 1, Exer-cises 2-4 to complete by building the figures withcolor tiles, recording the area of each figure, andwriting a rule for the area of the nth figure. Ifpossible, have two groups do each problem so thatdifferent strategies can be explored. After ampletime, have the groups share their strategies andconclusions with the whole group. Discuss using

Math Notes:

Middle school students aregenerally asked to find thenext three terms in asequence. In this lesson,participants will connectsequences to relations bygeneralizing rules for thenth terms of severalsequences.

Using color tiles to repre-sent the pattern is animportant concrete experi-ence for the participants.When discussing thepatterns, ask participantsto describe the figures intheir own words. Look forresponses like “one groupof two, two groups of two,three groups of two.” This

Also:

Grade 6

4A, 5, 7, 12AGrade 7

2D, 4, 7A, 14A, 15AGrade 8

3A, 4, 5, 14D, 15A, 16Algebra I

b.1A-C, 1E, b.2C-D, b.3A-B, b.4A-B, c.2F, d.1D



the Reason and Communicate questions. It isimportant throughout Stretching Sequences toencourage participants to find as many ways as theycan to represent each situation and make connec-tions between their representations.

4. Have the groups complete Activity 2 by graphingtheir data from Activity 1 on 1” grid paper and dis-playing the graphs on the wall. Then have thegroups compare their graph with the other graphsusing t-charts to record similarities and differences.Discuss as a whole group using the Reason andCommunicate questions.

5. Do Activity 3, Exercise 1 with the whole group.Make connections between alternate ways ofgeneralizing the rule for perimeter.

6. Assign each group the rest of the exercises fromActivity 3 to complete by building the figures,finding the perimeter of each figure, and writing arule for the perimeter of the nth figure. If possible,have two groups do each problem.

7. After ample time, have the groups share theirstrategies and conclusions by regrouping as follows:Have the participants in each group number off.Then have them regroup according to theirnumber. The new groups should have a personfrom each original group in them. Have participantsin the new groups take turns sharing their problemand strategies.

8. Discuss the activity with the whole group using theReason and Communicate questions. Also,discuss the regrouping strategy and how it mightbe used in classrooms.

9. Have the groups complete Activity 4 by graphingtheir data from Activity 3 on 1” grid paper anddisplaying the graphs on the wall. Then have thegroups compare their graph with the other graphs.Discuss as a whole group.

language helps build theprocess and leads to theidea of multiplying to getthe area.

In Activity 3, participantsreturn to the figures fromActivity 1, but this timethey will find the perim-eters of the figures insteadof the areas of the figures.The perimeter generaliza-tions are a bit trickier. Seethe Math Notes for thisactivity.


10. Pair participants. For Activity 5, assign some pairsExercise 1 and the other pairs Exercise 2. If time,have everyone do both parts. In this activity,participants are given a graph of figure numberversus area or perimeter of the figure and asked tocomplete a table. This leads them to developfigures to match the sequence using color tiles.Discuss the activity using the Reason and Commu-nicate questions. Have some pairs share theirstrategies and the figures they built.

Extensions: Reconstruct the sequences with the calculator tablefeature and graph.

Assessment: Have participants work in pairs. Each partner shoulddevelop a sequence and model the first three figureswith color tiles. Then have them complete a table as inActivity 1 for their partner’s sequence.

Notes:



Stretching SequencesReason and Communicate, cont:

• Fill in the figures and areas forFigures 4, 5, and 6 in Table 2. Whatgeneralizations can you make fromthis table? When the figure numberchanges by 1 unit, the area changesby two. This is the recursive routineused to fill in Table 1. Ask partici-pants to read the table horizontallyas well as vertically. By doing this,they should see that the area is 2times the figure number.• Can you use the same recursiveprocess you used in Table 1 to findthe 10th, 100th, and 1000th terms ofthe sequence? Yes, but it may takea long time.• How could you find the 10th termor 100th term in the sequence?Develop a rule describing therelationship between the number 10and the value of the 10th term, thenumber 100 and the value of the100th term, etc.• What is the relationship betweenthe figure number and the area?Describe it using a sentence. Thearea is twice the figure number.• Can you use this relationship tofind the areas for Figures 10 and100? Yes. 2(10)=20 and2(100)=200.Write a rule for the relationshipbetween the figure number and thearea. The area is twice the figurenumber. Area = 2(figure number).A=2n.• What does the n represent in therule? The n represents the figurenumber. We call n the variable.• What values are you putting intothe rule? We are inputting figurenumbers.• What are the values you areinputting called? Input values.Write “Input” above “Figure Number”in Table 2.• What values are we getting out ofthe rule? We are getting areavalues.• What are the values we get out ofthe rule called? Output values.Write “Output” above “Area” inTable 2.

Math Notes:

Middle school students are generally asked to find the next three terms in asequence. In this lesson, participants will move from sequences to relationsby generalizing rules for the nth terms of several sequences. The rule de-scribes the relationship between the position of the term in the sequence, orfigure number, and the actual value of the term in the sequence, or area.

Part of algebraic development is to move from a recursive description (add-ing two to the previous term each time to get the new area) to a closeddescription (2n), describing the nth term in the sequence. Here a table is usedto help make the shift. Also, this is an opportune time to introduce the lan-guage of “input and output values.” The position of the term or figure numberis the input value, and the term or area is the output value.


• How did you build the next figures in the sequence in Table 1? By adding

two tiles to the top of the previous figure.

• Using the previous term to get the next term is what kind of process?

Recursive


What are the next three terms in this sequence?

Table 1

Figure

Area 2, 4, 6, ____, ____, ____,

How can you find the area of the 10th, 100th, or 1000th figure?

Table 2 Figure Figure Number Area

1

2

3

4

5

6

10

100

n

Stretching SequencesTransparency



Math Notes:

See Reason and Communicate for the Transparency for discussion.

Sample Answers:

c. The area of the figure is twice the figure number.

d. A = 2n

e. A = 2(200)


Build each figure with color tiles, if possible. Then complete the table and answerthe questions that follow.

1.

a. How did you find the area of figure 10?

b. How did you find the area of figure 100?

c. Write a sentence to describe the relationship between the area of the figureand the figure number.

d. What rule can you use to find the area of any figure? Check yourconjecture with the first few figures.

e. How can you use this rule to create an equation to find the area of the 200th

figure?

Figure Figure Number Area Process

1 2

2 4

3

4

5

6

10

100

n

Stretching SequencesActivity 1



Math Notes:

The sequence in Exercise 2 can be thought of as (n + 1) groups of 2 tiles,2(n + 1). It can also be thought of as the sequence from Exercise 1 with2 tiles on top of each figure, 2n + 2. Make the connection between 2(n +1) and 2n + 2 in the model. Here we build intuition for the distributiveproperty in algebra as it comes up naturally in context.

(n + 1) groups of 2 tiles.

The sequences in Exercise 1 with 2 tiles on top. 2n + n.

Sample Answers:

c. The area of the figure is twice thefigure number plus 2. The area of thefigure is twice the quantity of the figurenumber plus 1.

d. A = 2n + 2 or A = 2(n + 1).

e. A = 2(150) + 2 or A = 2(150 + 1) = 302.


2.



c. Write a sentence to describe the relationship between the area of the figureand the figure number.


e. How can you use this rule to create an equation to find the area of the 150th

figure?


1

2

3

4

5

6

10

100

n

Stretching SequencesActivity 1, cont.


Stretching SequencesReason and Communicate:

• What do we call the output valuesgenerated in Exercise 3? Thesquare numbers.

Sample Answers:

c. The area of the figure is the figure number times the figure number. Thearea of the figure is the square of the figure number.

d. A n n A n= • = or 2.

e. A A= =( )( )300 300 3002 or = 90000


3.



c. Write a sentence to describe the relationship between the area of thefigure and the figure number.

d. What rule can you use to find the area of any figure? Check your conjec-ture with the first few figures.

e. How can you use this rule to create an equation to find the area of the300th figure?


1

2

3

4

5

6

10

100

n




• What strategies did you use to findthe area of the 10th figure?

• What strategies did you use to findthe area of the 100th figure?

• How can you use these physicalmodels to show the connectionbetween the expressions 2n+2 and2(n+1) or the expresions n2+n andn(n+1)? The sequence in Exercise 2can be thought of as the sequencefrom Exercise1, but with two tiles ontop of each figure, 2n+2. It also canbe thought of as (n+1) groups of 2tiles, 2(n+1). Likewise, the sequencein Exercise 4 can be thought of asthe square numbers, n2, (Exercise 3)plus n, n2+n, or it can be thought ofas n groups of (n+1) tiles, n(n+1).

• What relationship does your rulefrom Part (c) describe? The ruledescribes the relationship betweenthe two quantities figure number andarea.

Sample Answers:c. The area of the figure is the squareof the figure number plus the figurenumber. The area of the figure is thefigure number times the quantity ofthe figure number add one.

d. A n n A n n= + = +2 1 or ( ) .

e. A A= + = +250 250 250 250 12 or ( ) .

Math Notes:

Emphasize the algebraic thinking in this activity. Participants are general-izing patterns with rules and, within that process, are abstracting fromcomputation. Also, we are using equations to find specific areas andrules to describe entire relationships.

In Exercises 1 – 4, Parts a and b, equations are used to find specificareas, and in Exercises 1 – 4, Part d, rules (or functions) are used todescribe entire relationships.

The sequence in Exercise 4 can be thought of as n groups of (n + 1)tiles, n(n + 1) and n2 + n in the model.

1 • 2 12 + 1

2 • 3 22 + 2

3 • 4 32 + 3

n(n + 1) n2 + n


4.



c. Write a sentence to describe the relationship between the area of thefigure and the figure number.


e. How can you use this rule to create an equation to find the area of the250th figure?


1

2

3

4

5

6

10

100

n




• Do any of the graphs represent aproportional relationship? How canyou tell? Graph 1 represents aproportional relationship since it islinear and goes through the origin.

• Which graphs can be representedwith a line? Use string or a meterstick to test the graphs. Graphs 1and 2 can be represented with a line.

• What kind of relationship isrepresented by Graphs 3 and 4?Graphs 3 and 4 represent quadraticrelationships.

• Does it make sense to connect thedots or data points in these graphs?No, the data is discrete. For

example, a figure number of 11

2

does not make sense as an inputvalue. Recall that the input value isa figure number, not a dimension.

• Go back and look at the tablesfrom Activity 1. Which tables can begenerated recursively be repeatedaddition? The tables in Exercises 1and 2.

• Describe the graph that resultsfrom repeated addition. Linear.


1. Graph the data from Activity 1, Exercise 1.

Are

a

Figure Number


Are

a

Figure Number




Math Note:

Note the contrast between linear relationships in Exercise 1 and 2 andthe non-linear relationships in Exercises 3 and 4.



Are

a

Figure Number


Are

a

Figure Number




Sample Answers:

c. The perimeter is twice the figure number plus 4. The perimeter is thefigure number plus 2 plus the figure number plus 2.

d. P = 2n + 4 or P = n + 2 + n + 2 or P = n + n + 2 + 2

e. P = 2(200) + 4 or P = 200 + 2 + 200 + 2 or P = 200 + 200 + 2 + 2 = 404

An Alternate Process:

4 + 2 • 1, 4 + 2 • 2, 4 + 2 • 3, 4 + 2n


• What's the difference between

Activity 1 and Activity 3? In Activitiy1 you find the area of the figures. InActivity 3 you find the perimeter ofthe figures.


Complete the tables using the figures from Activity 1 or color tiles to model thefigures and answer the questions that follow.

a. How did you find the perimeter of figure 10?

b. How did you find the perimeter of figure 100?

c. Write a sentence to describe the relationship between the perimeter ofthe figure and the figure number.

d. What rule can you use to find the perimeter of any figure? Check yourconjecture with the first few figures.

e. How can you use this rule to create an equation to find the perimeter ofthe 200th figure?

1.Figure Figure Number Perimeter Process

1 6

2 8

3

4

5

6

10

100

n




Sample Answers:

c. The perimeter is twice the figure number plus 6.

d. P = 2n + 6

e. P = 2(150) + 6 = 306

Alternate Answers:

One can count half of the perimeter and multiply by 2: 2•4, 2•5, 2•6, etc.This can be written as 2(n+3). Also, one may think of it as the same asExercise 1 with 2 added to each term or (2n+4)+2=2n+6. One may countthe number of tiles on the top and bottom edges, 4, and then add thenumber of tiles on the sides: 4+4, 4+6, 4+8. This can be written as4+(2n+2)=2n+6.


• How can you visualize 2n + 6?

The top and bottom are constant.The two sides change in each figure,so add 2n each time.







2.Figure Figure Number Perimeter Process

1

2

3

4

5

6

10

100

n




Sample Answers:

c. Perimeter is 4 times the figure number. Perimeter is fhe figure numberplus the figure number plus the figure number plus the figure number.d. P = 4n or P = n + n + n + ne. P = 4(300) or P = 300 + 300 + 300 + 300 = 1200

Alternate Process:

Split the figure in half diagonally and count the semi-perimeter. Thenmultiply by 2 to get the perimeter: 2•2, 2•4, 2•6, etc. This can be writtenas 2•2n=4n. Also, one can look at the perimeter as multiplesof 4: 4•1, 4•2, 4•3, . . . , 4n


3.






Figure Figure Number Perimeter Process

1

2

3

4

5

6

10

100

n




• What strategies did you use to findthe perimeter of the 10th figure?

• What strategies did you use to findthe perimeter of the 100th figure?

• What equivalent expressions didyou or other groups use for eachsequence?

• What relationship does your rulefrom Part (c) describe? The ruledescribes the relationship betweenthe two quantities figure number andperimeter.

Math Notes:

Summarize the equivalent expressions that arise through participants’ work.

Emphasize different ways to generalize the patterns. For example, in Exer-cise 1, one could fill in the process column with 1+2+1+2, 2+2+2+2, 3+2+3+2,. . . , n+2+n+2 and then write the expression 2n+4. One could also fill in theprocess column with 3 groups of 2 or 3(2), 4(2), 5(2), . . . , (n+2)(2) and writethe expression 2(n+2).

Also, the sequence in Exercise 4 can be thought of as the sequence in Exer-cise 3 with 2 added to each term or 4n+2

Sample Answers:

c. Perimeter is 4 times the figure number plus 2.d. P = 4n + 2e. P = 4(250) + 2


4.






Figure Figure Number Perimeter Process

1

2

3

4

5

6

10

100

n




• How did you scale your graph?

Why? Answers will vary.

• Is there one correct way to scale

the graphs? No, but there may bemore appropriate ways. Comparethe different ways participants usedand point out any advantages ordisadvantages.



Perim

ete

r

Figure Number


Perim

ete

r

Figure Number




• Do any of the graphs represent aproportional relationship? Graph 4

• Which graphs can be representedwith a line? Use string or a meterstick to test the graphs. All thegraphs are linear.

• Does it make sense in this prob-lem to connect the graphs? No, thedata is discrete. For example, itwould not make sense to have figure

number 3 . Recall that the input

value is a figure number, not adimension.

3

4



Perim

ete

r

Figure Number


Perim

ete

r

Figure Number



Stretching SequencesMath Notes:

Recall from the Transparency thatparticipants were asked to read thetable vertically as well as horizontallyto recognize patterns. In this activity,reading the table in different ways todraw conclusions is useful. First, rec-ognizing vertically that as the figurenumber increases by one, the area in-creases by 4, leads to a recursive rou-tine of adding four to the previous areato get the new area. Second, usingthe increase of 4 in area to developthe process column (as shown in thesnapshot) enables some to recognizethe relationship between figure num-ber and area. Then, participants candescribe the relationship with a sen-tence and symbolic rule.

Developing the figures that fit the datamay take some time. One way todevelop the model in the snapshotfollows:

First, there is always an area of fourthat can be represented with 1 tileby 4 tiles. Then for Figure 1, thearea is eight, so add four more tilesor another group of 1 tile by 4 tiles.To get the next figure, add another1 by 4 group of tiles.

Representing the relationshipbetween the area of a figure and thefigure number as A=4(n+1) is usefulhere. Each figure has (n+1) groupsof four tiles. The four tiles can begroups of 1 by 4, 4 by 1, or 2 by 2tiles.

In Activity 1 — 4 participants find a ruleand a graph given a sequence offigures. In Activity 5, they reverse theprocess, creating a rule and asequence of figures given a graph.This is an example of doing and un-

doing.

Sample Answers:

Since Figure 1 has an area of 8, it can be modeled with 1 tile by 8 tiles.To get the second figure, add an area of four by adding four tiles, making1 tile by 12 tiles. Continue adding four tiles to the previous figure to getthe new figure.c. The area of the figure is 8 plus 4 times one less than the figure num-ber.d. A = 8 + 4(n - 1)

Using 2 tiles by 2 tiles to represent an area of four, one can model Figure1 with 4 tiles by 2 tiles. To get the new figure, add a group of 2 by 2 tilesto the top of the previous figure.c. The area of the figure is 2 times twice the figure number plus two.d. A = 2(2n + 2)

Area



• What information can you gatherfrom the graph? The areas ofFigures 1 - 4.

• What are the input and outputvalues? The input values are figurenumbers, and output values areareas.

• What strategies did you use tomodel the sequence with color tiles?

• What different models did youdevelop?

• What strategies did you use to findthe areas of Figures 5, 6, and 7?Most participants will develop arecursive routine by noticing that asthe figure number increases by one,the area increases by four. Thus,adding four to the previous area willgive the new area. Some partici-pants may extend the graph ordevelop their rule now as well.

• What strategies did you use to findthe areas of Figures 10 and 100?Most participants will move from therecursive routine by the 10th and100th figures. Using the processcolumn as shown in the snapshot isone way to find the areas of Figures10 and 100.

• How can you use the processcolumn to help you write a rule thatdescribes the area of Figure n?

• What sentence describes therelationship between the area of afigure and the figure number? Thearea is 4 times the figure numberplus 4.

• What rule can you write describing

the area of Figure n? A=4n+4 orA=4(n+1)

Area

Math Notes:

The following is one way to develop the model in the snapshot:

The first figure has an area of 8. This can be modeled with 2 tiles by 4tiles. Then, to get the second figure, add 4 more squares by addinganother group of 1 tile by 4 tiles. Keep adding a group of 1 tile by 4 tiles toget the next figure. Starting with a possible first figure and deciding howto add an area of 4 is the key to this development.

Sample answers:c. The area is 4 times the figure number plus 4.d. A = 4n + 4(n + 1)


a. What are the areas of Figures 5, 6, and 7?b. What are the areas of Figures 10 and 100?c. Write a sentence to describe the relationship between the figure number

and the area of the figure.d. Write a rule describing the area of Figure n.

1. Use the graph below to fill in the table.

Are

a

FigureFigure

NumberArea Process

1

2

3

4

5

6

7

10

100

n

1 2 3 4 5

4

8

12

16

20

0

Figure Number




• Since you are using the samegraph as in Exercise 1, what infor-mation do you already have? Theyknow everything except what thefigures look like.

• What strategies did you use tomodel the sequence with color tiles?

• What different models did youdevelop?

• Does anyone have a strategy forcompleting the table that has notbeen discussed?

• What sentence describes therelationship between the perimeterof a figure in this sequence and thefigure number? The perimeter isfour times the figure number plusfour.

• What rule describes the relation-ship between the perimeter of afigure in this sequence and thefigure number? P=4n+4 (same asPart 1)

Math Notes:

One way to develop the model in the snapshot follows:The first figure has a perimeter of 8. This can be modeled with 2 tiles by2 tiles. Then to get the second figure, add four more sides by addinganother group of 1 by 2 tiles. Keep adding a group of 1 by 2 tiles to getthe next figure.

Starting with a possible first figure and deciding how to add a perimeter offour to it is the key to this development.

Sample Answers:

Similar to the figures from Exercise 1, Figure 1 can be modeled with 1 tileby 3 tiles. Then to get figure 2, add two tiles making the figure 1 tile by 5tiles. Continue adding two tiles to the previous figure to get the newfigure.c. The perimeter is 2 plus 2 times twice the figure number plus 1.d. p = 2 + 2(2n + 1)= 4n + 4

Perimeter


2. Use the graph below to fill in the table.

a. What are the perimeters of Figures 5, 6, and 7?b. What are the perimeters of Figures 10 and 100?c. Write a sentence to describe the relationship between the figure

number and the perimeter of the figure.d. Write a rule describing the perimeter of Figure n.

FigureFigure

NumberPerimeter Process

1

2

3

4

5

6

7

10

100

n

Perim

ete

r

1 2 3 4 5

4

8

12

16

20

0

Figure Number



Pipe Cleaners

Institute Notes

Concept: Gather data to determine the relationships between thebase and height of rectangles with a fixed perimeter andexpress the relationship using words and symbols.

Overview: Participants will physically build rectangles with pipecleaners. Since the pipe cleaners are all the same length,the rectangles will all have the same perimeter. Partici-pants will investigate the relationship between the baseand height of each of these rectangles. Connectingmultiple representations, they will collect data in a table,graph the data, and build a symbolic rule relating baseand height for rectangles of a given perimeter.

TEKS Focus: 6.4—The student uses letters as variables in math ex-pressions to describe how one quantity changes whena related quantity changes.7.4—The student represents a relationship in numeri-cal, geometric, verbal, and symbolic form.

8.4—The student makes connections among variousrepresentations of a numerical relationship.

Materials: Pipe cleaners (light colored, one per participant),Markers, 1” grid paper, Peel-and-stick dots

Procedure: 1. Give each participant a pipe cleaner and ask eachparticipant to measure the length of the pipe cleanerto the nearest half-centimeter. Have the participantsin each group agree on a length and write it on thetransparency of Activity 1, Exercise 1a.

2. Instruct each participant to fashion a rectangle froma pipe cleaner so that the ends of the pipe cleanermeet at a corner of the rectangle. They are to workalone.

Also:

Grade 6

4A, 4B, 5, 8A, 8D, 10D,11C, 12A, 13AGrade 7

2B, 4A, 4B, 5, 8, 9, 13C,13D, 14A, 15AGrade 8

2A, 2B, 2C, 4, 5, 14C,15A, 16AAlgebra I

b.1A, 1B, 1C, 1E, b.2C,b.2D, b.3A, 3B, b.4A, 4B,c.2F, d.1D

Correct: Ends of pipecleaner meet in upper leftcorner.

Incorrect: Ends of pipecleaner meet on a side.


Pipe Cleaners

3. Have participants share their rectangle buildingstrategies with the other members of their group.Then have a member of one group share one of thegroup’s strategies with all of the participants. Havea representative from another group share adifferent strategy. Continue this until all differentstrategies used are mentioned. Possible strategiesinclude:• Guess-and-check bending• Folding the pipe cleaner in half first and then

folding the remaining halves.• Starting with an oval and forming a rectangle by

pinching the oval.

4. Ask participants to compare rectangles in theirgroups and change them, if necessary, so that allthe rectangles in the group are different from oneanother, i.e. short and wide, long and skinny, square.

5. Instruct participants to mark the corners of theirrectangles with a marker or piece of tape. Then theyshould straighten the pipe cleaner. Ask them to lineup the pipe cleaners in their group next to eachother and compare them. Discuss their observa-tions. Elicit the following two important points:• The pipe cleaners are all the same length.• The second marks are all in the same place, in the

middle.

6. Have participants do Activity 1 by measuring thelength of each piece of their rectangles to the near-est half-centimeter, filling in the table, and answeringthe questions.

Math Notes:

The idea is that you canuse the semi-perimeterfold as a starting point.This is important forgeneralizing a symbolicrule later.


7. Have participants do Activity 2 by graphing the datafrom the table in Activity 1 and answering thequestions. Have each group make a group graph on1-inch graph paper, display the graph on the wall,and compare their graph to the others.

Extensions: Build a table of heights, bases, and areas of therectangles with fixed perimeter. Graph area vs. baseand discuss the nonlinear plot. Which rectangle givesthe most area? (Square)

Assessment: Explain how you can move students from concreteexperiences to algebra via language.

Notes:

Pipe Cleaners


Pipe CleanersReason and Communicate:

• What strategies did you use tofind the sentence and rule todescribe how the lengths of thebase and height relate to the lengthof the pipe cleaner?

Math Notes:

a. Encourage participants to dothese problems numerically usingmental math.

c,d. For a pipe cleaner of length 20cm, the base plus the height is halfthe length of the pipe cleaner, or

base + height . Thus,

base + height = 10. Be sure to writethe verbal sentence and then the

rule B H+ = 10 .

Answers:

a. Answers will vary depending on the pipe cleaners used.

b. The length of the pipe cleaner represents the perimeter of the rectangle.

c. The base and height add to the same number, half the perimeter, eachtime.

d. Suppose that the length of the pipe cleaner is 30 cm. Then the base plus

the height is 1/2 the length of the pipe cleaner, 15 cm. B H+ = 15 .

e. For a base of 2 cm and pipe cleaner of length 30 cm, the height is 13 cm.

f. For a height of 4.5 cm and pipe cleaner length of 30 cm, the base is10.5 cm.

= =1

220 10( )


Pipe CleanersActivity 1

1. Measure the length of each side of your rectangle to the nearest half-centimeter. Fill in the table with all the data from your group’s rectangles.

a. What is the length of the pipe cleaner?

b. What does the length of the pipe cleaner represent in terms of a measure of arectangle?

c. For each base and height, find the sum. What do you notice?

d. Write a sentence and a rule to describe how the lengths of the base andheight compare to the length of your pipe cleaner.

e. For a base equal to 2 cm and a pipe cleaner of length 30 cm, how could youdetermine the height?

f. If the height is 4.5 cm and the pipe cleaner is 30 cm long, what is the base?How do you know?

Base Height

B H


Pipe Cleaners

Answers:

a. From Exercise 1, we know the base plus the height is one-half the lengthof the pipe cleaner. Add the first few base lengths to their correspondingheight lengths. The number you get is half the length of the pipe cleaner.Use this to find the missing heights.

b. The length of the pipe cleaner is twice the sum of the base and height, sothe length is 20 cm.

d. In this case, we know that half the perimeter is 10. Height is 10 minus thebase.

e. H B= -10


• What are B and H called? Vari-

ables.

• If you have a pipe cleaner of

length 30 cm, how can you describethe relationship among the base,height, and length of the pipecleaner?

The base plus the height is 15.Base plus height is 15.Base + Height = 15

B H+ = 15

We are “undoing” to develop:The height is 15 minus the base.Height is 15 minus BaseHeight = 15 – Base

H B= -15

• What are the big ideas we devel-

oped in this activity? Moving fromconcrete experiences to moreabstract representations in smallincrements with numerous experi-ences, writing equations frompatterns via language, and the ideaof undoing addition to write the

relationships like H B= -15 .

• What “habits of mind” did you

encounter in this activity? Movingfrom patterns to rules and doing

and undoing. These will ultimatelylead to a third habit of mind,abstracting from computation.

That is, understanding a b c+ =

implies a c b= − .

Math Notes:

c. Be sure to develop this carefullyto make the generalization inPart (d).

d,e. It is important here to use themeasure for the perimeter as anumber. Thus, height is 10 – base

or H B= -10 . Do not bog downusing the notation “0.5P” for thelength of the pipe cleaner. We willdevelop this later. The idea is forparticipants to move from languageto variables.


Pipe CleanersActivity 1, cont.

2. The data in the table below is from a different pipe cleaner. Complete the"Height" column.

a. Explain how you found the missing heights.

b. Can you tell what the length of the pipe cleaner is? If so, how?

c. Use the length from Part (b) to develop the process column, showing howto find the height if you know the base.

d. Write a sentence to describe how to determine the height if you know thebase.

e. Write a symbolic rule for your sentence.

Base Process Height

3 7

8 2

6 4

1

5

B


Pipe CleanersReason and Communicate:

• How are the table, graph, andsymbolic rule related?

• Why does the line slope down?

• In Activity 1, we chose the base asthe input variable. Did we have to?Explain your reasoning. Theassignment of dependence isarbitrary in this case. We couldeasily have labeled the axes theother way.

• What are reasonable values forthe base? Lengths of the base arerestricted to lengths greater than 0and less than one-half of theperimeter.

• What do the x- and y-interceptsmean on this graph? The pipecleaner is folded in half with either a0 base or 0 height, so these pointsdo not represent rectangles.

Math Notes:

In this activity, we are beginning tobuild intuition for multiple represen-tations.

When participants enter the data tobuild a scatter plot, you can use listoperations and your rule to build thesecond list. This is another way tocheck your rule and to build intuitionfor the concept of variables.

Answers:

1a. Answers may include: The line segment goes down from left to right.

b. (base, height)

c. In a rectangle with a perimeter of 30 cm and a base of length 2 cm, theheight is 13 cm long. Using a vertical line at 2, read 13 on the y-axis.

d. In a rectangle with a perimeter of 30 cm and a height of length 4.5 cm,the base is 10.5 cm long. Using a horizontal line at 4.5, read 10.5 on thex-axis.

e. For a rectangle with a perimeter of 30 cm, the height is 15 minus the

base: H B= −15 .

2a. In a rectangle with a perimeter of 30 cm and a base of length 1.5 cm,the height is 13.5 cm long. Enter 15 - x into the calculator and scrolldown the table until 1.5 appears in the x-column Read the correspondingoutput calue of 13.5 in the y-column.

b. In a rectangle with a perimeter of 30 cm and a height of length 7 cm,the base is 8 cm long. Enter 15 - x into the calculator and scroll down thetable until appears in the y-column. Read the corresponding input valueof 8 in the x-column.


Pipe CleanersActivity 2

1. Graph your data from Activity 1, Exercise 1 on the grid below.

a. Describe your graph.

b. What does the ordered pair (x, y) mean in this problem?

c. Use your graph to determine the height for a base of 2 cm.

d. Use your graph to determine the base for a height of 4.5 cm.

e. Label your graph with a rule.

f. Use the table and graph feature of your calculator to test the rule youconjectured.

2. Use your rule and the table feature of your calculator to answer thesequestions:a. For a base of length 1.5 cm, what would be the height?

b. For a height of length 7 cm, what would be the base?

3. Use the data in Activity 1, Exercise 2 to build a scatter plot on your graphingcalculator. Then show that your rule fits the data by graphing both the scatterplot and the rule.

He

igh

t

Base


Unit II: Reflect and Apply

Sample Responses:

1a. The physical model suggests what stays the same and what is chang-ing, i.e. the variable. For example, in "Pipe Cleaners" the perimeter of therectangles stays the same, but the lengths of the base and height change.Through patterns discovered while using the physical models, one can gen-erate rules to describe the patterns.

b. In Activities 1 - 4 of "Stretching Sequences," participants moved fromcolor tile figures to tables, graphs, and rules. In Activity 5, we "undo" thisprocess by moving from the graphs to the tables, rules, and figures.

c. In “Stretching Sequences” Activity 1, Exercise 1, we filled in theprocess column with 2(1), 2(2), 2(3), . . . , 2(10), 2(100). Then we usedthese computations to develop the rule for the nth term 2n.

d. "Find a rule" asks for a general description of the area of any such figure."Find an equation for the area of the 150th figure" asks for the specific in-stance of the area of the 150th figure. Note that here we are building up toparticipants recognizing the differences between "rule" and"equation". We will come back to this idea later in the institute.

Sample Responses, cont.:

3. Take the triangular shape, flip it andadd it back on itself to form arectangle of dimensions (n + 1) by n.(See snapshot.) The area of therectangle is n(n + 1) . Therefore, the

area of the triangle is n n +( )1

2.


Figure

Figure

Number1 2 3 4 n

Area

Process

Unit II: Reflect and Apply

1. Reflect on the activities in "Stretching Sequences" and "Pipe Cleaners."

a. Describe how you can use physical models and numerical sequences togeneralize rules.

b. Part of algebraic reasoning has to do with doing and undoing. Find anexample in this unit that illustrates doing and undoing.

c. Part of algebraic reasoning has to do with abstracting from computation.Find an example in this unit that illustrates abstracting from computation.

d. Consider the questions from Stretching Sequences, Activity 1:• What rule can you use to find the area of the figure?• How can you use this rule to create an equation to find the area of the

150th figure?Which of these questions asks for a general description of a situation?Which of these questions asks to solve for a specific instance of thatsituation?

2. Consider ways you might adapt these activities for your specific gradelevel. Construct a few of your own sequences for which your students cangeneralize expressions for the nth term.

3. Complete the table below and generalize a rule for the nth term.


Cross-Country Cycling

Institute Notes

Concept: Construct function rules and graphs using unit rates andstarting points, and solve related linear equations.

Overview: Participants will use the context of traveling at aconstant rate to graph the distance from points on acourse over time for cyclists. (The intuition for unitrate in miles per hour is already a relatively concretenotion for participants.) Participants will use the ideaof starting on a course and traveling at a certain rateto determine distance over time. Then they willdevelop the language and a function rule to describethe relationship. Using these function rules,participants will write equations that answer questionsabout the cycling situation. This lesson involvesdoing and undoing and the use of multiple represen-tations to make connections among tables, graphs,function rules, and equations. Participants will alsocompare proportional and non-proportional relations.

TEKS Focus: 6.4—The student is expected to draw and comparedifferent graphical representations of the same data.7.4—The student represents a relationship in numeri-cal, geometric, verbal, and symbolic form.8.4—The student makes connections among variousrepresentations of a numerical relationship.

Materials: Graphing calculator, 1" grid paper, Markers, Peel-and-stick dots

Procedure: 1. Start by having two participants walk next to eachother in a straight line at different rates. Connect thisexperience to the “moving” activities in Day 1, whereparticipants explored graphs of distance over timefor one mover. Talk briefly about moving at aconstant rate or speed and explain why accelerationand deceleration in the cycling activities can beignored because we are using average speeds.

2. Use the Reason and Communicate questions tolead participants, working in pairs, to mark thenumber line in Activity 1, Exercise 1. Then havesome pairs demonstrate how they marked their

Also

Grade 6

2C, 3, 4, 5, 10A, 11A, 12BGrade 7

2D, 4, 5 13A, 14Grade 8

2A, 2D, 3A, 5, 14A, 15Algebra I

b.1A-E, b.2A-D, b.3A-B,b.4A, c.1C, c.2A-G

Math Notes:Since each participant hadto start and stop, theywere not moving at aconstant rate becausethey were accelerating ordecelerating. The activi-ties in this section dealwith cycling over longperiods of time at constantrates. Because we aredealing with long periodsof time, it makes sensethat we can ignore theshort periods spentaccelerating and deceler-ating. Thus, we use their



number line on transparencies of Activity 1,Exercise 1.

3. Have participants complete Activity 1,Exercises 2 – 3a in pairs by filling in the table andplotting the points on the coordinate grid. After theyhave plotted the points in Exercise 3, work throughthe process columns of the tables with participants tofind the function rules (3b and 3c).

4. Have participants finish Activity 1 by writing functionrules to describe the relationships shown in thegraphs. Then discuss the activity using the Reasonand Communicate questions.

5. Have participants complete Activity 2 in pairs byexploring proportional and non-proportional relation-ships between distance from the beginning of thecourse and elapsed hours. Then discuss the activityusing the Reason and Communicate questions.

6. In new pairs, have participants complete Activity 3,Exercise 1 where one cyclist starts riding at the end ofthe course, which results in a line with a negative slope.Have some pairs share their strategy for marking thenumber line on the overhead. Discuss using theReason and Communicate questions.

7. Have participants complete Activity 3 with theirpartners. Then discuss the activity using the Reasonand Communicate questions.

8. Have participants complete Activity 4 by writingfunction rules to describe the relationships shown inthe graphs. Then discuss the activity using theReason and Communicate questions.

Extensions: Use graphing calculators to find the points of intersec-tion in Activity 3.

Graph the equations parametrically using a graphingcalculator. For example, recall the situation in Activity 1where Rebecca and Daryl are training on a 40-milecourse, Rebecca averages about 10 mph, and Darylaverages about 8 mph. Graphing parametricallycreates a "picture" of their bike ride.

To use parametrics to grapha “picture” of their bike rideon the course use thefollowing equations:1st Cyclist: X=1, Y=10T2nd Cyclist: X=2, Y=8T

average speeds in theactivities.



Assessment: Complete the following journal entry:Today I learned ________________________ aboutconnecting tables, graphs, function rules, and equations.

Notes:


Cross-Country CyclingReason and Communicate:

Possible guiding questions formarking the number line:• Where do Rebecca and Daryl startthe course? At distance 0.• If Rebecca averages about 10mph, about how far will she go inone hour? 10 miles• After one hour, how far willRebecca be from the beginning ofthe course? 10 miles• After two hours, how far willRebecca be from the beginning ofthe course? 20 miles• If Daryl averages about 8 mph,about how far will he go in one hour?8 miles• After one hour, how far will Darylbe from the beginning of the course?8 miles• After two hours, how far will Darylbe from the beginning of the course?16 miles• How could you mark the numberline to show Rebecca’s and Daryl’sdistances from the beginning of thecourse each hour?• What are some different marksused to show location?• What are some different ways toshow distance traveled on thenumber line?• How could you use color todifferentiate between Rebecca’sdistance traveled and Daryl’sdistance traveled?• What strategies did you use tomark Rebecca’s and Daryl’s distancefrom the beginning of the course?

Alternate Answers:

40

30

20

10

0R D

1

2

3

4

1

2

3

4

5

40

30

20

10

0R D

H=0, D=0H=0, R=0

H=1, R=10H=1, D=8

H=2, D=16

H=3, D=24

H=4, D=32

H=5, D=40

H=2, R=20

H=3, R=30

H=4, R=40


Rebecca and Daryl begin cross-country cycling as a hobby and train on a 40-milecourse. At first, Rebecca averages about 10 mph, and Daryl averages about8 mph.

1. Mark the number line to show Rebecca’s and Daryl’s distances from thebeginning of the course after each hour.

40

30

20

10

0R D

Cross-Country CyclingActivity 1



Answers:

a. Rebecca finishes the course in 4 hours.

b. Daryl finishes the course in 5 hours.

c. Rebecca’s distance from the beginning of the course is 10 times thenumber of elapsed hours, or Rebecca’s distance = 10 • elapsed hours:R H= 10 .

d. Daryl’s distance from the beginning of the course is 8 times the number of

elapsed hours, or Daryl’s distance = 8 • elapsed hours: D H= 8 .

Math Notes:

2a-b. Have participants answer thesequestions from the number line andtable. Do not use equations; every-thing in Exercises 1 and 2 should bedone numerically.

Mile Marker Mile Marker


Cross-Country CyclingActivity 1, cont.

2. Complete the tables to show Rebecca’s and Daryl’s distances from thebeginning of the course after each hour.

a. How long does it take Rebecca to finish the 40-mile course?

b. How long does it take Daryl to finish the 40-mile course?

c. Use the process column in the table to write a sentence and a function ruledescribing the relationship between Rebecca’s distance from the beginningof the course and elapsed time.

d. Use the process column in the table to write a sentence and a function ruledescribing the relationship between Daryl’s distance from the beginning ofthe course and elapsed time.

REBECCA

INPUT PROCESS OUTPUT

Hours Miles

0

1

2

3

4

5

H R

DARYL


Hours Miles

0

1

2

3

4

5

H D




Answers:

a. At 3 hours, Rebecca has traveled 30 miles.

b. How long does it take Rebecca to finish the course?

c. At 2.5 hours, Daryl has traveled 20 miles.

d. How long does it take Daryl to finish the course?

Math Notes:

The number line in Exercise 1 isnow the y-axis. We have “spread”the data over time. The one-dimensional graph is a graph ofdistance traveled. The graph ofdistance traveled in relation toelapsed time is not a picture of thesituation. Compare this to theparametric graphs in the extensions.

Note that these are proportionalrelations.

b. This equation answers thequestion “How long does it takeRebecca to finish the 40 milecourse?” Discuss different ways tofind the solution to the equation,including guess-and-check, findingthe value in the table, looking for theinput coordinate on the graph, andundoing using division. Then, haveparticipants write anequation in one variable to answerthe question "How long does it takeDaryl to finish the course?"



3. Label the axes, plot the points from both tables in Exercise 2, and label eachpoint with an ordered pair.

a. What does the ordered pair (3, 30) mean on this graph?

b. Write a question for which the following equation is an answer: 40 10= H .

c. What does the ordered pair (2.5, 20) mean on this graph?

d. Write a question for which the following equation is an answer: 40 8= H .

40

30

20

10

0

1 2 3 4 5



• How did you go from the graph tothe function rule in Exercise 4?Have participants share strategies.A simple strategy is to find thedistance at one hour. This gives youthe unit rate. Then since theselinear relations contain the origin,they are proportional relations, sothe function rule is y = unit ratetimes x.

• What do the ordered pairs (1, 15)and (1, 9) mean in the graph inExercise 4? The ordered pairsrepresent the locations of cyclist 1and cyclist 2, respectively, after onehour. These points are importantbecause we can use the coordinatesof the point (1, y) to create the unit

rate,y

1 , from the graph of a propor-

tional relation.

• What effect does the coefficienthave on the steepness of the line?The larger the coefficient, H, thesteeper the line.

Answers:

4. Cyclist 1:M H= 15 ; Cyclist 2: M H= 10

Math Notes:

This part of the activity is an example of doing and undoing. First theparticipants moved from a table to the rule to a graph. Now we go from thegraph to a rule.

You want participants to notice that the larger the coefficient of H, the steeperthe line. This foreshadows the concept of slope in algebra that the higher theslope, the steeper the line. The unit rate in these problems is the slope, m, of

the line y mx b= + .



4. The graph below shows the location of two cyclists on a fifty-mile trainingcourse. Write a function rule describing the relationship between distancefrom the beginning of the course and elapsed time for each cyclist. Thenuse your graphing calculator to check your rules by graphing them together.

1 2 3 4 5

30

20

10

0

40

50

Cyclist 1

Cyclist 2

Hours

Mile

s



• What were your strategies formarking the number line?

• Did you mark it differently this timeand why?

40

30

20

10

0R D

1

2

3

4

1

2

3

4

Alternate Answer:

1. Suppose Rebecca allows Daryl top start at the six mile mark. Rebeccaaverages 10mph and Daryl averages 8mph. Mark the number line to showwhere they are at each hour.



1. Suppose Rebecca allows Daryl top start at the six mile mark. Rebecca aver-ages 10mph and Daryl averages 8mph. Mark the number line to show wherethey are at each hour.

40

30

20

10

0R D



Answers:

a. Rebecca completes the course in 4 hours.

b. Daryl completes the course in 41

4 hours.

c. Rebecca’s distance from the beginning of the course is 10 times thenumber of elapsed hours, or Rebecca’s distance = 10 • elapsed hours:R H= 10 .

d. Daryl’s distance from the beginning of the course is 6 miles (for the headstart) plus 8 times the number of elapsed hours, or

Daryl’s distance = 6 + 8 • elapsed hours: D H= +6 8 .

Math Notes:

a. Note that this is a proportionalrelation.b. Note that this is not a proportionalrelation.




2. Complete the tables to show where they are at each hour.

a. How long does it take Rebecca to complete the 40-mile course?

b. How long does it take Daryl to complete the 40-mile course?

c. Use the process column in the “Rebecca” table to write a sentence and afunction rule for Rebecca’s distance from the beginning of the course overtime.

d. Use the process column in the “Daryl” table to write a sentence and afunction rule for Daryl’s distance from the beginning of the course overtime.

REBECCA


Hours Miles

0

1

2

3

4

5

H R

DARYL


Hours Miles

0

1

2

3

4

5

H D




Answers:

a. How long does it take Daryl to finish the 40-mile course?

b. After 3 hours, Rebecca and Daryl are both 30 miles from the beginning ofthe course. The point of intersection on the graph indicates that Daryl andRebecca are at the same place on the course at the same time.

c. 10 6 8H H= + . This equation describes the x-coordinate of the point wherethe two lines intersect.

Math Notes:

a. Discuss different ways to find thesolution to the equation, includingguess-and-check, find the value in thetable, and look for the input coordinateon the graph.



3. Plot the points from both tables and label each point with an ordered pair.

a. Write a question for which the following equation is an answer: 40 6 8= + H .

b. When are Rebecca and Daryl at the same place on the course? How canyou tell from the graph?

c. Write an equation to determine how long it takes for Rebecca and Daryl tobe at the same place on the course. What does this equation represent inthe graph?

40

30

20

10

0

1 2 3 4 5



Math Note:

Each of the cyclists are traveling at the same rate, and therefore, eachfunction rule had the same coefficient of H and each line has the sameslope and the lines are parallel.

Answers:

Cyclist 1: M H= +20 10

Cyclist 2: M H= +10 10

Cyclist 3: M H=10


• What are some similarities anddifferences between proportionalrelations and non-proportional linearrelations?

Similarities: In both proportionaland non-proportional linear relations,the difference between output valuesover one input unit is constant. Thiscommon difference is the unit rate orrate of change. They both have aconstant rate of change. For ex-ample in both Activities 1 and 2, theunit rates are 10 mph and 8 mph.Their graphs are all lines.

Differences: A proportionalrelation contains the origin, (0, 0); isof the form y mx= , and has a

constant ratio, y

x, for every ordered

pair, where

y

xm= . A non-propor-

tional linear relation does not containthe origin and is of the form

y mx b= + . In a non-proportional

linear relation, the ratio y

x is not

constant.

• Create a situation and question for

which the equation 9 40x = willanswer the question. Chris averages9 mph on a 40-mile course. Howlong will it take Chris to complete thecourse?

• How might students view solvingone variable equations differently ifthey are introduced in this way firstas opposed to solving algebraicallyby rote memorized rules?

• What strategies did you use to gofrom the graph to the function rule?

• How did you find the unit rate?

• How are the graphs similar? Theyhave the same rate.

• How are the graphs different?They start at different places.

• What happens when the rates arethe same? The cyclists never meeton the course. The lines are parallel.



4. The graph below shows three cyclists on a 50-mile course. Write a functionrule for each cyclist. Use your graphing calculator to confirm your rules bygraphing all three in the same calculator window.

1 2 3 4 5

30

20

10

0

40

50

Hours

Cyc

list 1

Cyc

list 2

Cyc

list 3M

iles



• How is Activity 3 different fromActivities 1 and 2? Daryl starts at theend of the course instead of at thebeginning.

• Did Daryl starting at the end of thecourse change the way you markedthe number line?

• How does Daryl starting at the endof the course affect the function rule?The rule is now: Daryl’s milestraveled equals his starting pointminus his rate times the elapsedhours.

Answers:

a. Rebecca completes the course in between 3 and 4 hours.

b. Daryl completes the course in between 4 and 5 hours.

c. Rebecca’s distance from the beginning of the course is 11 times thenumber of elapsed hours, or Rebecca’s distance = 11 • elapsed hours:

R H=11 .

d. Daryl’s distance from the end of the course is 40 miles minus 9 times thenumber of elapsed hours, or Daryl’s distance = 40 – 9 • elapsed hours:

D H= -40 9 .

Math Notes:

d. Have participants use the process column in the “Daryl” table to answerthis question.

Mile MarkerMile Marker



1. Suppose Rebecca starts cycling at the beginning of the course, and Darylstarts at the end of the course and moves toward the beginning. Rebecca isnow averaging 11 mph, and Daryl is averaging 9 mph. Mark the number lineand complete the table to show where each is on the course after each hour.

a. How long does it take Rebecca to complete the course?

b. How long does it take Daryl to complete the course?

c. Write a sentence and a function rule describing Rebecca’s distancefrom the beginning of the course in relation to elapsed time.

d. Use the pattern in the table to write a sentence and a function ruledescribing Daryl’s distance from the beginning of the course in relationto elapsed time.

40

30

20

10

0R D

REBECCA


Hours Miles

0

1

2

3

4

5

H R

DARYL


Hours Miles

0

1

2

3

4

5

H D




• How does Daryl starting at the endof the course affect the graph? Thegraph is going down from left to rightsince his distance from the begin-ning of the course is decreasing.

• How is a function rule, like R H=11

or D H= -40 9 , different from an

equation, like 11 40 9H H= - ? Afunction rule is a description of therelationship for every possible input.An equation describes the relation-ship for a specific input or outputvalue.

Answers:

a. After 2 hours, Rebecca and Daryl are both 22 miles from the beginningof the course. The point of intersection on the graph indicates that Daryland Rebecca are at the same place on the course at the same time.

b. 11 40 9H H= - . This equation describes the H coordinate of the pointwhere the two lines intersect where the two lines intersect, which is thenumber of hours when they meet on the course.


2. Plot the points from both tables and label each point with an ordered pair.


a. When are Rebecca and Daryl at the same place on the course? Howcan you tell from the graph?

b. Write an equation to determine how long it takes for Rebecca and Darylto be at the same place on the course. What does this equation repre-sent in the graph?

40

30

20

10

0

1 2 3 4 5

Mile

s

Hours



Answers:

a. Line 1: M H= +5 15 ; Line 2: M H= -45 5 . .

b. (2, 35)

c. Answers will vary.

d. 45 5 5 15- = +H H .


• What strategies did you use to findthe unit rate or speed?

• Has this activity changed the wayyou think of graphing and solvinglinear equations? How?

• How did you determine (or make

up) a situation to go with the graph?

• How would students develop a

situation?

• How is this different from previous

activities? Similar?



1. Consider the two function rules: y x y x= - = +45 5 5 15, . Fill in the tables with

some appropriate values and build the graph.

a. Make up a situation these functions could represent.

b. What is the point of intersection?

c. What does the point of intersection mean in your context?

d. What equation can you write to describe the point of intersection?

1 2 3 4 5

30

20

10

0

40

50

x

x

y x= −45 5

y x= +5 15



• What strategies did you use to findthe unit rate or speed?

Answers:

a. Line 1: M H= +15 10 ; Line 2: M H= -30 5

b. (1, 25)

c. Answers will vary.

d. 15 10 30 5+ = -H H .

Math Notes:

Discuss the process of abstracting from computation. In this lesson,participants generate and develop function rules and graphs from concreteunit rate problems. Through these experiences with concrete situations,

the abstract y mx= and y mx b= + , where m is the slope of the line and b isthe y-intercept, become more concrete representations for participants. Inother words, computations in the process columns are connected to rulesand graphs. After these experiences, the abstract variable representationsbecome more a part of participant’s cognitive structure.



2. Consider the two lines on the grid below. Make up a situation the graph couldrepresent.

a. Write a function rule describing the relationship between your variables.

Line 1: _______________ Line 2: ________________

b. What is the point of intersection?

c. What does the point of intersection mean in your context?

d. What equation can you write to describe the point of intersection?

Mile

s

Hours

1 2 3 4 5

30

20

10

0

40

50

Line 1

Line 2


Speed Trap

Institute Notes

Concept: Collect distance data using a motion detector and find atrend line for the data.

Overview: Participants will use data collection devices with motiondetectors to collect distance data for small battery-oper-ated vehicles that travel at a constant rate. They willobtain scatter plots of the vehicles’ distance from themotion detector over elapsed time while traveling awayand toward the motion detector and sitting still. Usingthis data, participants will find trend lines that describethe vehicles’ distance from the motion detector in rela-tion to time.

TEKS Focus: 6.13—The student uses logical reasoning to makeconjectures and verify conclusions.7.15—The student uses logical reasoning to makeconjectures and verify conclusions.8.16—The student uses logical reasoning to makeconjectures and verify conclusions.

Materials: Aluminum foil, Masking tape, Small battery-operatedvehicles (1 per group), Data collection devices with mo-tion detectors

Procedure: 1. Describe this scenario: A toy car starts at 1.5 feetfrom the motion detector and travels away at aconstant rate of 2 feet per second. Elicit the valuesto fill in a table as follows:

Using a graphing calculator, graph a scatter plot ofthe above data. Have participants write a sentenceand function rule to describe the relationship be-tween distance from the motion detector and time.

Also

Grade 6

2C, 5, 7, 10D, 11A, 11C,12AGrade 7

2D, 4, 5, 7, 9, 14, 15Grade 8

2A, 3B, 5A, 7, 12B, 14C,15AAlgebra I


Time Distance

0 1.5

1 3.5

2 5.5

10 21.5

T D =


Speed TrapThe distance from the motion detector is 1.5 plus 2times the number of seconds.Distance = 1.5 + 2 (number of seconds).

D T= +15 2. . Now graph the function rule over thescatter plot on the graphing calculator to show thatthe function rule fits the data. A line of best fit!

2. Do a demonstration of the data collection in front ofthe whole group. Tape a 3” x 3” sheet of aluminumfoil on the back of the vehicle. Set up the datacollection device with a motion detector to collectdistance data in feet for 3 seconds. Start the vehiclein front of the motion detector, moving away. Begincollecting data when the vehicle is about 1.5 feetfrom the motion detector. You should get a line ofdata that goes up from left to right. Repeat if neces-sary.

3. Have participants complete Activity 1 in groups of twoor more by collecting distance data as the vehiclesmove away from the motion detectors. Have partici-pants discuss their results and share their strategies,especially for Exercise 5, the police car scenario.

4. Have participants complete Activities 2 and 3 bycollecting distance data as the vehicles move towardand stay still in front of the motion detectors. Haveparticipants discuss their results and share theirstrategies.

5. Compile the trend lines from Activity 1 on a transpar-ency and use the transparency to answer questions2 - 4. Line up all the different vehicles in front of thewhole group and race them. Discuss the results.

Extensions: Do the activity collecting distance data in meters insteadof feet. Compare rates in meters per second versusfeet per second.

Instead of using the motorized vehicles, have a personwalk away and toward the motion detector at a constantrate. Find trend lines to fit the scatterplot of the walk.

Assessment: Transfer the data for a scatter plot of an unknownvehicle’s data that you have prepared in advance toparticipants’ calculators. Have them find a trend line.Ask them to discuss the distance of the vehicle from themotion detector and how it relates to the trend line.

Notes:

The aluminum foil is usedto create a bigger target.You can omit the foil if thevehicle is large enough.Do not collect data untilthe vehicle is up to speed.We want a linear scatterplot, so the rate must beconstant. If participantscollect data while thevehicle is accelerating ordecelerating, the graph willnot be linear.

Motion detectors have arange of detection. Do nottry to collect data out ofthat range.

The function rule inProcedure 1 is a line ofbest fit because it fits thedata exactly. The functionrules that participants findfor their cars' scatter plotsfrom the motion detectorswill be trend lines. Refrainfrom calling a trend line,found by estimation, a"line of best fit.".


Speed TrapReason and Communicate:

• What is the difference between thegraph that the motion detectorproduced and the graph of yourfunction rule? The graph that themotion detector produced is a scatterplot of data, a finite set of orderedpairs. The function rule represents aline, an infinite set of ordered pairs.

• Did the vehicles indeed travel at aconstant rate? How can you tell? Ifthe scatter plots are linear, then thecar was traveling at a constant rateover that time interval. If the scatterplot curves, the car was not travelingat a constant rate over that timeinterval.

• How did you find the function rule,the trend line, in Exercise 3? Sharestrategies. One way is by finding thestarting point by tracing to where timeis 0, finding the unit rate by tracing tothe point where time is 1, and writingthe ruley = starting point + unit rate • x.

• How do you think the graph andfunction rule will change if the car istraveling toward the motion detectoras it will in Activity 2? Encourageparticipants to make educatedguesses without giving them theanswer.

Answers: (A sample follows.)

1.

2. The distance from the motion detector is 1.7 plus 0.6 times the elapsed

time, D T= +1 7 0 6. . .

3.

Math Notes:

Note that instead of D for the distance or output variable, calculatorsrequire Y. Also, X is the required input variable instead of T for the num-ber of elapsed seconds.


Speed TrapActivity 1

Set up a data collection device and a motion detector to collect distance data for3 seconds. Use a small battery-operated vehicle that travels at a constant rate.Tape a sheet of aluminum foil to the back of the vehicle to make it a better target.Put the vehicle right against the motion detector traveling away from the motiondetector. Begin collecting data as soon as the vehicle is up to speed but notbefore it is about 1.5 feet away from the motion detector.

1. Sketch your graphing calculator’s graph of the vehicle’s distance whiletraveling away from the motion detector in relation to elapsed time. Use thetrace feature on your calculator to fill in the table.

2. Write a sentence and a function rule to describe the vehicle’s distance fromthe motion detector in relation to elapsed time.

3. Use your graphing calculator to graph your rule over the scatter plot. Adjustthe function rule if necessary.

TimeDistance from

Motion Detector

0

1

2

3

T D =

1.5 feetBegin collecting data here


Answers:

b. One would want the police car to end up at the same distance from themotion detector when time is 3 seconds. This distance for our example isabout 3.5 feet, and 3.5 feet in 3 seconds gives a unit rate of about 1.2 feetper second.

d. The police car is starting at the origin, so the police car’s distance from

the motion detector is zero plus 1.2 feet per second, or D T T= + =0 1 2 1 2. . .


• How far was your car from the

origin at 3 second? Trace on thescatter plot to find the distance.

• How far does the police car need

to go in three seconds to catch yourcar? The distance you found bytracing on your car's scatter plot to 3seconds.

• How can you find the average rate

the police car must travel to catchyour car in three seconds? If thedistance found above is 6 feet, thenthe police car needs to travel 6 feetin 3 seconds, so the rate is 6 feetper 3 seconds or 2 feet per second.


Speed TrapActivity 1, cont.

4. Imagine that a police car is at the origin at time 0.

a. Sketch the scenario below.

b. What average rate must the police car travel in order to catch your vehiclein three seconds?

c. Describe how you found your answer.

d. Write a sentence and a function rule describing the police car’s distancefrom the motion detector in relation to elapsed time.



• How did you find the function rule,the trend line, in Exercise 3? Haveparticipants share strategies. Oneway is by finding the starting pointby tracing to where time is 0, findingthe unit rate by tracing to the pointwhere time is 1, and writing theequation:y = starting point – unit rate • x.

• Look at the function rules youcreated in Activities 1 and 2. Whatis similar? What is different? Sinceparticipants used the same vehicles,the rates should be the same,except the rate for the car approach-ing the motion detector is negative.The starting points at t=0 should bedifferent because the vehicle inActivity 1 started close to the motiondetector, and the vehicle in Activity 2started far away from the motiondetector.

Answers: (A sample follows.)

1.

2. The distance from the motion detector is 4.8 minus (because the car isgoing toward the motion detector) 0.55 feet per second times the

elapsed time, or distance = 4.8 – 0.55 • elapsed time: D T= -4 8 0 55. . .

3.



Use the same set up as in Activity 1 except this time put the vehicle about 15 feetfrom the motion detector traveling toward the motion detector. Also tape thepiece of aluminum foil to the front of the car. Begin collecting data as soon as thevehicle is up to speed.

1. Sketch your graphing calculator’s graph of the vehicle’s distance whiletraveling toward the motion detector in relation to elapsed time. Use the tracefeature on your calculator to fill in the table.



TimeDistance from

Motion Detector

0

1

2

3

T D =



• At what rate was the vehicletraveling? Zero feet per second.

• How did you find the equation, thetrend line, in Exercise 3? Sharestrategies. One way is by finding thestarting point by tracing to any pointand writing the equation:y = starting point + unit rate • x. Asthe unit rate in this case is zero, theny = starting point + 0 • x or y =starting point.

• How do you write the function rulefor a horizontal line that contains thepoint (3,5)? (5,3)? (100,500)? y=5;y=3; y=500

• How do you write the function rulefor a horizontal line that contains thepoint (a,b)? y=b

• To get a horizontal line, the vehiclestayed still. How could one obtainthe graph of a vertical line using yourvehicle and motion detector? Onecannot obtain the graph because avertical line would mean that thevehicle was in more than one placeat a time, which is impossible.

Answers:

2. The distance from the motion detector is 5 feet, or distance = 5: D = 5.



Use the same set up as in Activity 1 except this time put the vehicle about 5 feetfrom the motion detector, with the piece of aluminum foil toward the motion de-tector. Collect the distance data for three seconds as the car is staying still.

1. Sketch your graphing calculator's graph of the vehicle standing still in front ofthe motion detector. Use the trace on your calculator to fill in the table.



TimeDistance from

Motion Detector

0

1

2

3

T D =


Answers:

2. The vehicle with the highest rate will win the race. Look for the largestcoefficient of T, the rate, in the trend line.

3. The vehicle with the lowest rate will probably lose the race. Look for thesmallest coefficient of T, the rate, in the trend line.

4. Vehicles with the same rate will tie. Look for trend lines with the samecoefficients of T, or rates.

6. Answers may include: The vehicles probably have different accelerationtimes. The batteries might be lower than when the group first obtained thetrend line. The trend line may have been inaccurate.


• How does this activity compare to"Movin' on Down the Line" and"Moving with Technology"? In"Movin' on Down the Line" and"Moving with Technology", wegraphed distance over time andanswered questions. In "SpeedTrap" we found trend lines. All of theactivities involved graphing distancein relation to elapsed time.

• How does this activity comparewith "Cross-Country Cycling". In"Cross-Country Cycling", we weregiven the rate and starting point andused them to find function rules andgraphs. In "Speed Trap", we startedwith the scatter plot of a vehicle'sdistance over time and we created atable and found a trend line (afunction rule). Both of the activitiesinvolved graphing distance inrelation to elapsed time.

Speed Trap



1. Obtain all the other groups' trend lines from Activity 1. Write them down,labeling each trend line with the vehicle it represents.

Let's race them!

Using the trend lines above, answer the following questions before the race.2. Which vehicle do you think might win the race and why?

3. Which vehicle do you think might lose the race and why?

4. Which vehicles do you think might tie and why?

After the race:

5. How do your predictions about the race compare to what happened?

6. What factors might have influenced the outcome?


Unit III: Reflect and Apply

Sample Responses:

1a. Answers will vary.

1b. In “Cross-Country Cycling,” we made number lines, tables, graphs, andfunction rules from a given problem situation. However, in Activity 5, wewrote a problem situation and function rule from a graph.

1c. From a graph or table, students can find the change in y when x changesby 1. This is the unit rate.

1d. Although most of us have learned to use the word equation to describeanything with an equal sign in it, sometimes it is important to distinguishbetween a general description of a situation and a specific instance of that

situation. For example, the function rule D H= +6 8 describes Daryl's dis-

tance from the starting point for any time H. The equation, 40 6 8= + H ,describes the point on the graph (4.25, 40) so Daryl finished the 4 milecourse in 4.25 hours.In the institute we use "function rule" as a description of the relationship forevery possible input and we use "equation" to describe the relationship forspecific input or output values.

Sample Responses, cont.:

3. D T D T= + = +8 5 10 8; .

Math Note:

3. In this case, two points are suffi-cient to generalize a function rule be-cause we are assuming that the func-tions are linear. Two points are notusually sufficient to generalize func-tion rules that are not linear.

8T


Unit III: Reflect and Apply1. Reflect on the activities in "Cross-Country Cycling" and "Speed Trap".

a. Describe how you can lead students from recursive thinking to writingfunction rules.

b. Part of algebraic reasoning has to do with doing and undoing. Find anexample in this unit that illustrates doing and undoing. Discuss howstudents can move from graphs to function rules and back.

c. Discuss how students can develop the abstract notion of unit rate in aconcrete way.

d. In Cross-Country Cycling, find an example where you are asked to find afunction rule. Write it here.

In Cross-Country Cycling, find an example where you are asked to writean equation. Write it here.

What is the difference between the above two examples, ie. between arule and an equation?

2. Consider ways you might adapt these activities for your specific gradelevel.

3. Generalize a function rule for each table.

Time Distance

0 8

3 23

T D=

Time Distance

1 18

3 34

T D=


Making Connections

Institute Notes

Concept: Formalize the input/output model for function andconnect multiple representations: tables, functionrules, equations, and graphs.

Overview: Participants will investigate the input/output model forbuilding function tables. Then they will connect tables,graphs, function rules, and equations in one variable.Finally, they will work backwards to determine functionrules for given data sets or graphs.

TEKS Focus: 6.5—The student uses letters to represent an unknownin an equation.7.4—The student represents a relationship in numeri-cal, geometric, verbal, and symbolic form.8.4—The student is expected to generate a differentrepresentation given one representation of data suchas a table, graph, equation, or verbal description.

Materials: 1" grid paper, Coffee stirrers or flat spaghetti, Graphingcalculator, Markers, Peel-and-stick dots

Procedure: 1. Many students and teachers are familiar with func-tion machines from elementary and middle schooltextbooks. Discuss the use of a function machineand input/output variables using the illustration atthe top of Activity 1. Review the notion with asimple example: input 3, output 8; input 4, output 9;etc. The rule is add 5. Then try a more compli-cated example, where there is a two-step rule: input1, output 3; input 4, output 9; etc. The rule is multi-ply by 2 and add 1.

2. Have participants complete Activity 1 in groups byusing language to move from tables to functionrules. Have half the groups do Exercise 1 and theother half do Exercise 2. Discuss using the Reasonand Communicate questions.

3. Have participants plot their data (Exercise 1 or 2)from Activity 1 on 1” graph paper using markers orpeel-and-stick dots. The graphs should be dis-played and discussed as a whole group.

Also:

Grade 6

2C, 4, 7A, 11A, 12A, 13AGrade 7

2B, 4B, 5, 5B, 7A, 13A,13B, 14A, 15Grade 8

2B, 4, 5, 5A, 7D, 14A,14C, 15A, 16Algebra I


Math Notes:

Emphasize that mental mathis to be used wheneverpossible.

Emphasize the use of orderedpairs. Note the labeling of theinput and output axes. Also,discuss the participants’ use ofdifferent scaling.


Making Connections

4. Then have participants complete Activity 2, transfer-ring their group’s graph to their activity sheet.Discuss Activity 2 using the Reason and Communi-cate questions.

5. Have participants complete Activity 3 in smallgroups by using tables to develop rules. Note thatparticipants should work independently at first; then,they should discuss their strategies with their group.

6. Discuss Activity 3 using the Reason and Communi-cate questions.

7. Use an overhead calculator and challenge partici-pants to guess a linear function using the askfeature.

8. Have participants complete Activity 4 by movingfrom graphs back to tables, function rules, andequations. For Exercise 3, participants shouldmodel the parachute drop first with the motiondetector.

Extensions: Have participants use a graphing calculator to gener-ate tables and have a partner guess the rule.

Assessment: Ask participants to create 3 different real-world situa-tions that can be modeled with a rule.

Notes:

In Activity 3, participantswork backwards todetermine the processand, hence, the rules usedto develop a table.


Making ConnectionsReason and Communicate:

• The variable T is called the outputvariable. Why is it called that? T isthe output variable because whenyou input some number of students,the total cost, T, is figured.

• How did you use mental math tocomplete the table? Recursively

Math Notes:

Some participants will use recursionto complete the table once they seethe pattern in the output column.This involves adding 150 to theprevious amount.

b. We include the input value of 0,though some may argue that it is notvery "reasonable." Sometimes thereare important values that helpdescribe the behavior of a function(the y-intercept) that might not bevery reasonable input values for thesituation the function describes.

c. Note the participants’ use ofdifferent letters for variables. Also,note the use of coefficient for theunit rate. Point out that the numberof students is the input variable.

d. Some may use an undoing of theprocess strategy such as

775 100 675- = , then 15 45 675( ) = .The numerical approach is animportant first step to equationsolving.

Discuss the importance of usinglanguage to generate rules andequations. It is important because itbridges concrete to abstract.

Answers:

a. No, you cannot have an input value of 70 because the bus only holds 60people.

b. {0,1, 2, 3, 4, . . . 60}

c. The total cost (T) is $100 for the bus plus fifteen dollars times the number

of students (N), or total cost = $100 + $15 • number of students: T N= +100 15

d. Guess and check yields: 100 + 15(45) = 775, so 45 students can go.

e. Enter 100 + 15x to graph y = 100 + 15x and create a table for thisfunction. Scroll down the table until 775 appears in the y-column. Readthe corresponding input value in the x-column, 45.

T


1. The school choir is planning a trip to Ocean Universe. The bus costs $100 torent and seats 60. Tickets to Ocean Universe cost $15 each. A functionmachine can be used to compute the total cost.

Output

$700

Input

40 students

Process

$100 + $15 40

Use mental math to complete the table below.

a. Can you have an input value of 70? Explain your reasoning.b. Make a list of input values that make sense in this situation.c. Write a sentence and a function rule describing the total cost of the trip in

terms of the number of students going. Use a calculator table to confirmyour rule.

d. Write an equation for the following problem: The choir only raised $775for the trip. How many people can go to Ocean Universe? Solve byguess and check.

e. Use a graphing calculator table created from your rule to find the solutionfor Part d.

Input Process Output

Number ofStudents

Total Cost

10

20

30

40

50

N

Making ConnectionsActivity 1



• How did you use mental math tocomplete the tables?

• In your graphing calculator table,which variable is the input variableand which is the output variable?"X" is the input variable, and "Y" isthe output variable.

• Create a situation in which onequantity is dependent upon anotherand identify the input and outputvariables.

• What are reasonable input andoutput values for your situation?Note situations that have discretedata sets for input and outputvalues. If your created situation islike the bus problem in the previousexercise, you do not assign part of aperson to a bus, nor do you orderhalf of a bus.

Answers:

a. The number of hours online is the input variable and monthly cost is theoutput variable.

b. For a 30-day month, the input values would be numbers of hours from 0to 30 hours.

c. Monthly cost (C) is ten dollars plus $ 0.25 times the number of hours (H),or monthly cost = $10 + $0.25 • number of hours: C=10 + 0.25H.

d. 10 + 0.25H = 15.75. Then 10 + 0.25 (23) = 15.75, so the bill includes 23hours of service.

e. Enter 100 + 0.25x to graph y = 100 + 15x and create a table for thisequation. Scroll down the table until 15.75 appears in the y-column.Read the corresponding input value in the x-column, 23.

C


2. U. S. ONLINE charges $10 per month plus $0.25 per hourfor Internet service. Your conscience says, “No more thanone hour per day!” and you always listen to your conscience.Build a table to show possible costs.

a. Identify the input and output variables and label them in your table.

b. What input values make sense in this problem?

c. Write a sentence and a function rule describing the monthly cost in terms ofthe hours used. Use a calculator table to confirm your rule.

d. Write an equation for the following problem: Your internet service bill is$15.75. For how many hours of use were you charged?

e. Use a graphing calculator table created from your rule to find the solutionfor Part d.

Making ConnectionsActivity 1, cont.


Number ofHours

Monthy Cost


Making Connections

Answers:

a. Answers will vary but may include that the points appear to be on thesame line and the line starts at $100.

b. The cost for 40 students is $700. Using a vertical line at 40 students,read $700 on the output axis.

c. Sixty students can go for $1,000. Using a horizontal line at $1000, read60 students on the input axis.

d. 1000 100 15= + N

e. C N= +100 15 . See Activity 1, Exercise 1c.

Math Notes:

a. Note this is a discrete data set. Discuss strategies for determining thescale.

d. Note the use of coefficient, how it describes the change in y per unitchange in x. Compare it to the graph.


• What strategies did you use tointerpret graphs using horizontal andvertical lines? See the answers forExercise 1b - c.

• How can one determine the rate ofchange from the graph? Look atthe change in output per input unit.




b. Use your graph to determine the total cost for 40 students. Explain howyou found your solution.

c. Use your graph to determine the number of students who can go on the tripfor $1000. Explain how you found your solution.

d. Write an equation in one variable for the problem in Part c.

e. Label your graph with a rule relating the total cost to the number ofstudents.

Tota

l C

ost

Number of Students




• What does a point on the ouputaxis mean in these problems? Thecost to have the service but to use itzero hours.

• How can you solve an equationgraphically? Give an example. Seeanswers for parts b and c.

• If you look at the tables in Activity1 recursively, each is an example ofrepeated addition. What kind ofgraph does repeated addition yield?Linear.

Math Notes:

a. Discuss how you can read the rateof change, 0.25 per one hour, from thegraph.

Answers:

a. Answers may include linear, starts at $10, and increases 0.25 everyhour.

b. The cost for 15 hours is $13.75. Using a vertical line at 15 hours, reads$13.75 on the output axis.

c. Twenty hours of access costs $15. Using a horizontal line at $15, read20 hours on input axis.

d. 15 10 0 25= + . H

e. C H= +10 0 25. . See Activity 1, Exercise 2c.




b. Use your graph to determine the cost for 15 hours. Explain how you foundyour solution.

c. Use your graph to determine the number of hours of service for a cost of$15. Explain how you found your solution.

d. Write an equation in one variable for the problem in Part c.

e. Label your graph with a rule relating the total cost to the number of hours.

Tota

l C

ost

Number of Hours



Making Connections

Answers:

1c. The total cost (T) is $7.50 times the number of t-shirts (N), or

total cost =$7.50• number of t-shirts: T N= 7 5. . Use the table feature toconfirm the rule with a graphing calculator.


• What strategies can you use forfinding the rate of change?

• What kind of relationship isillustrated in Exercise 1? Exercise1illustrates a proportional relationship.

Math Notes:

Stress the importance of usingmental math wherever possible.

a. Note the use of recursion to findvalues until 270. Then participantsare forced to compute the unit rate,7.5 per shirt, and multiply. Theyhave to guess-and-check or divideto find the last two numbers of t-shirts.

b. Illustrate the process on a trans-parency of Activity 3, Exercise 1.

c. Discuss using different variables,

e.g. y x= 7 5. for the calculator.


1. Sam sells t-shirts on the beach. Complete the table by yourself to find thecost for different numbers of t-shirts. Then compare with your group.

a. Explain your strategy for completing the table and compare it with others’ inyour group.

b. Use the second column to show the process.

c. Write a sentence and a rule describing how the total cost is related to thenumber of t-shirts. Confirm your rule with your calculator.


Number ofT-shirts

Total Cost

10 75

20 150

30

40

50

200

270

750

2250

N T



Making Connections

Answers:

c. The cost of the phone bill (C) is $15 plus $6 times the number of

hours(H), or cost = $15 + $6 • number of hours: C H= +15 6 .

d. 75 = 15 + 6 • 10.

e. 165 = 15 + 6 • 25.


• What kind of relationship isillustrated in Exercise 2? Exercise 2is linear but not proportional.

• Explain how you can use a tableto find the solution to an equation.Give an example.

• Describe other situations wherethe data in a table represents alinear function.

Math Notes:

a. Most will determine the process bystarting recursively and use the unitrate $6 per hour to work backwards tofind the start up, 15.

b. Develop the process carefully onthe transparency of Activity 3, Exer-cise 2.


2. WLT offers a different phone plan than its competitors. Complete the tableby yourself to begin discovering their plan. Then compare with your group.

a. Explain your strategy for completing the table and compare it with others' inyour group.


c. Write a sentence and a rule describing the cost of the phone bill related tothe number of hours. Confirm with your calculator.

Write an equation for the output value of:d. 75

e. 165


Number of HoursCost of Phone

Service

1 21

2 27

3

4

5

12

75

165

H C



Making Connections

Answers:

c. The height (H) is 20 feet minus 2 feet times the number of seconds (T),

or height = 20 – 2 • number of seconds: H T= -20 2 .


• Compare this activity withExercise 1 and Exercise 2.Exercise 1 is proportional, andExercises 2 and 3 arenonproportional. In Exercises 1 and2, we went from table values tofunction rules. In Exercise 3 wewent from a verbal rule to a table toa function rule.

• Could you have written a functionrule directly from the verbal descrip-tion?

• Did the table values help you findthe function rule?


3. Suppose a toy parachute is dropped from a height of 20 feet and falls at arate of 2 feet per second. Complete the table by yourself. Then comparewith your group.

a. Explain your strategy for completing the table and compare it with others' inyour group.


c. Write a sentence and a rule describing the height in relation to the time.Confirm your rule with your calculator.


Time (seconds) Height (feet)

0 20

T H



Making Connections

Answers:

d. The amount (A) is 500 minus 50 times the number of weeks (W), or

amount = 500 - 50 • number of weeks: A W= -500 50 .

f. A = - =500 50 8 100( ) .

g. 0 500 50= - W . The amount is 0 when the number of weeks is 10.


• Describe your strategies for usingthe graph to find the rule describingthe relationship.


1. Record the data from the graph in the table. Label the input and outputvariables.

a. Make up a scenario that the graph can represent.b. Explain your strategy for completing the table and compare it with others' in

your group.c. Use the second column to show the process.d. Write a sentence and a function rule describing the amount in terms of the

number of weeks. Confirm your rule with your calculator.e. Write an equation in one variable for each input value, where the input

value is the solution.f. Find the amount at 8 weeks.g. Write and solve an equation for the problem: When is the amount 0?


W A


1 2 3 4

100

400

300

0

Number of Weeks

200

500A

mo

un

t


Making Connections

Answers:

d. The total cost (T) is $5 plus 5

2 times the number of hours (H), or

total cost = $ 5 + 5

2 • number of hours:

T H= +5

5

2.

f. T = + =5

5

212 35( ) .

g. $ .42 50 5

5

2= + H . H= 15 hours.


• How can you find the rate ofchange?

• How does the starting point (y-intercept) help you find the rule?

• Since repeated addition yields alinear model, what can you say aboutthe table for a line? It has repeatedaddition.


2. Record the data from the graph in the table by yourself. Label the input andoutput variables.

a. Make up a scenario that the graph can represent.b. Explain your strategy for completing the table and compare it with others' in

your group.c. Use the second column to show the process.d. Write a sentence and a function rule describing the total cost in relation to

time. Confirm your rule with your calculator.e. Write an equation in one variable for each output value, where the input

value is the solution.f. Find the cost for 12 hours.g. Write and solve an equation for the problem: When is the cost $42.50?


H T


2 4 6 8 10

10

20

30

0

Number of Hours

Tota

l C

ost


Making Connections


3. Use a data collection device with a motion detector and aplastic grocery sack to simulate a parachute drop.

• Attach a paper clip to the handles of the grocery sack asshown.

• Set the motion detector on the floor facing up.

• Throw the parachute in the air directly over the motion detector. Estimatethe time, T, it takes the parachute to fall from its maximum height to thefloor.

• Throw the parachute again; this time collect distance data with the motiondetector for T seconds. Start the data collection when the parachute is atits maximum height.

• Repeat if necessary.

a. What do you think the graph of the parachute's distance from the motiondetector over time will look like? Sketch your conjecture.




• How is the parachute activity likeExercises 1 and 2? In Exercises 1and 2, we read data points from agraph and put them in a table to helpfind a rule. In the parachute drop,we collected data with technology,which gave us a scatter plot ofpoints. We then read data pointsfrom the scatter plot to fill in a tableand find a rule.

• How did you find the rate ofchange? We are looking for thechange in y per unit change in x.Trace to find the difference in heightfor one unit time.

• What is the meaning of the rate ofchange? The rate of change is thevertical distance traveled by theparachute in relation to a specificinterval of time.

Sample Data:

b.

c. 7.087 feet in the air

d. 3.315 feet per second

e. y = 7.087 - 3.3x


b. Sketch the graph you obtained of the parachute's distance from the motiondetector over time.

c. Where is the parachute at time 0?

d. Estimate the speed in feet per second.

e. Use your parachute’s starting point at time zero and speed to fit the data witha rule that describes the relationship.

f. Refine your rule for a good fit, if necessary.



Grids Galore

Institute Notes

Concept: Explore situations numerically using combination

charts and solve for unknowns given two equations.

Overview: Using a typical coin combination problem, participantsexplore the situation using a numeric approach with acombination chart. Then participants will explore awages-earnings situation by using a combination chartand building a three-dimensional combination chart.Finding patterns in the chart, they will apply thesepatterns to combination charts for other situations.Then participants will combine rods of different,unknown lengths to match a given length. Using thegiven length, they will set up two equations that canthen be solved numerically using a combination chart

to find the lengths of the rods.

TEKS Focus: 6.4 – The student uses tables and symbols to

represent and describe relationships.7.4 – The student represents a relationship in numeri-cal, geometric, and symbolic form.8.5 – The student uses tables and algebraicrepresentations to solve problems.

Materials: Cuisenaire rods, Linking cubes, 1" grid paper, Mark-

ers, Dimes, Quarters

Procedure: 1. Briefly review the previous activities in the institute,emphasizing that much of what we have done hasbeen describing the relationship between twovariables. "Grids Galore" is different in that herewe will model using algebraic thinking in differentproblems.

2. Use the reason and communicate questions toguide the discussion of typical middle school coinproblems, algebraic thinking, and the money prob-lem and grid in Activity 1, Exercise 1. Have partici-pants fill in the grid, write sentences for cells, andfind patterns for the combinations of dimes andquarters. On the overhead, place stacks of coins inthe grid to represent the appropriate values.

Also:

Grade 6

2c, 3, 5, 8, 11aGrade 7

2d, 5a, 9, 13aGrade 8

2d, 9, 14aAlgebra I

b.1A-E, b.3A, 3C, b.4A,c.2G, c.3A-C, c.4A


Grids Galore3. Discuss Cameron’s summer earning possibilities in

Activity 1, Exercise 2. Mention that he wouldprobably prefer to work at the pool with his friends,but he wants to do some yard work to make moremoney per hour. Start filling in the chart, explain-ing as you go, on a transparency of Activity 1,Exercise 2.4. Ask participants to fill in the rest ofthe chart, noting patterns they find. Have themshare the patterns. Use the Reason and Commu-nicate questions to elicit any patterns they did notfind, especially the diagonal and the “chess-knight”pattern. Also, ask them how to find the unit rate

from a first row or first column entry.

5. Use the Reason and Communicate questions towrite sentences and equations for some of thevalues.

6. Have each group build a three-dimensional grid onone-inch graph paper using linking cubes to createstacks representing the combinations of poolhours and yard work hours. Ask a few participantsto share their grid making strategies. Explore thethree-dimensional grid using the Reason andCommunication questions.

7. Ask participants to think of other situations thatcould be modeled with a combination chart.

8. Have participants label cells, write equations, andfind the indicated values to complete Activity 2.Do the first grid with them. Use the Reason andCommunicate questions to discuss.

9. End Activity 2 by discussing the kinds of problemsthat lend themselves to combination charts, theadvantages of using a combination chart tonumerically explore these problems, and the typesof algebraic thinking skills being developedthrough the use of these charts.

Note:

In the game of chess, theknight (the piece thatlooks like a horse head)can move up/down aspace and over twospaces or up/down twospaces and over onespace.


10. Use the Reason and Communicate questions toguide the participants through Activity 3, Exercise1, demonstrating how to find possible combina-tions of rods to get each length. Write equationsto model the situation. Set up a combination chart,displaying the equations and use the chart to findthe lengths of the rods.

11. Have participants work together in small groups tocomplete Activity 3 by finding the lengths of otherrods and writing their own problems. Circulateand ask guiding questions. Bring the whole grouptogether to share strategies.

Extensions: Solve the following by guess and check with the wholegroup:

1g + 5r = 394g + 1r = 42

Assessment: Fill in the grid below and label it with an appropriate

problem situation.

Notes:

31

30

?

0 ?

Grids Galore


Grids GaloreReason and Communicate:

• What are a few typical middleschool coin combination problems?

• How could you structure a coincombination problem to emphasizethe use of algebraic thinking? Oneway is to ask, “What are the possibletotal amounts you can get combiningquarters and dimes?”

• How could you use the grid toexplore the problem?

• What sentence could you writeabout the lower left cell? Zeroquarters and zero dimes is $0,0(0.25) + 0(0.10) = 0.

• What sentence could you writeabout the cell for 2 dimes and 3quarters? Three quarters and 2dimes is $0.95,3(0.25)+ 2(0.10) = 0.95.

• What patterns do you notice in thegrid? Do not spend a lot of timediscussing this now. Just get partici-pants thinking. As you move left toright, the amount increases $0.10.As you move up a column, theamount increases $0.25.

• What kinds of questions does thischart answer? This chart gives thetotal amount for different combina-tions of up to six dimes and sixquarters.


Grids GaloreActivity 1

1. What amounts of money can you get using only dimes and quarters? Use thegrid below to explore.

6

5

4

3

2

1

0

0 1 2 3 4 5 6

Num

ber

of Q

uart

ers

Number of Dimes


Grids Galore

Notes:

In the game of chess, the knight (the piece that looks like a horse head) canmove up/down a space and over two spaces or up/down two spaces andover one space.


• What kind of questions does this chart answer?

• What does the number in cell ____ represent? The amount Cameronearns if he works ___ hours at the pool and does ____ hours of yardwork.

• What pattern do you see vertically?

• What pattern do you see horizontally?

• What pattern do you see diagonally?

• If you only know the value 24 in the fourth cell on yard work hours, howcould you find the unit rate for yard work hours? 24/3 = $8 per hour

Reason and Communicate, cont:

• If you only know the value 30 inthe fifth cell on swimming poolhours, how could you find the unitrate for working at the pool? 30/5 =$6 per hour

• Does the rate change as youmove from cell to cell? No, the rateremains constant; only the amountof money earned changes as youmove from cell to cell.

• Try this: Start in a cell and go upone and over two. Now do it againand again. What pattern do youfind? The “chess-knight” move.This move always produces aconstant "difference" between thetwo cells. (See "Notes".)

• Try this: Start in a cell and go uptwo and over one. Now do it againand again. What pattern do youfind? Another “chess-knight” move.This move always produces aconstant "difference" between thetwo cells. (See "Notes".)


Grids Galore


• Write a sentence and an equation for cells containing 38, where Pstands for the number of hours worked at the pool and Y stands for thenumber of hours Cameron did yard work. If Cameron works 5 hours atthe pool and does 1 hour of yard work, he will make $38: 5P + 1Y = 38.Also, if Cameron works 1 hour at the pool and does 4 hours of yard work,he will make $38: 1P + 4Y = 38.

Consider the following: 4P + 5Y = 64.• What does P represent? The rate of money earned at the pool, $6 perhour.• What does 4P represent? Four hours at the pool times the pool rate,which is the money earned at the pool for 4 hours of work.• What does Y stand for? The rate of money earned doing yard work,$8/hour.• What does 5Y represent? Five hours doing yard work times the yardwork rate, which is the money earned doing yard work for 5 hours.• What does the whole sentence 4P + 5Y = 64 mean in the problem?Find the cell(s) to which it refers. If Cameron works 4 hours at the pooland 5 hours doing yard work, he will make $64. It refers to the cell in the4 hour pool column and the 5 hour yard work row.

Reason and Communicate, cont:

After you build the three-dimensionalgrids:• Where would you be on the grid ifyou want to make more moneyworking a certain number of hours?If you drew a diagonal line on thegrid where pool hours equals yardwork hours, then the cells above theline would have more yard workhours than pool hours and henceyou would make more money for acertain number of hours.• Where would you be on the grid ifyou want to be at the pool with yourfriends more than make money? Ifyou drew a diagonal line on the gridwhere pool hours equals yard workhours, then the cells below the linewould have more pool hours thanyard work hours and, hence, youwould be at the pool more than inthe yard.

• Focus on the first row of linkingcubes. What equation could youwrite for the set of ordered pairs(number of pool hours, amountearned)? Amount earned =6 • number of pool hours.

• Focus on the first column of linkingcubes. What equation could youwrite for the set of ordered pairs(number of yard work hours, amountearned)? Amount earned =8 • number of yard work hours.


Grids GaloreActivity 1, cont.

2. Cameron can earn $6 an hour working at the neighborhood pool. He canearn $8 an hour doing yard work for his neighbors. How much money couldhe earn for different combinations of pool hours and yard work hours?Explore using the chart below.

6

5

4

3

2

1

0

0 1 2 3 4 5 6

Yard

work

hours

Pool hours


Grids Galore

Answers:

1a. The cell with “42” in it represents “Forty-two dollars is earned for 1 hourtutoring and 3 hours babysitting.” The cell with “32” in it represents “Thirty-two dollars is earned for 1 hour tutoring and 2 hours babysitting.”

b. 1T + 3B = 42 1T + 2B = 32

c. 36

2a. The cell with “6” in it represents “Six dollars is earned for 1 hour at thepool and 0 hours babysitting.” The cell with “15” in it represents “Fifteendollars is earned for 2 hours at the pool and 1 hour babysitting.”

b. 1P + 0B = 6 2P + 1B = 15

c. 9


• Why find the indicated cell? Whatquestion could you be answering byfinding the value of the indicatedcell? For Exercise 1, you did onehour of tutoring and 2 hours ofbabysitting and earned $32. Thenyou did one hour of tutoring and 3hours of babysitting and earned $42.What should you charge for 3 hoursof tutoring?

• What do you need to know to fill inthe other values in the chart? Youneed to know the unit rate for thehorizontal change and the unit ratefor the vertical change.

• In the "After School Earnings" grid,how did you find the indicatedvalue? Subtract 32 from 42 to getthe unit rate for babysiting, which is$10/hour. Subtract 10 from 32 twiceto get $12 which is the unit rate fortutoring, $12/hour. Therefore, youwould earn 3($12) = $36 for 3 hoursof tutoring.

• In the "Summer Earnings" grid,how did you find the value for whichyou were asked? Go over by 6 toget 12 for 2 hours of working at thepool. To get to 15 from 12, you goup by 3, so you must be earning $3/per hour for babysitting. Therefore,for 3 hours of babysitting, you earn$9.



For each grid,a. Write a sentence to describe the values in the given cells.b. Write equations for the values.c. Find the value of the indicated cell.

Summer Earnings

Hours

Babysitting

Hours at the Pool

2.

1.

5

4

3 42

2 32

1

0 0 ?

0 1 2 3 4 5

After School Earnings

Hours

Babysitting

Hours Tutoring

5

4

3 ?

2

1 15

0 0 6

0 1 2 3 4 5


Grids Galore

Answers:

3a. The cell with “12” in it represents “12 units for 1 horizontal and 2 verti-cal.” The cell with “18” in it represents “18 for 2 horizontal and 1 vertical.”

b. 1H + 2V = 12 2H + 1V = 18

c. 24

4a. The cell with “25” in it represents “25 units for 1 green and 2 purple.” Thecell with “45” in it represents “45 units for 3 green and 1 purple.”

b. 1G + 2P = 25 3G + 1P = 45

c. 70


• What did you label the axes inExercise 3? Why?

• In Exercise 4, how did you label thecell containing “25”? Two purple rodsand 1 green rod measure 25 units.

• In Exercise 4, how did you find thevalue for which you were asked?One way is to look at the “chess”move. Starting at 0, go over one andup two to get to 25. Start at 45, goover one and up two to get to “?”, sothe indicated value is 45 + 25 = 70.Another way is to go from 25 to 45 bygoing over two and down one. Nowstart with 45 and go over two anddown one to get 65 in the cell for 5green rods. If 5 green rods measure65, then one green rod measures 13.The "45" cell is 3 greens and 1purple, so subtract the 3 greens,45 - 3(13) = 45 - 39 = 6. One purplemeasures 6. Then, the missing valueis 4(13) + 3(6) = 70.

• What types of algebraic thinkingskills are being developed throughthe use of these charts? We developdoing and undoing as we go fromproblem situations to charts tosentences to equations and back andforth in between all of these repre-sentations. We also develop doingand undoing as we "undo" the totalsin the chart using subtraction anddivision. We develop patterns torules as we find patterns in the chartand use them to write sentences andequations. We develop abstractingfrom computation as we gain theperception that multiplication by theunit rate represents a quantity; 4Prepresents the amount earned fordoing 4 hours of pool work.

• How did you find the indicatedvalue in the grid in Exercise 3? Movediagonally from 12 to 18 by adding 6.Move diagonally from 18 to theindicated cell by adding 6 again toget 24.



Combine Lengths of Purple

and Green Rods

Num

ber

of P

urp

le R

ods

Number of Green Rods

4.

3. ?

?

?

5

4

3

2 12

1 18

0 0 ?

0 1 2 3 4 5

5

4

3 ?

2 25

1 45

0 0

0 1 2 3 4 5



• How did you find the length of theyellow and red rods in Exercise 1?Share strategies.

• How did you find the length of thelight green and red rods in Exercise2? Share strategies

Math Notes:

The lengths of the line segments aregiven in different, non-standard unitsfor each exercise. This is so thatparticipants do not simply measurethem. We want the participants tofind the length of the rods withrespect to lengths of the line seg-ments.

For Exercises 1 and 2 we restrict thenumber of rods participants may usebecause we want to limit the equa-tions they find.

Answers:

1. One light green rod and 2 red rods is 28 lems. One light green rod and1 red rod is 20 lems.

1g + 2r = 281g + 1r = 20

The light green rods are 12 lems long and the red rods are 8 lems long.

2. Two yellow rods and 2 red rods is 14 picos. Three yellow rods and 1 redrod is 17 picos.

2y + 2r = 143y + 1r = 17

The yellow rods are 5 picos long and the red rods are 2 picos long.


In each of the following, line up the color rods along each line. Write a sentenceand an equation to represent the relationship between the number of color rodsand the measure of the line given. (The measurements are in a different non-standard unit for each exercise.) Use the accompanying grids to find the lengthof the color rods based on the lengths of the segments. Show your work.

1. Choose from 3 light green rods and 3 red rods.

2. Choose from 3 yellow rods and 4 red rods.


28 lems

17 picos

20 lems

14 picos



• If you have not already done so,how could you set up the rods inExercise 3 in a different way?

• How does this affect theequations? The equations willchange.

• How does this affect the answer?It does not affect the answer.

• What did you think when you foundthat there was more than one way toset up the rods in Exercise 3?

• How do you think students willreact?

• What things did you have toconsider when writing your ownproblems? Share strategies.

Math Note:

The measurements are given in adifferent nonstandard unit for eachexercise. This is so that participantsdo not simply measure them. Wewant the participants to find thelength of the rods with respect tolengths of the line segments.

For Exercises 1 and 2 we restrict thenumber of rods participants may usebecause we want to limit the equa-tions they find. Exercise 3 allowsparticipants to use any number of redand light green rods. They can findseveral different equations for eachsegment. Compare the differentsystems and grids that arise.

Answers:

3. Five red rods and 1 light green rod is 39 zibs. Also, two red rods and3 light green rods is 39 zibs.

5r + 1g = 392r + 3g = 39

Four light green rods and 1 red rod is 42 zibs. Also, two light green rodsand 4 red rods is 42 zibs.

4g + 1r = 422g + 4r = 42

Seven red rods is 42 zibs.7r = 42

The red rods are 6 zibs long and the light green rods are 9 zibs long.



3. Use red rods and light green rods.

4. Write your own problem like the above for use in your classroom. Use adifferent combination of color rods than above. Write the problem and show asolution.

42 zibs

39 zibs


Unit IV: Reflect and Apply

Sample Answers:

3. A sample context: We are redecorating our bathroom with new vinyl andcarpet.If we order 3 square yards of carpet and 2 square yards of vinyl, it costs$67. If we order 2 square yards of carpet and 3 square yards of vinyl, itcosts $63.

How much does the carpet cost? How much does the vinyl cost?3c + 2v = 672c + 3v = 63

The carpet costs $15 per square yard. The vinyl costs $11 per squareyard.


Unit IV: Reflect and Apply

1. Reflect on the activities in "Making Connections" and "Grids Galore".

a. Explain how you can lead students from their recursive thinking towriting function rules for relationships.

b. Part of algebraic reasoning has to do with doing and undoing. Find anexample in this unit that illustrates doing and undoing. Discuss howconnections are made among representations.

c. Part of algebraic reasoning has to do with abstracting from computation.Find an example in this unit that illustrates abstracting from computation.

Discuss how students can abstract y mx b= + from concrete experiences.

d. Describe how students can use combination grids to explore problemsnumerically.

2. Consider ways you might adapt these activities for your specific grade level.Create a few of your own application problems for your students to generalizefunctions and systems and make connections.

3. Using the following grid, create a context for the grid, write two equations forthe grid, and explain how to find the solutions with the grid.

63

67

0


Swimming Pool

Institute Notes

Concept: Open-ended investigation of relationships between quan-tities, both linear and non-linear.

Overview: Participants are given a pattern of square swimmingpools to investigate. They can use physical models, pic-tures, tables, graphs, and function rules to investigatethe relationship between such quantities as the numberof blue tiles in relation to the pool number. Participantswill organize and present their findings to the group. Theywill then compare a linear growth model (the number ofwhite tiles in relation to the pool number) and a qua-dratic model (the number of blue tiles in relation to thepool number).

TEKS Focus: 6.4 – The student uses variables in mathematical

expressions to describe how one quantitiy changeswhen a related quantity changes.

7.4 – The student makes connections among various

representations of a numerical relationship.

8.4 – The student makes connections among various

representations of a numerical relationship.

Materials: Two colors of color tiles, 1” grid paper, Markers, Peel-and-stick dots

Procedure: 1. Discuss the situation in the Activity. Build the firstthree square swimming pools with color tiles on theoverhead projector.

2. Explain that each group will present their findingsfrom at least one of their investigations. Give them atime frame in which to work. Circulate and askguiding questions.

3. Have each group present their findings from at leastone of their investigations as time permits. Discusstheir strategies. Ensure that one group presents thelinear model of white tiles in relation to pool numberand one presents the quadratic model of blue tiles inrelation to pool number. Compare the two modelsusing the Reason and Communicate questions.

Note:

This activity is adaptedfrom “Experiences withPatterning” by JoanFerrini-Mundy, GlendaLappan, and ElizabethPhillips, Teaching ChildrenMathematics: AlgebraicThinking Focus Issue,February 1997.

Also

Grade 6

2c, 5, 6cGrade 7

2f, 5Grade 8

2a, 5Algebra I

b.1A-E, b.2C-D, b.3A-B,b.4A-B, c.1A, c.2A-C, 2G,c.3A-B, d.1D, d.3B-C


Swimming Pool

4. Point out in the groups’ presentations where they usedalgebraic expressions to represent a quantity. Forexample, in the fourth swimming pool, n = 4, the sidelength for the outside square is 6 or n+2. This is thehabit of mind, abstracting from computation.

5. Discuss how this activity could be used in teachers’classrooms and at what level.

Extensions: Look at some of the relationships that deal with fractionsif no group has already. For example: the ratio of thenumber of blue tiles to the total number of tiles in rela-tion to the pool number, the ratio of the number of whitetiles to the total number of tiles in relation to the poolnumber, and the ratio of the number of blue tiles to thenumber of white tiles in relation to the pool number.

Assessment: Give students a table of numbers of tiles for swimmingpools that are constructed differently than the ones inthe activity. Ask students to construct possibleswimming pools that fit the numbers in the table.

For Example:

Notes:

PoolNumber

Number of WhiteTiles

Number of BlueTiles

1 10 2

2 14 6

3 18 12


Swimming PoolReason and Communicate:

• Recall the Stretching Sequencesactivity, where we looked at relation-ships between figure number andarea and figure number and perim-eter. How does that activity com-pare to the "Swimming Pool" activ-ity? In both activities we are lookingat relationships between variables.The figure number in "StretchingSequences" is sort of analogous tothe pool number in the "SwimmingPool" activity.

Answers:There are a myriad of possibilities here. See the next few pages forexamples of investigations. Encourage participants to write down asmany quantities as they can think of as they build and color their swim-ming pools. Encourage them to look for even more as they begin toinvestigate.

Pool 4


Swimming PoolActivity

Imagine a square swimming pool that can be modeled as follows. What will thenext swimming pool in the pattern look like?

As you complete the instructions below, think about the different quantities thatare changing in this situation. How could you investigate possible relationshipsbetween these quantities?

1. Use color tiles to build some of the swimming pools in the pattern. Write anyobservations here.

2. Draw the swimming pools you created above on a large piece of 1” graphpaper.

3. Fill in the table with different quantities from the situation that you might wantto investigate the relationships between.

Pool 1 Pool 2 Pool 3


0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

0 1 2 3 4 5 61

1

1

1

1

3

5

7

9

1 1

Swimming PoolReason and Communicate:

Ask the following about the relation-ships participants investigate:

• Is the relationship linear?

How do you know? Look for aconstant rate of change per unit of x.

• Is is proportional? Emphasize

that even though all proportionalrelationships are linear, not all linearrelationships are proportional.

• How can you express the relation-

ship between the variables?

• Can you describe the relationship

geometricallly? Algebraically?Graphically?

Draw triangles as modeled below todiscuss rate of change.• What is the length of the base ofthe triangles on both the linear andthe non-linear graphs? They are 1unit in length.• What is the length of the height ofthe triangles in the linear graph?They are all 4. The rate of change isconstant.• What is the length of the height ofthe triangles in the non-linear graph?The lengths vary. The rate ofchange is changing.

Answers:

You can take as much time as you have here to let participants go in asmany directions as they want. If they run out of ideas, let them circulate andsee what others are doing.

Nu

mb

er

of

Blu

e T

iles

Pool Number

0

5

1 0

1 5

2 0

2 5

3 0

0 1 2 3 4 5 6

1

4 1

4 1

4 14 1

4

Nu

mb

er

of

Wh

ite

Tiles

Pool Number

4. As a group, decide on two quantities between which you want to investigatethe relationship. Investigate, organize using a table and graph, and beprepared to present your findings. Use a graphing calculator whereappropriate.

5. As a group, decide on a different relationship between quantities that youwant to investigate. Investigate, organize using a table and graph, and beprepared to present your findings. Use a graphing calculator whereappropriate.


Swimming PoolActivity, cont.

4. As a group, decide on two quantities between which you want to investigatethe relationship. Investigate, organize using a table and graph, and be pre-pared to present your findings. Use a graphing calculator where appropriate.

5. As a group, decide on a different relationship between quantities that youwant to investigate. Investigate, organize using a table and graph, and beprepared to present your findings. Use a graphing calculator where appropri-ate.


Swimming PoolPossible Relationships

Pool Number Total Number of Tiles

1 9

2 16

3 25

4 36

5 49

6 64

n n2 + 4n + 4 = (n + 2)2

Total Number of Tiles in Relation to Pool Number

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

0 1 2 3 4 5 6

There are at least two good ways to see the total number of tiles. First, the total number oftiles (n2+4n+4) = number of white tiles (n2) + number of blue tiles (4n+4) . Also, the whole poolwith the border has dimensions of n+2 tiles by n+2 tiles, so the amount of all the tiles is(n+2)(n+2) = (n+2)2. This is a good time to note that (n+2)2 = n2+4n+4.

To

tal N

um

ber

of

Tiles

Pool Number

Pool NumberNumber of Blue

Tiles

Number of White

Tiles

Total Number of

Tiles

1 1 8 9

2 4 12 16

3 9 16 25

4 16 20 36

5 25 24 49

6 36 28 64

n n2 4n + 4 (n2 + 4n + 4) = (n+2)2


Swimming PoolPossible Relationships, cont.

Pool Number Number of Blue Tiles

1 1

2 4

3 9

4 16

5 25

6 36

n n2

Number of Blue Tiles in Relation to Pool Number

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

0 1 2 3 4 5 6

The pool floor's dimensions are n by n tiles, so the number of blue tiles is the same as the areaof the surface of the water, which is the area of a square with side n, or n2. Another way torepresent the number of blue tiles is the total number of tiles minus the number of white tiles.The total number of tiles can be represented as (n + 2)(n + 2). The number of white tiles canbe represented as 4n + 4.

Therefore, the number of blue tiles = +( ) +( ) − +( ) = + + − − =n n n n n n n2 2 4 4 4 4 4 42 2.

Number of White Tiles in Relation to Pool Number

Pool Number Number of White Tiles

1 8

2 12

3 16

4 20

5 24

6 28

n 4n + 4

To see that the number of white tiles is 4n + 4, separate the white tiles into two parts. First,each pool has 1 tile in each of the 4 corners, for a total of 4 tiles. If these corner tiles areremoved, then you can see the number of white tiles left is the same as the number of units inthe perimeter of the surface of the water. Since the dimensions of the surface of the water aren by n, the perimeter of the surface of the water is 4n units (4 sides, all n units long). Thusthere are 4n white tiles without the corner tiles. Add the two parts together, 4 + 4n, and you getthe total number of white tiles. Another way to represent the number of white tiles is the total

number of tiles minus the number of blue tiles, ( )n n n n2 24 4 4 4+ + − = + .

Nu

mb

er

of

Blu

e T

iles

Pool Number

0

5

1 0

1 5

2 0

2 5

3 0

0 1 2 3 4 5 6

Nu

mb

er

of

Wh

ite

Tiles

Pool Number



Pool Number

00.10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6

Number of White TilesTotal Number of Tiles

in Relation to Pool Number

Number of Blue Tiles

Total Number of Tiles in Relation to Pool Number

Pool Number

1

2

3

4

5

6

n

Number of White TilesTotal Number of Tiles

Num

ber

of W

hite

Tile

s

Tota

l N

um

be

r o

f T

ile

s

8

90 89≈ .

20

360 56≈ .

12

160 75≈ .

16

250 64≈ .

24

490 49≈ .

28

640 44≈ .

4 4

2 2

n

n

++( )

Pool Number

1

2

3

4

5

6

n

Number of Blue Tiles

Total Number of Tiles

1

90 11≈ .

4

160 25≈ .

9

250 36≈ .

16

360 44≈ .

25

490 51≈ .

36

640 56≈ .

n

n

2

22( ) +

Nu

mb

er

of

Blu

e T

ile

s

To

tal N

um

be

r o

f T

ile

s

00 .10.20.30.40.50.60.70.80.9

1

0 1 2 3 4 5 6

Pool Number


Pool Number

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6

Number of Blue TilesNumber of White Tiles

in Relation to Pool Number


Num

ber

of B

lue T

iles

Num

ber

of W

hite T

iles

Pool Number

1

2

3

4

5

6

n

Number of Blue TilesNumber of White Tiles

n

n

2

4 4+( )

1

80 13≈ .

4

120 33≈ .

9

160 56≈ .

16

200 80≈ .

25

241 04≈ .

36

281 29≈ .

Perimeter of Pool in Relation to Pool Number

Pool Number Perimeter of Pool

1 9

2 16

3 25

4 36

5 49

6 64

n (n + 2)2

01 02 03 04 05 06 07 0

0 1 2 3 4 5 6 7

Pool Number

Perim

ete

r


Cover Up

Institute Notes

Concept: Develop the cover-up method for equation solving.

Overview: Participants will link the idea of guess and checkequation solving to the cover-up method. The cover-up method can be used to lead students to the usualundoing algorithms for equation solving.

TEKS Focus: 6.11D—Students use mental math to solve problems.7.13D—Students use mental math to solve problems.8.14D—Students use mental math to solve problems.

Materials: Overhead projector, Graphing calculator

Procedure: 1. Do the first few problems from Activity 1, Exercise 1with the whole group. Have participants completeExercise 1 by using substitution or guess-and-checkand graphs to find solutions.

2. Have participants discuss the different ways theyused mental math and guess-and-check inExercise 1 of Activity 1.

3. Review the method for solving using a table andgraph on a graphing calculator. Then have partici-pants complete Activity 1, Exercise 2 by solvingusing a table and graph on a graphing calculator.

4. Have participants complete Activity 1, Exercises 3and 4, which are similar to Activity 1, Exercises 1and 2. Discuss using the Reason and Communicatequestions.

5. In Activity 2, develop the cover-up method using theTransparency with the first two problems. Thenhave participants complete Activity 2, Exercise 1a-f.

6. Using the Transparency, work through the thirdproblem, which is Activity 2, Exercise 1i, withparticipants. Then have them complete Activity 2using the cover-up method. Discuss using theReason and Communicate questions.

Math Notes:

The equations used in theseactivities come from real-world examples developedin Swimming Pools andMaking Connections.

Also

Grade 6

1B, 1E, 2, 5,Grade 7

1C, 2, 5, 15AGrade 8

1B, 2, 4, 5, 16AAlgebra I

b.4B


Cover Up

Extensions: Show the identities n n n n2 1+ = +( ) in the table for

y x x= +2 and y x x= +( )1 .

Assessment: Explain how one can move from the cover-up methodto undoing algorithms (using inverse operations).

Notes:


Cover UpReason and Communicate:

• What kind of strategies did youuse to solve equations by guess andcheck?

• Do these equations look familiar?Where have you seen them? Theseequations are from previous activi-ties in the institute. See if you canfind them!

Math Notes:

2. Make the connection betweenthe input value in the table and onthe graph and the solution to theequation.

3. Compare the pairs of similarequations (a and b, c and d, etc.)

It is important for students to useguess-and-check or subsitution tosolve equations before they develop"undoing" algorithms.

Answers:

1a. 40 b. 25c. 40 d. 65e. 5 f. 10g. 5 h. 8i. 10 j. 49

2. An example of the solution displayed in a table and graph.

3a. 8 b. 12c. 5 d. 8e. 5 f. 8g. 6 h. 10i. 6 j. 10k. 4 l. 3

4. Answers vary.


Cover UpActivity 1

1. For each equation, find a real-world or tile model scenario based on yourprevious activities. Then solve by guess-and-check.

a. 100 15 700+ =x b. 100 15 475+ =x

c. 15 0 25 25+ =. x d. 15 0 25 31 25+ =. .x

e. 20 2 10- =x f. 20 2 0- =x

g. 4 1 24n +( ) = h. 4 1 36n +( ) =

i. 2 1 4 26n +( ) + = j. 2 1 4 104( )n + + =

2. For each equation in Exercise 1, use the table feature of your calculator tosolve. Then solve graphically.

3. For each equation, find a real-world or tile model scenario based on yourprevious activities. Then solve by “guess and check,” substituting your guess.

a. n2 64= b. n

2 144=

c. n n +( ) =1 30 d. n n +( ) =1 72

e. n n2 30+ = f. n n2 72+ =

g. n +( ) =2 642

h. n +( ) =2 1442

i. n n2 4 4 64+ + = j. n n2 4 4 144+ + =

k.

10025

x= l.

10033

1

3x=

4. For each equation in Exercise 3, use the table feature of your calculator tosolve. Then solve graphically.


Cover Up

Math Note:

Develop the equations on the transparency a line at a time to illustrate thecover-up method.


Equation n + 4 = 14

Some number plus 4 is 14.

What is it?____ + 4 = 14

It is 10. n = 10

Equation 2n = 20

Two times some number is 20.

What is it?2(____) = 20

It is 10. n = 10

Equation 2 (n + 1) + 4 = 104

Some number plus 4 is 104.

What is it?____ + 4 = 104

It is 100, so 2 (n + 1) = 100

Two times some number is

100. What is it?2 (___) = 100

It is 50, so n + 1 = 50

A number plus 1 is fifty.

What is it?___ + 1 = 50

It is 49, so n = 49

Now check. 2 (49 + 1) + 4 = 104

Cover UpTransparency


Cover UpReason and Communicate:

• How is cover-up different fromguess and check? Guess andcheck tends to be a forward pro-cess, while cover-up leads to“undoing” the operations.

• How can we move from cover-upto the standard undoing algorithms?Cover-up shows how one undoesaddition and multiplication bysubtracting and dividing.

• What do we use for undoingalgorithms in Algebra I? We useproperties of equalities that showgeneralized arithmetic. For ex-ample, if a = b and c = d, thena – c = b – d.

• What “habits of mind” are involvedhere? Primarily undoing is involved,but we are going toward abstract-

ing from computation to generalizeproperties of equality.

Answers:

1a. 40 b. 25c. 40 d. 65e. 5 f. 10g. 5 h. 8i. 10 j. 49

2. Problems a, b, g, h, k, and l can be done with the cover-up method.


1. Use the cover-up method to solve each equation. Record your results.

a. 100 15 700+ =x b. 100 15 475+ =x

c. 15 0 25 25+ =. x d. 15 0 25 31 25+ =. .x

e. 20 2 10- =x f. 20 2 0- =x

g. 4 1 24n +( ) = h. 4 1 36n +( ) =

i. 2 1 4 26n +( ) + = j. 2 1 4 104( )n + + =

2. Which equations below can be solved using the cover-up method?

a. n2 64= b. n

2 144=

c. n n +( ) =1 30 d. n n +( ) =1 72

e. n n2 30+ = f. n n2 72+ =

g. n +( ) =2 642

h. n +( ) =2 1442

i. n n2 4 4 64+ + = j. n n2 4 4 144+ + =

k.

10025

x= l.

10033

1

3x=

Cover UpActivity 2


Unit V: Reflect and Apply

Answers:

3. x = 4.

Math Note:

3. The problem could be worked inthe following manner:What number subtracted from 14 is9? Five.What number divided into 15 is 5?Three.What number subtracted from 7 is 3?Four.So x = 4.


Unit V: Reflect and Apply

1. Reflect on the activities in "Cover Up" and "Swimming Pool".

a. Contrast the way you learned equation solving with the guess-and-checkand cover-up methods.

b. Discuss how you can use color tiles to make connections between

expressions, like n +( )22 and n n2 4 4+ + .

c. How can you as the teacher facilitate the development of algebraicthinking using open-ended explorations like "Swimming Pool"?

2. Consider ways you might adapt these activities for your specific gradelevel. Create a different swimming pool for which your students couldinvestigate relationships. Construct a few of your own equations for yourstudents to solve.

3. Solve the following using the cover-up method: 14

15

79-

-=

x.


Ups and Downs

Institute Notes

Concept: Distinguish between linear and exponential growth anddecay.

Overview: Participants will compare linear and exponential growthand decay, contrasting repeated addition with repeatedmultiplication. First they will use a table to develop therelationship recursively, and using manipulatives, theywill develop a concrete representation. They will thendevelop the growth and decay function rules. Finally,they will use M&M’s, a calculator program, and motiondetectors to gather data and fit the data with exponentialmodels.

TEKS Focus: 6.5 – The student uses letters to represent an unknown

in an equation.

7.14a – The student communicates mathematical ideas

using language, efficient tools, appropriate units, andgraphical, numerical, physical, or algebraic mathemati-cal madels.

8.15a – The student communicates mathematical ideas

using language, efficient tools, appropriate units, andgraphical, numerical, physical, or algebraic mathemati-cal madels.

Materials: 1” grid paper, Peel-and-stick dots, Markers, Color tiles,Paper cups, M&M's, Graphing calculator, Motion detector,Ball (Basketballs, raquetballs, and golf balls work well.)

Procedure: 1. Do Activity 1 with the whole group by first demonstrat-ing how to physically model the growth with color tileson1" grid paper. Place the appropriate number of colortiles on the grid paper for each time increment. Place apeel-and-stick dot on the upper left corner of each gridsquare where the top color tile sits in each column. Thenremove the color tiles and discuss the resulting graphs.Have participants transfer the graphs to the grids onthe Activity page. Then have them fill in the table withthe data from their graph.

Math Notes:

In the same way we movefrom repeated addition toget the linear model, wemove from repeatedmultiplication to exponen-tial notation and theexponential model,

y a bx= • .

Also

Grade 6

3, 4, 11, 12Grade 7

4, 5, 13, 14bGrade 8

4, 5, 14, 15bAlgebra I

b.1A-E, b.2C-D, b.3A-B,b.4A-B, c.1A, c.2A-C, 2G,c.3A-B, d.1D, d.3B-C


2. Use the Reason and Communicate questions tocontrast linear and exponential growth in Activity 1.Build the two rules by moving from recursion vialanguage as shown in the snapshot:Height = 5 + 2 • Weeks, Number of Bacteria = 2Hours.

3. Have participants use the color tiles, peel-and-stickdots, and 1" grid paper to create the graph and fill inthe table in Activity 2.

4. Use the Reason and Communicate questions tocontrast linear and exponential decay in Activity 2.Build the two rules, by moving from recursion vialanguage as shown in the snapshot:Food Level = 47 – 6 • Days,

Grams = 32 • ( 12

)Hours.

5. Explain how to do the growth experiment inActivity 3. Pass out the M&M's in cups to the partici-pants. Have participants collect their individual dataand then use an overhead calculator to enter all oftheir combined data. Take a numeric approach and,using mental math, estimate how close the numbersare to doubling each time. Then create a scatterplot and find a function rule to fit the data.

6. Discuss the participants’ results of the M&M growthexperiment using the Reason and Communicatequestions.

7. Explain how to do the decay experiment in Activity 3.Have participants collect their individual data andthen use an overhead calculator to enter all of theircombined data. Again, use mental math to estimatehow close the numbers are to halving each time.Then create a scatter plot and find a function rule tofit the data.

8. Discuss the participants’ results of the M&M decayexperiment using the Reason and Communicatequestions.

Ups and Downs


9. Drop a ball and discuss return height. Prompt forresponses on parts a and b of Activity 4, Exercise 1.Create a scatter plot of maximum height in relationto bounce number and find a rule with a calculator.

10. For Exercise 2 in Activity 4, do a whole group ballbounce experiment first. Then have each group doa ball bounce experiment on their own. Refer to thesnapshot example to develop the model. Discussusing the Reason and Communicate questions.

Extensions: Run a program on a graphing calculator or computer tosimulate doubling. The program should take an initialvalue and multiply it by a randomly chosen factor closeto two. The program should then repeat this, obtainingthe next value by multiplying the previous by a randomfactor close to two. Produce a scatter plot of the data to“see” the simulation. Ask students to create their owngrowth or decay scenario. Then have students “tweak”the calculator program to simulate their problem.

Assessment: Give students graphs of linear and exponential growthand decay and ask students to label the graphs cor-rectly. Have them fill in tables with the correspondingdata, find the rate of change for the linear graphs, andshow that the rate of change is not constant for theexponential graphs.

Notes:

Ups and Downs


Ups and Downs

Answer:2. 14 weeks

Math Notes:

Draw triangles to illustrate the constant rate of change, as shown in thesnapshot. In the linear growth of the flower, the change is 2 cm per week.This is a constant change in centimeters per week.

A constant rate of change yields a line and a changing rate of change does not.


• What operation did you repeat tocomplete the table for the botanist’sflower? Addition.

• What kind of graph do you get withrepeated addition? Linear.

• In this situation, is the linear graphincreasing or decreasing?Increasing.

• Are the graphs obtained fromrepeated addition always increasing?No, only if the addend is positive,which in this case it is.

• What would the graph look like ifthe addend was negative?Decreasing.

• What function rule can you gener-alize for the height?Height = 5 + 2 •Weeks, H = 5 + 2W .

Number of

Weeks

0

1

2

3

4

5

W

Process

5 + 2 (1)

5 + 2 (2)

5 + 2 (3)

5 + 2 (4)

5 + 2 (5)

5 + 2 (W)

Height (cm)

5

7

9

11

13

15

17


Ups and DownsActivity 1

1. A botanist plants a new variety of flower that is 5 cm in height. She expectsthe average growth rate to be 2 cm each week. Track the growth of the flowerin the table and the graph. Label the axes.

2. When does the plant reach 33 cm? Show the solution recursively on thehomescreen.

Number of

WeeksProcess Height (cm)



• What operation did you repeat tocomplete the table for the biologist’sbacteria culture? Multiplication.

• What notation can we use withrepeated multiplication? We useexponents to denote repeatedmultiplication as shown:

• What kind of graph do you get withrepeated multiplication?Exponential.

• In this situation, is the exponentialgraph increasing or decreasing?Increasing.

• Are the graphs obtained fromrepeated multiplication alwaysincreasing? No, only if the factor isgreater than one, which in this case itis.

• Which grows faster? The bacteriagrows faster after a while because it

is exponential growth.

• What function rule can yougeneralize for the number of bacte-ria? Number of Bacteria = 2Hours,B = 2H.

• Since it is assumed that the floweris growing at the same rate everyweek, we say it has a constant rateof change. Does the bacteria growat a constant rate? No, the rate ofchange is changing. (Refer to theMath Notes.)

Ups and Downs

0 1 = 20

1 2 = 1 2 = 21

2 4 = 2 2 = 22

3 8 = 2 2 2 = 23

4 16 = 2 2 2 2 = 24

Answers:

4. 20 weeks.

5. The growth rate of the plant is constant, but the growth rate of thebacteria is changing.

Math Notes:

Draw triangles to illustrate the changing rate of change, as shown in thesnapshot. In the exponential growth of the bacteria, the rate for the firstweek is 1 cm per week, the rate for the second week is 2 cm per week,the rate for the third week is 4 cm per week, and so on. Although thebacteria is constantly doubling, the change in number of bacteria perweek is not constant. A changing rate of change is not linear.

•

•

• •

• • •

Number of

Weeks

0

1

2

3

4

5

W

Process

20

2 = 21

(2)(2) = 22

(2)(2)(2) = 23

(2)(2)(2)(2) = 24

(2)(2)(2)(2)(2) = 24

2w

Number of

Bacteria

1

2

4

8

16

32

2w


Ups and DownsActivity 1, continued

3. A biologist grows bacteria. He begins with 1 unit of bacteria and expects theculture to double in size each week. Track the growth of the culture in thetable and the graph. Label the axes.

4. When does the bacteria population reach 1 million? Show the solutionrecursively on the homescreen.

5. How does the growth rate of the plant compare with the growth rate of thebacteria?

Number of

WeeksProcess Number of

Bacteria


Ups and Downs

Answer:

2. The feeder is emptied in between the 7th and 8th days.

Math Notes:

Draw triangles to illustrate the constant rate of change, as shown in the

snapshot. In the linear decay of the hummingbird food, the change is -6millimeters a day. This is a constant change in millimeters per day.

Stress here that repeated subtraction is really the same thing as repeatedaddition of a negative number. Therefore, they have the same graphicalresult, linear.


• What operation did you repeat tocomplete the table for the zoologist’semptying bird feeder? Subtraction, whichcan be thought of as repeated addition ofnegative numbers.

• What kind of graph do you get withrepeated addition of a negative number?Linear.

• In this situation, is the linear graphincreasing or decreasing?Decreasing.

• Why? The graph is decreasing be-cause you are adding a negative number.

• What function rule can you generalizefor the food level?

Level = 47 – 6 • Days, L = 47 – 6D.

47 - 6(0) = 47

47 - 6(1) = 41

47 - 6(2) = 35

47 - 6(3) = 29

47 - 6(4) = 23

47 - 6(5) = 17

Number of

Days

0

1

2

3

4

5

D

Process

47 - 6 (0)

47 - 6 (1)

47 - 6 (2)

47 - 6 (3)

47 - 6 (4)

47 - 6 (5)

Food Level

(mm)

47

41

35

29

23

17

47 - 6 (D)



1. A zoologist fills a hummingbird feeder to 47 mm. During the peak humming-bird season, he estimates that the feeder is emptied at an average rate of6 mm a day. Track the amount of food in the feeder in the table and the graph.Label the axes.

2. When is the feeder empty?

Number of

DaysProcess Food Level

(mm)


Ups and DownsReason and Communicate:

• What operation did you repeat tocomplete the table for the scientist’sradioactive substance? Division,which can be thought of as multipli-cation by a fraction between zeroand one.

• What notation can we use withrepeated multiplication? We useexponents to denote repeatedmultiplication as shown:

• What graph do you get withrepeated multiplication? Non-linear,a curve.

• In this situation, is the exponentialgraph increasing or decreasing?Decreasing.

• Why? The graph is decreasingbecause we are multiplying by afraction between 0 and 1.

• Which decreases or decays faster,the bird food or the radioactiveelement? The radioactive sub-stance decays faster over timebecause it is an exponential decay.

Answers:4. 9 hours.

5. The decay rate of the food is constant, but the decay rate of the radioac-tive substance is changing.

Math Notes:

The half-life of a radioactive substance is the amount of time that it takes forhalf of the substance to decay. The change in the amount of carbon-14makes it possible to calculate the approximate age of the fossil. Carbon-14(a radioactive carbon, with a half-life of 5600) and carbon-12 (a stable car-bon) are both present in all living organisms. When an animal dies, the

carbon-14 decays, but the amount of carbon-12 remains the same. The

change in the amount of carbon-14 makes it possible to calculate theapproximate time period of decay.

In the exponential decay of the radioactive substance, the rate for the firstweek is 16 grams per hour, the rate for the second week is 8 grams perhour, the rate for the third week is 4 grams per hour, and so on. Althoughthe radioactive substance is constantly halving, the change in grams perhour is not constant.

0

1

2

3

4

32 = 32 1

2

16 = 32 1

2

= 32

1

2

8 = 32 1

2

1

2

= 32

1

2

4 = 32 1

2

1

2

1

2

= 32

1

2

2 = 32 1

2

1

2

1

2

1

2

= 32

1

2

0

1

2

3

4

47 - 6(0) = 47

47 - 6(1) = 41

47 - 6(2) = 35

47 - 6(3) = 29

47 - 6(4) = 23

47 - 6(5) = 17

Number of

Hours

0

1

2

3

4

5

W

ProcessAmount of

Radioactive Substance

(grams)

32

16

8

4

2

1

32 1

2

w

32 1

2

32 1

2

= 32

1

2

321

2

1

2

= 32

1

2

321

2

1

2

1

2

=32

1

2

321

2

1

2

1

2

1

2

=32

1

2

0

1

2

3

4

321

2

1

2

1

2

1

2

1

2

=32

1

2

5


Ups and DownsActivity 2, cont.

3. A scientist knows that the half-life of a certain radioactive substance is onehour. She has a sample of 32 grams of the substance. Track the amount leftof the radioactive substance as it decays in the table and the graph. Labelthe axes.

4. When does the substance have 0.0625 grams left?

5. How does the decay rate of the bird food compare with the decay rate of theradioactive substance?

Number of

HoursProcess

Amount of

Radioactive

Substance (grams)


Ups and Downs


• What would you expect thepopulation to be at each roll? Twicethe previous amount. The probabilityof getting an M for each M&M is one-half, so you would expect to get abouttwice the previous amount each time.

• How close were your results towhat you expected?

• What kind of graph did you get?Increasing exponential.

Answers:

3. A rule that would fit the sample class data above is y x= •120 2 .

Technology Notes:

After each participant has filled in the "Your Data" table, use a graphingcalculator to sum the data in lists and then fill in the "Class Data" table.



1. Start with 4 M&Ms in a cup. Shake the cup and roll out the M&Ms. Count thenumber with “M” showing, multiply the number by 2, and add that many M&Msto the population. Record the population and repeat 4 times.

(For example, when you rolled out your 4 M&Ms, three had “M” showing.(2)(3) = 6, so add 6 to the population of 4, which gives you 10. Roll out the 10.Four had “M” showing. (4)(2) = 8, so add 8 to the population of 10, whichgives you 18.)

Your Data Class Data

Year Population Year Population

0 4 0

1 1

2 2

3 3

4 4

5 5

2. Combine your data with the rest of the class' and fill in the table above. Enterthe combined data into a calculator and create a scatter plot. Sketch thescatter plot.

3. Find a function rule to fit the data. Use your calculator to confirm your rule.


Ups and DownsReason and Communicate:

• What would you expect thepopulation to be at each roll? Half ofthe previous amount. Theprobability of getting an "M" for eachM&M is one-half, so you wouldexpect to get about one-half of theprevious amount each time.

• How close were your results towhat you would expect? What kindof graph did you get? Decreasingexponential.

• What are some reasons that yourdata might not be quite what youexpected? The "M"s may be hard tosee on some of the colors likeyellow. The lighting may be bad.Your sample size may not be verybig. (As a general rule, the largerthe sample size, the more likely it isthat the data will fit the expectedprobabilities.)

Answers:

6. A rule that fits the sample class data about is yx

= •18721

2.


Ups and DownsActivity 3, cont.

4. Fill your cup with M&M’s. Then count them. Put the M&M's back in the cup,shake the cup and roll out the M&M's. Remove the M&M’s that have an "M"showing. Record the remaining population. Repeat until no M&M’s are left.

Your Data Class Data

Year Population Year Population

0 0

1 1

2 2

3 3

4 4

5 5

5. Combine your data with the rest of the class' and fill in the table above.Enter the combined data into a calculator and create a scatter plot. Sketchthe scatter plot.

6. Find a function rule to fit the data. Use your calculator to confirm your rule.


Ups and Downs

Answers:

1a. 1001

2•

x

. b.

2. a.

b.


• At what height is the ball forbounce number 0? The ball is at100 cm when it is dropped.

• What is the height of each succes-sive bounce? Each successivebounce height is 50% of the previ-ous bounce height.

• What function rule can you writefor the data? Height is 100 times50% raised to the time,height = 100 times 0.5time, or

ht

= ( )100 0 5.

Math Notes:

To find the percent return forExercise 2, Part a, divide eachmaximum height by the previousheight. Use mental math to approxi-mate the average percent return.Use this average to find the expo-nential rule:

yx

= ( )start percent return .



1. Assume that you have a ball that bounces at a 50% return rate. If you dropthe ball from 100 cm, how high will the ball bounce the first time? Fill in thetable for successive bounces. Create a scatter plot of bounce height inrelation to bounce number.

BounceMaximum

Height

0 100

1

2

3

4

5

a. Write a function rule for your data.b. Build a scatter plot of your data with your calculator and confirm that your

rule fits the data.

2. Use a motion detector with an appropriate program to collect the distance fromthe floor to a bouncing ball.

• Hold the ball at least 1.5 feet under the motion detector.• Collect data for about 6 seconds.• Repeat if necessary to get good data.

a. Trace to find the return height of each bounce and record the data below.Build a scatter plot of bounce height in relation to bounce number.

b. Use the percent return to find an exponential function rule to fit the data.

BounceMaximum

HeightPercent Return

Maxim

um

Heig

ht

Bounce Number


Looking Back, Looking Forward

Institute Notes

Concept: Connect institute materials to your classroom.

Overview: In this lesson, participants do two things in grade-levelgroups. First, they go back to each Reflect and Apply,Exercise 2 and discuss their ideas for application in theclassroom. Second, they look at their own texts and adaptquestions to promote algebraic thinking.

TEKS Focus: 6.3, 6.4, 6.5, 6.11, 6.12, 6.137.3, 7.4, 7.5, 7.13, 7.14, 7.15

8.3, 8.4, 8.5, 8.14, 8.15, 8.16

Materials: Blank transparencies, Overhead pens, Presentation pador computer with projection device

Procedure: 1. Assign participants to grade specific groups: 6th, 7th,and 8th.

2. Assign Activity 1. Have groups make presentations tothe whole group.

3. For Activity 2, go over Exercise 1 with the whole group,brainstorming ways to make algebraic connections.

4. Assign Exercise 2 and invite groups to make presen-tations.

Extension: Which of the activities in the institute could you do usingspreadsheets to build algebraic reasoning? One exampleis to use spreadsheets to build the grids in Grids Galore.After students have used mental math to complete somegrids, then have them use spreadsheets to complete gridswith more complicated numbers. Have them look for pat-terns and answer questions based on the spreadsheetgrids.


Assessment: Journal entry:

1. The key things that I learned about algebraic rea-soning are . . .

2. One of the ways that I will incorporate ideas from theinstitute is . . .

3. What about algebraic reasoning do I want to learnmore about?

Notes:




Review your suggestions for implementing the institute activities in yourclassroom from all the Reflect and Apply, Exercise 2 items throughout theinstitute.

1. Discuss your strategies and applications with other members in your gradelevel group. Prepare to deliver a joint presentation of your collective workfor the whole group.

2. Disucss your strategies and applications in your vertical team. Prepare todeliver a joint presentation of your collective work for the whole group.


Looking Back, Looking ForwardActivity 1

Review your suggestions for implementing the institute activities in your class-room from all the Reflect and Apply, Exercise 2 items throughout the institute.

1. Discuss your strategies and applications with other members in your gradelevel group. Prepare to deliver a joint presentation of your collective workfor the whole group.

2. Disucss your strategies and applications in your vertical team. Prepare todeliver a joint presentation of your collective work for the whole group.



Math Notes:

1. These questions were adapted from middle school math books on thestate adoption list.

a. This is from an 8th grade book.

b. This is from a 6th grade book.


Looking Back, Looking ForwardActivity 2

1. How could you rewrite these problems to be problem situations that promotealgebraic reasoning?

a. A coastline is changing at a rate of -3 3

4 inches per year. At that rate, by

how many inches will the coastline have changed in 75 years?

b. Scott has a flowerpot in the shape of a cylinder. The container has a 5"radius and an 8" height. To determine the amount of potting soil to buy, heneeds to know the volume of the flowerpot. What is the volume, to thenearest cubic inch?

2. Review your textbook for examples of algebraic reasoning or situations thatlend themselves to algebraic reasoning. Specifically, find some problems thatyou can change to problem situations that enhance your teaching of algebraicthinking. Identify the algebraic habits of mind discussed during the institute:patterns to rules, doing and undoing, and abstracting from computations.


Materials

Consumables Amount

Large sketch pads 1

Markers 5-6 per group

Dry-erase pens 1-2 per group, several for presenter

Coffee stirrers or flat spaghetti 1 per participant

1" grid paper several per group

Peel-and-stick dots several per group

Pipe cleaners, light-colored 1 per participant

Aluminum foil 1 strip per group

Masking tape 1 roll

Paper cups 1 per participant

M&M's about 60 per participant

Non-consumables Amount

Ruler or measuring tapes 1 per participant

Stop watch or watch with second hand 1 per group

100' or 150' Measuring tapes 1 per group

Data Collection Cards 1 per group

Secret Instruction Cards 1 per group

Data collection device 1 per group

Motion sensor 1 per group

Graphing calculators 1 per participant

Set of color tiles 1 per group

Small battery operated vehicles 1 per group

Balls (Basketball, racquetballs, and golfballs work well.)

1 per group

Set of overhead color tiles 1 for presenter

Overhead graphing calculator 1 for presenter

Overhead projector 1 for presenter (2 is best)

Blank transparencies several

Set of linking cubes 1 per group

Dimes and quarters 18 of each

Presentation pad or computer with

projection device

1 for presenter


References

Barnes, M. Investigating Change. An Introduction to Calculus for AustralianSchools. Curriculum Corporation, 1991.

Berner, L., McLaughlin, P., & Verzoni, K. "Algebra Affinity." Mathematics Teach-ing in the Middle School 2 (February 1997): 248-249.

Brueningsen, C., Brueningsen, E., & Bower, B. Math and Science in Motion:Activities for Middle School. Texas Instruments, Inc., 1997.

Browning, C. & Channell, D. Graphing Calculator Activities for Enriching MiddleSchool Mathematics. Texas Instruments, Inc., 1997.

Day, R. & Jones, G. "Building Bridges to Algebraic Thinking." MathematicsTeaching in the Middle School 2 (February 1997): 208-212.

Driscoll, M. Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10.Heinemann, 1999.

Ferrini-Mundy, J., Lappan, G., & Phillips, E. "Experiences with Patterning."Teaching Children Mathematics 3 (February 1997): 282-288.

Fouche, K. "Algebra for Everyone: Start Early." Mathematics Teaching in theMiddle School 2 (February 1997): 226-229.

Kennedy, P.A. Developing Algebraic Thinking: A Laboratory Approach for Begin-ning Algebra. Willow, New York: DistanceEd.com, 1999.

Kennedy, P. A., McGowan, D., Schulz, J. E., Hollowell, K. A., & Jovell, I. HRWAlgebra One Interactions: Course 1. Orlando: Holt, Rinehart and Winston, 1998.

Kennedy, P. A., McGowan, D., Schulz, J. E., Hollowell, K. A., & Jovell, I. HRWAlgebra One Interactions: Course 2. Orlando: Holt, Rinehart and Winston, 1998.

Leutzinger, L., ed. Mathematics in the Middle, National Council of Teachers ofMathematics & National Middle School Association, 1998.

McCoy, L. "Algebra: Real-life Investigations in a Lab Setting." MathematicsTeaching in the Middle School 2 (February 1997): 220-224.

Murdock, G., Kamischke, E., Kamischke, E. Discovering Algebra: An Investiga-tive Approach. The Preliminary Edition. Berkley, CA: Key Curriculum Press, 1999.

Patterson, A. "Building Algebraic Expressions: A Physical Model." MathematicsTeaching in the Middle School 2 (February 1997): 238-243.


Owens, D. Research Ideas for the Classroom: Middle School Mathematics.National Council of Teachers of Mathematics, 1993.

Reeuwijk, M. & Wijers, M. "Student's Construction of Formulas in Context."Mathematics Teaching in the Middle School 2 (February 1997): 230-236.

Ruopp F., Cuoco, A., Rasala, S. & Kelemanik, M. "Algebraic Thinking: A Profes-sional-Development Theme." Teaching Children Mathematics. 3 (February 1997): 326-329.

Schultz, J.E., Kennedy, P.A., Ellis, W., Hollowell, K.A., Jovell, I. Algebra I.Orlando: Holt, Rinehart and Winston, 2001.

Stacey, K. & MacGregor, M. "Building Foundations for Algebra." MathematicsTeaching in the Middle School 2 (February 1997): 252-260.

Willoughby, S. "Activities to Help in Learning about Functions." MathematicsTeaching in the Middle School 2 (February 1997): 214-219.

Willoughby, S. "Functions: From Kindergarten Through Sixth Grade." TeachingChildren Mathematics 3 (February 1997): 314-318.

Yackel, E. "A Foundation for Algebraic Reasoning in the Early Grades." Teach-ing Children Mathematics 3 (February 1997): 276-280.


Appendix C—Graphing Calculator Keystrokes

Texas InstrumentsTI-73

TI-80

TI-82

TI-83 and 83+

CasioCFX-7400

CFX9850

How To Build a Table on the TI-73

To demonstrate how to build a table on the TI-73, let’s use the following example. You want to look at theX and Y values for the equation Y=X2.

Action Keystrokes:Turn the calculator ON. Press …Go to Y= Screen Press oIf you have equations entered, you will want to clearthem first.

Press

Place the cursor beside Y1= Use arrows if necessaryEnter equation Press Ø°

To display the Table:Make sure the equation is selected by highlighting thedesired equation

Press

Show Table Press yιTo change the Table Range and have the table start at a new number (example: -2)

Press y φPress αΖ

Change (delta)table to .5 Press # ΡView Table Press −ι




Press�

Place the cursor beside Y1= Use arrows if necessaryEnter equation Press �ϒ


Press ⊆

Show Table Press ψιTo change the Table Range and have the table start at a new number (example: -2)

Press ψφ Press αΖ

Change (delta)table to .5 Press #ΡView Table Press −ι




Press�



Press ⊆


Press ψφ Press αΖ


How To Build a Table on the TI-83 or TI-83 Plus

To demonstrate how to build a table on the TI-83 or TI-83 Plus, let’s use the following example. You wantto look at the X and Y values for the equation Y=X2.


Press �



Press ⊆


Press ψφPress αΖ


How to Produce a Scatterplot on the TI-73

Action Keystrokes:Turn the calculator ON. Press …Enter Stat Editor Press 3Enter x data in L1 Press Ζβ

Press ΘβPress ΣβPress Κβ

Enter Y data in L2 Press ∀Press ΛβPress ϑβPress ΘβPress [β

To graph the data:Enter stat plot editor(Make sure all plots are off, if not, press [4] [enter].)

Press −ε

Enter Stat Plot 1 Press ΨTurn plot ON Highlight ON and Press βSelect the scatter plot Highlight scatter plot and press βSelect L1 as our X list Place cursor next to X list

Press −ϖSelect L2 as our Y list Place cursor next to Y list

Press −ϖΖ

Graph Plot Press (ϑfor ZoomStat


Action Keystrokes:Turn the calculator ON. Press …Enter Stat Editor Press �Select Edit Press ℵEnter x data in L1 Press ℑ⊆

Press ∂⊆Press ÷⊆Press ↑⊆

Enter Y data in L2 Press ∼Press →⊆Press ←⊆Press ∂⊆Press ℜ⊆

To graph the data:Enter stat plot editor(Make sure all plots are off, if not, press [4] [enter],then [2nd] [plot] again.)

Press ψ [statplot]

Enter Stat Plot 1 Press ℵTurn plot ON Highlight ON and Press ⊆Select the scatter plot Highlight scatter plot and press ⊆Select L1 as our X list Highlight L1 and press ⊆Select L2 as our Y list Highlight L2 and press ⊆Change Window Press π�ℵ⊇

Press ��ℵ⊇Graph Plot Press σ






Press ψ [statplot]

Enter Stat Plot 1 Press ℵTurn plot ON Highlight ON and Press ⊆Select the scatter plot Highlight scatter plot and press ⊆Select L1 as our X list Highlight L1 and press ⊆Select L2 as our Y list Highlight L2 and press ⊆

Graph Plot Press θ→for ZoomStat

How to Produce a Scatterplot on the TI-83 and TI-83 Plus





Press ψ[statplot]

Enter Stat Plot 1 Press ℵTurn plot ON Highlight ON and Press ⊆Select the scatter plot Highlight scatter plot and press ⊆Select L1 as our X list Cursor down to X list, press ψ [L1]

([L1] is the [1] key.)Select L2 as our Y list Cursor down to Y list, press ψ [L2]

([L2] is the [2] key.)


How to generate a Random Number on the TI-73

To generate a random number between 0 and 1:Action Keystrokes:

Turn the calculator ON. Press …Start from clear Home Screen Press −λ:Select Random command (Rand) by highlighting PRB (probability)

Press 1∀∀Press Ψto select rand

Run rand command Press β

To generate a random integer between 2 and 8 (2 and 8 were chosen as examples):Turn the calculator ON. Press ⊥Start from clear Home Screen Press −λ:Select Random Integer command (RandInt<) byhighlighting PRB (probability)

Press 1∀∀Press Ζto select randInt(

Enter Lower Limit Press ΖϒEnter Upper Limit Press ΚΕRun randInt( command Press β

***NOTE: To run command again, press ⊆.

How to generate a Random Number on the TI-80


Turn the calculator ON. Press …Start from clear Home Screen Press ψλ�Select Random command (Rand) by highlighting PRB (probability) and selecting rand

Press 1∀∀Press Ψto select rand


To generate a random integer between 2 and 8 (2 and 8 were chosen as examples):Turn the calculator ON. Press ⊥Start from clear Home Screen Press −λ:Select Random Integer command (RandInt<) byhighlighting PRB (probability) and selecting randInt

Press 1∀∀Press Ρto select randInt(

Enter Lower Limit Press ΖϒEnter Upper Limit Press ΚΕRun randInt( command Press β***NOTE: To run command again, press ⊆.

How to generate a Random Number on the TI-82, TI-83 and TI-83 Plus


Turn the calculator ON. Press …Start from clear Home Screen Press ψλ�Select Random command (Rand) by highlighting PRB (probability) and selecting rand

Press 1∀∀∀Press Ψto select rand


To generate a random integer between 2 and 8 (2 and 8 were chosen as examples):Turn the calculator ON. Press …Start from clear Home Screen Press −λ:Select Random Integer command (RandInt<) byhighlighting PRB (probability) and selecting randInt

Press 1∀∀∀Press Ρto select randInt(

Enter Lower Limit Press ΖϒEnter Upper Limit Press ΚΕRun randInt( command Press β

(The TI-82 does not perform Random Integer generations)***NOTE: To run command again, press ⊆.

How to produce a Linear Regression on the TI-73

In this example, we will plot the data for x=2,4,6,8 and y=9,7,4,3.Action Keystrokes:

Turn the calculator ON. Press …Enter Stat Editor Press 3Enter x data in L1 Press Ζβ

Press ΘβPress ΣβPress Κβ

Enter Y data in L2 Press ∀Press ΛβPress ϑβPress ΘβPress [β

To graph the data:Enter stat plot editor(Make sure all plots are off, if not, press [4] [enter].)

Press −ε

Enter Stat Plot 1 Press ΨTurn plot ON Highlight ON and Press βSelect the scatter plot Highlight scatter plot and press βSelect L1 as our X list Place cursor next to X list

Press −ϖΨSelect L2 as our Y list Place cursor next to Y list

Press −ϖΖ

Graph Plot Press (ϑfor ZoomStat

To perform the linear regression using the scatter plot data and graph above:Action Keystrokes:

Perform the linear regression and highlight CALC(calculation)

Press −ϖ∀∀∀Press Ρto select LinReg (ax+b)

Select L1 and L2 as your lists Press −ϖΨϒPress −ϖΖ

Run linear regression Press β

Graph regression: Enter the Y= Editor Press &Clear all previous equations Place cursor next to Y1= and press :Enter equation for linear regression into Y1= Press −}[ to select statistics

Press ∀∀ to highlight EQ (equations)Press Ψ to select RegEq (regressionequation)

Graph Equation Press ∗






Press ψ[statplot]

Enter Stat Plot 1 Press ℵTurn plot ON Highlight ON and Press ⊆Select the scatter plot Highlight scatter plot and press ⊆Select L1 as our X list Highlight L1 and press ⊆Select L2 as our Y list Highlight L2 and press ⊆Change Window Press π�ℵ⊇

Press ��ℵ⊇

Graph Plot Press σ



Press �∼Press •

Select L1 and L2 as your lists Press ψ [L1]′Press ψ [L2](L1 and L2 are found above the ℵand ℑ keys.)

Run linear regression Press [⊆

Graph regression: Enter the Y= Editor Press οClear all previous equations Press �Enter equation for linear regression into Y1= andselect statisitics

Press �ℑto select statisticsPress ∼∼ to highlight EQ (equations)Press • to select RegEq (regressionequation)

Graph Equation Press σ






Press ψ[statplot]

Enter Stat Plot 1 Press ℵTurn plot ON Highlight ON and Press ⊆Select the scatter plot Highlight scatter plot and press ⊆Select L1 as our X list Highlight L1 and press ⊆Select L2 as our Y list Highlight L2 and press ⊆




Press �∼Press •

Select L1 and L2 as your lists Press ψ [L1]′Press ψ [L2](L1 and L2 are found above the ℵand ℑ keys.)

Run linear regression Press [⊆


Press �•to select statisticsPress ∼∼ to highlight EQ (equations)Press ← to select RegEq (regressionequation)


How to produce a Linear Regression on the TI-83 and TI-83 Plus





Press ψ[statplot]

Enter Stat Plot 1 Press ℵTurn plot ON Highlight ON and Press ⊆Select the scatter plot Highlight scatter plot and press ⊆Select L1 as our X list Cursor down to X list, press ψ [L1]

([L1] is the ℵ key.)Select L2 as our Y list Cursor down to Y list, press ψ [L2]

([L2] is the ψ key.)

Graph the Plot Press θ → for ZoomStat



Press �∼Press ∂

Select L1 and L2 as your lists Press ψ[L1]′Press ψ [L2](L1 and L2 are found above the ℵand ℑ keys.)

Run linear regression Press ⊆


Press �• to select statisticsPress ∼∼ to highlight EQ (equations)Press ℵ to select RegEq (regressionequation)


HOW TO USE THE CFX7400

MAIN MENU

**The icon that is highlighted is the one that is currently “selected”.**To enter the mode that is highlighted, press [EXE], or the number in the bottom right corner of the icon.

What does each Icon do?

ICON MEANING

RUN USED FOR ARITHMETIC AND FUNCTIONAL CALCULATIONS

STAT USED TO PREFORM SINGLE AND PAIRED VARIABLE STATISTICAL CALCULATIONS, AND TO DRAW STATISTICAL GRAPHS.

LIST USED FOR STORING AND EDITING NUMERIC DATA.

GRAPH USED TO STORE GRAPH FUNCTIONS AND TO DRAW GRAPHS USING THE FUNCTIONS.

TABLE USED TO GENERATE A TABLE OF NUMERIC VALUES FROM THE DIFFERENT FUNCTIONS.

PRGM USED FOR WRITING, READING, AND EXECUTING PROGRAMS.

CONT USED FOR ADJUSTING THE DISPLAY CONTRAST.

MEM USED TO CHECK HOW MUCH MEMORY IS BEING USED AND HOW MUCH IS REMAINING. ALSO TO RESET CALCULATOR TO INITIAL PARAMETERS.

BASIC KEYS[MENU] Will always bring you back to the MAIN MENU screen.

[QUIT] Will back-out of the last step performed, to leave an option you have chosen whileremaining in the icon mode.

[F1…F4] Function keys- control menu found on the bottom line of the calculator screen.Differ depending on which icon you are operating in.

[SHIFT] Activates yellow operation written above keys.

TABLE ICON

Table Menu

The table menu makes it possible to generate numeric tables from functions stored in memory. Since theTable Menu uses the same list of functions used in the Graph Menu there is no need to reenter the samefunctions into the different modes.

From the Table Mode you can do the following:• Specify the range and increment of values assigned to variables for table value generation.• Assign the list values to variables.• In addition to graphing the stored function, you can also plot table values generated by the table menu.• Table values can be assigned to a list.

The Soft Menu

SEL - [F1] draw or non-draw status

DEL - [F2] delete function

RANG - [F3] choose a specific numerical range for X

TABL - [F4] draw the table

How To Create A TablePress [MENU]Go to TABLE icon

If equations exist please delete-highlight the equation and press [F2] (DEL), [F1] (YES)

Enter this equations: Y=X+2 [EXE]

To change the range:

[QUIT], [F3], Enter new range *[-5, 5], -5[EXE], 5[EXE], [QUIT], [F4] (TABL)

STAT ICON

Stat Menu

In this mode you can input statistical data into lists and calculate the mean, maximum and other statisticalvalues. You can calculate single variable and paired variable statistical data. You will also be able toperform regression calculations and graphs.

How to Create a Scatter Plot and Linear Regression:

From the main menu screen select the STAT mode in order to input the statistical data. The “soft function”keys are as follows:

[F1] (GRPH) … Graph[F2] (CALC) … Statistical calculation[F3] (SRT*A)… Sort in Ascending order[F4] (SRT*D)… Sort I Descending order

By pressing the [4] key you can access additional choices

[F1] (DEL)……..Deletes a single data item[F2] (DEL*A)….Deletes all data[F3] (INS)………Insert a data item

Press [4] to return the previous menu.

Example:

Input the following data:LIST 1 0.5, 1.2, 2.4, 4.0, 5.2

LIST 2 -2.1, 0.3, 1.5, 2.0, 2.4

Use the cursor keys to move the highlighting to any cell.

After the data is entered into the lists, press [F1] (GRPH) todisplay the graph menu.

[F1] (GPH 1)……DRAWS GRAPH 1[F2] (GPH 2)……DRAWS GRAPH 2[F3] (GPH 3)……DRAWS GRAPH 3PRESS [4] FOR MORE CHOICES

[F1] (SEL)……GRAPH SELECTION[F4] (SET)……GRAPH SETTINGS

The default graph type setting is for a scatter diagram withLIST 1 data as x-axis values and LIST 2 data as y-axisvalues

By using the set command you can change the GRAPH TYPE,LISTS USED, FREQUENCY, and MARK TYPE.

To plot a scatter diagram press [F1] (GPH 1).

Once the graph has been drawn the following soft menusappear:

[F1]……Linear regression[F2]……Med-Med graph[F3]……Quadratic regression graph

PRESS [4] FOR MORE CHOICES

[F1]……Logarithmic regression[F2]……Exponential regression[F3]……Power regression[F4]……Paired-variable statistical results

To display a linear regression for the data above press:

[4] [F1] (X)

This will display the data used in writing the logarithmic regression.

From this screen you are given two choices:

[F3] (COPY)……Stores the displayed regression formula as a graphfunction

[F4] (DRAW)……Graphs the displayed regression formula

Run Menu

Many functions can be run in this menu including the random number selection.

How to Create a Random Number

Press the [MENU] key Press the [OPTN] keyGo to the RUN icon

Press [F4] (PROB) Press [F4] (RAN#), [EXE] key

Numbers will be randomly generated to the tenth decimal place, which is the default mode.To fix the decimal to another place value:

Press the [SHIFT] key Press [F1] Fix - choose the numberthen the [MENU]/SET UP key of decimals by desired selectingand down arrow to Display the appropriate [F1-6] key

Press the [EXIT] key and continue to press the [EXE] key.

HOW TO USE THE CFX9850

Main Menu

Menu Descriptions

Icon Mode Name Description

RUN Use this mode for arithmetic calculations andfunction calculations, and for calculationsinvolving binary octal, decimal and hexadecimalvalues.

STATistics Use this mode to perform single-variable(standard deviation) and paired-variable(regression) statistical calculations, to performtests, to analyze data and to draw statisticalgraphs.

MATrix Use this mode for storing and editing matrices.

LIST Use this mode for storing and editing numericdata.

GRAPH Use this mode to store graph functions and todraw graphs using the functions.

DYNAmic graph Use this mode to store graph functions and todraw multiple versions of a graph by changing thevalues assigned to the variables in a function.


TABLE Use this mode to store functions, to generate anumeric table of different solutions as the valuesassigned to variable in a function change, and todraw graphs.

RECURsion Use this mode to store recursion formulas, togenerate a numeric table of different solutions asthe values assigned to variables in a functionchange, and to draw graphs.

CONICS Use this mode to draw graphs of implicit functions.

EQUAtion Use this mode to solve linear equations with twothrough six unknowns, quadratic equations, andcubic equations.

PRoGraM Use this mode to store programs in the programarea and to run programs.

LINK Use this mode to transfer memory contents orback-up data to another unit.

CONTrast Use this mode to adjust the color contrast of thedisplay.

MEMory Use this mode to check how much memory isused and remaining, to delete data from memory,and to initialize (reset) the calculator.

KNOW THESE THREE THINGS TO MAKE THE 9850 EASY TO USE1. [MENU] key - Shows all main topics. Eliminates having to know where hidden

functions are.2. [EXIT] key - Takes you back one screen, but not to the main menu.3. Function keys [F1] through [F6] - Read bottom of screen for options pertinent to the

icon you are in. The menu bar at the bottom of the screen changes in every icon.

Table MenuThe table menu makes it possible to generate numeric tables from functions stored in memory.Since the Table Menu uses the same list of functions used in the Graph Menu there is no need toreenter the same functions into the different modes.


TABLE Use this mode to store functions, to generate anumeric table of different solutions as the valuesassigned to variable in a function change, and todraw graphs.

The Soft MenuSEL - [F1] draw or non-draw statusDEL - [F2] delete functionTYPE -[F3] graph type menuCOLR - [F4] graph colorRANG - [F5] choose a specific numerical range for XTABL - [F6] draw the table

How To Create A TablePress [MENU]Go to TABLE icon

If equations exist please delete-highlight the equation and press [F2] (DEL), [F1] (YES)Example:Enter this equations:Y=X+2 [EXE], [F6] (TABL)

To Change the Range: [EXIT], [F5] (RANG): Enter new range *[-5, 5]: -5[EXE], 5[EXE], [EXIT]

Statistics MenuThis menu will allow input of statistical data into lists, perform single-variable and paired-variablestatistical calculations, perform tests, analyze data and draw statistical graphs.


STATistics Use this mode to perform single-variable(standard deviation) and paired-variable(regression) statistical calculations, to performtests, to analyze data and to draw statisticalgraphs.

The Soft MenuGRPH [F1] – Graph menu which allows graphing of data as a scatter diagram, xy line, normalprobability plot, histogram, med-box, mean-box, and many more

CALC [F2] – Statistical calculation menu which allows single-variable and paired-variablestatistical calculations

TEST [F3] – Test menu that performs z tests, t tests, etc.

INTR [F4] – Confidence interval menu

DIST [F5] – Distribution menu

[4] [F6] - Turn the soft menu page

SRT-A [F1] and SRT-D [F2] – Sorts data in ascending/descending order

DEL [F3] and DEL-A [F4] – Deletes highlighted data or all data

INS [F5] – Inserts new cell at highlighted cell

[4] [F6] - Turn the soft menu page

How to Create a Scatterplot and Linear RegressionPress [MENU]Go to STAT icon

Delete any data in the lists. Press [F6] (>), [F4] (DEL-A), [F1] (YES)Example:Enter the following into List 10 500 1000 1500 2000 2500 3000 3500

Enter each data point followed by [EXE].

Press right arrow to go to List 2.Enter the following into List 2.32 27 23 18 14.5 9 3.5 -3Enter each data point followed by [EXE].

Graph the Data:Press [F1] (GRPH)

Press [F6] (SET)Down arrow to Graph Type, Press [F1] (Scat), [EXIT]

Press[F1], (GPH1)

To Draw a Linear Regression:Press [F1] (X), equation, correlation coefficient

Press [F6] to DRAW ([F5] will COPY to Graph Menu)

Troubleshooting:If the screen is not fitting to the data, press [SHIFT] [MENU] and check that the “Stat Wind” is setto “Auto”.

How to Create a Random NumberPress the [MENU] key Press the [OPTN] keyGo to the RUN icon

Press [F6] [4] - Turn the soft menu page Press [F3] (PROB)

Press [F4] (RAN#), [EXE] key

Numbers will be randomly generated to the tenth decimal place, which is the default mode.To fix the decimal to another place value:Press the [SHIFT] key Press [F1] Fix - choose the numberthen the [MENU]/SET UP key of decimals by desired selectingand down arrow to Display the appropriate [F1-6] key

Press the [EXIT] key and continue to press the [EXE] key.The decimal place was changed in Set Up: Default is Normal 1, [F1] (Fix), [F2] (1) and [F2] (0)

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