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Report Series Published by SRI International

Research collaborators:

University of Massachusetts, Dartmouth; Virginia Polytechnic Institute and State University; The University of Texas at Austin; and the Charles A. Dana Center at the University of Texas at Austin

Scaling Up SimCalc Project

Jennifer Knudsen

Technical Report 07 June 2010

Design and Development of Curriculum Units and Professional Development

SRI InternationalCenter for Technology in Learning333 Ravenswood AvenueMenlo Park, CA 94025-3493650.859.2000

Authorization to reproduce this publication in whole or in part is granted. While permission to reprint this publication is not necessary, the citation should be: Knudsen, J. (2010) Design and Development of Curriculum Units and Professional Development (SimCalc Technical Report 07). Menlo Park, CA: SRI International.


Prepared by:

Jennifer Knudsen, SRI International

Acknowledgments

This report is based on work supported by the National Science Foundation under Grant No. 0437861. Any opinions, findings, and conclusions or recommendations expressed in this report are those of the authors and do not necessarily reflect the views of the National Science Foundation.

We thank J., Roschelle, N. Shechtman, G. Estrella, G. Haertel, K. Rafanan, P. Vahey, S. Carriere, L. Gallagher, H. Javitz, T. Lara-Meloy, M. Robidoux, S. Empson, S. Hull, L. Hedges, S. Goldman, H. Becker, J. Sowder, G. Harel, P. Callahan, F. Sloane, B. Fishman, K. Maier, J. Earle, R. Schorr, and B. McNemar for their contributions to this research. We thank the participating teachers and Texas Educational Service Center leaders from regions 1, 6, 9, 10, 11, 13, 17 and 18; this project could not have happened without them. We thank and remember Jim Kaput, who pioneered SimCalc as part of his commitment to democratizing mathematics education.

4 © 2010 SRI International — Design and Development of Curriculum Units and Professional Development


The SimCalc research team conducted two experiments to test whether a combination of professional development and SimCalc-based curriculum materials could be used by a wide variety of teachers to support their students’ learning of conceptually complex mathematics. The experiments were conducted with about 150 seventh- and eighth-grade teachers from across the state of Texas. This technical report describes the content of the units, their features, the training that accompanied the units and issues in its development.

Unit ContentTwo curriculum units were designed for these experiments, one for seventh-graders and one for eighth-graders. Each of the curriculum units addresses Texas state standards and also includes topics that were more challenging than those in the standards. Managing the Soccer Team addresses seventh-grade standards on rate and proportionality and includes multi-rate functions and the meaning of slope. Designing Cell Phone Games targets eighth-grade standards for linear function and introduces average rate through further exploration of multi-rate functions. A detailed description of each unit—the storyline, the activities, the main mathematical ideas, and the learning goals—is appended. What follows is a more compact description.

Both units follow a similar progression. Each begins with simple analyses of motion at a constant speed, and follows a learning progression that culminates in the more complex topics. Moving from qualitative to quantitative analyses, the unit develops graphical, tabular and symbolic forms of linear functions. In the seventh grade units, the functions are directly proportional, so that each point represents the same rate of change. In the eighth grade units, proportional and non-proportional functions are distinguished. Each

of these topics is found in the Texas state standards. The progression then continues in each unit to address multi-rate functions—piecewise linear functions. The eighth grade unit continues on to explore average rate. Each of these topics is developed in contexts that can be represented in the SimCalc MathWorlds®’ “world” window, a place where motion or growth can be shown with characters and other representations of real-world phenomena. Motion and non-motion contexts are important in both units: both to help students generalize across contexts and to provide another opportunity to learn complex content.

A tour through Managing the Soccer Team will illustrate one of the progressions in some detail:

Early in the soccer team unit, students observe a simple line graph and a simulation of a soccer player running along a straight line (Exhibit 1). The graph and simulation are linked so that as the runner moves forward, the graph of that motion builds. In the graph window, the x-axis represents time and the y-axis represents distance traveled. At first, students just identify the starting and ending time of the runner, correlating the runner’s position in the world window and the graph window. A series of questions in the student workbook guides

5 Technical Report 07 June 2010

students from this initial interpretation of the graph to the understanding that the entire line represents the speed of the runner, the ratio of distance to time.

In the middle of the unit, Run, Jace, Run introduces symbolic algebraic notation to represent the same type

of motion (Exhibit 2). Students express the relationship between time and distance first in a graph, then in a table, then in words, and finally in an equation. Both table and language representations of motion enable students to write the equation of the line. Through more activities, students develop facility in connecting representations and translating from one to another.

Exhibit 1. Activity from Early in the Unit


Back to the Office is an activity that parallels Run, Jace, Run: it presents the same content in a non-motion context. Students solve problems involving, for example, best buys on soccer cones and setting ticket prices for the team’s games. Back to the Office re-introduces content in a non-motion context, allowing students to generalize but also to learn what they may have missed the first time around. There are a number of these activities throughout the unit.

On the Road consists of a series of stories about the troubled history of the team’s trips from its hometown to Dallas. In each trip, a bus and a van traveled between the two cities, stopping, slowing and speeding up—and sometimes turning around and going the other way. The activity allows students to build correspondences between multi-rate, piecewise linear graphs and verbal descriptions of motion. Slope is dealt with qualitatively. This activity is where the unit starts to move beyond state standards for seventh grade. But the unit’s storyline and contexts remain familiar, as does the core concept of rate.

Exhibit 2. Activity Midway through the Soccer Unit


Unit FeaturesIn addition to detailing the mathematics learning progression, the unit designers used a set of features to support teachers and students in their use of the unit.

Both the teacher notes and the student materials were designed to be helpful to teachers in planning and implementing each unit. Teachers who might never open their teacher notes could instead use the student workbook as a kind of default guide or lesson plan. Important questions for the class to consider are written into the student book—in fact, every important question that the developers could think of was treated this way. On the other hand, the workbook is not a prescriptive script—teachers were free to write their

own lesson plans, augmenting or highlighting different parts of the student workbook. A third option was to use the teachers’ notes to adapt premade day-by-day lesson plans. These plans provided extra questions to ask students, and included sample student responses. A suggested pacing chart helped teachers plan for how to complete the unit in whatever amount of time they might have—with the recommended time being 10 days.

Other features of the units were designed to help students directly. SimCalc MathWorlds® software allows animated versions of objects and people in motion in contexts. The theme of each unit provides continuity across these contexts and the storyline provides detail that the software could not capture.

Exhibit 3. Activity from Second Half of Unit


Realistic numbers are used so that students could check speeds, prices and other rates against their knowledge of how expensive uniforms are or how fast people run. The text uses simple sentence structure and consistent vocabulary, never going beyond a fifth-grade reading level, in order to accommodate those with low-level reading skills and those just learning to read English in making sense of the context and the math. To help guide and organize students’ work, the workbook uses graphical conventions to indicate various kinds of activities and content. For example, definitions appeared inside boxes on the page, as did other critical content information. The amount of white space left after a question indicated the type and length of an expected answer. Simple graphics served as implicit indices for the activities. Even the fact that the workbook contained all the student activities physically bound together provided another organizational aid to students. The workbooks contained as much color as the budget would allow, appealing to media-savvy students who are used to plentiful use of color.

Professional DevelopmentThe seventh-grade and eighth-grade experiments used two different professional development (PD) models. The first was designed to best ensure that the researchers’ and developers’ intentions were reflected in the PD and therefore reached teachers. The second was designed to be better aligned with professional development practices in Texas. The workshops offered in the PD were less than a week long, about as much as could be expected from teachers for training in teaching a replacement unit.

For Managing the Soccer Team, the goals of the 5 days of training were to

• Provide teachers with a mental image of the unit as a whole, as well as direct experience with most activities.

• Improve teachers’ content knowledge, both in terms of the mathematics in the unit—going deeper—and the mathematics that might come after the unit—going farther.

• Allow teachers to develop comfort with the software

• Model teaching strategies consistent with a SimCalc approach.

• Show how the unit fit in with local customs and addressed state standards.

The workshop had three parts. In the first part, a well-known local expert in math professional development led the teachers in transitioning from a view of proportionality as a/b = c/d—a relationship among four numbers—to a broader view of proportionality as a linear function of the form y = kx—relating an infinite number of pairs of numbers. This understanding was key to considering rate and proportionality in the SimCalc context, where functions are the primary object of study. Additionally, this transition was advocated by influential state education groups; materials similar to theirs were used in the workshops. For the next two days, a member of the SRI team led teachers through the Managing the Soccer Team unit, using their workbooks and the software. The leader modeled a small set of SimCalc strategies such as “predict, check, explain,” in which teachers were encouraged to predict what would happen in SimCalc MathWorlds® simulations before running them and then explain the differences between what they predicted and what actually happened. To boost teachers’ comfort with the software, the leader gave demonstrations and encouraged teachers to play with the software, as well as use it in the unit activities. In the third part of the workshop, teachers used additional activities and the software to explore calculus concepts, learning about the relationship between graphs of velocity and speed. Teachers learned this more advanced mathematics core to SimCalc, both so that they could understand where the mathematics in the soccer unit could lead and so that they could experience new content, in the same way that Managing the Soccer Team is likely to be for their students.

The eighth grade experiment used a two-tiered system of training—a “train the trainers” approach. The same pair of professional developers led a 2-day training for local professional development experts, who then delivered a 3-day training session for teachers in the


regions of Texas that they served. These local experts were provided with curriculum, teacher notes and presentation slides. They were asked to do a training much like that described for the second two parts of the seventh-grade study.

Balancing ConstraintsEvery curriculum is designed within constraints—which might be about the people, the resources or the politics in a given situation. Designing the Scaling Up SimCalc units was no different—but with the added constraint of the experimental setting in which the materials were used.

Designing for the experiment and for other constraints was a balancing act. The units, of course, needed to well represent the SimCalc approach to teaching and learning—that is what the experiments were designed to test. Typically, SimCalc is used to help young students, middle through high school, learn the fundamental ideas of calculus by providing real-life interpretations for differentiation and integration (hence the same SimCalc). But these topics were more advanced that those in the middle school mathematics standards in Texas, where the experiments took place. The teachers, administrators and policymakers associated with the experiment wanted to ensure that materials used by thousands of students would address the most important topics in their standards. Finding a balance between typical SimCalc topics and state standards-based content resulted in the content described in this report. By inching past the standards, with multi-rate functions, we managed to stay within the comfort zone of local education leaders as well as many teachers, while still pushing into the “precalculus” territory of SimCalc.

Another part of the experimental design was recruiting a wide variety of teachers in terms of background, experience, pedagogical style, and mathematics knowledge. In designing the teacher supports described, this wide variety was considered. The materials needed to meet the needs of teachers who struggled with

the math, without over-constraining teachers with advanced content knowledge. The teachers with advanced content knowledge, though, might be still developing pedagogical skills, and so both the skillful questioner and the teacher struggling to keep order in class needed to be supported. Finally, other work had revealed that many of today’s teachers do little written planning before delivering their lessons. So the training included time and structure to help teachers do that planning, an important part of teaching particularly for novice teachers.

Pedagogy is often embedded in curriculum and made explicit during training. Traditionally, SimCalc materials presented fairly complex and open-ended problem situations in which students could gradually develop mathematical insights. This was not the commonly endorsed pedagogy at the time, and not many teachers would have had exposure to such methods. So the experiment’s units were more highly structured than customary SimCalc materials, and the PD leaders did not push teachers to adopt wholly new practices.

ConclusionThis report described several aspects of the curriculum and professional development used in two scaling up experiments. Paper materials and software served to guide students in an exploration of real-world contexts and associated mathematics representations, focusing on rate, proportionality, and linear function. Developers of the unit took into account not only the mathematics that could be learned using a SimCalc approach, but also state standards that were essential to address. Teachers were supported in their classroom use of the materials through a set of teacher notes and professional development that focused on teachers’ mathematics learning and effective implementation of the unit. The developers took into consideration then-current state standards and assessments in their curriculum and PD design.

10 © 2010 SRI International — Design and Development of Curriculum Units and Professional Development 11 Technical Report 07 June 2010

App

endi

x A

Mat

h C

onte

nt in

the

Man

agin

g th

e So

ccer

Tea

m a

nd D

esig

ning

Cel

l Pho

ne G

ames

Tabl

e A

1. M

anag

ing

the

Soc

cer T

eam

Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

1• M

anag

ing

the

Soc

cer T

eam

• A R

ace

Day

• Ano

ther

Rac

e D

ay

• Inf

o Q

uest

In a

gra

ph o

f obj

ects

in m

otio

n at

a

cons

tant

spe

ed in

the

coor

dina

te

plan

e, th

e x-

axis

typi

cally

repr

esen

ts

time;

the

y-ax

is, d

ista

nce;

and

th

e sp

eed

can

be fo

und

from

any

po

int o

n th

e lin

e re

latin

g tim

e an

d di

stan

ce.

1. F

ind

the

spee

d of

an

obje

ct m

ovin

g at

a c

onst

ant

rate

, sta

rting

at 0

tim

e an

d 0

dist

ance

, by

calc

ulat

ing

from

the

endp

oint

s an

d la

ter f

rom

any

po

int o

n th

e lin

e.

2. Id

entif

y th

e x-

axis

, y-a

xis,

line

gra

ph, a

nd g

iven

m

eani

ng fo

r the

m in

spe

cific

con

text

.

Stu

dent

s ar

e ap

poin

ted

as

tem

pora

ry te

am m

anag

ers

whe

n th

e cu

rren

t man

ager

mys

terio

usly

le

aves

tow

n. T

he fi

rst t

ask

is to

tim

e pl

ayer

s m

akin

g da

shes

, st

raig

ht-li

ne ru

ns d

esig

ned

to

impr

ove

play

ers’

spe

ed.

Stu

dent

s al

so fi

nd in

form

atio

n ab

out t

he re

al-li

fe s

peed

of

runn

ers

and

othe

r obj

ects

and

cr

eatu

res.

2• I

sabe

lla Im

prov

es

• Fas

ter T

han

Max

• Pra

ctic

e R

uns

For g

raph

s of

obj

ects

in m

otio

n at

a

cons

tant

spe

ed (f

ollo

win

g th

e co

nven

tions

abo

ve):

If di

stan

ce is

con

stan

t, as

the

amou

nt o

f tim

e de

crea

ses,

the

spee

d in

crea

ses.

Ste

eper

line

s re

pres

ent f

aste

r sp

eeds

.

1. U

nder

stan

d re

latio

nshi

ps a

mon

g sp

eed,

tim

e,

and

dist

ance

, as

repr

esen

ted

by g

raph

s an

d nu

mbe

rs o

r des

crib

ed in

wor

ds.

2. D

escr

ibe

patte

rns

in a

gra

ph.

3. G

iven

a ti

me/

dist

ance

gra

ph o

f the

mot

ion

of tw

o ob

ject

s (s

ingl

e ra

te o

nly

each

), de

term

ine

the

dist

ance

bet

wee

n ob

ject

s at

a g

iven

tim

e.

4. H

ave

a qu

alita

tive

unde

rsta

ndin

g th

at g

reat

er

slop

es (i

nfor

mal

ly, s

teep

er) r

epre

sent

fast

er

mot

ion,

in ti

me/

dist

ance

gra

phs.

5. D

raw

tim

e/di

stan

ce g

raph

s fo

r tw

o m

ovin

g ob

ject

s (e

ach

at a

con

stan

t rat

e), g

iven

suf

ficie

nt

info

rmat

ion.

(e.g

.; tim

e, e

nd p

oint

s, re

lativ

e tim

es)

Stu

dent

s ex

plor

e th

e re

latio

nshi

ps

amon

g lin

es o

f das

hes

of d

iffer

ent

spee

ds a

nd d

urat

ion.


Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

3• R

un, J

ace,

Run

• Run

, Jac

e, R

un,

Rev

isite

d

Giv

en a

n ob

ject

mov

ing

at a

co

nsta

nt s

peed

and

a g

raph

and

ta

ble

repr

esen

ting

rela

ted

dist

ance

s an

d tim

es, e

ntrie

s in

the

tabl

e ca

n be

foun

d gi

ven

the

num

bers

in tw

o ro

ws.

Eac

h ro

w in

the

tabl

e ca

n be

us

ed to

cal

cula

te th

e sp

eed.

Eve

ry

poin

t on

the

asso

ciat

ed li

ne g

raph

ca

n be

use

d to

cal

cula

te th

e co

nsta

nt

spee

d/st

eepn

ess

of th

e lin

e. F

rom

th

e ta

ble

or th

e gr

aph,

it is

pos

sibl

e to

writ

e an

equ

atio

n th

at is

als

o a

repr

esen

tatio

n of

the

cons

tant

spe

ed

and

that

can

be

used

to fi

nd th

e di

stan

ce a

t any

tim

e.

In p

revi

ous

less

ons,

spe

ed c

ould

be

cal

cula

ted

usin

g th

e tim

e an

d di

stan

ce a

t the

end

poin

t of a

gra

ph.

Now

spe

ed is

a ra

te th

at re

late

s ea

ch ti

me

to a

dis

tanc

e an

d ca

n be

ex

pres

sed

as a

n eq

uatio

n w

ith tw

o va

riabl

es.

1. A

ssum

ing

a co

nsta

nt s

peed

(pro

porti

onal

re

latio

nshi

p), c

ompl

ete

entri

es in

a ti

me/

dist

ance

ta

ble

by a

var

iety

of v

alid

met

hods

reas

onin

g ac

ross

and

dow

n ta

ble

colu

mns

.

2. U

se th

e ta

ble

to w

rite

an e

quat

ion

that

rela

tes

time

to d

ista

nce.

3. Id

entif

y gr

aphs

, equ

atio

ns, a

nd ta

bles

re

pres

entin

g th

e sa

me

prop

ortio

nal r

elat

ions

hip.

One

stu

dent

’s ru

nnin

g is

use

d to

exp

lore

and

uni

te ta

bles

, eq

uatio

ns, a

nd g

raph

s.

4• B

ack

At t

he O

ffice

The

sam

e id

eas

as in

less

on 2

are

de

velo

ped

in th

e co

ntex

t of m

oney

an

d ob

ject

s.

1. A

ssum

ing

a co

nsta

nt ra

te re

latin

g tw

o va

riabl

es

(pro

porti

onal

rela

tions

hip)

, com

plet

e en

tries

in

a ta

ble

by a

var

iety

of v

alid

met

hods

reas

onin

g ac

ross

and

dow

n ta

ble

colu

mns

.

2. U

se th

e ta

ble

to w

rite

an e

quat

ion

that

rela

tes

the

two

varia

bles

.

3. D

escr

ibe

a si

tuat

ion

that

cou

ld b

e re

pres

ente

d by

a g

iven

equ

atio

n (u

sing

dis

cret

e, n

onm

otio

n co

ntex

t).

4. Id

entif

y gr

aphs

, equ

atio

ns, a

nd ta

bles

re

pres

entin

g th

e sa

me

prop

ortio

nal r

elat

ions

hip.

Man

agin

g th

e te

am in

clud

es o

ff-fie

ld a

ctiv

ities

, too

. Thi

s le

sson

us

es p

robl

ems

abou

t buy

ing

socc

er c

ones

and

uni

form

s,

for e

xam

ple,

to d

evel

op th

e sa

me

idea

s of

pro

porti

onal

ity

and

func

tions

as

in th

e ea

rlier

ac

tiviti

es. T

his

also

gen

eral

izes

th

e id

eas

pres

ente

d ea

rlier

; th

ey a

pply

in m

ore

than

mot

ion

cont

exts

.

Tabl

e A

1. M

anag

ing

the

Soc

cer T

eam

(Con

tinue

d)


Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

5• S

lope

and

Rat

eTh

e de

finiti

ons

of ra

te, u

nit r

ate,

an

d sl

ope

can

be c

onne

cted

to

the

prev

ious

con

text

s an

d re

pres

enta

tions

. A tr

aditi

onal

way

to

cal

cula

te s

lope

is b

y cr

eatin

g tri

angl

es b

ased

on

rise/

run

as re

late

d to

a li

ne g

raph

.

1. U

nder

stan

d an

d us

e de

finiti

on o

f uni

t rat

e.

2. C

onne

ct u

nit r

ate

to s

lope

. Und

erst

and

slop

e as

a n

umer

ical

des

crip

tion

of th

e re

latio

nshi

p be

twee

n an

y tw

o po

ints

on

a lin

e.

This

act

ivity

is a

pur

e m

athe

mat

ics

time-

out t

o gi

ve m

athe

mat

ical

vo

cabu

lary

for t

he c

once

pts

expl

ored

mor

e in

form

ally

ear

lier.

6• O

n th

e R

oad

• Roa

d Tr

ips

Gra

phs

with

con

nect

ed li

ne

segm

ents

at d

iffer

ent s

lope

s ca

n re

pres

ent a

n ob

ject

mov

ing

at

diffe

rent

spe

eds

(with

x-a

xis

as ti

me,

y

as d

ista

nce)

• Pos

itive

slo

pe re

pres

ents

mov

ing

ahea

d (in

the

posi

tive

dire

ctio

n.)

• Neg

ativ

e sl

ope

repr

esen

ts

mov

ing

back

war

d (in

the

nega

tive

dire

ctio

n.)

• Whe

n th

e sl

ope

is 0

, the

re

pres

ente

d ob

ject

is n

ot m

ovin

g.

• The

leng

th o

f the

0 s

lope

line

se

gmen

t giv

es th

e pe

riod

of ti

me

the

obje

ct is

sta

ndin

g st

ill.

1. In

terp

ret g

raph

s re

pres

entin

g m

ovin

g ob

ject

s w

ith tw

o or

mor

e ra

tes;

giv

e co

ntex

tual

sto

ry

asso

ciat

ed w

ith e

ach

grap

h. G

iven

sto

ry, c

reat

e gr

aph.

2. M

ake

thes

e co

nnec

tions

/ int

erpr

etat

ions

:

• P

ositi

ve s

lope

mea

ns m

ovin

g ah

ead.

• N

egat

ive

slop

e m

eans

goi

ng b

ack.

• 0

slo

pe m

eans

sta

ndin

g st

ill.

• T

he le

ngth

of t

he 0

slo

pe li

ne in

dica

tes

how

long

st

andi

ng s

till.

The

team

has

a m

yste

rious

ly

bad

hist

ory

of m

akin

g tri

ps to

th

e st

ate

cham

pion

ship

s. T

he

trips

are

take

n w

ith b

oth

a bu

s an

d a

van.

Stu

dent

s ex

plor

e an

d cr

eate

sto

ries

that

mat

ch g

raph

s re

pres

entin

g th

ese

trips

.

7• G

raph

s an

d M

otio

nN

o ne

w m

athe

mat

ics

cont

ent

This

act

ivity

pro

vide

s pr

actic

e in

m

otio

n co

ntex

ts, i

nclu

ding

sin

gle-

an

d m

ultip

le-r

ate

mot

ions

and

th

e co

ordi

natio

n of

gra

phs

and

narr

ativ

es. I

t als

o he

lps

stud

ents

le

arn

to in

terp

ret “

on s

ight

” cer

tain

ty

pica

l mul

tirat

e lin

ear g

raph

s.

Rec

ogni

ze ty

pica

l gra

ph s

hape

s fo

r:

• Sta

ndin

g st

ill

• Goi

ng o

ne ra

te, t

hen

goin

g at

a fa

ster

rate

for t

he

sam

e am

ount

of t

ime

• Goi

ng o

ne ra

te, t

hen

goin

g at

a s

low

er ra

te fo

r the

sa

me

amou

nt o

f tim

e

• Goi

ng fo

rwar

d an

d th

en b

ackw

ard

at th

e sa

me

rate

Tabl

e A

1. M

anag

ing

the

Soc

cer T

eam

(Con

tinue

d)


Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

8 - 9

• Sal

ary

Neg

otia

tions

• Sum

mer

Job

Adv

ice

• All

Abo

ut M

PG

• How

Far

On

ow

Muc

h? M

PG

• Sui

ting

Up

No

new

mat

hem

atic

s co

ncep

ts.

Exe

rcis

es a

nd p

robl

em s

olvi

ng u

se

conc

epts

intro

duce

d ea

rlier

.

Stu

dent

s ha

ve d

one

so w

ell a

s in

terim

man

ager

that

they

are

of

fere

d th

e jo

b, w

ith tw

o ch

oice

s of

sal

ary.

Afte

r the

y de

cide

whi

ch

choi

ce is

bes

t, th

ey th

en a

dvis

e ot

her s

tude

nts

who

are

faci

ng

sim

ilar i

ssue

s in

mak

ing

sum

mer

jo

b ch

oice

s.

10• M

athe

mat

ical

ly

Spe

akin

gIf

you

can

writ

e a

form

ula

y =

kx

whe

re k

is a

real

num

ber a

nd y

and

x

repr

esen

t var

ying

qua

ntiti

es, w

ith

a si

ngle

y v

alue

for e

ach

x, th

en w

e ca

n sa

y th

at th

e qu

antit

ies

x an

d y

vary

pro

porti

onal

ly.

A un

it ra

te te

lls h

ow m

any

y yo

u ge

t fo

r one

x.

Use

the

defin

ition

s of

uni

t rat

e an

d pr

opor

tiona

lity

to

iden

tify

thes

e co

ncep

ts in

the

curr

icul

um.

Tabl

e A

1. M

anag

ing

the

Soc

cer T

eam

(Con

tinue

d)

Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

1• W

orki

ng a

t Tex

Sta

r G

ames

• Cel

l Pho

ne G

ames

an

d D

esig

n

• Yar

i, th

e Ye

llow

S

choo

l Bus

• Our

Firs

t Cel

l P

hone

Gam

e

Mot

ion

can

be re

pres

ente

d on

a

time/

dist

ance

gra

ph. L

ines

can

re

pres

ent i

deal

ized

mot

ion

(with

out

chan

ges

in s

peed

).

1. F

ind

the

dist

ance

that

an

obje

ct m

oves

, its

spe

ed,

and

for h

ow m

uch

time

it tra

vels

giv

en a

tim

e/di

stan

ce g

raph

of t

hat m

otio

n re

pres

ente

d as

a lin

e.

2. C

alcu

late

spe

ed fr

om ti

me

and

dist

ance

—fro

m

the

endp

oint

s of

the

grap

h an

d fro

m a

ny p

oint

on

the

grap

h (b

egin

ning

leve

l).

3. C

ompa

re th

e gr

aph

of th

e m

otio

n of

an

obje

ct

“idea

lized

” as

a lin

e w

ith a

gra

ph th

at is

mor

e ac

cura

te.

Stu

dent

s ar

e hi

red

at th

e ce

ll ph

one

desi

gn fi

rm T

exS

tar G

ames

to

use

mat

h to

impr

ove

the

busi

ness

. The

stu

dent

s us

e m

ath

to d

esig

n ho

w th

e ga

mes

wou

ld

wor

k.

Tabl

e A

2. D

esig

ning

Cel

l Pho

ne G

ames


Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

2• C

ontro

lling

C

hara

cter

s w

ith

Gra

phs

Gra

phs

can

be m

athe

mat

ical

re

pres

enta

tions

of m

otio

n, fr

om

whi

ch s

tude

nts

can

com

pare

po

sitio

ns, s

peed

s, a

nd s

tarti

ng

plac

es o

f mov

ing

obje

cts.

For

a

posi

tion/

time

grap

h, s

teep

er li

nes

repr

esen

t fas

ter s

peed

s.

1. F

ind

spee

d us

ing

mul

tiple

met

hods

in a

gra

ph—

unit

rate

, “sl

anty

-nes

s” (i

nfor

mal

ver

sion

of

slop

e), a

nd e

nd p

oint

s.

2. U

se m

ath

know

ledg

e to

cha

nge

the

spee

d of

ob

ject

s in

the

softw

are

by c

hang

ing

the

grap

h.

3. U

nder

stan

d th

e co

nnec

tion

betw

een

info

rmal

sl

ope

and

spee

d.

4. In

terp

ret w

ord

prob

lem

s ab

out s

peed

s, s

tart,

and

en

d tim

es u

sing

gra

phs.

Stu

dent

s us

e gr

aphs

to c

ontro

l the

sp

eeds

of c

ars

in d

esig

ning

a ro

ad

rally

gam

e.

3–4

• Con

trolli

ng

Cha

ract

ers

with

E

quat

ions

• One

to A

noth

er

• Bet

ter G

ames

Equ

atio

ns a

re a

noth

er fo

rm o

f m

athe

mat

ical

repr

esen

tatio

n us

ed to

re

pres

ent f

unct

ions

(wor

d us

ed o

nly

info

rmal

ly),

as a

re ta

bles

.

The

equa

tion

y =

kx +

b is

a c

omm

on

form

for w

ritin

g lin

ear f

unct

ions

, w

here

y a

nd x

are

var

iabl

es a

nd k

an

d b

are

cons

tant

s. If

x is

tim

e an

d y

is d

ista

nce,

k u

sual

ly re

pres

ents

sp

eed

and

b is

the

star

ting

poin

t of

the

obje

cts.

Tabl

es c

an h

elp

us w

rite

equa

tions

.

One

can

tran

slat

e am

ong

grap

hs,

tabl

es, a

nd e

quat

ions

.

1. W

rite

equa

tions

of t

he fo

rm y

= k

x fo

r mov

ing

obje

cts

star

ting

from

a fi

xed

orig

in (t

ime

0,

posi

tion

0).

2. W

rite

equa

tions

of t

he fo

rm y

= k

x +

b fo

r mov

ing

obje

cts

that

sta

rt at

diff

eren

t non

-zer

o po

sitio

ns

3. C

ompl

ete

tabl

es, e

quat

ions

, and

gra

phs

base

d on

suf

ficie

nt in

form

atio

n ac

ross

all

thre

e.

Stu

dent

s us

e eq

uatio

ns to

con

trol

the

mot

ion

of c

hara

cter

s in

cel

l ph

one

gam

es, i

nclu

ding

thos

e w

ith

thre

e ch

arac

ters

.

5• W

ende

lla’s

Jou

rney

Mul

tiseg

men

t gra

phs

can

repr

esen

t an

obj

ect m

ovin

g at

diff

eren

t spe

eds.

You

can

tell

a st

ory

to e

xpla

in th

e m

otio

n of

an

obje

ct o

ver t

ime

to

mat

ch s

uch

a gr

aph.

Flat

(inf

orm

al s

lope

0) l

ines

repr

esen

t st

andi

ng s

till.

Line

s th

at a

re s

lant

ed d

ownw

ard

(info

rmal

ver

sion

of n

egat

ive

slop

e)

repr

esen

t mov

ing

back

war

d.

1. C

reat

e gr

aphs

that

repr

esen

t for

war

d m

otio

n at

sl

ower

and

fast

er s

peed

s, b

ackw

ard

mot

ion

and

no m

otio

n.

2. C

reat

e st

orie

s th

at m

atch

suc

h gr

aphs

. Cre

ate

grap

hs th

at m

atch

sto

ries

abou

t mot

ion.

Stu

dent

s cr

eate

jour

neys

for a

ga

me

char

acte

r tha

t is

a do

g fe

atur

ed in

a g

ame

abou

t mot

ion.

Tabl

e A

2. D

esig

ning

Cel

l Pho

ne G

ames

(Con

tinue

d)


Less

onA

ctiv

ities

Mai

n

mat

hem

atic

al id

eas

Goa

ls

(stu

dent

s w

ill b

e ab

le to

)S

tory

line

6• M

oney

Mat

ters

Mul

tiseg

men

t gra

phs

can

be u

sed

to

show

oth

er k

inds

of a

ccum

ulat

ion

over

tim

e, s

uch

as m

oney

in a

ban

k ac

coun

t.

1. In

terp

ret m

ultis

egm

ent g

raph

s in

a ti

me/

mon

ey

situ

atio

n.

2. P

redi

ct u

sing

mul

tiseg

men

t gra

phs.

Stu

dent

s he

lp T

exS

tar u

nder

stan

d its

cas

h ba

lanc

e.

7• M

athe

mat

ical

ly

Spe

akin

g: G

raph

s to

Kno

w

• Cra

b Ve

loci

ty

Velo

city

is s

peed

with

dire

ctio

n.

Neg

ativ

e ra

tes

indi

cate

bac

kwar

d m

otio

n. P

ositi

on c

an b

e ne

gativ

e,

whe

re 0

is a

defi

ned

star

ting

poin

t th

at o

bjec

ts c

an b

e on

eith

er s

ide

of.

1. S

ketc

h gr

aphs

that

sho

w, f

or e

xam

ple,

in

crea

sing

and

then

dec

reas

ing

rate

s of

mot

ion,

in

crea

sing

and

then

dec

reas

ing

rate

s of

sav

ings

.

2. R

epre

sent

bac

kwar

d m

otio

n w

ith n

egat

ive

rate

s an

d do

this

in e

quat

ions

.

Stu

dent

s us

e eq

uatio

ns a

nd

grap

hs to

con

trol c

rabs

in a

gam

e in

whi

ch c

rabs

mov

e ab

ove

and

belo

w th

e w

ater

line.

8• W

olf a

nd R

ed

Rid

ing

Hoo

dN

o m

atte

r how

cha

ract

ers

mov

e, if

th

eir m

otio

n gr

aphs

hav

e th

e sa

me

endp

oint

then

they

mee

t at t

he s

ame

time

in th

e sa

me

plac

e.

1. M

ake

qual

itativ

e pr

edic

tions

of a

sin

gle

rate

that

ge

ts a

n ob

ject

to th

e sa

me

plac

e at

the

sam

e tim

e as

an

obje

ct th

at is

mov

ing

at tw

o di

ffere

nt

rate

s.

2. U

se e

quat

ions

to re

pres

ent a

sin

gle

rate

that

get

s an

obj

ect t

o th

e sa

me

plac

e at

the

sam

e tim

e as

an

obje

ct th

at is

mov

ing

at tw

o di

ffere

nt ra

tes.

Stu

dent

s an

alyz

e a

gam

e in

whi

ch

play

ers

try to

kee

p R

ed R

idin

g H

ood

from

bei

ng e

aten

by

a w

olf.

9• S

ecre

ts o

f Ave

rage

R

ate

Rev

eale

d

• Pro

blem

Sol

ving

• Pro

blem

s fro

m th

e Te

xSta

r Lun

chro

om

Aver

age

rate

is th

e si

ngle

rate

that

co

uld

“sta

nd in

” for

the

rate

s to

be

aver

aged

; it w

ould

acc

ompl

ish

the

sam

e th

ing.

For e

xam

ple,

if a

n ob

ject

is tr

avel

ing

at s

ever

al ra

tes

then

the

aver

age

rate

is

a s

ingl

e ra

te th

at w

ill ge

t the

obj

ect

to th

e sa

me

plac

e at

the

sam

e tim

e.

1. U

nder

stan

d th

e co

nnec

tion

betw

een

rate

s an

d th

eir a

vera

ge.

2. S

olve

pro

blem

s in

volv

ing

aver

age

rate

s—to

find

av

erag

e ra

tes,

mis

sing

rate

s, a

nd to

acc

ount

fo

r diff

eren

t am

ount

s of

tim

e fo

r var

ious

rate

s in

ca

lcul

atin

g th

eir a

vera

ge.

3. U

se g

raph

s to

est

imat

e or

cal

cula

te a

vera

ge ra

tes.

Stu

dent

s so

lve

prob

lem

s fro

m th

e bu

lletin

boa

rd a

nd c

onve

rsat

ion

in

the

lunc

hroo

m a

t wor

k.

10• L

inea

r R

elat

ions

hips

: P

ropo

rtion

al a

nd

Non

-pro

por-

tiona

l

• Tex

Sta

r Gam

es:

Goi

ng F

ull T

ime

Pro

porti

onal

rela

tions

hips

can

be

exp

ress

ed a

s y

= kx

, lin

ear

rela

tions

hips

can

be

expr

esse

d as

y

= kx

+ b

. The

refo

re, p

ropo

rtion

al

rela

tions

hips

are

line

ar re

latio

nshi

ps

whe

re b

= 0

. Whe

n b

is n

ot =

to 0

, th

en th

e lin

ear r

elat

ions

hip

is n

on-

prop

ortio

nal.

All

func

tions

that

are

not

lin

ear a

re a

lso

non-

prop

ortio

nal.

Dis

tingu

ish

betw

een

prop

ortio

nal a

nd n

on-

prop

ortio

nal e

quat

ions

usi

ng g

raph

s or

equ

atio

ns o

r fro

m s

ituat

ions

exp

ress

ed in

wor

ds.

Stu

dent

s ex

plai

n to

Tex

Sta

r why

th

e w

ork

they

hav

e do

ne in

the

past

2 w

eeks

qua

lifies

them

for

a po

sitio

n; b

oth

mat

hem

atic

al

and

nonm

athe

mat

ical

wor

k ar

e in

clud

ed.

Tabl

e A

2. D

esig

ning

Cel

l Pho

ne G

ames

(Con

tinue

d)

Sponsor: The National Science Foundation, Grant REC - 0437861

Prime Grantee: SRI International. Center for Technology in Learning

Subgrantees: Virginia Polytechnic Institute and State University; University of Massachusetts, Dartmouth; The University of Texas at Austin; and The Charles A. Dana Center at The University of Texas at Austin

SRI InternationalCenter for Technology in Learning333 Ravenswood AvenueMenlo Park, CA 94025-3493650.859.2000

www.ctl.sri.comwww.math.sri.com

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