Functions Defined by Data

MATH IN FLIGHT

First Class Postage Rates Year Postage

(cents)Year Postage

(cents)

1885 2 1976 13

1918 3 1978 15

1919 2 1981 18

1932 3 1982 20

1958 4 1985 22

1963 5 1988 25

1968 6 1991 29

1971 8 1995 32

1974 10 1999 33

(Source of data: Associated Press, 7/2/2000.)

Postage Stamp

0

5

10

15

20

25

30

35

1860 1880 1900 1920 1940 1960 1980 2000 2020

Year

Cost

of O

ne F

irst-C

lass

Sta

mp

Piecewise Solution1885 To 1958

y = 0.0274x - 49.857

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1880 1900 1920 1940 1960 1980

Year

Cost

of S

tam

p (c

ents

)

1963 To 1999

y = 0.8984x - 1761.4

0

5

10

15

20

25

30

35

40

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Cost

of S

tam

p

• The year is 2006. The current price of a first-class stamp is $0.39. Does this fit the trendline?

• From the data, what do you predict the price of a stamp will be in 2010?

Path of Object

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100 120 140

Time (minutes)

Dist

ance

(fee

t)

What is this?

Mystery Data

• Dist vs. Time data• How fast is the object moving?• Is the speed changing?• Connections to distance vs. time graph?• (For fun): what is this object?

Path of Object

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100 120 140

Time (minutes)

Dist

ance

(fee

t)

Calculator Lists: TIMEA and TDIST

Time(s) Total Dist(ft) Time(s) Total Dist(ft) Time(s) Total Dist(ft)

0 0 12 1564 24 2834

1 137 13 1682 25 2926

2 275 14 1800 26 3017

3 412 15 1919 27 3105

4 545 16 2029 28 3193

5 679 17 2136 29 3277

6 809 18 2243 30 3361

7 942 19 2346 31 3441

8 1068 20 2453 32 3521

9 1198 21 2552 33 3601

10 1324 22 2643 34 3677

11 1442 23 2743 35 3754

Path of Object

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 100 120 140

Time (minutes)

Dist

ance

(fee

t)

Change in Distance wrt Time

0

20

40

60

80

100

120

140

160

0 20 40 60 80 100 120 140

Time (sec)

Chan

ge in

Dis

tanc

e (ft

)

Time (s) Distance

(ft) Time (s) Distance

(ft) Time (s) Distance

(ft)1 137 13 118 25 92

2 138 14 118 26 91

3 137 15 119 27 88

4 133 16 110 28 88

5 134 17 107 29 84

6 130 18 107 30 84

7 133 19 103 31 80

8 126 20 107 32 80

9 130 21 99 33 80

10 126 22 91 34 76

11 118 23 100 35 77

12 122 24 91 36 72

Calculator Lists: TIMEB and DDIST

http://flightaware.com/live/

• Flight Aware Live Tracker Web Page• Select a plane type – Boeing 767• Select a flight – Delta Flight -

– Cincinnati to Atlanta• Select STATUS to see details of flight

32.5

33

33.5

34

34.5

35

35.5

-85 -84.5 -84 -83.5 -83 -82.5 -82 -81.5 -81 -80.5

FLIGHT PATH

0

50

100

150

200

250

300

350

400

450

0 10 20 30 40 50 60 70

Distance Traveled (miles) vs. Time (min.)

TRIPy = 6.8707x - 0.2906

-50

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60

Time (min)

Tota

l Dis

tanc

e

Trip

Linear (Trip)

Altitude (ft) vs. Time (min)

0

5000

10000

15000

20000

25000

0 10 20 30 40 50 60

Time (min)

Altit

ude

(ft)

Piecewise FunctionAltitude

0

5000

10000

15000

20000

25000

0 2 4 6 8 10 12

Time (min)

Altit

ude

(ft)

Altitude

y = 2172.4x + 923.53

0

5000

10000

15000

20000

25000

0 2 4 6 8 10 12

Time (min)

Altit

ude

(ft)

Altitude

Linear (Altitude)

For 0 < t < 10 sec.

Altitude (ft) vs. Time (min)

0

5000

10000

15000

20000

25000

0 10 20 30 40 50 60

Time (min)

Alti

tude

(ft)

Middle Piece10 < t < 42

0

5000

10000

15000

20000

25000

0 5 10 15 20 25 30 35 40 45

Time (min.)

Altit

ude

(ft)

2nd Piece

y = 22000

0

5000

10000

15000

20000

25000

0 5 10 15 20 25 30 35 40 45

Time (min)

Altit

ude

(ft)

10 < t < 42

Finish with Model of Descent

0

5000

10000

15000

20000

25000

0 10 20 30 40 50 60

Time (min)

Alti

tude

(ft)

Atlanta to DullesFlight From Atlanta, GA to Dulles, Washington DC

0

100

200

300

400

500

600

0 10 20 30 40 50 60 70 80 90 100

Time (minutes)

Gro

und

Spee

d (m

ph)

Atlanta to Dulles

0

5000

10000

15000

20000

25000

30000

35000

40000

0 10 20 30 40 50 60 70 80 90 100

Time (minutes)

Altit

ude

(ft)

Collect Live Data

• Students can collect live data• Model the data they collect with

mathematical equations.• Linear to Quadratic to Other models• Rate of Change – Average to

Instantaneous for Precalculus to Calculus

Thank YouDiane LeightyMathematics CoordinatorPowhatan County Public Schoolsdiane.leighty@powhatan.k12.va.us

Powerpoint will be posted to website:www.powhatan.k12.va.us/teachers/dleighty/Dleighty.htm