Visual Calculus

University of Texas at El PasoSpring - 2002

Ana FrancoVeronica HerreraAmra Kanlic Gabriel Mendoza Alister NgAlfredo RodriguezLina Yick

Introduction Hypothesis Intervention - Cone problem Video- Classroom activity Pretest Posttest Solutions Results Acknowledgements Discussion

Hypothesis

Calculus concepts can be taught in middle schools.

Optimization

InterventionOn a hot, sunny El Paso day, you and your buddies decide to buy some ice cream. When you get to the ice cream shop, there’s a sign that reads,

"SALE! Eat ice cream by making your own waffle cone for only $1.00!" .

The store sells you a circular waffle and allows you to roll it up into a cone, which will be filled with ice cream for a $1. Of course you want the most amount of ice cream in your waffle cone.

Intervention Supplies• Cone (circular disk)

• Container with 500 ml of Salt

• Ruler

• Measuring spoons

Video

14 26514 25014 20513.9 26515 23512.25 20512 21514 24210 18015.5 157.512 23512.5 23512 23514 23513.5 21014.5 21512.5 23514.5 21511.5 24011.5 24011 20513.5 24311 20512.5 23514 25012 220

d (cm) V (ml)1234567891011121314151617181920212223242526

S # Students’ Data

11 12 13 14 15 16 17

160

180

200

220

240

2601

2

3

4

5

67

8

9

10

11 1213

14

15 16

17

18

1920

21

22

23

2425

26

d (cm)

V (ml)

r2 +h2 = 82

h=±

"

64 - r2h

8

r V= 13

h p r2

V H

rL

= 13

p r2

"

64 - r2

h

r

8

8 cm diameter

88

8

8

Finding Equation

(1)

(2)

(3)

(4)

Substitute the value of h into equation (3).

Graph of diameter v.s volume

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

25

50

75

100

125

150

175

200

225

250

V ik d

2

y{= p

12d2

$64 - d2

4

d (cm)

V (ml)

Students’ graph

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

25

50

75

100

125

150

175

200

225

250 12

3

4

5

6

7

8

9

10

111213 14

1516

1718

19 20

21

22

2324

25

26

d (cm)

V (ml)

V' H

rL

= p3

i

k2 r

"

64 - r2 - r3è

64 - r2

y

{

= p3

i

k 2 r H

64 - r2L

- r3è

64 - r2

y

{

= p3

i

k - rH

3 r2 - 128L

è

64 - r2

y

{

r =±$

1283

=± 8$

23

3 r2 - 128= 0

Take the derivative with the respect to r.

To solve

Thus,

V' HrL=0, 64 - r2 π0.

V

i

k 8$

23

y

{ ª 206.37 cm3 = 206.37 ml

r = 8$

23

ª 6.532 cm

d= 2 r ª 13.064 cm

Calculus Solution

Pretest

Figure 1.

Three firefighters are called about a fire at a house (H), figure 1. The fire station is located at F. Each firefighter Ana, Bobby, and Roberto leave the fire station with an empty bucket. They each pick a path to the ocean to fill their bucket with water. Then they proceed to the burning house.

Figure 2.

Pretest Graph

a) Who ran the shortest distance to the ocean? b) By person, rank the distance they had to run to the ocean (from shortest

to longest). 1.2.3.

c) Who ran the shortest total distance to the house? d) By person, rank the total distance they ran from the fire station to the

burning house (from shortest to longest). 1.2.3.

e) Is there an optimal way to run to shorten your distance from the station to the house? Explain.  

f) Which way would you run to the ocean to create the shortest path from the station to the burning house (create your own path)? Label your path on figure 1 and also on figure 2. Explain.

Questions

PosttestFour fighter fighters: Ana(A), Bobby(B), Edith(E) and Roberto (R) are called about a fire at a house (H), figure 1. The fire station is located at F. Each firefighter leaves the fire station with an empty bucket.

Figure 3.

Figure 4.

Posttest Graph

a) By using figure 2, who ran the best way to the house? Why?

  b) To save the house from the fire, which way would you

run? If there is not a point on the graph, then mark your own point and explain this path on figures 1 and 2.

Questions

Pretest Solution

Ocean

F

H

B A RX

Posttest Solution

Results

Acknowledgements Alister’s 8th grade classes at St. Joseph Erandi Perez and O. Perez

References Lott, W. Johnny and Smith, Paul. (1979). Reflections on

putting out a fire. School Science and Mathematics, 79 (5), 434-38.

Sobel A. Max and Maletsky M. Evan. (1999). Teaching mathematics : a sourcebook of aids, activities, and strategies. Boston, MA: Allyn and Bacon.

http://www.math.utep.edu/Student/alfredo/opt/optimization.html

Discussion

Any questions?

Thank you.