visual calculus university of texas at el paso spring - 2002 ana franco veronica herrera amra kanlic...
DESCRIPTION
Hypothesis Calculus concepts can be taught in middle schools. OptimizationTRANSCRIPT
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.