Casio FX-CG10 Manuel utilisateur

Fostering Advanced Algebraic Thinking with Casio Technology

CHAPTER 1—POLYNOMIAL FUNCTIONS

Mathematics curricula often emphasize the practical applications of quadratic

functions, yet there is limited justification provided for studying higher-order

polynomial functions. In this chapter, we do just that!

We motivate the study of higher-order polynomial functions by introducing a classic

investigation into maximizing the volume of a cone—to hold ice cream, of course.

We then progress to using polynomial functions to model trends in data. For

example, a quadratic is used to model the results of a study on the empowerment of

women in Bangladesh, a much different use of the quadratic function than generally

found in algebra curricula. Higher-order polynomial functions are also used to

model the average nightly hotel rates in US cities and the number of lives saved

using airbags, seatbelts, and motorcycle helmets.

Features of polynomial functions, including intervals of increase and decrease and

relative minima and maxima, are the focus of these investigations. The goal is for

students to identify these features, and interpret the features with respect to the

data. We are hopeful that the investigations will help students better understand

polynomial functions and develop a greater understanding of the role these

functions can assume in identifying trends in data.



Chapter 1: Polynomial Functions

Investigation 1.1—Homemade Waffle Cones

Did You Know?

Italo Marchiony sold homemade ice cream from a pushcart on Wall Street. Often,

his customers wandered off with the ice cream dishes, or worse yet, broke them.

To curb these expenses, Italo baked edible waffle cups with sloping sides and flat

bottoms. He patented his idea in 1903.

Reference: http://www.makeicecream.com/hisoficecrea.html

The local scholarship committee is planning to sell ice cream cones at high school

events. The scholarship committee is interested in serving the ice cream in waffle

cones, and, the committee has a waffle maker that will make circular waffles 21

centimeters in diameter.

Let’s investigate how the scholarship committee can design a waffle cone with the

greatest volume using a circular waffle, with a diameter of 21 centimeters. To do so,

cut out a circle (from paper, not waffle yet) that is 21 centimeters in diameter.

Remove a sector from the circle, and construct a cone. Be sure to tape or glue your

cone together, as we will be investigating the dimensions of the cone that will

maximize its volume.

a. What is the slant height of the cone? Did you need to use a ruler to

determine this measurement? Explain.

b. Without measuring, write an equation for the radius of the base of the cone

you constructed in terms of the arc length you removed from the circle.

[Hint: Write an equation for the circumference of the base of the cone first.]

c. Again, without measuring, write an equation for the height of the cone you

constructed in terms of the arc length you removed from the circle.



Fostering Advanced Algebraic Thinking with Casio Technology

Investigation 1.1—Homemade Waffle Cones (continued)

d. Write an equation for the volume of the cone in terms of the arc length you

removed.

e. Determine the arc length that must be removed to construct a cone with the

maximum volume.

f. What would be the dimensions of the cone that has a maximum volume?

Construct another cone from a circle of diameter 21 cm, but now construct a cone

with maximum volume.

g. Do you recommend the scholarship committee make waffle cones with these

dimensions? Why or why not?



Chapter 1: Polynomial Functions

Sample Solution—Homemade Waffle Cones

a. What is the slant height of the cone? Did you need to use a ruler to

determine this measurement? Explain.

The slant height of the cone is the radius of the circle it was constructed from,

so the slant height is approximately 10.5 centimeters.

b. Without measuring, write an equation for the radius of the base of the cone

you constructed in terms of the arc length you removed from the circle.

[Hint: Write an equation for the circumference of the base of the cone first.]

The circumference of the base of the cone is the circumference of the paper

circle less the arc length removed. Since the diameter of the paper circle is 21

centimeters, the circumference is 21

centimeters. Therefore, the

circumference of the base of the cone is 21

centimeters less the length of the

arc removed, a, or 21

a centimeters.

To find the radius of the base of the cone, we recall the formula for the

circumference of a circle is C = 2

r. Solving this equation for the radius, r,

gives the circumference divided by 2

. So, the radius of the base of the cone

is r =

21 a= 10.5 



Fostering Advanced Algebraic Thinking with Casio Technology



Sample Solution—Homemade Waffle Cones (continued)

c. Again, without measuring, write an equation for the height of the cone you

constructed in terms of the arc length you removed from the circle.

The height of the cone can be determined using the Pythagorean Theorem, as

the radius of the base of the cone and the height of the cone are two legs of a

right triangle in which the slant height is the hypotenuse. Therefore, (radius of

cone)2+ (height of cone)2= (slant height of cone)2, or (height of cone)2=

(slant height of cone)2(radius of cone)2. Substituting the expressions from

parts a and b:

(height of cone)2= (10.5)2

510 ¸

§

(height of cone)2= (10.5)2



ª¸



510

(height of cone)2= 2

510

aa. 

height of cone = 2

510

aa. 

Note: we are only considering the positive root.

d. Write an equation for the volume of the cone in terms of the arc length you

removed.

Recall, the volume of a cone is V = hr 2

. Using the expressions from parts b

and c, the volume of the cone we constructed is:

V = 2

510

aa.a

.

§



Chapter 1: Polynomial Functions

e. Determine the arc length that must be removed to construct a cone with the

maximum volume.

Table:

Sample Solution—Homemade Waffle Cones (continued)

xPress pto return to the Main Menu, then 7to access the Table

mode.

xInput the equation for the volume of the cone as a function of the arc

length removed from the circle into Y

. [Notice: This may be

challenging for students, as they must be very careful to use grouping

symbols to ensure the order of operations are adhered to. Yet, the

natural display of the Casio PRIZM calculator allows the students to

quickly assess whether or not they entered the function correctly.]

xNow, press y(SET) to set the input values for x (a, arc length) to be

used in the table. Let’s create a table of arc lengths from 0 to 75

centimeters, inclusive. To do so, input the following values: S

START:

0l, next to E

END:

75l.

xLet’s set the S

STEP

value to be 5, meaning the x-values in the table will

be increasing by 5. Press 5l.



Fostering Advanced Algebraic Thinking with Casio Technology



From the table, we determined that the arc length to be removed should be

greater than 5 centimeters and less than 15 centimeters. To be more precise,

we either need to change the settings of our table, or use a different

representation—such as the graph or the equation. It is important that

students are able to move flexibly between and within representations, and

that they also learn to identify the strengths and weaknesses of each

representation when solving problems.

Graph:

To construct a graph using the calculator:

xPress pto return to the Main Menu, then 5to access the Graph

mode. Notice the function used in the Table mode appears in the Graph

mode as well.

xSet the view window, by press Le

(V-Window)

. The length of the

arc removed must be greater than or equal to 0. We have chosen to set

the input values for the view-window as 0 to 75. To do so, input the

following values: X

Xmin:

0l; X

Xmax:

75l; s

scale:

5l.

xSimilarly, the volume of the cone must be greater than or equal to 0; so

we have chosen to input the following values: Y

Ymin

:0l;

Xmax:

500l; s

scale:

100l.

xPress dto return to the equation screen, and press u(DRAW) to

draw the graph of the equation in the view window you defined.



Sample Solution—Homemade Waffle Cones (continued)

Chapter 1: Polynomial Functions

Sample Solution—Homemade Waffle Cones (continued)

xTo determine the arc length that will maximize the volume of the cone,

press Ly(

G-Solv)

for the graph solver, then press w(MAX). The

calculator will identify the maximum of the graph selected.

Equation:

To determine the arc length that must be removed to maximize the volume of

the cone symbolically requires knowledge of the concept of derivative. This is

beyond our expectations of students studying advanced algebraic concepts.

Yet, notably, graphing technology often provides students with access to

mathematical concepts that they may not be prepared to investigate

symbolically.



Fostering Advanced Algebraic Thinking with Casio Technology

f. What would be the dimensions of the cone that has a maximum volume?

Since the arc length of the cone, with a maximum volume, is approximately

12.1 cm, we need to substitute this into the expressions from parts b and c to

determine the radius and the height of this cone. Recall, from part a, we know

the slant height is 10.5 centimeters.

We chose to enter the equations from parts b and c into the Table mode of

the calculator to approximate the dimensions of the cone with maximum

volume, as follows:

Sample Solution—Homemade Waffle Cones

The radius and the height of the cone are approximately 8.6 and 6.1

centimeters, respectively.

g. Do you recommend the scholarship committee make waffle cones with these

dimensions? Why or why not?

If the goal is to make waffle cones with the greatest volume, the committee

should make the waffle cones with these dimensions. However, students may

also want to discuss if these dimensions result in a cone that “fits” the human

hand. Some containers are manufactured to be easy for the consumer to

handle or use, and others are manufactured with the goal of being

aesthetically appealing.



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