Whatcom Math Project
Fish Tank Investigation
Course: Precalculus
Grade Level: 10-12
College Readiness Standard Name(s) and Number(s):
Standard 1: The student uses logical reasoning and
mathematical knowledge to define and solve problems.
Standard 7: The student accurately describes and applies
concepts and procedures from algebra.
Standard 8: The student accurately describes and applies
function concepts and procedures to understand mathematical relationships.
Student Attributes:
Perseveres when faced with time-consuming or complex tasks.
 Is willing to work on problems that require time and thought, particularly problems
that cannot be solved by mimicking a previously seen example.
Student Learning Outcomes − College Readiness Standard Component and
Number:
1.3 Evaluate potential solutions for appropriateness, accuracy, and suitability to the
context of the original problem.
7.2 Combine and simplify algebraic expression that contain polynomials
8.2 Represent advanced functions (cubic) using and translating among words, tables,
graphs, and symbols.
8.3 Analyze and interpret features of a function.
8.4 Model situations and relationships using a variety of functions (cubic)
Learning objectives:
 Evaluate a cubic to generate a graph.
 Given an algebraic representation of a polynomial, find the intercepts, degree,
max/min values.
 Understand the relationship between the degree of a polynomial and the number
of roots. Interpret the multiplicity of roots graphically.
 Determine and interpret the meaning of intercepts, zeros, extrema, and trends.
 Evaluate potential solutions for appropriateness, accuracy, and suitability to the
context of the original problem.
Prerequisite Skills:
Students should be able to
 use a graphing calculator (reasonable window) to graph a cubic function.
 find the volume of a rectangular solid.
 factor polynomials.
 determine complex zeros from the factored form of a polynomial.
Estimated Time For Completion: 2 – 55 minute periods
Material for Students:
Graphing calculators
Fish tank problem handout
2 colored pencils
Teaching Aids:
Document Camera recommended
Geometer’s Sketchpad recommended for visualizing Fish Tank
Projection of Sketchpad file for Demonstration or Computer-Lab
References:
None
I. PREASSESSMENT:
1. Find the volume of a rectangular solid with a height of 4 feet, width of 5 feet, and
length of 7 feet.
2. Find the volume of a rectangular solid with a height of 3x feet, width of 4x feet,
and length of (2x-7) feet. Show the steps you used to determine the volume.
3. Find the length, x, in the following:
x
3y
21
4. Find the length, x, in the following:
x
3y
29
5. Completely factor the following:
x 2  4x  5
6. Completely factor the following:
x 3  x 2  12x
7. Completely factor the following:
9x 4  25x 2
2y
PREASSESSMENT ANSWERS:
1. Find the volume of a rectangular solid with a height of 4 feet, width of 5 feet, and
length of 7 feet.
140 cubic feet
2. Find the volume of a rectangular solid with a height of 3x feet, width of 4x feet,
and length of (2x-7) feet. Show the steps you used to determine the volume.
24x 3  84x 2 cubic feet
3. Find the length, x, in the following:
x
3y
21
x  21  3y
4. Find the length, x, in the following:
x
3y
29
x  29  5y
5. Completely factor the following:
x 2  4x  5
(x  5)(x  1)
6. Completely factor the following:
x 3  x 2  12x
x(x  4)(x  3)
7. Completely factor the following:
9x 4  25x 2
x 2 (3x  5)(3x  5)
2y
II.
INTRODUCTION:

Open the fishtank.gsp file (Geometer’s Sketchpad Application required to open
file). Notice there are three tabs on the bottom of the file. The first tab titled
“Fishtank” is a good opener for the investigation.

Project the file to the large group (or grouped in a computer lab) and demonstrate
the variable aspect of “x” by adjusting the length of x.

After a visual inspection, ask if anyone can guess what value of x will give the
largest fresh-water volume. If projecting the image, students could volunteer to
adjust x for the class until they think they have maximized the volume of the
fresh-water.

Encourage students to generalize what is happening with the different volumes.
See if anyone can think of a reasonable domain for x.
“Does it ‘work’ if “x” is 10?”
“What about if “x” is -2?”

Explain that engineers and designers don’t always have given dimensions to
work with – that they must often maximize or minimize volumes or areas in order
to meet the demands of clients’ specifications.
III.
LESSON:
Day 1:
1. Give the pre-assessment to students individually.
2. Go over the pre-assessment together as a class.
a. Call on students to share out their answers.
b. Ensure that factoring is a comfortable skill.
15 min
20 min
3. Hand out the student handout.
20 min
a. Read together the introductory paragraph.
b. Use the introduction above (with the Geometer’s Sketchpad file) to get
students thinking about the problem.
c. A common misconception is that students think the first step is to try and
solve for x right away. Encourage the students to think about x truly as a
variable (as opposed to something to immediately solve for) – a number
that varies in order that different scenarios can be visualized or
contemplated.
Day 2:
1. Group students into mixed-ability groups of 2 to 4.
5 min
a. Groups are effective for the difficulty level of the problem and for the
troubleshooting of the graphing portion.
b. Have a student (or two) summarize for the class the problem that was
presented the previous day.
2. Allow students to begin the handout.
35 min
a. Again, reiterate the idea of x being a variable as observed in the
Geometer’s Sketchpad demo.
3. Share out solutions
15 min
a. If you have a document camera, project various groups’ graphs for
discussion.
If time permits (or next day): Explore the Extensions.
IV.
APPLICATION:
1. The application in this lesson is the handout in which students work in their
groups to explore the fish-tank.
2. Monitor student progress.
3. Common misconceptions / pitfalls:
a. When asked to find a volume, students want to immediately find a
constant, instead of an expression.
b. Not using parentheses will get students into trouble.
c. Coming up with a friendly graphing window can be a challenge.
Encourage students to think about what possible volumes could be.
Student Handout – Fish
Tank Investigation
You are designing a fish-tank.
The tank must fit in a 10 feet
deep by 16 feet wide space.
You have both salt-water fish
and some fresh-water fish.
You decide to divide the tank
into four sections as shown:
1. What would be a reasonable domain of x for this problem?
2. Use your knowledge of algebra and geometry to determine an equation that
represents the volume of salt-water S(x). Answer in polynomial form.
3. Use your knowledge of algebra and geometry to determine an equation that
represents the volume of fresh-water F(x). Answer in polynomial form.
4. Verify your answers to #2 and #3 with the following: Use the length, width, and
height of the tank to determine the volume of the whole tank T(x). Now add the
volumes you found in #2 and #3. Do they add to the total volume?
5. Take your salt-water equation and factor it. Determine the types and values of
the zeros of the function.
6. Take your fresh-water equation and factor it. Determine the types and values of
the zeros of the function.
7. Graph both the salt-water equation S(x) and the fresh-water equation F(x) on the
same screen of your grapher. Use colored pencils to graph your results below
after checking your scale with your instructor:
8. For what length, x, does the salt-water have a maximum value? Use the “Calc”
“Maxima” feature of your calculator.
Student Handout - Answers
You are designing a fish-tank.
The tank must fit in a 10 feet
deep by 16 feet wide space.
You have both salt-water fish
and some fresh-water fish. You
decide to divide the tank into
four sections as shown:
1. What would be a reasonable domain for this problem?
x : 0  x  8
2. Use your knowledge of algebra and geometry to determine an equation that
represents the volume of salt-water S(x). Answer in polynomial form.
S(x)  (3x)(2x)(10  x)  (3x)(16  2x)(x)  12x 3  108x 2
3. Use your knowledge of algebra and geometry to determine an equation that
represents the volume of fresh-water F(x). Answer in polynomial form.
F(x)  (3x)(2x)(x)  (3x)(10  x)(16  2x)  12x 3  108x 2  480x
4. Verify your answers to #2 and #3 with the following: Use the length, width, and
height of the tank to determine the volume of the whole tank T(x). Now add the
volumes you found in #2 and #3. Do they add to the total volume?
Total Volume = 16 10 3x  480x
T (x)  S(x)  (12x 3  108x 2 )  (12x 3  108x 2  480x)  480x
YES
5. Take your salt-water equation and factor it. Determine the types and values of
the zeros of the function.
S(x)  12x 2 (x  9)
Double zero at x = 0
Zero at x = 9
6. Take your fresh-water equation and factor it. Determine the types and values of
the zeros of the function.

F(x)  12x x 2  9x  40
Zero at x = 0

1
 79i
2
Two imaginary zeroes at
1
x2   79i
2
x1 
7. Graph both the salt-water equation S(x) and the fresh-water equation F(x) on the
same screen of your grapher. Use colored pencils to graph your results below
after checking your scale with your instructor:
Fresh – water
Volume
(m3)
Salt – water
Length of x
(m)
8. For what length, x, does the salt-water have a maximum value? Use the “Calc”
“Maxima” feature of your calculator. (6.00, 1296)
V.
ASSESSMENT: No calculators
f x =is-the
x+3
x-1 (in
x-3

1. What
equation
factored
form) of the following graph?
15
10
5
-5
5
-5
-10
-15
-20
a. y  (x  3)(x  1)(x  3)
b. y  (x  3)(x  1)(x  3)
c. y  (x  3)(x  1)(x  3)
d. y  2(x  1)(x  3)2
2. What is the degree of the following graph?
2
-5
5
-2
-4
a. 1
b. 2
c. 3
d. 4
3. What is the leading coefficient for the following graph?
10
5
2
-2
a. -2
b. -1
c. 1
d. 2
4. The dimensions of the following tank
are in feet. The volume of fresh-water
for the tank can be expressed as:
a. (3x)(2x)(10  x)  (3x)(16  2x)(x)
cubic feet
b. (3x)(2x)(x)  (3x)(10  x)(16  2x)
cubic feet
c. 480x cubic feet
d. 48x(10  x) cubic feet
x
5. At most, how many zeros can the following function have?
2(x  4)(3x  2)(5  3x)2
a. 1
b. 2
c. 3
d. 4
6. Which of the following is not possible?
a. A cubic function with only one real zero.
b. A quartic function with no real zeros.
c. A cubic function with no real zeros.
d. A quartic function with four distinct zeros.
7. Simplify: (3x)(2x)(10  x)  (3x)(16  2x)(x)
a. 96x 3
b. 12x 3  108x 2
c. 54x 3  42x 2
d. 6x 3  54x 2  48x
8. Which of the following is a factor of: 5x 4  20x 2
a. x 2  4
b. x  5
c. x  4
d. x  2
9. How many real and complex zeros does y  x 2  9x  40 have?
a. 2 real, 0 complex
b. 1 real, 1 complex
c. 0 real, 2 complex
d. 2 real, 2 complex
10. 2 Points: Sketch the function y  (x  3)(x  2)2 indicating x- and y-intercepts on the
graph.
20
-5
5
-20
11. 2 Points: Show algebraically that -2 is a zero of: f (x)  x 3  4x 2  3x  18
12. 2 Points: Indicate the degree of the polynomial and the types of zeros for the
following function. Be sure to indicate the number of each type of zeros.
40
30
20
10
-4
-2
2
4
-10
13. 2 Points: Factor completely the following cubic equation, including the two
imaginary factors: 3x 3  12x 2  39x
14. 4 Points: Draw the graph of a function that meets the following requirements:





Is a quintic function
Has exactly two distinct, real zeros
Has exactly two complex zeros
Contains one double zero
Has a positive leading coefficient
15. 4 Points: Calculator O.K. A box has dimensions (x  1) by (6  x) by (2x  3) .
Graph the volume of the box with respect to x over the indicated domain. From
your graph determine the following:
10 0
50
10
-50
a. What is a reasonable domain for x?
b. Why are values outside this domain not reasonable?
c. Approximately what is the value of x when the box’s volume is maximized?
Show this point on the graph.
ASSESSMENT SOLUTIONS
1. What is the equation (in factored form) of the following graph?
(CRS 8.2f: Given a graph or graphical features, including degrees, intercepts,
asymptotes, and/or holes (discontinuities), generate an algebraic representation of a
f x = - or
x+3
x-1function.)
x-3 
polynomial
rational
15
10
5
-5
5
-5
-10
-15
-20
a. y  (x  3)(x  1)(x  3) (incorrect leading term)
b. y  (x  3)(x  1)(x  3) (opposite of correct zeros)
c. y  (x  3)(x  1)(x  3)
d. y  2(x  1)(x  3)2 (no double zeros)
2. What is the degree of the following graph?
(CRS 8.3e: Understand the relationship between the degree of a polynomial and the
number of roots; interpret the multiplicity of roots graphically.)
2
-5
5
-2
-4
a. 1 (appears to only pass through x-intercept once)
b. 2 (touches x-intercept twice)
c. 3
d. 4 (incorrect)
3. What is the leading coefficient for the following graph?
(CRS 8.3: Analyze and interpret features of a function.)
10
5
-2
2
x
a. -4 (does not take the double zero into account, but knows that it must be
negative)
b. -2
c. 2 (misunderstands that leading coefficient must be negative)
d. 4 (does not take the double zero into
account)
4. The dimensions of the following tank are
in feet. The volume of fresh-water for the
tank can be expressed as:
(CRS 8.2d: Algebraically construct new
functions using addition and subtraction,
multiplication, division, and composition.)
a. (3x)(2x)(10  x)  (3x)(16  2x)(x) cubic feet (volume of salt-water)
b. (3x)(2x)(x)  (3x)(10  x)(16  2x) cubic feet
c. 480x cubic feet (volume of total)
d. 48x(10  x) cubic feet (volume of back half)
5. At most, how many zeros can the following function have?
2(x  4)(3x  2)(5  3x)2
(CRS 8.3b: Identify y-intercepts and zeros using symbols, graphs, and tables.)
a. 1 (incorrect)
b. 2 (incorrect)
c. 3 (doesn’t take double root into account)
d. 4
6. Which of the following is not possible?
(CRS 1.3c: Evaluate potential solutions for appropriateness, accuracy, and
suitability to the context of the original problem.)
a. A cubic function with only one real zero. (possible)
b. A quartic function with no real zeros. (possible)
c. A cubic function with no real zeros.
d. A quartic function with four distinct zeros. (possible)
7. Simplify: (3x)(2x)(10  x)  (3x)(16  2x)(x)
(CRS 7.2a: Find the sum, difference, or product of two polynomials, then simplify
the result.)
a. 96x 3 (adding all terms as though cubic)
b. 12x 3  108x 2
c. 54x 3  42x 2 (combining cubes and squares)
d. 6x 3  54x 2  48x (neglecting final x)
8. Which of the following is a factor of: 5x 4  20x 2
(CRS 7.2b. Factor out the greatest common factor from polynomials of any degree.)
a. x 2  4 (similar to x 2  4 which is a factor)
b. x  5 ( -5 is a factor of -20)
c. x  4
(-4 is a factor of -20)
d. x  2
9. How many real and complex zeros does y  x 2  9x  40 have?
(CRS 8.3b: Identify y-intercepts and zeros using symbols, graphs and tables)
a. 2 real, 0 complex (adds to 2, error with finding discriminant)
b. 1 real, 1 complex (misunderstanding that complex zeros come in pairs)
c. 0 real, 2 complex
d. 2 real, 2 complex (misunderstanding that quadratic can only have 2 zeros)
10. 2 Points: Graph the following showing x and y intercepts: y  (x  3)(x  2)2
(CRS 8.2g: Sketch the graph of a polynomial given the degree, zeros, max/min
values, and/or initial conditions.)
20
-5
5
-20
2-point response: The student shows understanding of the concept of graphing
polynomials by doing all of the following:
 Shows a zero at -3
 Shows a double zero at 2
 Shows a y-intercept of 12

Graph is a cubic that ends in the positive direction
1-point response: The student shows understanding of the concept of graphing
polynomials by doing two of the following:
 Shows a zero at -3
 Shows a double zero at 2
 Shows a y-intercept of 12
0-point response: The student shows little or no understanding of the concept of
graphing polynomials.
11. 2 Points: Show algebraically that -2 is a zero of: f (x)  x 3  4x 2  3x  18
(CRS 7.1d: Know what it means to have a solution to an equation.)
f (2)  (2)3  4(2)2  3(2)  18
f (2)  8  16  6  18
f (2)  0
2-Point Response: The student shows understanding of evaluating by doing the
following:
 Writes the original equation with -2 substituted for x
 Shows that f(-2) = 0
1-Point Response: The student does one of the above requirements.
0-Point Response: The student shows little to no understanding of the concept or
procedure.
12. 2 Points: Indicate the degree of the polynomial and the types of zeros for the
following function. Be sure to indicate the number of each type of zeros.
(CRS 8.3: Analyze and interpret features of a function.)
40
30
20
10
-4
-2
2
4
-10
2-Point Response: The student writes the following:
 5th Degree
 One real zero
 4 non-real zeros
1-Point Response: The student writes any one of the above requirements.
0-Point Response: The student shows little to no understanding of the concept.
13. 2 Points: Factor completely the following cubic equation, including the two
imaginary factors: 3x 3  12x 2  39x
(CRS 7.2j: Factor polynomials over the complex numbers)
step 1  (3x)(x 2  4x  13)
step 2  (3x)(x  1  i)(x  1  i)
2-Point Response: The student shows understanding of factoring over complex
numbers by doing the following:


Finding the real factor 3x
Finding the two imaginary factors (x – 1 + i) (x – 1 – i)
1-Point Response: The student shows some understanding of factoring over complex
numbers by doing one of the following:

Finding the real factor 3x

Finding the two imaginary factors (x – 1 + i) (x – 1 – i)
0-Point Response: The student shows little to no understanding of factoring over
complex numbers.
14. 4 Points: Draw the graph of a function that meets the following requirements:
(CRS 8.2g: Sketch the graph of a polynomial given the degree, zeros, max/min
values, and/or initial conditions.)





Is a quintic function
Has exactly two distinct, real zeros
Has exactly two complex zeros
Contains one double zero
Has a positive leading coefficient
4-Point Response: The student shows the following:
 is a quintic function
 has two distinct real zeros
 has 2 complex zeros
 contains a double zero
 has a positive leading coefficient
3-Point Response: The student shows three or four of the above requirements.
2-Point Response: The student shows two of the above requirements.
1-Point Response: The student shows one of the above requirements.
0-Point Response: The student shows little to no understanding of the concept.
15. 4 Points: Calculator O.K. A box has dimensions (x  1) by (6  x) by (2x  3) .
Graph the volume of the box with respect to x over the indicated domain. From your
graph determine the following:
(3.5, 112)
10 0
50
10
-50
a. What is a reasonable domain for x? 1  x  6
b. Why are values outside this domain not reasonable?
x values outside the domain create non-boxes.
c. Approximately what is the value of x when the box’s volume is maximized?
Show this point on the graph. (3.5, 112)
(CRS 8.1b: Determine the domain and range of a function.)
(CRS 8.4a: Choose a function suitable for modeling a real world situation.)
(CRS 8.3c: Identify extrema and trends using graphs and tables.)
4-Point Response: The student does five of the following;
 Shows a reasonable graph
 Gives a domain of 1  x  6
 Explains that values of x outside the domain give non-boxes.
 Gives a value of about x = 3.5 (0.5) for a x.
 Gives a value of about V = 112 (8.0) for a maximum volume.
 Marks the maxima (approximately (3.5, 112)) on the graph.
3-Point Response: The student shows four of the above requirements.
2-Point Response: The student shows three of the above requirements.
1-Point Response: The student shows two or three of the above requirements.
0-Point Response: The student shows little to no understanding of the concept.
VI.
EXTENSIONS:
Things to ponder
1. Why do you think mathematicians don’t call the maximum point a “vertex”?
A vertex is the point at which the function changes direction. The
vertex in a cubic function is the point of inflection.
2. What algebraic method do we use for quadratics to find the maxima (in this case
the vertex)? Do you think there might be a similar method for cubics?
For quadratics algebraic methods for finding maxima/minima
include:

Completing the square
b
 h , k
2a
For a cubic…
3. If your cubic doesn’t factor nicely like the ones in this example, how might
somebody go about finding the x-intercepts (zeroes)? Think about the method
used for quadratics – might there be something similar for cubics? How about for
quartics?
For quadratics, there is the quadratic formula:
b  b 2  4ac
x
2a
For cubics, there is a cubic formula:
The quartic formula is insane. Google “quartic formula.”
4. For quadratics, another way to find the maxima (or minima) is to average the
b
zeros. Prove that this leads to h  
2a
 b  b 2  4ac  b  b 2  4ac  2b

b
2a
2a
 2a 
2
2
2a
Would the maxima of a cubic also be the average of the zeros?
No – visual inspection