Effectively Incorporating Popular Media Into A Mathematics Curriculum Overview

Effectively Incorporating Popular Media Into
A Mathematics Curriculum
Roseann Y. Casciato
Pittsburgh Allderdice High School
Overview
Rationale
Objectives
Strategies
Classroom Activities
Annotated Bibliography/Resources
Appendices-Standards
Overview
In the seminar Media-Math, I was given the opportunity to explore mathematics
through the popular media. We looked at mathematics concepts that were evident
in cartoons, newspapers, television shows, movies, and the internet. Before
taking this seminar, I never realized how prevalent mathematics is in the world or
the ways I could incorporate it into my classroom. What is even more astonishing
is the fact that most people would not think that mathematics can truly be
pleasurable for our students. Even when lessons are not enjoyable, some students
rush through procedures to try and solve problems without always understanding
the mathematics that are involved and consider „doing math‟ painful. We must
also recognize that mathematics does not come easy to every student all of the
time. As mathematics teachers, we are constantly asked “When will we ever
need to use this?” I am optimistic that after trying some of the lessons in this
curriculum unit, all teachers will be able to answer this question. As mathematics
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teachers we are so driven by the school district‟s curriculum and state
assessments, that we rarely have time to incorporate additional activities into our
classrooms. This curriculum unit will give all of us the opportunity to include
exciting activities throughout the school year in our classes. Whether you teach
advanced placement, gifted, scholars or mainstream students, you will be able to
make use of and benefit from this unit.
As a teacher of an Advanced Placement Calculus class, it is expected that I
add enrichment into the existing curriculum once the students have taken the
Advanced Placement exam. One of the benefits of teaching an Advanced
Placement course is that I have the flexibility to add other topics of interest once
the students have taken the exam. Our Advanced Placement AB calculus exam is
usually scheduled every year for the first week of May, which allows the
opportunity to supplement the curriculum with enrichment. My students are
always looking for new and exciting opportunities, so I think they will be very
receptive to these new activities. I am excited to be able to try something new in
class and to incorporate reading and writing into my mathematics classes.
Problem solving activities where students are expected to read a problem
situation, solve the problem for a correct solution, and then write why they solved
it the way they did, is the extent to which reading and writing is built-in the
curriculum.
The advantage of teaching this unit to my CAS Algebra 2 students is that I
can treat it as an extension to the gifted curriculum. “The term „mentally gifted‟
includes a person who has an IQ of 130 or higher, when multiple criteria as set
forth in Department Guidelines indicate gifted ability. Determination of gifted
ability is not based on IQ score alone. A person with an IQ score lower than 130
may be admitted to gifted programs when other educational criteria in the profile
of the person strongly indicate gifted ability. Determination of mentally gifted
includes an assessment by a certified school psychologist” (CAS Centers for
Advanced Study Teacher Handbook 7). “The CAS program has been designed to
meet the needs of the gifted students for individualized, accelerated and enriched
learning. All CAS classes emphasize an inquiry approach to learning, problem
solving techniques and the higher cognitive skills of analysis, synthesis and
evaluation” (CAS Centers for Advanced Study Teacher Handbook 3). The CAS
class sizes are limited to 10 to 18 students which facilitates more individualized
attention. In addition to the regular course work for each class, students are
required to complete a Long Term Project (LTP). The LTP should focus on a
topic that interests the students, as they have to research it in-depth, perform a
minimum of 30 hours outside of the classroom, and present the project to the
class.
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Each of the lessons I developed for this curriculum unit will be
independent. If another teacher would like to use a portion of my curriculum unit
they will be able to incorporate it into one of their lessons very easily. From my
own experience of trying to teach what is expected for a particular subject, it is
sometimes very difficult to try to add anything else into your school year. The
advantage to this curriculum unit is that if you have a day or two within the year,
maybe during the week of a holiday break or during standardized testing, you can
try one of my assignments and do as much or as little as you see fit. Also, once
you do one or two lessons, you may decide to assign some of the other lessons as
homework for students to do outside of the classroom. These are still beneficial
enrichment activities and you do not have to use all of your class time to complete
them. I will be incorporating cartoons, movies, and television into my lessons.
Rationale
Do you think you would be interested in this curriculum unit dealing with
mathematics as seen through popular culture? Would you like to expose your
students to the mathematics that is around them? Can the media be used to help
engage our students mathematically? I am hoping the answer to all of these
questions is yes and I believe you will enjoy using some of these lessons in your
classroom. I attempted to ensure that the lessons in this unit could be used at any
time in any course curriculum to make connections between mathematics and the
real world. I cannot stress the importance of this connection and believe that once
students are exposed to this unit, they will be able to make mathematical
connections in their everyday lives. I expect this unit to reach some students who
may not have previously been engaged in their studies. If I am able to spark an
interest in my students, accomplishments are endless.
With the high expectations of high school mathematics teachers, meeting
the time constraints of completing the curriculum, and preparing our students for
state exams, I am certain you will find this curriculum unit refreshing. It is not
like a traditional curriculum unit and my hope is that all of our students will be
stimulated by these lessons.
When I first heard about the seminar Media-Math, I was thrilled to
participate in it. Our leader, Dr. Marty Hildebrandt, described our seminar “by
examining television shows, movies, advertisements, music, the internet and the
printed page you will discover ways to motivate your students as well as make
math relevant to their everyday lives.” Who wouldn‟t be excited to take part in
this seminar? Who wouldn‟t want help in motivating their students? We also
included a great deal of problem solving techniques throughout many of the
activities in the seminar. “Successful problem solvers are resourceful, seeking out
information to help solve problems and making effective use of what they know.
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Their knowledge of strategies gives them options. If the first approach to a
problem fails, they know how to reconsider the problem, break it down, and look
at it from different perspectives – all of which can help them understand the
problem better or make progress toward its solution” (Principles and Standards
for School Mathematics 334). As a high school mathematics teacher, I am
looking forward to incorporating more problem solving activities into my
classroom. This will be accomplished through warm-ups, class work or
homework assignments. Students will be given open-ended problem solving
activities; they may work individually or in groups. I have noticed that when
students work in groups, the answers may not always come to them quickly, but
they persevere until they determine the solution.
The first eight lessons in this curriculum unit deal with
mathematics that is found in cartoons (comic strips). “Short comic strips of one
to four panes can be useful for explaining a new concept to students. Putting a
humorous spin on a new idea, like showing the value of pie carved into pumpkins,
can help students remember the concept. They can also be created by students to
help them understand something new. The effort they put into finding a creative
or funny way to explain the material helps make mental connections that will
make it easier for them to remember it later” (Andrew 2009). In creating these
lessons, I incorporated topics ranging from Algebra 1 through Calculus. It was
important for me to have a broad range of topics so that teachers reading this
curriculum unit had a variety to choose from. I used the books Cartoon Corner
and Math, Science, and Unix Underpants to choose the cartoons for these lessons.
In using Cartoon Corner, I used most of their activities but added questions to
some of the cartoons while removing others. I plan on allowing my students to
work with a partner or in groups of three on the cartoon lessons. All of the
lessons should be able to be completed in a regular class period. A benefit to
these lessons is that you can use as much or as little as you like. Some are fun
warm-ups just to get the students excited for class while others have a specific
purpose and were chosen because the unit of study from the textbook deals with
that particular topic in the cartoons. It is advantageous to look through each
cartoon lesson to see what you can incorporate in your classroom. The eight
mathematical topics present in the cartoons are: sequences and series, parabolas,
circles, Pythagorean Theorem, Venn Diagrams, probability and odds, slope, and
data analysis.
Why cartoons, you may ask? There are many benefits to using cartoons in
the classroom. Cartoons:


are motivational
fun for students
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



engaging
relevant
encourage creativity
take math out of the classroom
Some of the cartoon warm-ups created for this unit will ask students to examine
cartoons and determine the mathematical topic present, whether it makes sense,
and if the mathematics in the cartoon is correct. If it is not correct, students will be
asked to correct and determine if the cartoon is being portrayed honestly, and
determine if the media skewed it at all. They will also be expected to create a title
for a cartoon or they will have only some of the words in the cartoon and they will
have to complete the cartoon, as well as create their own cartoon.
The next few lessons created for this curriculum unit deal with
mathematics that is depicted in movies. I am looking forward to having my
students look at different movies which show mathematics in them. I chose Cast
Away, Die Hard with a Vengeance, and Stand and Deliver. The reason I chose
these movies is that because somewhere in the movies, mathematics was shown.
Cast Away and Die Hard with a Vengeance actually has characters working out
mathematical problems in order to survive. I plan on writing lessons which first
will have students determine if the mathematics are correct and then I will have
the students work on the mathematical problems together. “When students work
together in a less stressful environment, their confidence grows. Making
connections and communicating their ideas enable students to acquire a better
understanding of the underlining concepts” (NcNamee 464).
One of the main reasons I chose Stand and Deliver for this curriculum unit
was because it dealt with taking the Advanced Placement AB calculus exam.
“Jaime Escalante is a mathematics teacher in a school in a Hispanic
neighborhood. Convinced that his students have potential, he adopts
unconventional teaching methods to try and turn gang members and no-hopers
into some of the country's top algebra and calculus students” (Murray Chapman,
International Movie Data Base). Escalante taught at Garfield High School in East
Los Angeles, which was a poor public school. The year was 1982, 18 of
Escalante‟s calculus students passed the Advanced Placement exam, and the
Educational Testing Service found the test scores to be suspect. The Educational
Testing Service asked that 14 students retake the exam. Twelve of the 14 students
took the exam again and passed. The students in this calculus class are portrayed
as coming from underprivileged families and are poorly prepared for the calculus
course. The movie takes us through the beginning of the school year and the
struggles that Escalante has while he prepares his students for the Advanced
Placement Exam. We also see a wide variety of personalities among the students.
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The first time I watched this movie I was surprised to see the students working so
hard in order to prepare for the Advanced Placement exam. There were moments
in the movie where some of the students did not want to work to their potential,
but ended up rising to the challenge. But as a teacher, and working on this
curriculum unit, I ask, is what I am watching factual? Did Jaime Escalante have a
group of students enrolled in his class with little or no mathematical knowledge
and was he able to miraculously help them to pass the exam? I struggle daily
teaching this course to students who have been on the “advanced mathematics
track” and it is very difficult for them. Was it as easy as it seemed in the movie?
Escalante stated that the movie Stand and Deliver was based on 90 percent
truth and 10 percent drama. After researching this movie, I was surprised to find
out how many facts were eliminated. Escalante was portrayed as a hard working
teacher who was able to take unmotivated students from arithmetic to calculus in
one year. What the movie did not include was:
 Escalante had established a program with the East Los Angeles College,
where students could take a seven-week summer session to help prepare
them for higher level mathematics courses.
 Escalante and his principal were pivotal in getting feeder schools to offer
algebra in eighth and ninth grades, so by the time they were in twelfth
grade, they could take calculus.
 One of the major points to the success at Garfield dealt with who taught
the prerequisite courses to calculus. The movie never mentioned the
prerequisites. Escalante actually hand-picked teachers to teach the
courses. They prepared lessons together all the while Escalante was
ensuring the students would receive the mathematics necessary to be
successful in calculus.
 The teachers are portrayed as getting along in the movie. However, there
was talk that Escalante wanted too much power and that he was creating
two mathematics departments, one for his group of students and then
another with the other students in the school.
One major part of the movie that attributed to the success at Garfield dealt with
tutoring. Escalante had a key to the building so he could tutor students before and
after school. As a teacher I wish that everything that contributed to the success of
the program had been included in the movie. I know what it is like to teach that
course every day and have the pressures of covering all the topics in a timely
manner while still allowing enough time to review and practice for the exam. I
realize that every detail cannot be included, but feel that the omitted details are
misleading.
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When thinking about how to incorporate television shows into this unit, I
thought the show Number3s would be perfect. This series debuted in January of
2005 and was on the air for six seasons. “Numb3rs follows FBI Agent Don Eppes
(Rob Morrow) and his genius brother, Charlie (David Krumholtz), who helps
authorities solve cases with mathematics” (TV Series Finale). “The two brothers
take on the most confounding criminal cases from a very distinctive perspective.
Inspired by actual events, the series depicts how the confluence of police work
and mathematics provides unexpected revelations and answers to the most
perplexing criminal questions” (CBS.com). With the show being on the air for so
long, I had more than enough episodes to choose lessons from. When trying to
determine which show I would like to incorporate in this curriculum unit, I tried
to think about a mathematical topic that could be used in a number of
mathematics classes and decided on parabolas. Parabolas are seen in some
Algebra 1 classes, explored in great detail in Algebra 2, and then revisited again
in Pre-Calculus and Calculus. What I like about parabolas is that students truly
enjoy learning about the real world applications they can use with them. This is a
great way to talk about projectile motion. Students are taught about the maximum
height an object travels, the time it takes to reach that height, how long it was in
the air, what happens when the object is not at ground level, etc. Not only are
students learning how to solve parabolas using Algebra, but then they are
expected to try solving problems with their graphing calculators. This is when it
is exciting for students. When you simulate the path that an object takes using a
graphing calculator, you usually get that „ah‟ moment of excitement from your
students. Students really like learning about parabolas, and I hope they enjoy the
two lessons that are included in this curriculum unit.
Not only is the way students think and perceive things important, but I
also would like them to make connections between mathematics and the real
world. Being a mathematics teacher is not only about doing problems from a
textbook, but to also provide a thorough explanation about the topics you are
teaching and to make connections to real world situations. I am looking forward
to teaching these lessons with movies because it is an opportunity for me to
connect how mathematics is perceived in the real world.
When I was growing up I thoroughly enjoyed learning mathematics and
“research has shown that many students actually like mathematics in elementary
grades. Unfortunately, this enjoyment seems to decrease as students advance to
high school” (Lewkowicz February 2003: 92). I am hoping this unit of study will
enlighten some, if not all, of my students. I do not want them to just sit in my
classroom, watch me work out problems, and then practice the same concepts at
home. I want them to explore and investigate other topics and to recognize the
mathematics that is around them in the world.
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Objectives
My main goal in this curriculum unit is to have students uncover the area of
mathematics that is found within the media. When I teach, I always try to
incorporate real life application problems in my lessons so students understand
they are not just “doing math.” Students need to make the connection that
mathematics exists all around us. This unit works extremely well in that students
will learn about a new topic and see how it relates to them every day.
Being a mathematics teacher is very challenging. In fact, at times it can be
quite discouraging because not every student embraces mathematics or are always
successful. “American high school students are not performing up to expectations
in mathematics. Only 23 percent of twelfth graders in the United States scored at
the “proficient” level or above on the 2005 National Assessment of Educational
Progress (NAEP), and significantly smaller percentages of African-American and
Hispanic students, compared with white students, achieved proficiency, according
to the National Center for Education Statistics” (Hart and Martin January 2009:
378). These statistics speak volumes. For only 23 percent of our high school
students to score at the proficient level is dispiriting. We have to make changes in
how we reach our students. It is apparent that what we have been doing for years
is not working. As teachers it is vital for us to try and reach our students in
whatever means we can. I believe this curriculum unit will help expose our
students to mathematics outside of the classroom and in turn get students excited
about mathematics. “Robert Oppenheimer (1904 – 1967), the famous theoretical
physicist known as the father of the atomic bomb, was once asked how he became
such a great scientist. He replied that early on he had teachers who afforded him,
as he put it, the joy of discovery. It truly is a joy to discover something for
yourself, even if it were previously discovered by someone else. Discovery
learning should be a significant part of the mathematics curriculum. Activities
that lead students to discover some interesting and perhaps unexpected results are
fun for both students and teachers” (Fairbairn August 2008: 62). I am optimistic
that my students will enjoy trying something new in class other than the major
emphasis being the district‟s curriculum. The main objectives for this curriculum
unit are for my students to:







recognize the mathematics that exists in their everyday lives
make connections between mathematics and the real world
embrace and have fun with mathematics
gain more confidence in mathematics
become better problem solvers
be able to work successfully with others
appreciate writing in a mathematics class
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

be able to communicate orally and by writing
expand their inductive, deductive, explicit, and recursive reasoning
skills
It is imperative that I incorporate the Pennsylvania State Standards
(Appendix 1) into my lesson plans. During this unit, the seven mathematics
standards will be addressed in this curriculum unit. Students will be expected to
use number systems, compute and solve practical problems, apply the concepts of
patterns, formulate and solve problems, communicate the mathematical processes
used, understand and apply basic concepts of algebra and geometry to solve
theoretical and practical problems, evaluate, infer, and draw appropriate
conclusions from charts, tables, and graphs, showing the relationships between
data and real world situations, and lastly, make decisions and predictions based
upon the collection and interpretation of statistical data and the application of
probability. It also is important that I incorporate the communication standard.
“In high school, there should be substantial growth in students‟ abilities to
structure logical chains of thought, express themselves coherently and clearly,
listen to the ideas of others, and think about their audience when they write or
speak. Consequently, communication in grades 9 to 12 can be distinguished from
that in lower grades by higher standards for oral and written exposition and by
greater mathematical sophistication” (Principles and Standards for School
Mathematics 348 – 349).
Strategies
This unit of study will be comprised of cartoon activities, movies and television
shows or excerpts of them, group discussions, and writing assignments. I believe
it is important for students to become active participants in the classroom to
enhance their learning process. To do this, I will:




foster an environment of respect to help students feel comfortable
as they learn
offer history of the movies or television shows to give students
background information
guide students to be more self sufficient
ask thought provoking questions
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Classroom Activities
Lesson 1 – Mathematics in a Cartoon – A Sheepish Problem – Cartoon Corner
I plan on incorporating the cartoon “A Sheepish Problem” as an activity for the
first day when teaching the topics of sequences and series. (See Appendix 3 for
the cartoon and lesson.) Students have been exposed to number and picture
patterns, but the complete chapter dealing with sequences and series is a new
lesson for Algebra 2. Students will be expected to determine the cartoon caption
(I will make sure to remove the caption before giving the students the cartoon),
evaluate an expression, determine a rule, solve problems algebraically, solve
problems algebraically if they were to change the mathematics in the cartoon, and
then complete a problem solving activity. This is a good introduction to
sequences and series because when teaching this in Algebra 2, students “define
explicit rules that generate number sequences whose terms have a common
difference or a common ratio, and they use summation notation to represent and
find the sum of the terms of a series. They use rules for the sum of arithmetic
series, finite geometric series, and infinite geometric series. Also students define
recursive rules for generating arithmetic and geometric sequences and they
investigate how to use iteration to generate a sequence recursively given a
function rule” (Boswell et al. 792). This is a cute cartoon to show to an Advanced
Placement BC Calculus class before beginning the chapter on sequences and
series.
Lesson 2 – Mathematics in a Cartoon – Why Weight? – Cartoon Corner
I plan on incorporating the cartoon “Why Weight?” as an activity when teaching
the topic of parabolas. (See Appendix 4 for the cartoon and lesson.) Students
will be expected to complete a table of values, make a graph, use a graphing
calculator, predict quadratic equations, explain their findings, complete
transformations, determine an equation of a parabola given two points, and justify
their answers. In Boswell‟s et al. Algebra 2 textbook, students “will learn how to
graph quadratic functions written in standard form, vertex form, or intercept form,
how to graph quadratic inequalities, and how to use the graph of a quadratic
inequality to solve it” (234). Students will benefit from this cartoon lesson as you
begin teaching about parabolas.
Lesson 3 – Mathematics in a Cartoon – Keeping Track – Cartoon Corner
I plan on incorporating the cartoon “Keeping Track” as an activity for the first day
when teaching about circles. (See Appendix 5 for the cartoon and lesson.) There
is only one section in our Algebra 2 textbook dealing with circles so this cartoon
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is a nice segue into a circle lesson. Due to the nature of this topic, geometry
teachers will especially enjoy using this cartoon lesson. Students are expected to
determine the diameter using the equation for circumference, calculate the
difference between three runners‟ circumference, and calculate surface area. I
added another activity to this lesson that I included when teaching Geometry from
Michael Serra‟s Discovering Geometry textbook. The activity is called Racetrack
Geometry. If you were to ask someone which lane they would want to run in on a
racetrack, most people would choose the inner lane because the distance is the
shortest. This task asks students to “design a 4-lane oval track with straightaways
and semicircular ends. The semicircular ends must have inner diameters of 50
meters so that the distance on one lap in the inner lane is 800 meters. Draw
starting and stopping segments in each lane so that an 800 meter race can be run
in all four lanes” (Serra 286).
Lesson 4 – Mathematics in a Cartoon – Running Patterns – Cartoon Corner
I plan on incorporating the cartoon “Running Patterns” as a warm-up when
dealing with the Pythagorean Theorem in my Algebra 2 class. (See Appendix 6
for the cartoon and lesson.) Due to the fact that this topic is taught in Geometry
which most students have already completed before coming into Algebra 2, I
really do not want to go through the whole lesson. After the students have read
the cartoon, I will ask if they thought the route that Jason ran was the shortest and
to explain why, then I will have them complete question number four so they
recall the sides of the special right triangle are: a, a, a√2 if the triangle angles
have measurements of 45°, 45°, and 90°. Because this has a football theme to it,
if your school or city has a big game coming up, you could always treat this as
warm-up to get your students excited for the game. It‟s always nice to be able to
make connections outside of the classroom. Geometry teachers can use this
complete lesson as a launch in their classrooms to get students ready to learn more
about the Pythagorean Theorem.
Lesson 5 – Mathematics in a Cartoon – Perfect Pumpkins – Cartoon Corner
I plan on incorporating the cartoon “Perfect Pumpkins” as a warm-up when
working in the chapter dealing with Venn Diagrams. (See Appendix 7 for the
cartoon and lesson.) In Algebra 2 we use Venn Diagrams to illustrate the
probabilities of overlapping events. If students have seen Venn Diagrams before
this warm-up, it will move along a little faster than if they have not seen Venn
Diagrams before, but in either instance, I do not anticipate that it will take that
long to complete. If you do not have a unit on Venn Diagrams, you might choose
to use this lesson on or near Halloween, as students always enjoy holiday
activities.
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Lesson 6 – Mathematics in a Cartoon – Prime Guess – Cartoon Corner
I plan on incorporating the cartoon “Prime Guess” as an activity for the first day
when teaching the topics of probability and odds. (See Appendix 8 for the
cartoon and lesson). The lesson from the textbook is geared more toward prime
numbers, but dealing with high school students I removed some of the questions
and added another. I began the lesson by asking what the difference is between
probability and odds. Students were expected to find the probability and odds of
something occurring, then they are asked to explain why a certain number trick
worked, and create their own number trick.
Lesson 7 – Mathematics in a Cartoon – The Aging Process – Cartoon Corner
I plan on incorporating the cartoon “The Aging Process” as an activity for the first
day when teaching the topic of slope. (See Appendix 9 for the cartoon and
lesson.) I was glad there was a cartoon dealing with slope because we constantly
see slope in Algebra 1 through Calculus. Everyone can make use of the lesson by
using as much or as little as they see fit. In my Algebra 2 class, I plan on having
students complete the whole lesson of calculating and plotting points, then finding
the slopes of the line segments created from their points, and finally creating their
own cartoon similar to this one. Each group of students will then compare their
cartoons with each others to determine which set of graphs match each of the
cartoons created.
Lesson 8 – Mathematics in a Cartoon – Graphs Alive – Cartoon Corner
I plan on incorporating the cartoon “Graphs Alive” as a warm-up when working
in the chapter dealing with slope and/or data analysis. (See Appendix 10 for the
cartoon and lesson.) For my Algebra 2 class this is truly a warm-up exercise
which should not take a complete class period to finish. Students are asked
questions about the horizontal and vertical axes and slopes of lines from a cartoon
dealing with a line graph. They are also asked to create a graph from an event
that occurred in their life and then write a story about it. When writing their story,
you can have them complete it for homework if you need to move along in class.
This is a great warm-up for Algebra 1, Geometry, and Algebra 2 classes.
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Lesson 9 – Mathematics in a Cartoon – Math, Science, AND Unix Underpants
I plan on incorporating the four cartoons below as warm-ups in my Advanced
Placement AB Calculus class. Unlike the first 8 lessons, I do not plan on having
any assignments attached to these warm-ups. We will read the cartoons and
discuss the mathematics that is evident. Students will be expected to complete the
mathematics in the cartoons below and determine if the answers make sense.
This warm-up is appropriate for Pre-Calculus and Calculus classes.
This warn-up is appropriate for Pre-Calculus and Calculus classes.
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The cartoon above will be a great warm-up after you teach the concept of
integrals and area under the curve.
The cartoon above can be given at any time in my Advanced Placement AB
Calculus class as well as any other mathematics class. Students will be pleasantly
surprised as to what the coded message reveals.
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Lesson 10 – Mathematics in a Movie – Die Hard with a Vengeance
The first assignment dealing with mathematics in movies incorporates the movie
Die Hard with a Vengeance. The two main characters, Detective John McClane
and Zeus, are playing a game of “Simon Says” with a villain. The villain is
making the two characters solve puzzles in a specific amount of time. If they fail,
he sets off a bomb. In one scene, the two main characters are given the following
task to complete: within five minutes, they have to weigh on a scale exactly four
gallons of water to diffuse a bomb. However, the catch is that they are given two
plastic bottles to use -- one five gallons and one three gallons. I will divide my
class into partners and give them the same assignment to see who can complete
the task in five minutes. After the exercise, we will have a group discussion about
the mathematics involved in solving the problem. I also will show the class the
clip which explains the problem situation (1:00:16 – 1:00:45) then the movie clip
showing John McClane and Zeus successfully solving the problem (1:02:28 –
1:03:05). Due to some vulgarity in the movie, you cannot show the complete clip
in its entirety. We‟ll then discuss how realistic the students think this problem
situation is.
Lesson 11 – Mathematics in a Movie – Cast Away
The movie Cast Away has a few scenes where the main character, Chuck Nolan,
is stranded alone on a tropical island after an airplane crash. During this time, he
uses math to help him survive. I will discuss two specific situations with my class.
Situation 1: Movie: Cast Away – movie clip (1:14:19 – 1:15:17).
The main character Chuck Nolan states: “in route from Memphis for 11 ½ hours,
475 mph.” He then draws a straight line marking where he thinks searchers
would think he would be, versus where he actually believes he is. From the point
where he ventured out to, he then draws a circle to determine the area in terms of
a circle.
X______________X___________________________________
think
venture out
we are
of radio contact
here
and flew around
the storm 1 hour
so that is a distance
of 400 miles
He continues talking: “400 squared is 160,000 times Pi (π). 160000 X 3.14
equals 502,400. That‟s a search area of 500 square miles. That‟s twice the size of
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Texas. They may never find us.” In this scene, Chuck Nolan uses his knowledge
of the area of a circle to determine the size of his search area. I would like my
class to determine if Chuck Nolan used the correct formula for the area of a circle.
Did Chuck Nolan calculate the area correctly? Did rounding π to 3.14 make a
significant difference in the search area? Why or why not? We will discuss the
importance and significance of rounding. Nolan mentions that the search area is
twice the size of Texas. How big is Texas? How could you calculate the square
miles of Texas? Is this an accurate analogy to use in this movie?
Situation 2: Movie: Cast Away – movie clip (1:24:25 – 1:26:01).
Again we see Chuck Nolan talking to himself. He states: “22, 44 lashings. We
have to make rope again. Eight lashings make structural, 20 feet a piece 160.
Gives us a month and a half. High tides, March and April. Need 424 feet of good
rope plus 50 miscellaneous around 475 feet of good rope. If average 15 feet a day
and build it, stock it, launch it, not much time.”
Assignment: I will first show the movie clip to the class. I will then break the
class up into small groups. Each group will discuss the scene and determine if it
makes sense to them. They will decide if the mathematics that Chuck Nolan used
were accurate. Was his approach realistic? Once each group has had enough
time to discuss the scene, we will discuss this scene together as a whole class.
Lesson 12 – Mathematics in a Movie – Stand and Deliver
Movie: Stand and Deliver – movie clip (33:35 – 35:40)
I will have the question written on the board and have my class try to solve it.
The main character Jaime Escalante states this problem situation. “Juan has five
times as many girlfriends as Pedro. Carlos has one girlfriend less than Pedro. If
the total number of girlfriends between them is 20, how many does each gigolo
have?”
I would like my students to try the problem and see if they are able to successfully
solve it. Once someone has answered it correctly, we will discuss the
mathematics necessary for solving the problem. I will then show the movie clip,
which shows several students answering the question incorrectly before one girl
answers it correctly. At that point, we will discuss how she was able to solve the
problem at the same time she entered the classroom. Was anyone in my class able
to solve it that quickly? Do they think this was reasonable or realistic? The class
will have to explain why.
40
Lesson 13 – Mathematics in a Television Show – Numb3rs Episode 104
“Structural Corruption” (Activity Title: Exploring Parabolas)
Introduction: “When a Cal Sci student apparently commits suicide by jumping
off of a bridge, Charlie investigates and suspects foul play. He believes the
parabolic path followed by the student terminates farther from the bridge than it
should for a jumper. While Charlie was ultimately wrong in his assumption of
foul play, he was correct that the body would follow a parabolic path in its
descent. This activity is intended as a means for students to discover the effects
the parabola‟s coefficients have on its graph” (Numb3rs Activity.)
(See Appendix 11 for complete lesson.)
Lesson 14 – Mathematics in a Television Show – Numb3rs Episode 224 “Hot
Shot” (Activity Title: Parabolic Food Flight)
Introduction: “In „Hot Shot’” Larry is in his office catapulting grapes with a
spoon to practice for the Physics Department food fight. This catapulting action
is similar to work that was done in the Middle Ages, when Galileo Galilei found
that the path of a projectile can be modeled by a parabola. As Larry practices
launching a grape with a spoon, things that he might consider are how hard he hits
the spoon to launch the grape, where his target is, any obstacles between him and
the target that he would like to miss, and so on. This activity allows students to
approach the problem mathematically and to review algebraic solutions of
equations as well as practice curve fitting to determine a parabolic path of a
projectile” (Numb3rs Activity.)
(See Appendix 12 for complete lesson.)
41
Annotated Bibliography/Resources
“About Numb3rs.” CBS.com. 2010. 22 May 2010
<http://www.cbs.com/primetime/numb3rs/about>.
This is a CBS website where you can view information about all of their
television shows.
Amend, Bill. Math, Science, and Unix Underpants. Kansas City, Missouri:
Andrews McMeel Publishing, LLC, 2009.
This book is a collection of FoxTrot comic strips.
Andrew, Tammy. “Math Comic Strip Activity.” Suite101.com. 1 Nov 2009. 3
April 2010 <http://classroomactivities.suite101.com/article.cfm/math_comic_strip_activity>.
This is an online article describing how to better assist students in creating
cartoons in a classroom.
Boswell, Laurie, Kanold, Timothy D., Larson, Ron, Stiff, Lee. Algebra 2.
Illinois: McDougall Littell, 2007.
Algebra 2 is mathematics textbook.
Cast Away. Prod. Hanks, Tom, Rapke, Jack, Starkey, Steve, Zemeckis, Robert.
DVD. Twentieth Century Fox and DreamWorks LLC, 2006.
A movie about a FedEx systems engineer whose plane crashes and he finds
himself trying to survive on a deserted island alone.
Chapman, Murray. “Plot Summary for Stand and Deliver.” The International
Movie Database. 1988. 20 April 2007.
< http://www.imdb.com/title/tt0094027/plotsummary>.
This site is an internet movie database. It offers many categories of interest.
Die Hard with a Vengeance. Prod. McTiernan, John, Tadross, Michael. DVD.
Twentieth Century Fox, 1995.
A movie about a New York detective who becomes a target of a terrorist who
threatens to blow up the city if he does not get what he wants.
“Discussion Questions and Projects for Use with Any Film.” Teach With Movies.
4 June 2005. 26 May 2007
<http://www.teachwithmovies.com/members/standard-questions.htm>.
Teach with Movies is a website that offers lesson plans and learning guides for
movies.
42
Fairbairn, Donald M. “What Did One Angle Say to the Other Angles?”
Mathematics Teacher Volume 102, Number 1 August 2008: 62.
Mathematics Teacher is an official journal published by The National Council of
Teachers of Mathematics. It is published nine times a year and you must be a
member in order to receive the magazine. There are problem solving activities in
the magazine as well as mathematical articles.
Hart, Eric W. and Martin, W. Gary. “Standards for High School Mathematics:
Why, What, How? .” Mathematics Teacher Volume 102, Number 5
January 2009: 378.
Mathematics Teacher is an official journal published by The National Council of
Teachers of Mathematics. It is published nine times a year and you must be a
member in order to receive the magazine. There are problem solving activities in
the magazine as well as mathematical articles.
Jessness, Jerry. “Stand and Deliver Revisited: The untold story behind the
famous rise – and shameful fall – of Jaime Escalante, America‟s master
math teacher.” July 2002.
<http://www.reason.com/news/show/28479.html>.
This is an online article discussing the untold story behind the movie Stand and
Deliver.
“Learning Guide To: Stand and Deliver.” Teach With Movies. 2005. 5
September 2006. <file://C:\twmnet-com\members\Guides\stand-anddeliver.html>.
Teach with Movies is a website that offers lesson plans and learning guides for
movies.
Lewkowicz, Marjorie L. “The Use of “Intrigue” to Enhance Mathematical
Thinking and Motivation in Beginning Algebra.” Mathematics Teacher
Volume 96, Number 2 February 2003: 92.
Mathematics Teacher is an official journal published by The National Council of
Teachers of Mathematics. It is published nine times a year and you must be a
member in order to receive the magazine. There are problem solving activities in
the magazine as well as mathematical articles.
McNamee, Rick. “Making Connections and Communicating Ideas.”
Mathematics Teacher Volume103, Number 6 February 2010: 464.
Mathematics Teacher is an official journal published by The National Council of
Teachers of Mathematics. It is published nine times a year and you must be a
member in order to receive the magazine. There are problem solving activities in
the magazine as well as mathematical articles.
43
National Council of Teachers of Mathematics. Cartoon Corner. Virginia:
NCTM, 2007.
A book published by the National Council of Teachers of Mathematics which
includes humor based mathematical activities.
National Council of Teachers of Mathematics. Principles and Standards for
School Mathematics. Virginia: NCTM, 2000.
A book published by the National Council of Teachers of Mathematics which
includes mathematical understanding, knowledge, and skills that students should
acquire from Pre-K through grade 12.
“Numb3rs: Will the CBS Show Be Cancelled? ABC and FOX Think So!” TV
Series Finale. 12 March 2010. 3 April 2010 <http://tvseriesfinale.com/tvshow/numb3rs-cancelled-season-seven/>.
A website devoted to TV show endings, reunions and revivals.
Pittsburgh Public Schools Committee Members. CAS Centers for Advanced
Study Teacher Handbook. Pittsburgh: Pittsburgh Public Schools, March
2005.
A Teacher Handbook published by the Pittsburgh Public Schools which contains
information on Gifted Education.
Serra, Michael. Discovering Geometry – An Inductive Approach. California:
Key Curriculum Press, 1989.
Discovering Geometry is a mathematics textbook.
44
Appendices - Standards
Appendix 1
Mathematics Content Standards
1. All students use numbers, number systems, and equivalent forms
(including numbers, words, objects and graphics) to represent theoretical
and practical situations.
2. All students compute, measure and estimate to solve theoretical and
practical problems, using appropriate tools, including modern technology
such as calculators and computers.
3. All students apply the concepts of patterns, functions and relations to solve
theoretical and practical problems.
4. All students formulate and solve problems and communicate the
mathematical processes used and the reasons for using them.
5. All students understand and apply basic concepts of algebra, geometry,
probability and statistics to solve theoretical and practical problems.
6. All students evaluate, infer and draw appropriate conclusions from charts,
tables and graphs, showing the relationships between data and real-world
situations.
7. All students make decisions and predictions based upon the collection,
organization, analysis and interpretation of statistical data and the
application of probability.
45
Appendix 2
Communication Standard for Grades 9 - 12
Instructional programs for prekindergarten through grade 12 should enable all
students to




Organize and consolidate their mathematical thinking through
communication
Communicate their mathematical thinking coherently and clearly to
peers, teachers, and others
Analyze and evaluate the mathematical thinking and strategies of
others
Use the language of mathematics to express mathematical ideas
precisely (NCTM 348)
46
Appendix 3
A Sheepish Problem – Cartoon Corner
1. What do you think the title should be under the cartoon? Explain why.
2. If n is equal to 200, what is the value of n – 1 and n + 1?
3. Suppose that n is an even number; how would you represent the next even
number greater than n? The previous even number less than n?
47
4. Write a mathematical sentence that could be used to determine the value of n
if the sum of the three numbers in the cartoon is 159. What are the three
numbers? Show your work below.
5. This mathematician usually counts to at least 3000 before falling asleep. At
the rate of one number per second, about how long would it take him to fall
asleep? Show your work below.
6. What should the labels on the sheep read if the mathematician counted by 2s,
by 3s, and so forth?
7. Pretend that the sheep are all lined up waiting to be counted. Billy the Bully
sheep doesn‟t want to wait his turn. He‟s 46th in line to start, but every time a
sheep is counted, Billy jumps ahead of 2 sheep. How many sheep will be
counted before Billy? Draw a picture to help you decide.
48
Appendix 4
Why Weight? – Cartoon Corner
1. Fill in the values in the table then make a graph from those values.
x
–3
–2
–1
0
1
2
3
x2
49
2. Use a graphing calculator to graph y  x 2  2 and y  x 2  7 . Sketch and
then predict what the graph of y  x 2  5 will look like. Check it on the
graphing calculator.
3. Predict an equation that will produce the graph below, and check it on a
graphing calculator.
Prediction:
Equation:
50
________________________
________________________
4. Start with the graph of y  x 2 , and write an equation that will create each
transformation. Check your equation on your graphing calculator.

Move the graph 3 units to the right

Move the graph 2 units left and 3.5 units up
51
5. Given the coordinates of two points, (1, 3) and (2, - 1), find other points that
will result in a parabola containing the given points. After you find the other
points, determine an equation.
52
Appendix 5
Keeping Track – Cartoon Corner
1. What is the diameter of the table in feet if 478 laps equal a mile? (Use 3.14
for π, and round your answers to tenths).
2. Suppose that three lanes, each 3 feet wide, surround the table. Runners run
down the center of each lane. After ten laps, how much farther would the
runner in the outside lane travel than the runner in the inside lane?
53
3. If the table above is 3 feet tall, how many square feet of fabric are needed to
make the tablecloth? (Use 3.14 for π, and then round to tenths)
Extension to this lesson:
Special Project – Racetrack Geometry – Discovering Geometry Pages 286 – 287
If you had to start and finish at the same line, which lane of the racetrack below
would you choose to run in? Sure, the inside lane. If the runners in the four lanes
were to start and finish at the line shown, the runner in the inside lane would have
an obvious advantage because that lane is the shortest. For a race to be fair,
runners in the outside lanes must be given head starts.
54
Extension: Your task in this special project is to design a 4-lane over track with
straightaways and semicircular ends. The semicircular ends must have inner
diameters of 50 meters so that the distance of one lap in the inner lane is 800
meters. You determine a width for the lanes. Draw starting and stopping
segments in each lane so that an 800 meter race can be run in all four lanes.
55
Appendix 6
Running Patterns – Cartoon Corner
1. Do you think Jason‟s path was shorter than running the coach‟s play? Why?
2. Use a calculator to express 10 2 to the nearest tenth.
3. On a separate sheet of paper use a scale of 1 centimeter equals 1 yard. Place a
dot in the lower-left corner of the paper to represent Jason‟s starting point.
Put a second dot at Jason‟s position at 10 yards out and a third dot at his final
position of 10 yards to the right. Measure the distance between Jason‟s
starting position and final position, and use your scale to express your answer
to the nearest tenth of a yard. Was Jason‟s math correct stating his distance
was 10 2 ? How do you know? Show your work used to verify this.
56
4. Use a protractor and your scale drawing to measure 45 degrees from the same
starting point. Measure 10 2 centimeters along the diagonal. Is Jason in the
same final position?
5. Assume Jason can move only forward horizontally or vertically in two-yard
increments. How many different paths can he take to get to his final position
at 10 yards out and 10 yards to the right? Use graph paper, and try easier
examples of 2 yards out and 2 yards to the right; 4 yards out and 4 yards to
the right. Look for a pattern. What conjecture can you make?
57
Appendix 7
Perfect Pumpkins – Cartoon Corner
1. Sal has 210 pumpkins, and every second pumpkin is either too big or too
small. How many pumpkins are either too big or small?
2. Every third pumpkin is too oblong. How many pumpkins are too oblong?
3. Every fifth pumpkin is too flat. How many pumpkins are both too oblong and
too flat?
58
4. How many pumpkins are too big or too small, too oblong, and also too flat?
5. Use the Venn Diagram to help organize your thoughts. The circles represent
pumpkins that are too big or too small, too oblong, and too flat.

The center section is for pumpkins with three imperfections. Place the
answer to question 4 in the center of the diagram.

The remaining sections of the diagram are for the number of pumpkins
with one or more imperfections. Complete the diagram by writing the
correct number in each section. Some, but not all, of the numbers can be
found in problems 1 – 3.
6. How many perfect pumpkins are in Sal‟s pumpkin patch?
7. Sal agrees to sell all the pumpkins in his patch. Perfect pumpkins are $5.00
each. Pumpkins that are too big or small are reduced 10 percent. Pumpkins
that are too oblong are reduced 20 percent. Pumpkins that are too flat are
reduced 30 percent. Pumpkins with more than one defect get more than one
discount. A pumpkin that is too small and too flat gets a 10 percent discount
plus a 30 percent discount, for a total discount of 40 percent. What is the total
value of the pumpkins in Sal‟s pumpkin patch?
59
Appendix 8
Prime Guess – Cartoon Corner
1. How many counting numbers occur between 1 and 100?
2. What is the difference between probability and odds?
3. What is the probability that Katy will randomly pick 23?
4. What are the odds that Katy will randomly pick 23?
60
5. Start with any number from 1 to 100.

Add 6. Multiply by 2. Subtract 8. Divide by 2. Subtract your “start
number.” Multiply by 12. Subtract 1. What is your answer? Compare
your answer with the class. What do you notice?

Use algebraic methods using the variable n to show why the number trick
works. Show your work.
6. Write your own number trick making your answer the starting number.
61
Appendix 9
The Aging Process – Cartoon Corner
1. Why might the mother have aged much faster than her son from his age 13 to
age 17?
2. Assume the mother was 24 years old when the boy was born. Assume also
that before age 13, the boy and mother aged at the same rate – one year for
one year. How old would the mother be when the boy turned 17?
3. Assume the boy‟s life calmed down and the mother aged only 2 years for each
year for the boy, from his age 17 to 21. How old would she be when he
turned 21?
4. Assume that after the boy turned 21, the mother and boy returned to a normal
relationship and each gained one year while the other was gaining one year.
How old was the boy when his mother retired at age 72?
62
5. Draw a line graph to show the ages of the mother and her son over the years.
6. Determine the slopes of the four line segments that make up the graph.
7. Create a cartoon similar to this one. Compare your cartoon with other groups
in class and then try to match the cartoons with the graphs.
63
Appendix 10
Graphs Alive – Cartoon Corner
1. What kind of graph is pictured in the cartoon?
2. What do the vertical and horizontal axes represent?
3. What is happening in the business to cause the graph line to slope up?
4. What is happening in the business to cause the graph line to slope down?
64
5. What happened when the cartoon character was on vacation? Why might this
situation have occurred?
6. Think of an event in your life that could be shown graphically. (It might be
that you were saving for something but had to spend your money suddenly, or
the speed of the car/bus as you ride to school in the morning, etc.) Below,
draw a graph of what happened over time. Label both axes properly. Notice
that you can start the graph below zero!
7. Write a story of the event shown in your graph in question 6. Be sure to
explain what happens when the slope of your line changes.
65
Appendix 11
NUMB3RS Activity: Exploring Parabola
Episode: “Structural Corruption”
Topic: Parabolas
Grade Level: 9 – 10
Objective: Students will investigate parabolas of the form y = a(x – b)2 + c and
state the effect each coefficient has on the parabola‟s graph.
Time: 20 minutes
Materials: TI-83 Plus/TI-84 Plus graphing calculator
Episode: “Structural Corruption”
Numb3rs Activity
Student Pages
Name: ____________________________________________ Date: __________
NUMB3RS Activity: Exploring Parabolas
When a Cal Sci student apparently commits suicide by jumping off of a bridge,
Charlie investigates and suspects foul play. He believes the parabolic path
followed by the student terminates farther from the bridge than it should for a
jumper. While Charlie was ultimately wrong in his assumption of foul play, he
was correct that the body would follow a parabolic path in its descent.
What exactly is a parabolic path? A parabolic path follows a certain trajectory
that can be described by an equation of the form y = ax2 + bx + c. The path made
by a baseball in a game of catch, a cannon fired at a target, or even an object
thrown off a bridge are parabolas.
66
1. Suppose the person jumping off the building followed the equation
y  16 x 2  200 where x is time (seconds) and y is distance above the ground
(feet). Complete the table below and graph the parabolic path.
x
0
1
2
3
4
y
This equation can also be graphed using your graphing calculator. Press Y= and
enter y  16 x 2  200 in equation Y1. Now press WINDOW and use the
following settings: Xmin = 0, Xmax = 4, Ymin = 0, and Ymax = 200. Now press
GRAPH.
2. The highest (or lowest) point a parabola reaches is called its vertex. What is
the vertex for the parabola above? ____________________
3.
The x-coordinate of the points where the parabola crosses the x-axis are
called the x-intercepts. What are the x-intercepts of the equation above?
___________________
While any equation can be graphed by plotting point, it would be much easier if
the graph could be determined directly from the equation. Use the
Transformation Graphing App to determine if the vertex and shape of the
parabolas can be obtained directly from the equation.
67
Press the APPS key and select Transfrm (press any key to start the App). Once
this is done, press Y=. Notice that to the left of each equation, there is a new
symbol. This shows that the Transformation Graphing App is running. (It also
means that you can only graph one equation at a time while the App is running).
This App allows the changing of the coefficients in an equation to explore how
each value affects the graph.
Example: Enter A(X – B)2 + C into Y1. (Remember to use the ALPHA key to
type letters). Press ZOOM and select 6:ZStandard to display the following:
The equation is shown at the top left of the window, and the values of A, B, and C
are shown below. Use the up and down arrow keys to select different values for
A, B, and C. Use the right and left arrow keys to change the values by an
increment. Observe how the graph changes. When finished, choose values to
match the screen above.
4.
What happens to the graph when the B coefficient is changed?
5.
What happens to the graph when the C term is changed?
6.
What happens to the graph when the A coefficient is changed?
7.
Using the Transformation Graphing App, find the vertex of the parabola
when B = 2 and C = 5. (Use any nonzero integer for A).
68
8.
Using integers, describe what effect the value of A has on the graph.
9.
Experiment with values of A and decide how to make the graph wider.
10. Explain why a horizontal line is formed when A = 0.
11. Use the Transformation Graphing App to find an equation whose graph has
zeros at – 3 and 5, has a vertex at (1, 8), and opens downwards.
69
Appendix 12
NUMB3RS Activity: Parabolic Food Fight
Episode: “Hot Shot”
Topic: Parabolas and curve fitting
Grade Level: 9 – 12
Objective: Solve simultaneous equations; fit a curve to data
Time: 30 minutes
Materials: TI-83/84 Plus Calculator
Episode: “Hot Shot”
Numb3rs Activity
Student Pages
Name: ____________________________________________ Date: __________
NUMB3RS Activity: Parabolic Food Fight
In “Hot Shot,” Larry is in his office catapulting grapes with a spoon to practice
for the Physics Department food fight. This catapulting action is similar to work
that was done in the Middle Ages, when Galileo Galilei found that the path of a
projectile is parabolic in nature. These parabolic paths can be modeled with
quadratic functions.
As an exercise (and not recommended for actual practice for a food fight), this
activity reviews algebraic solutions of equations as well as practices curve fitting
to determine a parabolic path of a projectile. In the graph shown below, the point
marked L represent the thrower (Larry), the point marked B is a point that the
path passes through and is slightly above an obstacle that the path must miss, and
the point T is the target that Larry tries to hit.
70
1.
a. Identify the coordinates of points marked L, B, and T.
L = __________
B = __________
T = __________
b. One way to write the equation of a parabola (as a quadratic function) is
y  ax 2  bx  c , where a is nonzero. Substitute the values of x and y
from part a into the equation to determine three equations in a, b, and c.
c. Using substitution, solve the system of three equations form part b to find
the values of a, b, and c.
d. What is the quadratic equation that determines the path of the object being
thrown?
71
Another method of finding the solution of three equations in three unknowns is to
use the TI-84 Plus calculator to solve the matrix equation AX = B, where A is the
coefficient matrix, X is the variable matrix, and B is the constant matrix. In the
example below, the system of equations on the left could be represented by the
matrix equation on the right.
2a  3b  c  7
 1a  2b  3c  4
a b c  5
3
1   a  7 
2
 1 2  3 b   4

   
 1  1  1  c  5
A
X
B
The matrix equation can be solved using the TI-83/84 Plus using the following
steps. (You may need to first clear some existing matrices from your calculator‟s
memory.)
 Press ` ù.
 Go to the Edit menu and select Matrix A.
 Define Matrix A as a 3 x 3 matrix, and enter the coefficients of a, b, and c
above.
 Follow the same directions to create a 3 x 1 matrix for B, using the
constraints shown on the right-hand side of the equations.
 To solve the equation, find A-1B (where A-1is the inverse of matrix A).
Press ` ù 1 to select matrix A, and press i. Then press ` ù 2 to select
matrix B. Press e to obtain the values seen at the right, which are
approximations of the solution to the system of equations.
2.
Use your calculator to solve the system of equations from Question 1. How
do the calculator‟s answers compare to those obtained by hand?
72
3.
Define new matrices A and B as in Question 2, but use the command
randint(1,10) for each of the entries. (To find randint(press m < 5.) Solve
your new system of equations. What does it mean about the points if there are
no solutions? What does this mean about matrix A?
4.
Use your calculator to determine a quadratic regression equation for the
points from Question 1. Enter the x-coordinates of the points in list L1 of your
calculator, and enter the y-coordinates of the points in list L2. To find the
quadratic regression equation, use the command QuadReg L1,L2. (To find
QuadReg, press S, go to the CALC menu, and select 5:QuadReg.) How
does this equation compare to the solution you obtained in Questions 1 and 2?
73