Academic Honesty in Question

Cheating? Really? Are you positive?

Cheating is sometimes a hard call and other times it is not even questionable.

Here’s my story:

One day I decided once to give my students extra time on an exam the next day because I had made it too long .  I was trying to be nice but maybe not one of my wisest decisions.  Looking back, it changed the way I wrote tests and administered them in the future – but I’m getting ahead of myself.

Someone took a copy of the test when he/she left class and shared it with friends. It was very evident that this had happened and a couple of students came to see me when they found out this out. It was also obvious that not all students saw it but most of them knew about it.  I couldn’t eat or sleep while I was trying to figure out what to do.  (Now students don’t even have to take the physical test if they can get pictures with their mobile devices.)  I gathered my facts and took them to the principal.  I laid out my plan, got his input, and made sure I had his support for what I was going to do. (Don’t forget to do this if you get in this situation! )  Next, I sent a letter home to parents, wrote another exam, and required all students to take it over since I couldn’t tell for sure who all were involved. It wasn’t a popular decision with all students or parents, but in my opinion, it was the most fair thing to do. The funny thing is, that some of the students I know were involved  had the parents who were most outraged that their child had to take the test over.  My whole body was a mess for a long time after and the relationship that I had with that class was back to the “prove to me you’re worthy of my trust.”   BTW, did I mention it was an AP Calculus class with potential valedictorians in the class, and that it was when I had a student teacher?

To make things worse

Now there are all sorts of other complications.

from (check out the conversation on this page if you need more to think about!)

from (check out the conversation on this page if you need more to think about!)

Here are some articles to check out and maybe help you think this through.  Warning: Don’t expect that all of your questions will be answered.

  1. Digital Tools Raise Questions about what is and what is not cheating by Katie Ash (August 21, 2013) –
  2. Read the stats about Harvard’s freshmen in the class of 2017 By  (Posted Thursday, Sept. 5, 2013 )
  3. Moving from Cheating to Academic Honesty By Eugene Bratek –
  4. Cheating Runs Rampant:  No Child Left Behind has unleashed a nationwide epidemic of cheating. Will education reformers wake up?  by Daniel Denvir ( May 25, 2012) –

What would you do if you saw or suspect students of cheating? 

Posted in Calculus, Common Core, Math Education, Technology | Tagged , , , | 1 Comment

Read a Test to Students without Saying a Word!


I’ve always thought there would be some very cool educational applications for QR codes.  I’ve make QR scavenger hunts that are fun for students while being great learning activities.  There is an example of a Quadrilateral Properties Scavenger Hunt Activity at:

NAPMATHqrCodeBut today… I think that I have very helpful application for ELL / ESL students,  students with special needs, and for teachers that need to be in many places at once.  My students didn’t like to leave the classroom to have tests read to them.  They didn’t like being singled out.  My mission was to figure out how to get them the help they needed while keeping them in the classroom with their peers.

QR Codes, Mobile Devices, Assessments, and Empowering Students

With more and more students and schools having access to mobile devices, technology can help out when students need a test read for them.  They can listen to the questions as many times as they need to and move at their own pace.

Prepare the Test

What is needed to MAKE the quiz or test:

  1. Create your test as usual, leaving a place to enter a QR code.  See/download a sample assessment below.
  2. Access or another site that allows you to record the question and save it to a website as RecordMP3 does.  RecordMP3 is very user friendly and does everything you need it to do with ease!  You can use it for other things, but that’s another post.  Record your voice, reading a question as clearly as possible. (Okay you got me! You do have to talk but not during the test and your voice can be “on demand” by multiple students at once!!!!)
  3. Use a microphone to make your recordings as clear as possible.  Make a document (and/or add the website to the test) and save each questions’ websites – make sure to label them to make them easy to find again in order to create your QR codes.
  4. Access  a QR Code Generator like that will allows you to enter the web address for your each of your recordings and generate the QR code for each of your recordings/questions.  Copy the web site for each questions’ recording and paste it into the QR generator and generate a QR code for the web site for the recording of each question, one at a time.
  5. Paste the QR code into the document and continue until all questions have a QR code.  Save the document each time as you enter codes.

 SampleQuizDownload this sample quiz as a

Give the Assessment

What students need to TAKE the quiz or test:

1. A copy of the quiz or test and writing device (if taking this as a paper/pencil assessment).

2. A device (such as an iPad, iPhone, Android, etc.) with a QR Reader such as the

3.  Earphones

Practice makes perfect

Now it’s time for students to take the assessment.  Before students take an assessment for the first time, give them some practice examples with QR codes until they are comfortable with the process.  You could use the sample quiz or make some review questions that take them to websites, to voice recordings, or other activities on the web.

Individualize the experience

To allow the students to remain in class during the assessment, have them use headphones so they won’t bother students near by.  Now you have EMPOWERED your students to complete their assessments on their own!!!!

One final idea for all your assessments!

Oh… one final thought – wouldn’t QR codes be great for making different forms of a test for all of your students, with or without voice recordings? Try for making QR codes for just text entries.  You might still need to include the figures for some of your questions, but Classtools makes it easy to get the QR codes!

Websites used:

Additional resources for QR codes

Posted in Algebra, Common Core, Math Education, Technology | Tagged , , , , , , | 1 Comment

Origami + Math + Mobiles

image001So exactly how do origami and math relate to each other? The connection with geometry is   clear and yet multifaceted; a folded model is both a piece of art and a geometric figure. Just unfold it and take a look! You will see a complex geometric pattern, even if the model you folded was a simple one. As a beginning geometry student, you might want to figure out the types of triangles on the paper. What angles can be seen? What shapes? How did those angles and shapes get there? Did you know that you were folding those angles or shapes during the folding itself?  Can you come up with any relationships between a fold and something you know in geometry?

purple3Any basic fold has an associated geometric pattern. Take a squash fold – when you do this fold and look at the crease pattern, you will see that you have bisected an angle, twice! Can you come up with similar relationships between a fold and something you know in geometry?

On the other hand, if you are a person who likes puzzles, there are a number of great origami CubeOrigamichallenges that you might enjoy trying to solve. These puzzles involve folding a piece of paper so that certain color patterns arise, or so that a shape of a certain area results. But let’s continue on with crease patterns… For instance, the traditional crane unfolded provides a crease pattern from which we can learn a lot. Pick a point (vertex) on the crease pattern. How many creases originate at this vertex? Is it possible for a flat origami model to have an odd number of creases coming out of a vertex on it’s crease pattern? How about the relationship between mountain and valley folds? Can you have a vertex with only valley folds or only mountain folds?  

image06How about the angles around this point? You can really impress your teacher (or your students) with this…of course, you will need to understand it first! There is a theorem called Kawasaki’s Theorem, which says that if the angles surrounding a single vertex in a flat origami crease pattern are a1, a2, a3, …, a2n, then:

a1 + a3 + a5 + … + a2n-1 = 180      and        a2 + a4 + a6 + … + a2n = 180

In other words, if you add up the angle measurements of every other angle around a point, the sum will be 180. Try it and see!   Can you see that this is true, or, even better, can you prove it?

The study of origami and mathematics can be classified as topology, although some feel that it is more closely aligned with combinatorics, or, more specifically, graph theory.  You can investigate these connections further on your own.


Origami, a Japanese word, combines the word oru (to fold) and the noun maki (paper).

A Bit Of HistoryOrigami origin is unclear. Some historians claim the origin of origami began as a Japanese tradition of folding important documents/certificates. “Origami Tsuki” means “certified or “guaranteed”. The phrase stems from their ancient custom of folding certain special documents – such as diplomas for Tea Ceremony masters, or masters of swordsmanship – in such a way as to prevent unauthorized copies from being made.  Others claim origami came about after the invention of paper and made by Cai Lun in AD 105. This was the first usage of the word “origami” traced in Japan. The word “origami” came to be used occasionally for another kind of ceremonial folding, namely for “tsutsumi”, or formal wrappers, by the beginning of the 18th century. However, its use for recreational origami of the kind with which we are familiar did not come until the end of the nineteenth century or the beginning of the twentieth. Before that, paperfolding for play was known by a variety of names, including “orikata”, “orisue”, “orimono”, “tatamgami” and others. Exactly why the switch came to “origami” is not clear, but it has been suggested that the word was adopted in the kindergartens because the written characters were easier for young children to write. Whatever the origin, Japan has fully practiced the art. It is so valued in Japan that it has become part of religious ceremonies.

What is origami? Origami is the art of folding paper and creating three-dimensional figures of: people, animals, object, and abstract shapes. The material for creating origami is a piece of thin paper; although any paper may be used. The paper is normally cut into a 15 cm. square that is plain white on one side, and decorated on the other (color or decoration). Some creative origami artists try to experiment with cardboard, cloth, wire mesh, sheet metal, and even pasta. I bet there a million other possible ways to be creative in origami.

The four most common bases of origami are the kite base, fish base, bird base, and the frog base. Bases are the starting shapes for different figures. Adding additional folds you can create figures of virtually any shape. Some of the folds specialize in modular origami, or making multiple copies of a simple single shape and forming the pieces to make an elaborate structure.  See the cube to the right.

In 1999, Joseph Wu provided this simple yet encompassing definition.

Origami is a form of visual / sculptural representation that is defined primarily by the folding of the medium (usually paper). 

Here’s another interesting theorem:  
 Every flat-foldable crease pattern is 2-colorable.

In other words, suppose you have folded an origami model which lies flat. If you completely unfold the model, the crease pattern that you will see has a special property. If you want to color in the regions of your crease pattern with various colors so that no two bordering regions have the same color, you only need two colors. This may remind you of the famous map-maker’s problem: what is the fewest number of colors you need to color countries on a map (again, so that two neighboring countries aren’t the same color)? This is known as the Four Color Theorem, since the answer is four colors. As an interesting aside, this theorem was proven in 1976 by American mathematicians Appel and Haken using a computer to check the thousands of different cases involved. You can learn more about this proof, if you like.

But back to our theorem since I know that it is buggin’ you (look to the right!!).  Can you see that you need only two colors to color a crease pattern? Try it yourself! You will see that anything you fold (as long as it lies flat) will need only two colors to color in the regions on its crease pattern.   Here’s an easy way to see it: fold something that lies flat. Now color all of the regions facing towards you red and the ones facing the table blue (remember to only color one side of the paper). When you unfold, you will see that you have a proper 2-coloring!


image021   Mobiles are not just to hang on a baby’s crib. They’re modern art.  Mobiles were first made by Alexander Calder, an American artist who started as a boy making wire and wooden toys.  Calder created a whole circus of animals of and performers with wire and cloth.     image020    Mobiles are sculptures with parts that move. Mobiles mean movement. You’re mobile when you move and jump and play. Mobiles move when they are suspended freely in space.


Mobiles are shapes – circles, triangles, rectangles and squares – floating in space, suspended by a string or wire. If the shapes are balanced, they will spin and float, turn and twirl.  Touch them or blow on them, and watch them move.


Generally mobiles hang from a ceiling, but some are mounted on pedestals.  All parts of a mobile should swing freely so that the movement of the mobile is maximized!


Links – More about Origami

purple3Links – More about Mobiles

Download the Make your own custom made origami mobile.

Posted in Geometry | Tagged , , , , , | 1 Comment

Not Permanent but Perfect – a Post-It®

PostItPadsI LOVE a Post-It®.  I never leave home without a pad of  them in my purse or computer bag. You’ll find them on my desk at work and at home.  They’ve become a valuable tool for marking up materials that I’m presenting in a workshop, noting important ideas from another workshop or book, or when I’m looking for new activities in books. magazines, online, or just watching television.

So, I was just thinking about another idea we had when I was working with teachers and putting together a workshop for the Common Core State Standards – printing on a Post-It®!  We wanted teachers to see the vertical progression of the objectives in each of the standards.  Printing each objective on a Post-It® gave groups of teachers the opportunity to order and re-order them easily.  It worked!  How do you do it?  The YouTube Channel, SecretsofTeaching ( has a quick 1 minute how-to video.

Ideas for using a printed Post-It®

  1. Print student names and pictures on a Post-It® for
    1. Seating charts – easy to rearrange and for a substitute.
    2. Make group assignments or choosing partners.
  2. Print historical or fictional figures for a matching activity on a bulletin board.
  3. Print a sequence of events and have students place them on a timeline.
  4. Make Post-It® Note signs and messages – quick and easy to display/change, once the letters are printed.
  5. Start an in-class Twitter Board – Have students “tweet”  thoughts/ideas/questions on a TwittterPost-itpost-it and stick it on the board at the end of class or during an activity! Print post-its® with 140 blanks for this activity! Students will need to learn how to say what they are thinking in 140 characters or less!

Post-it® Templates –

Use Post-it®s with these activities for all grades and subjects

Progress Folder

Here are some other ideas I’ve used in my classroom and thanks to teachers on the web for photos of their creative classroom activities.

  1. Document student progress – pick a color to indicate a mastery level (i.e. yellow – Demonstrates understanding and can apply knowledge, pink – Demonstrates understanding, or blue –  Demonstrates limited understanding)  place it in the student’s box on the chart, noting important information that documents the achievement like a test score, project completed, take a photo and print a post-it with their completed art work or assignment, etc.  Glue a chart inside a file folder for each unit.  Grab students’ post-its later to make a folder for parent conferences.  Download the Word document for this chart:   Progress template
  2. If a student finishes early, have the student pick an activity on a post-it® from a board of activities –
  3. Exit Ticket Chart- and
    1. Math – give students an “answer” to a problem and have them write an equation with a solution of that “answer.”
    2. Any class – have  students complete a post-it® with what “stuck with them” and place it on the “exit slip board” as they leave class.
    3. Language Arts – Have students write a sentence and label each word with its part of speech.
  4. Have students make a prediction – place their prediction in a specific place in the classroom, on a poster or bulletin board, in a file folder to make it easy to take home and organize, etc.
  5. Use a gallery walk to view ideas, pictures, and/or problems and have students put their ideas on a post-it® below each “station.”

Please share your Post-It® ideas for the classroom.

For even more ideas, visit the Post-it® Activity Center at

Posted in Common Core, Math Education, Technology | Tagged , , , , , , | 1 Comment

What’s Math Got to do with it? – Video/Writing Assignment

Students always ask, “What jobs use math?” “When are we ever going to use math?” We know that the answer is almost all of them. However, when I started teaching, it became more important to have examples to share with students. As the Internet became more available (thanks to Al Gore LOL when he stated in the forward to an Internet Guide,“I called for creation of a national network of information superhighways.”),  it became easier to find examples.

Movie PopcornVideos, Internet, and the Assignment

I wanted to get students thinking about this and came up with an assignment to help students start investigating the answer to these questions. Students in ANY math class can do this assignment and the same video can be viewed by different classes and you’ll get different observations. Little did I know how powerful the assignment was going to be, how much I would learn, and how fast it was to grade. Students needed to watch two-four videos and papers per semester.  For the first two, I took students time in the computer lab.  After that, students did this as an assignment outside class time with little, if any, resistance. Papers were saved to their class Moodle site to make accessing their papers easier than via emails.  A shared drive or a DropBox would be great alternatives to Moodle for this assignment. For students without computer access at home, there were computers available before and after school and during the day in the media center.

Below is the actual assignment and the grading rubric.  Please customize #2 before giving it to students. I changed it to Algebra concepts for Algebra, Geometry concepts for Geometry, Calculus concepts for Calc classes, etc.

What’s Math Got to Do with It?

Each week when there is a movie to watch (on the Internet) and each movie is about 5 minutes long on the average. Please submit these on your math class moodle site.

Take a few minutes (these will be short movies) and watch the movie and do a little research (use a book, a magazine, another Internet site, etc.) and then write a paper at least one page long if word processed or 2 pages if hand-written. Please do not hand in a list of questions and answers.

1. What’s MATH got to do with the subject of the movie? (2 points)

2. Be more specific and talk about What course concepts (i.e. replace this with “Algebra” concepts if you’re in Algebra, Geometry if you’re in geometry, Calculus if you’re in Calc, etc.) appear to be related to the subject of the movie? For ideas, look in your math book after watching the movie. Be specific and use correct mathematical vocabulary. (3 points)

3. Would you like to have this job? Why or why not? (1 point)

4. What occupations (2 or more) that use this math might be applicable to the math and the subject of the movie? (2 points)

5. What did you find most interesting, what was new to you, and/or did it make you think differently about math in the “real world”? (2 points)

Please remember to cite your sources. There must be at least one other source in addition to the movie website – another related website or magazine/newspaper article or other print material.(-1 if they are not given!)

If your paper does not meet the following requirements -1 point .
*Word-processed written paper must be at least one page long using a 12-point or smaller
font (such as Times New Roman), not more than 1″ margins, and double-spaced. (submit these on Moodle)
paper must be two pages hand-written.

Reading papers can be time-consuming but these went really quickly and were easy to grade.  The time invested reaped great rewards!


Download the assignment as a Word Document: MovieAssignment

Download the evaluations sheet as a Word Document: MovieEval

What I learned from student papers:

(examples of student work)

  1. Students stopped asking the question about when were they going to use the math. They started realizing that until they knew what they were going to do, they needed to be prepared to meet any math challenges and learn the math. Some of the people in the movies said that they didn’t know that they were ever going to use the math that they used in their jobs.
  2. Some students didn’t even know some of the careers they saw in the movies existed and got excited that they were REAL jobs.
  3. I learned more about my students, their experiences, their hopes and dreams, their interests, and more. I used this information in interactions with students, later assignments and when developing future projects.
  4. I realized that English teachers probably learned lots more than I had about my students by reading their papers.  This might be why they were probably perceived as more “interested in them as a person” than math teachers.

Sites for good videos – Here are some recommendations.

  • My all-time favorite: The Futures Channel– Do your students ever ask “What’s math got to do with it?” Here is a site that has videos that you can use to answer some of their questions. The “stars” are real people – some famous, some not, some young, some not, but all interesting and current topics. Movies are about 2 – 6 minutes in length – just perfect to begin or end a lesson. The videos change from week to week so sign up for their free newsletter that keep you up to date on the movies that are new for the week. (Just so you know, they are generally accessible for two weeks and then they are replaced so use them fast!) The Futures Channel uses media technologies to link scientists, explorers, and visionaries with today’s learners and educators. Videos on this site are very high quality and there are often lessons and activities to go along with the videos, making them easy to use in the classroom.
  • Math at Work Mondays – Look on this site on Mondays (and other days) for Math at Work Mondays for interviews (some videos) with people at work and how they use math in their jobs.
  • Head Rush Cool Jobs  – While this site is technically about cool jobs in science, there are several videos that relate to mathematics. For example, there is one video where a skate park designer describes how he uses shapes, angles, and trigonometry to create his skate parks. It is a great video for geometry classes.
  • Mathematics for Economics -ME:TAL or Mathematics for Economics: Enhancing Teaching and Learning is an organization that creates videos and lessons illustrating to students the use of mathematical topics in the business world. For example, there is a video that shows two industries that use linear programming. The website is great to show students real life applications.
  • Careers Information – There are many career profiles/interviews at this site
  • Maths in Work videos – This includes many links to jobs including Designing Aircraft, Listening to Music, Experimenting with the Heart, Revolutionizing Computing, Beating Traffic, Scanning the Unseen Cat scan, Packing It In, Unearthing Power Lines, What to do with a Maths Degree, and much more  Learn a lot about 40+ careers and answer when math is used in each, including possible salary predictions
  • WeUseMath – Learn a lot about 40+ careers and answer when math is used in each, including possible salary predictions
  • 10 Amazing Jobs You Could Land with the right STEM Education
Posted in Algebra, Calculus, Common Core, Geometry, Technology | Tagged , , , , , , , , , , | 2 Comments

What’s Math Got to do with it? Student work

A few excerpts from student “Movie” papers

These are randomly chosen parts of students’ papers that I had on my computer.  They are typical  kinds of things I read in papers but by no means could ever let you know all of the amazing things I learned from my students when I read their papers for this assignment.  I do hope, however, that they will encourage you to try this assignment with your students.  Papers get better after they’ve watched several movies and written several papers.  Enjoy!


aviationJeff L:  There are many different math concepts involved with aviation.  Man physics ideas are taken into account, such as air resistance forces, gravitational potential energy from being higher in the air, and kinetic energy from motion.  Also, math goes in to determining what shapes the wings and body of the flying object will take so that the path that the air takes as it travels over the object will help force the object upward and keep it in flight.  Math would be used to measure each of these parts, in designing and building the objects, and calculating what forces act upon the object as it flies.

… One thing that I found to be the most interesting about this subject [aviation] is how easily you can get an airplane to travel at a somewhat quick pace.  I started to wonder what the fastest airplane is and how fast it has traveled before.  NASA had a project to build such a plane, called the X-43, and they got it to break the speed record recently at a speed of Mach 9.6, or almost 7,000 mph.

Roller Coasters:

RollerCoasterKelly S:  They use basic math concepts such as speed and velocity. This is important because they need to know how fast the roller coasters are going at specific times. They can calculate speed and velocity by algebra.  They also use conversions such as the one involved with the conversion of potential energy to kinetic energy. This type of conversion is important for understanding how the cars at the top of a loop with react.  Also, it involves extraneous variables, also known as stressors.  These have to observed and monitored carefully for the safety of the passengers. They want to be sure that the presence of these stressors does not affect the outcome of the independent variable. They also use modeling in conjunction with the conversions. They start by building the scale models of the large coasters before they start construction. This helps save a lot of money in resources.

…  People working in the car industry would be very concerned with this same type of math.  They would want to be able to test their cars and clock exact speeds and also want to be sure that their products are safe.  Also, people who work in NASCAR would be concerned with the same principles of clocking the speeds and positions of the cars.  They want their timing to be exact, which would make them very interested in the math.

I found it interesting that so much math went into building a roller coaster.  I never thought about how dangerous they were too.  If someone calculates one little equation wrong, that could mean that the car flies off the tracks because it was going to fast.   Also, I think it would be cool to try new ideas and come up with creative thrilling rides.

The Forester

foresterKylor S: I love to camp with my grandparents. When we go camping one of our favorite things to do is to take hikes through the trails. My grandpa is great at knowing what types of trees and plants are along the trail. I thought this movie was pretty cool about how foresters use math to manage these areas. I never really thought about that before; I guess I just thought that these areas just happened.  This forester in this movie did a good job of explaining how her job dealt with decision making and problem solving to decide which trees to leave and which trees to harvest. The planting of a certain area can be done naturally where certain trees that are left put down seeds or it can happen by design where a certain number of trees are planted realizing that some may not make it.

Geometry terms that help Tami the forester in this movie were ratio, percentage, proportion, diameter, hypothesis and conclusion.  In these cases the diameter of the tree and the tree height are used to determine which trees will be left to produce seeds.  The percentage of trees are then determined based on a hypothesis of what trees are already in the area and what percentage might not survive. This hypothesis is based on a volume per acre. The lumber off each are of land that consumes an acre is expressed as a board foot. A board foot is one inch thick by 12 inches long by 12 inches wide. Atypical logging truck that you might have passed on the interstate can typically carry 4000 board feet.  Tami explained how the ratio of the types of trees is important when planting; in the example from the movie for every 6 Ponderosa pines they planted 16 Douglas firs. Tami also explained that in burn areas they overplant because they hope to get an 85% survival rate. Tami also explained that math helps them figure that when they plant approximately 22,000 trees this works about to about 170-180 trees per acre.

…  I thought it was interesting how much math Tami and other foresters use. I think it is neat that the decisions she makes can last over a hundred years and she uses problem solving and math to help her. Different forests can have different percentages of trees and you need to know what those are and to do that you need special equipment. The stuff in her pack like a compass, bear spray, snow shoes, chainsaws, and the space blanket just aren’t the things one normally takes to work!

Water Supply

waterT. T: I chose the movie water supply,which i thought would be very interesting being that I’ve always wondered where and how tap water was available.  I would have never imagined that so much math was used in every day work. Dealing with the water supply has a constant use of math .one of the big things is volume how much water a tank can hold at a time .speed how much time it will take for the process to keep up with the supply and demand .although the job is very interesting i don’t think i would like to go into this field of work because it has a lot to do with machines. i don’t like dealing with things that are bigger than i am ,and i cant swim so the thought of being around 1.5 billion gallons of water is scary.

Posted in Uncategorized | 1 Comment

Eating Volumes: Delicious & Unexpected Lesson in Calculus

Yesterday I gave a workshop at the iMathination Conference in St. Charles,IL.  The conference is full of hands-on math workshops for teachers, mostly from the city of Chicago  – amazing teachers!  I was asked to give a hands-on workshop for pre-calculus/calculus teachers and immediately thought of volume and prepared the workshop.  In the hour and 15 minute session, I planned to explore volumes of solids created with known cross-sections with oranges, cupcakes, and ending with making a Styrofoam model for them to take back to their classrooms.  If time allowed, ideas for hands-on activities for volumes of solids of revolution, but we didn’t get that far.


What I did not expect was that in my workshop were:  1 Calculus teacher, 1 pre-calc teacher, 1 art teacher, some science teachers, math teachers that did not teach calculus, and a middle school teacher who had never had calculus (and hated geometry)!  So, not only were there not a group of Calculus teachers, but I was intrigued why people came to this session.  Was it food in the title?  Curiosity?  Last choice on their list and other sessions were full?    No matter what the reason, this was my chance to see if I could make the ideas of calculus accessible to non-math/non-calculus people!!!!!

Area_formulasWe started with something they were familiar with – visualizing a deck of cards.  They told me that they could find the area of a card.  If we put them into a stack, they could see that the deck of cards also had volume, introducing the idea of summing the areas to find the volume.

Each person was given 1/2 of an orange and a knife. I asked them, “if I were to ask you what the base of your orange was, what would it be?”   All of them quickly identified the cut side of the orange and placed it on the little cutting mat in  front of them.  I asked them to make a cut – but not through the center.  They looked at the surfaces created and they correctly identified them as semi-circles. So, if you cut your half of an orange into more pieces in the same way and VERY thin, each piece would have an area that could be calculated but together they could be stacked and made 1/2 an orange which was more likely to be identified to have volume.  We repeated the idea with a Little Debbie’s chocolate cupcake (no Hostess ones anymore – sad) and again they identified them as having trapezoidal cross-sections.  We stopped and reviewed basic area formulas.

The Calculus

We looked at models using the concept of Riemann sums in 3-D to realize that if we cut the slices thin enough and put them back together that the model began to smooth out and look more like the original “orange.” They also realized that the model with thinner slices was getting  closer and closer to having the same volume as the orange.


It was time to introduce the calculus.  With the help from Audrey Weeks Calculus in Motion files for Geometer’s Sketchpad, we watched the shapes being generated from one side of the circle to another using squares vs. semi-circles.  It was easy to visualize the sides of the square going from sides of length zero to 4 (twice the radius) and back to zero, creating the solid with a base of a circle and cross-sections of squares.  Next we decoded the notation of a defiinite integral that would calculate the volume of the solid and how it related to what we saw.  integral

The area function A(x) was written in terms of x and is called the integrand.  The limits of integration showed where the square started and where it ended up as it moved along the x-axis. The dx represented the infinitely thin thickness of each cross-section,  The integral sums those areas calculating the volume of the solid.  I realize that it might be a bit simplified but it made the calculus make sense with what they saw.  I didn’t teach them how to integrate – just how to set up the integrals.  There are sites on the Internet that will do the  integration like the Definite Integral Calculator and I think it is more important to know how to set them up and take the mystery out of the notation!

OrangeCrossSectionsThe Models

What I saw was awesome.  Everyone asked great questions, showing that they were invested in the process, and intent on understanding how the integral was set up to find the volume of the “smoothed out” solid made of infinitely thin cross-sections.  The artist was relating the models and math to pottery and was trying to figure out how she could make the math accessible to her students.  They all made models of solids with known cross-sections using styrofoam.  They chose their cross-sections and I saw them making  squares, equilateral triangles, semi-circles, and isosceles right triangles on a circular base.



I will admit, that in the early years of my career, this workshop might have not made my list of workshops because I never saw the usefulness or the fun in calculus – scary to even type that.  Even as a math major, Calculus was intimidating.  My calc teacher in college was less than helpful (and was fired)!  Once I wasn’t afraid of it anymore – with help from my friend Carol, a Calculus grant from HP and Oregon State in 1993, and having to teach Calculus,  it became my goal to make sure that others never become afraid of  MATH!  I believe in the power of math and that all people need to find and use the power too!

The materials needed for this activity:

  • 1/2 orange and a cupcake (preferably one with a flat top!)
  • 8″ x 8″ or 12″ x 12″ squares of styrofoam – 2 per person: one to cut and one to mount the finished solid on – ours were 3/4″ thick and the home improvement store cut them for me for free!   4′ x 8′ sheets run around $10.
  • ruler, compass, cutting surface (cutting mat or board) per person
  • knife per person – just make sure to keep track of them in the classroom.  Collect them 5 minutes before the bell and make sure you have all of them before they leave class!!!
  • glue and/or toothpicks to put the solids together
  • wet wipes and paper towels
  • optional: tablecloths to cover the surfaces to make clean up easier

I love the dollar store for these materials.

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