It’s #throwbackthursday again–time for another oldie, but goodie! Back in March 2009, my AP Calculus students completed a Pi Day project in which they found the volume of an object using the disc method of finding volume by integration. It is still one of my favorite math projects, and it only took about a week of class time! See below for a synopsis of the project phases and tips for how to facilitate this project with your Calculus students (and ideas for adapting the project for lower grades).

**Project Timeline **(One/two 90-minute class periods per phase)

- Preliminary Phase |
**Object Selection**| Students must choose a receptacle that has a circular cross section. They are not told specifically about the cross section, but just that they must be able to use the disc method to find its volume. Additionally, the object’s shape must be complex enough to require a piecewise, non-linear function to model its profile (i.e. no cylinders, partial spheres/cones, etc.) Object must be approved by teacher before student can proceed to next phase. - Phase One |
**Trace & Plot**| Students are given graph paper to trace the profile of the object, using the x or y-axis as the center line. Then, they must identify a minimum of 50 ordered pairs to model the object’s shape and record them on a printed t-chart or spreadsheet. - Phase Two |
**Piecewise Modeling**| - Phase Three* |
**Find Volume**| Using the appropriate formula based on the axis of symmetry, students are now ready to find the volume of their object. My students were required to show their work for this phase, but they were also allowed to check their work with their graphing calculators. This is particularly important to prepare them for the AP exam, on which they will be required to find integrals by hand and using their calculators. *This did not take a full class period, but it could be extended to a full class period if you allow students check each others’ work. - Phase Four |
**Poster + Presentation**| Students created and presented a poster that summarized their work. Students were required to include a photo or drawing (photo preferred) of their object, a printed graph of their piecewise model (most students used Geogebra for this, but Desmos would be appropriate too), their volume calculations, and possible sources of error for their calculations. The Error part is important so that students think through issues such as the thickness of their object or mistakes in their modeling methods/ estimation.

**Tips**

- Keep suitable objects on hand to assign to students who do not bring their own or bring inappropriate objects–with no circular cross section or too simple in shape. Be sure to add this to your rubric and deduct points if the student is unable to bring an appropriate shape to class in a timely manner or select an appropriate shape from your set.
- Remind students that while 50 points is the minimum that they have to find, there is no maximum. Also, encourage students to identify non-integer ordered pairs. The more points they identify, the more accurate their regression models will be.
- Inspect piecewise function graphs before Phase 3. Require students whose graphs do not closely resemble their objects’ profiles to identify more ordered pairs.
- If you require that students calculate volume in square centimeters, they can check their work by measuring its actual volume in milliliters. My students discovered this on their own, and most of them added this information to their final posters. Even I was surprised at how accurate their calculations were!

**Adaptions**

**Pre-Algebra/Algebra 1/Geometry**| Phase One Only | Students could simply trace and identify points for any object (circular cross section not important) Then, using Geogebra or Desmos, they could plot the points and adjust for any errors. If you print out the plotted points, students could “connect the dots” to see their shape come to life! For Geometry students, students could choose a real-life object that conforms to a specified shape or choose from a list of shapes provided by you.**Algebra 1/Algebra 2**| Phases One & Two | Students who are familiar with linear or quadratic functions can choose an object whose shape or profile can be modeled with a single function. This could be introduced not only for real objects but also for photos of objects/shapes, as shown to the right.**Algebra 2/ Pre-Calculus**| Phases One, Two & Four | Advanced students may be required to find and object or photo that can only be modeled with a piecewise or other specific advanced functions (i.e. exponential, logarithmic, trigonometric, cubic, etc.). Students at this level could also present their findings with a poster or slide show.

For this week’s #throwbackthursday post, I am sharing part of a post I wrote back in January 2013 highlighting an activity I did with my Algebra students involving making laundry soap on the cheap.

Each kid went home with a baggie full of homemade soap and a mind full of fractions, decimals and questions to ask about why they use the things they use in their homes on an everyday basis. For me, this activity was born from the idea that I was spending too much on laundry soap! My best friend found a recipe online titled “MAKE A YEAR’S WORTH OF LAUNDRY SOAP FOR $30.00!” and the rest, as they say, is history.

I have shared this recipe with everyone I knew because it was so easy to make, it was so inexpensive, and it was just amazing to replace something that I mindlessly spent hundreds of dollars a year on when it was unnecessary. Here’s how I adapted it into an amazing classroom activity:

We took the recipe with ingredients of given weights in ounces. In order to make three batches of the soap with my ingredients (for three classes) we divided each weight by three, introducing fractions into the activity, and measuring everything out. The entire class participated in this activity by physically measuring the ingredients, calculating how much to add and even performing additional operations to make up for measuring mistakes. Measurement is not only a big part of the Geometry curriculum, but it is also a major weakness for most of the students I’ve taught. After making the soap, we engaged in a discussion where we compared the price of our soap to the price of store-bought soap–especially the super-expensive “pods” that seem to be gaining popularity. At the conclusion of the activity, I had the students write reflections on what they learned. Overwhelmingly, the students expressed that they enjoyed the activity and wished that they had the chance to do math in this way more often. We also watched a documentary on Netflix called Chemerical, which follows a family of five who commits to ridding their home of all chemical cleaners to replace them with homemade cleaners made with simple, inexpensive ingredients like baking soda and vinegar.

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Polygraph is a game, similar to “Guess Who”, where a “picker” selects one out of sixteen objects (graphs or shapes) and another student, the “guesser”, asks yes/no questions to narrow down which of the sixteen objects his or her partner chose. All you have to do is select the activity, generate a code, and Desmos does all of the work of pairing up students (randomly) and then they start playing! I had a group of younger students at Math Monday last night, so I had them play Polygraph: Shaded Rectangles. As you can see below, the students asked questions that required both the picker and the guesser to think deeply about fractions and their visual representations. There are other Polygraphs where students are presented with fractions in number form. Although most of the Polygraph activities are focused on graphing functions, there are a few that involve basic fractions and polygons that can be used with younger students.

Another favorite Desmos game for my younger kids is Tile Pile. In this activity, students begin by tiling a 4 square foot section of floor. Then they use estimations and a t-chart/table to predict the number of tiles that would be needed to cover larger floor sizes. By creating a ratio table, students are building a table of values that could be graphed to form a linear function. Although the activity is listed as appropriate for sixth graders, I was able to engage third and fourth graders with this fun activity.

All in all, don’t dismiss Desmos as just an app for graphing. There are many dynamic activities available at teacher.desmos.com for all math teachers, including elementary and middle school educators.

]]>For my first *#throwbackthursday* post, I am sharing an article I wrote for the Cooperative Catalyst blog on November 30, 2011 titled “Love, Logarithms and Learning at its Best”. Enjoy!

I have believed in the power of educational technology to connect teachers and classrooms for a long time, but this experience really enabled me to show my students and peers the transformative potential that edtech has when placed in the hands of enthusiastic kids.

This story started when I won a grant to purchase a “digital kiosk”– a 40-inch television to mount outside of my classroom door to show off my students’ digital media products. My original vision for the TV was to show a never-ending slideshow of student video projects, online posters and other electronic student work. Once the TV was installed, I quickly realized that there was more that could be done. I began adding screenshots of my syllabus to the slideshow, and even found “math rap” videos on YouTube related to what we did in class. Neither my students nor I had any idea just how many math rap videos were out there! A school in Ohio, Westerville South High (WSHS), had created a video called “Teach Me How to Factor” that really stood out because of the videographers’ great editing, attention to mathematical detail, and overall high quality product. Immediately, my students began asking when would we be making one of our own. At the time, the idea seemed so far-fetched that I dismissed the requests and simply continued to share the awesome videos from WSHS with my classes. Early this school year, we discovered another WSHS hit–“Do the Quad Solve!”

I call this video a “hit” not only since it has many thousands of “hits” on YouTube, but because my students really loved it. Every time it came on, whether on the digital kiosk in the hall or when I showed it in class after the kids’ constant requests, it was an instant “math party”! In October of this year, around the time we were about to start a unit on logarithms in my advanced math classes, I heard that we could sign up for our state edtech conference’s student video contest. I knew exactly what we were going to do.

My first step was to contact David Schultz and Tyler Winner, math teachers and masterminds of the WSHS math rap videos to let them know that we loved their videos so much that we wanted to make one of our own. I was hoping we could schedule a Skype call or something to get my kids amped about doing this project. Although we were unable to make the video call happen, Schultz and Winner filmed a short video shout-out to my kids. After seeing all the WSHS videos, my kids knew exactly who they were and felt flattered and excited that they took time to send a message just for us!

In my excitement to hurry and start working on our video, I overlooked the following sentence in the contest rules: “Music is an important element of the movies and we will not accept movies that feature music unless you have legal permission to utilize.” Needless to say, I started freaking out the moment I saw these words as my students had already started recording and we certainly did not have any type of permission, legal or otherwise, to use the song we chose to parody. Being the techie I am, I turned to Google as my first line of defense to try and address this issue. I found our song’s artist, Kourtney Heart (aka K. Hart), a local star who recently signed with Jive Records, via Twitter and Facebook. I contacted her through both, and was even able to find email addresses for her management team online. We continued to work on the video, although I was unsure of how things were going to pan out. My plan was to simply do an “a capella” version of the song minus the copywritten music if things did not work out.

But, they did. Not only was Kourtney’s manager gracious enough to provide the instrumental for our song and a permission letter to submit for the contest, but he also arranged for Kourtney to show up and surprise my students once the video was completed. Needless to say, my kids were stoked when Kourtney showed up on campus. As usual, we had a math party to celebrate, but this time, we had our own soundtrack!

What you see in the above video is the result of hours of hard work on the part of several students who created this great product. I was blown away by the previously hidden talents of kids who stepped up to film, edit, sing, dance, choreograph, write lyrics, and recruit students and teachers to participate in this project. I’ll never forget the day after I told my students what we were trying to do– a kid walked in my classroom first thing in the morning carrying a laptop and the biggest microphone I’d ever seen saying: “this has got to be done right, Mrs. B!” The kids’ school spirit and enthusiasm comes through on screen, just as Mr. Schultz and Mr. Winner’s students’ excitement did, which made us fall in love with their videos. My students and I worked for weeks at lunch and after school as to not interfere with the flow of my classes. Now I can’t get through a class period without someone yelling “put the video on!” or hearing the kids humming and singing about logarithms!

Needless to say, we won the contest. I knew we had a hit on our hands the moment we began this journey, but I never expected it to have the impact it did on me and my kids. And our school. And our community. I teach at a large urban “turnaround” school where there is a lot going on that often overshadows what our kids have to offer beyond “data”. Every day since the release of the video I meet a new student, parent, teacher or complete stranger in person or online who has been touched by our video. I understand that I am a math teacher and that it’s important to teach content, but I think that what we’ve done with this video will stay with many of my students long after they’ve forgotten the math I taught them. We set out to reach a lofty goal, and not only did we reach it, but we met some great people along the way and connected with others who share our passion for what we’re doing. These lessons, I believe are equally , if not more, valuable than the math my kids will learn in my class.

**First Things First: Doing the Math**

Needless to say, the students in my after-school math program were not feeling very “math-y” at the start of our last session before Christmas. Nevertheless, I required each child to measure the dimensions of the (empty/rinsed) orange juice carton that would be used as the base for their gingerbread houses and calculate the base perimeter and area. Older students (middle school) also had to find the carton’s surface area and volume. Did I mention that the dimensions were not whole numbers? And we did not round! My volunteers and I closely watched for measurement errors and assisted the children with using a ruler properly and calculating measurements with attention to accuracy and precision. After about thirty minutes, the students were so into their measuring and calculating, we had to remind them that we had another table set up with candy and icing for them to decorate their houses!

**Building the Houses**

Setting up separate areas for decorating and measurement helped the kids to focus on the math until it was time to decorate the houses. To make a gingerbread house out of a milk carton, coat the carton with icing (the layer should be very thin to reduce the mess) and cover the carton with graham crackers. You may even be able to find gingerbread-flavored crackers! Then they can decorate their houses with candy, marshmallows and icing. It was perfect–our learners went home for the holidays with a sweet Christmas treat and a head full of numbers and units!

]]>“Unit or modular origami is a specialized type of paper folding that breaks with tradition by allowing the folder to use more than one square of paper to for geometric objects. “

What I like best about unit origami as a math teacher is that you can show your students how to fold a single origami pattern and use this as a basis for teaching a broad variety of 2D and 3D shapes, including all five of the platonic solids. Additionally, there are lots of different units that can be folded and assembled to make myriad models.

When teaching basic Euclidean geometry, it is important to note that each fold of a piece of paper–a *plane*, can be used to highlight a geometric concept. For example, if a module requires students to fold their paper in half, teachers can ask questions such as: “what is the relationship between the **areas** of the two shapes you just created?” or more simply: “what is the relationship between the two shapes you just created? what are these shapes called?” As students continue to fold, there will be more and more geometric objects and relationships for them to observe and describe, including:

- angle bisectors
- parallel and perpendicular lines
- isosceles and right triangles
- supplementary and complementary angles

Check out the book link above or simply do a Google search for “modular origami” to get started!

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