What is a Sandwich?

After the heated sandwich discussion in class, I decided to revisit the subject.  When asked to give a general definition of a sandwich, our group said a sandwich is 2 pieces of bread with stuff inside.  This definition seemed to cover what we believed to be all sandwiches, but little did we know we had left out a lot of options.  When thinking of the list: grilled cheese, BLT, pizza, hot dog, bagel
 (as seen to the right), taco, burger, and corn dog, I decided that only the grilled cheese and BLT were logical sandwiches.  After later realization, I noticed that my definition fit for the bagel and burger as well.  I thought that was crazy so I went back to change my definition to something that made more sense, but I couldn't figure out a way to exclude the two odd food items.  After talking in class I realized that I was looking at the question all wrong.

After coming home and doing some research, I found that Dictionary.com defines a sandwich as "two or more slices of bread with a layer of meat, fish, cheese, etc. in between each pair."  When looking at this definition, I can poke a lot of holes into this opinion.  I used to work at

Subway Sandwich Shop, and we were known for specializing in sandwiches.  We have a special way of making the bread which looks like a long bread roll when finished.  To open it, we stick the knife into one side, cut one long edge until we are evenly on the opposite side, and then fold the bread open.  In the end, there is only one piece of bread that contains all of the fillings.  This completely proves that Dictionary.com's definition is false.  To add to our group's definition, I would say that a sandwich is one or more pieces of bread with stuff inside.  Looking back at the list of possible sandwiches, this new definitions would include the grilled cheese, BLT, hot dog, bagel, burger, corn dog, taco, and pizza when eaten in taco format.  After determining my results, I thought this could't be right, but then I decided to relate it to mathematical shapes.

When remembering our assignment about deciding what shapes fell into what
category, I remembered how we learned that a square is a rectangle, but a rectangle is not a square.  I think that the proposed sandwich question forces us to think exactly the same way as when looking at shapes.  A square is defined as a figure with 4 equal straight lines and 4 right angles.  A rectangle is defined as a figure with 4 straight sides including 2 pairs of equal sides parallel to each other and 4 right angles.  When looking at these definitions, a square could count as a rectangle because it has 4 straight sides and 2 pairs of equal sides parallel to each other and 4 right angles.  It doesn't matter that both pairs are the same thing because it still counts as a rectangle.  A rectangle, however, could not count as a square because rectangles only have 2 pairs of equal sides parallel to each other. A result of this theory is that even though a square can fall into the category of a rectangle, a person always calls it a square as soon as they see it.  The same thing applies with sandwiches, when a person looks at a piece of pizza they automatically think of it as a piece of pizza even though it can technically be placed in the category of a sandwich.  By looking at all sandwiches (and shapes) differently, a person's mindset can completely change and open up to the possibilities  the world holds.  To end this post I would like to leave you a little Oprah.

Adventures with Kaleidocycles

Throughout the time our math class spent diving into the fun exercise of kaleidocycles, I learned a lot of new ways to teach a math lesson plan.  I went from majorly struggling at understanding how
to properly fold paper to because able to show and tell students how to properly create their own kaleidocycle.  During the first day of the lesson, I cut, folded, and taped a poor excuse of of an origami project.  I was extremely unsure of the instructions, and the informational video went to fast to stay in my head.  I tried my best to fold in all the right places, but it never looked right.  It wasn't
until a classmate put my kaleidocycle together that I even knew how I was supposed to be working.  I once my classmate had everything put together, I taped everything together as well as I could before it broke.  I decided to use spots as a design, but unfortunately the marker did not show on top of the tape, so only half of my spots showed up.  Overall, my first attempt at making a kaleidocycle was an extremely unfortunate occurrence, but it help me realize what I was supposed to do in order to give students proper instructions the following week.

The first session at West Michigan Academy of Arts & Academics was very interesting because I walked in nervous that I was going to mess up in front of students.  Once we sat with a group of students, a teacher told me that the students at my table were 'special needs'.  That statement made me way more excited walking into the lesson because I knew I would have to put a little more effort to

have a great experience.  The two little boys were incredibly nice and enthusiastic about creating kaleidocycles.  When they started to feel down about messing up a step they just needed an extra boost of support from my classmates and I, and we were happy to provide the help.  Throughout the session, I made myself a new kaleidocycle that turned out a lot better than the first.  I liked my spot design, and was excited to try it on a decently made kaleidocycle before I taped it up.  While working with students, I discovered that this project is better done on card stock, a person can check where they need to fold by looking at the back of their project, and students can create a design that tells a story throughout the movements of the kaleidocycles.  After leaving this session, I felt confident that I could show the next session's students how to properly assemble a kaleidocycle without making a mistake.

The final session at WMAAA was a great experience that went as well as I would ever want my classroom to be.  I used all of the previously learned methods to help the girls at my table create their own perfect kaleidocycle.  The girls enjoyed the process of making this project, and were excited about the outcomes.  They had a lot of fun figuring out how to design their kaleidocycles, and then
they put a lot of effort into designing.  I made a third kaleidocycle so the students could follow along.  This project was definitely my best, and it looks the neatest with its striped design.  I was so glad that I could finally complete a successful kaleidocycle, and i'm glad I got the chance to help students enjoy the experience.  This whole lesson really helped me understand the importance of trying hard and not giving up to give students their best chance of succeeding.

Future Math Lesson Plan

In a future class taught by myself, I would love to use physical tools to help my students learn new concepts.  By placing manipulative tools in a math lesson, students develop a fun outlet for expanding their knowledge.  Using tools allow for a kinesthetic learning process, and can make a seemingly boring topic into a topic that will always be remembered.  In our current 221 class, using all of the different resources that we have available have really helped me grasp certain mathematical concepts, so just think of the wonders in can do for young students eager to learn!  

After researching different lesson plans that use manipulative tools, I found one for 7th graders on a site entitled CPALM.  This lesson goes over a fun way to teach students how to find the surface area of a cube or rectangular prism by using wrapped gift boxes.  The purpose of this lesson plan follows Michigan Standard 7.G.B.6 "Solve real-world and mathematical problems involving area volume and surface area of two-dimensional and three-dimensional objects composed of triangles, polygons, cubes, and right prisms".  By giving students boxes nicely wrapped in fun paper, students will be able to take the information already taught about surface area and apply it to finding the surface area of the boxes.  For the main activity, students will be split up into 4 groups and then sent to one of four stations.  They will rotate around to each station and measure the different sized boxed located on the tables.  They will document the surface areas they find as a group for each box and write it on their worksheet.  After

all of the students have had a turn at each station they will return to a class discussion comparing their answers to see if their needs to be further explanation.  Once the class decides on their answer, the teacher will reveal the true surface areas for each box.  The students will receive a review worksheet to take home as homework.  This will provide students with a conclusion assignment to recap everything taught in class.

This lesson plan is a great example of how a teacher can use manipulative tools to increase the enjoyment of learning a topic.  In my classroom, I would use this lesson as a closure activity to the unit of surface area, because it wraps up the topic in a creative way while providing an opportunity for students to use their already learned knowledge.  By using gift boxes, it provides a different way to measure a cube or rectangular prism besides a picture on a piece of paper.  These tools give students the ability to physical touch and see how using surface area in everyday use would take place.  This activity livens up a classroom, and I would purposely try to schedule the lesson around the holiday season to creative a festive tone to the class.  I am excited that I found this lesson plan, and if I ever find myself teaching surface area to my students, I will remember this as use it as the final activity of my unit.

Homework Review Worksheet 

Worksheet Used During Activity

Pentomino Fun

The closest I could ever get
to a full rectangle.

Games can enhance a person's use of their mind, and when a teacher places a game in front of his or her students, their imagination goes crazy.  Luckily, our class had the pleasure of fiddling around with the fun, new toy Pentominoes.  This is an exciting way to learn about perimeter and area, as well as practice patience and memory.  Pentominoes provided our class with an entertaining way to spend our class time.  After the joy of having our own completed set of Pentominoes, we had the chance to get roped into a problem solving game.  My favorite puzzle was trying to figure out how to fit all the Pentominoes into a single rectangle.  I will say... I spent a lot of time invested on this problem with no success in the end.  I tried fitting pieces together to form the outer corners of a rectangle with the hopes of pushing the new masses together to form one full rectangle, but had no luck.  After doing this process, I made a new shape with a few small adjustments.

Cross formed with Pentominoes

Although I was unable to create a rectangle using the Pentominoes, I was able to form a cross using most of the pieces.  I concluded that it was interesting that the pieces left over after making the cross were all of the mirror images that were determined different shapes by the class.  After spending a long time trying to form a rectangle, the success of forming any new shape made me extremely proud.  I had to give up after this point on forming a rectangle or else I would have been sitting there all day.  I will be keeping my Pentominoes on hand for whenever I have a break and need a fun activity to mess around with.

When using Pentominoes in a classroom setting, I think it is important for the teacher to know the skill level of his or her students.  For example, using area and perimeter wouldn't be for younger students, but they can use the pieces to form pictures or patterns.  Older students can use the Pentominoes to do complex puzzles.  When using Pentominoes to teach about perimeter and area, the exercises taught in our class will be a great tool to help ease students into the subject.  Without putting actual measurements on a shape, you give students the chance to understand what perimeter and area are before they have to solve a mathematical equation for an answer.  Pentominoes give students a chance to learn math while having fun solving puzzles.  These tools feel more like a game than classwork.  If your students are anything like me, they will become addicted with solves new problems and old ones.  Pentominoes are a classroom tool that can be used by the average teacher that will give students an enjoyable relationship with math problems.

I hope you enjoyed my position on Pentominoes and how they are a fun resource for the average classroom teacher to have on hand for his or her students to use.  Pentominoes are a great way to advance students' knowledge of area and perimeter as well as improve their problem solving skills.  These tools can be enjoyed by all age groups, and for all sorts of problems.  Pentominoes are a great tool for all teachers.  By having an open mathematical mindset, teachers can help students achieve great goals.

Problem Solving

Throughout my many years completing various math courses and projects, I have struggled with doing well.  I would only understand what was being taught when I had mountains of additional help from outside sources.  My mom took me to tutors, I spent my lunch practicing math problems, and all summer I would complete activity booklets.  Although I had a daily struggle with math, it became one of my favorite subjects.  It gave me pleasure to solve a complex math problem.  All of the time and energy spent focusing on one, single problem gave me the largest amount of satisfaction one could hope for.  When I finally understood a problem I thought I could accomplish anything until I couldn't solve a new problem the next day.  Although there was some disappointment, the tiny amount of success only fueled my energy in each problem to come.

During my senior year of high school, I was lucky enough to be part of the teacher cadet program and work with my former second grade teacher.  I loved being back in her classroom.  I had to choose a main topic to  help with so, of course, I chose mathematics.  Everyday when I went to the classroom, it was my job to take the advanced students into the hallway to work through their more difficult problems.  As a small group, we worked through every problem no matter how difficult.  One of the students had been homeschooled and tested into a high school level math course.  At first glance, her problems were similar to the ones I was currently struggling with.  I was terrified that she would ask me for help, and I would look like a complete idiot by not knowing anything about the answer.  I began to make copies of her worksheets and would practice the problems alongside her.  We would both ask each other questions and give each other advice.  Through this process we both were able to work through the problems in front of us.  In the end, we both learned valuable mathematical skills by working together to solve the problems.