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Paper Plates & Fuzzy Sticks

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As the school year enters fourth quarter, teachers are blessed/cursed by the arrival of nice weather. For those of us in the northeast, we rejoice the end of winter and celebrate the boost to the spirit from the new spring. But along with that comes the battle to keep the students on task and motivated as they can sense that the end of the school year can't be that far away. I have an extra challenge as my classroom looks out on the tennis court and track, and I am always competing with teens enjoying gym class for the attention of my students. One of the last units that we cover in Geometry is our unit on Circles. Having taught Geometry for a number of years, I have always been on the lookout for new ways to engage students with the many terms, concepts, and relationships that come with the Circle unit. While it is always easy for me to immediately go to the Chromebooks and have the students use Desmos and/or GeoGebra to investigate, I decided to go low-tech (again) this ye
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Geometry Modeling Set From the Thrift Store

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A couple of weeks ago, I had some time to kill between dropping my daughter off at a friend's home and picking up my son from musical rehearsal. So I decided to visit the local thrift store. I've made out well there over the years, picking up toys for the kids, a record player for my parents, and a variety of other items for the home. But on this day, I found myself drawn to the toy section. This is funny because my own children are (sadly) of an age where they have outgrown toys. Scanning the shelves, there were a number of jigsaw puzzles, "well-loved" toys, and many games. Among the many boxes, I found this: Over the past few months, I have spent many hours constructing polyhedra structures using straws and wire. Here are a few of my creations: So when I came across the Discovery Kids 77-Piece Build & Play Construction Fort for the low-low price of $3.99, I just had to have it. Little did I know, it would quickly become a favorite source of math
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Why isn't ASS a member of the "Fab Five"?

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I have taught Geometry for many years, and a big part of the curriculum has been proving congruent triangles. If you teach this, you know about what I refer to as the "Fab Five": Angle-Side-Angle (ASA) Side-Angle-Side (SAS) Side-Side-Side (SSS) Angle-Angle-Side (AAS) Hypotenuse-Leg of a Right Triangle (HL Rt. △) I introduce this list with accompanying diagrams illustrating each method. Here is a sample: After generating the list of the "Fab Five", I emphasize to the students that each method requires three parts from each triangle. This, in turn, always results with students asking: "Hey Mr. K. What about AAA and ASS?", the latter of which usually produces a number of giggles. So I ask them: "If you know that three angles in one triangle are congruent to the three corresponding angles in another triangle, why are the triangles not necessarily congruent?" After a little guidance, students will recall our work from earlier in
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Pre-Calculus and the Witch of Agnesi

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This year, a colleague announced that he needed to take a short leave from school. This is the second teacher in our department who has needed to take time from work this year. Due to an unfortunate shortage of qualified substitutes, teachers in our department have been asked to take on another teaching section. Some teachers are eager to do it, thanks to the bump in pay (after all, who couldn't use a few extra bucks in the bank?). However, some teachers are reluctant to teach an additional period, citing an already busy personal and/or professional schedule. So when I was asked to teach one of my colleague's classes, I was hesitant. But wanting to help out (and with a son looking at tuition payments next year), I agreed. I knew the leave would only be a few weeks and most of it was during the month of November, a month with days off for Election Day, Veterans' Day, and the Thanksgiving break. So when I was asked to cover his Honors Pre-Calculus class, that combination
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What's Your Favorite Shape?

I have taught high school geometry for many years now. But as a result of my MTBoS explorations and work with my colleagues in the NYS Master Teacher Program, I have become more aware and fascinated by shapes and their structures. I have also tried to increase student awareness of the beauty and powerful properties of the myriad of shapes that we study and don't study. It is still early in the school year, and my students' understanding and sense of shapes is still developing. So at the end of a recent quiz, I asked the students: "What is your favorite shape? Why?" Here are their unedited responses. Enjoy!  I will try to remember to ask them the same question at the end of the year, after working with a variety of shapes and their properties. Thanks for reading!
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Exploring the Triangle Inequality Theorem

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I know... sounds exciting right? Please bear with me... I was inspired today! As a geometry teacher, I am always looking for ways to visualize and demonstrate geometric concepts. Often times, I utilize tools like GeoGebra or Desmos. But frequently, geometry can utilize simple resources. This morning, I came across a tweet from Halcyon Foster: I really like how this utilizes simple materials that most geometry teachers have - paper, ruler, scissors, and dice. It looks like the idea is to roll the dice, then take a strip of paper and measure and divide it into lengths (in this case 2cm, 5cm & 6cm). Then fold the strip to try to make a triangle. It's simple and effective, and serves as a great way to investigate the triangle inequality theorem. It inspired me to think of a way to "kick it up a notch". So I turned to one of my favorite geometry tools - bendy drinking straws! Here is what I put together: Supplies: Bendy straws Scissors Ruler Sharpie
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Getting to Know...

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After making the change to CC Geometry a few years ago, and after much #MTBoS influence, I took a chance and made a big change in the way I start my school year. Last year, I didn't give out a course outline and require the Ss to have it signed and returned. I didn't lay out the rules and requirements of my course and classroom. Instead, I started with "One Cut Shapes". The inspiration for this came from a PRESENTATION that I took part in last year with Dan Anderson and some other NYS Master Teacher colleagues: It was an interesting investigation that I got into last summer: So last year, I started the first day of class with: "Hi, I'm your geometry teacher, Mr. Kaercher. Let's play with one-cut shapes." It was a freeing experience and probably different than the first-class-of-the-year-experiences that the Ss had that day. I'm starting this year the same way, but I have also revamped my first Geometry unit on "Es
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