Right Triangle Trigonometry with Clinometers

I've been going out of my way this year to make my geometry classes more applicable. At the first of the year my kids really struggled, so I've tried to re-invent the way I teach. I'm doing a lot more application and less theory. I'm not saying that's the way it should be done, just that this is what works with my particular group of students.

Today we went outside and measured the heights of some of the taller objects around campus. We used homemade clinometers to do this. Since we only had one measuring tape, my co-teacher went outside and pre-measured distances from the building to the point where kids were supposed to measure their angles of elevation. (Note: next time, we'll spray only non-permanent surfaces like gravel. That green 10 ft mark may never come off the sidewalk.)

Anyway. The kids loved it. It didn't seem like work. Each class also discovered that their measurements didn't "make sense". They were too short. They decided that they measured from eye level up, so we needed to add on their heights to determine final measurements. This went great. Even my students who usually really struggle enjoyed it.

Clinometers and Trigonometry Handout

Some Pictures...

AP Statistics Recruitment

It's that time of year again - the time when kids start registering for the next academic year. This year, we have lots of kids that met math benchmarks and thus can take non-remedial senior level math. Lots of these kids are torn between pre-calculus, AP Calculus, and AP Statistics. Many have room to take two courses, but few will. With this time of year comes many "discussions" about the kind of math each student needs to prepare them for their chosen career field. 95% of our kids don't do their research, and thus are swayed by friends or other teachers about which class to take. I often hear from adults that statistics isn't important, so I set out to prove them wrong. I looked up requirements for popular majors at two of the colleges to which we feed the largest amount of kids. There is a third large college to which we feed, but their academic requirement website was nearly impossible to navigate today. I'll try again later.

On these two colleges' websites, I search for curriculum maps for common majors. It turns out that almost all majors (give or take a few) at Morehead State require statistics (Accounting, Business, Nursing, Biology/Pre-Med, Chemistry, Criminology, Pre-Physical Therapy, and more). At the University of Kentucky, part of "The Core" academic requirements for ALL majors includes one course on statistical logic and reasoning, and then a second on statistical inference. I must admit that I was quite surprised at UK's incredibly heavy emphasis on statistics, and I was also shocked that I didn't know this. If I don't know, how in the world are my kids going to know?

With this in mind, I printed out the common major curriculum maps and "The Core" curriculum map from the University of Kentucky. I then highlighted all the required math classes in the sequences and wrote the generic name of the course (I mean... really... what is MAT 305?) on the paper. I then made big red arrows that say statistics to point out all the statistics courses. With the help of my student TA, we taped all these documents outside my door and made a pretty informative recruitment wall.

(Please excuse the decorative ink splatter on the wall behind the papers. Some cherub decided to add their own flourishes to the hallway paint job.)

Following Up Pythagoras (It Went Great!)

I started off my geometry classes' reintroduction to the Pythagorean Theorem with a simple bell ringer. It was a 3-4-5 triangle, with a missing hypotenuse. With some figuring, they remembered the Pythagorean Theorem and applied it. A student said "The answer is 5." I agreed and asked why. Why was it 5? The response I received was that a^2+b^2=c^2. Yep. That's true - when you have a right triangle. Why does a^2+b^2=c^2? My students decided they didn't know.

So, we proved it. ( I didn't call it that because my kids hate proofs.) I used this awesome 3-4-5 template from Jennie at mathfoldables.blogspot.com (HERE). I didn't use the two layers like she recommends. One worked just fine. We discussed the right triangle formed in the center, its side lengths, finding the area of the squares, and how this all relates to the Pythagorean Theorem. Students really understood and were able to draw connections between what they had colored (see picture below) and the concept behind the Pythagorean Theorem.

It turns out that this introductory coloring activity was a great transition to the Converse of the Pythagorean discovery activity. I modeled completing the INB page for a 3-4-5 triangle. We also completed a 5-7-9 triangle as an example. From there, students were instructed to use any of their squares to form additional triangles and complete the form. This was a great way to spiral in the Triangle Inequality Theorem we'd just learned in the last chapter. Students would ask if x, y, and z would work, and I'd ask them... how could we figure this out? They use those skills and really started to see patterns in their Pythagorean Theorem inequalities and the triangle types.

Tomorrow we'll finish up our tables and discuss takeaways. This was a great activity, and after a tiny bit of productive struggle, the students caught on well.

Teaching the "WHY" of the Pythagorean Theorem

Every year, there are two things I can reliably predict about my new batch of Geometry kids: (1) They know enough about transformations and (2) They know the Pythagorean Theorem. Now, let's expand on the second. They "know" the Pythagorean Theorem - well kinda. What they really know is that a^2+b^2=c^2. They have no idea where it came from, why it works, who Pythagoras is... none of the good stuff.

So, this year, instead of just reviewing how to plug and chug with the Pythagorean Theorem, I decided I want them to re-discover (and thus re-learn) the Pythagorean Theorem. I set out on a virtual journey to find the perfect Pythagorean Theorem activity - one that shows kids WHAT they are doing when they use the Pythagorean Theorem, not just HOW to do it. I searched high. I searched low. The #MTBoS came to the rescue, as usual. I had happened upon Dan Meyer's Pythagorean Theorem discovery activity during a google search, but Lisa Bejarano's blog posts about it (HERE and HERE) helped me to re-think how it might be successfully implemented in my classroom. Jacqueline (@_Cuddlefish_) also sent along some really awesome links, especially an informal proof of the Pythagorean Theorem based on Perigal's proof of the Pythagorean Theorem (HERE). 

I decided to re-work Dan Meyer's Pythagorean Theorem idea into an INB page that would provide a little more support for my students. It covers both the Pythagorean Theorem and its converse. Here is what I came up with:

Pythagorean Theorem and its Converse INB page (Click for Google Drive Download Link):

FRONT:

INSIDE:

I plan to still use the square blocks included in Dan Meyer's Pythagorean Theorem Discovery (HERE - Week 25), but I'm going to white out the area calculations and have students calculate those themselves.

Triangle Midsegments & Exterior Angle Speed Dating

I've learned so much from the #MTBoS. So, so much. One of the best things I've ever learned, though, has to be Kate Nowak's Speed Dating (See post HERE). It's just amazing and does so much for kids on so many levels. With our three thousand (perhaps two thousand ninety-nine) snow days, we've needed to review a lot of topics. This was the perfect way to do it. I primarily wanted my kids to review the triangle midsegment theorem and the exterior angle of a triangle theorem. I also threw in some isosceles and equilateral triangle questions from the pre-snowpocalypse era.

I got comments like "Ms. Boles, I really learned a lot today. I really did."  "I like this game."  "When can we do this again?" "We should do this everyday!" I said no to the last one. However, I really loved all of the positive feedback I was hearing.

Since I've done speed dating for three years, I've learned quite a bit about how best to facilitate it. First, count the number of students in your smallest class and half it. My smallest class was 22, so I wrote 11 questions that really hit the main ideas I wanted reviewed. I obviously have to write more questions, but the rest are usually very similar to the first 11. This way, students don't have to make it all the way around the room to review all necessary topics. In my class of 34, speed dating all around the room would take far too long.

Midsegment & Triangle Exterior Angle Speed Dating Index Cards (Click for Google Drive DL)