Calculus for Geometry Students

In Geometry, we’re beginning a chapter on areas of polygons, and the first lesson is area of a rectangle. Pretty exciting, huh? My students are mostly ninth-graders, with a few tenth-graders, and I thought they might enjoy seeing how the area of a rectangle is used to estimate the area under a curve, i.e. a Riemann Sum.

I used a Geogebra activity created by Alex Kasantsidis to demonstrate a simple Riemann sum. We discussed how the sum of the rectangles can either overestimate or underestimate the area under the parabola, and how we can get a better approximation of the area by increasing the number of rectangles used.

Riemann

Then, I had my students work through an activity (you can download it here) to estimate the area under the curve y = 12 – x^2 for x = -1 to x = 3 using eight rectangles. After averaging the left-hand and right-hand sums, they came up with 38.5. The actual area is 38.66…, so with only eight rectangles they achieved very good results!

What my students enjoyed even more, though, was the satisfaction of learning calculus-level mathematics. Hopefully, this activity allayed some of the apprehension they might have when they hear the word “calculus”.

 

Algebra + Area = A Nice Proof

In my Geometry classes, it is time to work with right triangles. Rather than simply present my students with the Pythagorean Theorem, I decided to have them prove it without them knowing that’s what they were doing.

I based the following activity on a proof I found at Alexander Bogomolny’s very useful site, Cut-The-Knot.

A Proof

  1. What is the total area of the four triangles below? (They are all congruent)

 4 rt. triangles

 

 

 2. If the triangles are rearranged into the square pattern below,

Pythagorean Proof Square

(a) In terms of a and b, what is the area of the “hole” in the middle?

 

 

(b) What is the overall area of the square pattern?

 

 

3. How can you use the results of parts 1 and 2 to prove a famous theorem?

 

I like this proof, because it’s very visual as well as algebra-based. The most challenging part for my students was figuring out the area of the square “hole”. Many assumed the side of the square was 1/2 of a, which is not correct. The satisfaction and excitement they had as they saw the pieces of the proof come together were well worth the effort, though! They were all familiar with the Pythagorean Theorem, but this was the first time any of them had actually proved it.

Click here for a pdf of the worksheet and here for the solution.

NCTM Comes to Music City

The National Council of Teachers of Mathematics held a regional conference in Nashville, TN last week. Since I live in Music City, this was too good an opportunity to pass up! I’ve attended several NCTM conferences, both national and regional, and this was by far the best and most technologically savvy one.

To begin, I installed the conference app on my phone, which made planning and organizing my experience a breeze. It synced the sessions I wanted to see with my calendar, so I received reminders and updates for each one. The app also promoted interaction among attendees through its Activity Feed. We could also keep up with what was happening via Twitter using the hashtag #NCTMregionals.

The first session I went to was “Hands On Trigonometry”, by Elizabeth Petty. She led us through a lesson on the unit using paper plates and twizzlers to demonstrate what a radian is. We then folded the paper plates in half to create an x-axis, then thirds to mark the 30 and 60-degree points on the circle. It was an excellent example of using a hands-on activity to reinforce a difficult math concept.

The next session, “Keeping It Real: Authentic Real-World Math Lessons”, was presented by Ginny Stuckey of Mathalicious. Their mission is to develop lessons that use real-world data and situations to spur critical thinking in students. The lesson Ms. Stuckey demonstrated involved how municipal fines can quickly become impossible to pay off for low-income offenders.

Next up was one of the best sessions of the conference, “Using Manipulatives and Investigations to Teach Geometry”, by Christine Mikles. Ms. Mikles uses the CPM Geometry text, which is full of hands-on activities. Here’s an example of using two mirrors and a protractor to learn about central angles in regular polygons:

Polygon with mirrors

The last session I attended on Thursday was “Slices of Calculus”, by Nina Otterson. As a veteran calculus teacher, I was very excited to learn a new way to teach how to calculate the volumes of solids of known cross-sections. Ms. Otterson’s approach makes a lot of sense: she has her students learn what ratio of a square’s area is an equilateral triangle, an isosceles right triangle, and a semicircle. Then, they find the volume of the solid using a square, which is easy, and apply the appropriate ratio. Her students build models of each type of solid. Here’s one I built using equilateral triangles whose base is a pair of intersecting parabolas:

Solid with Triangle cross-sections

Friday’s first session was “The Math Department I’ve Always Wanted: Twitter As My PLC”, by Michael Felton. It was an excellent presentation on how math teachers can use Twitter to ask questions of other teachers, get great ideas for lessons, and get feedback on their own lessons. Michael is part of the “Math Twitter Blogosphere” (#MTBoS on Twitter), where some of the most innovative teaching is being developed today. If a teacher needs some ideas on how to teach practically any math topic in a creative way, he or she can find it at MTBoS.

Next, I went to a session sponsored by CPM (College Prep Mathematics): “CCSS Math Practices? Trust CPM’s 25 Years of Writing Experience”. This company is a nonprofit textbook publisher run by math teachers. Their series of books stresses the importance of hands-on explorations to teach math concepts. I was very impressed with the passion and excitement of the teachers who use these books.

Finally, I went to another session by Michael Felton, “Desmos and Modeling”. I’ve used Desmos‘ online grapher for several years now (I even did a presentation on it at a TAIS conference last year). They have moved far beyond a simple function grapher, though. Their site now includes all kinds of activities that teachers can use in the classroom. The one Michael demonstrated involved matching transformed sinusoidal curves, and it was a lot of fun. Go to https://teacher.desmos.com/ to see all the fantastic lessons they offer. Teachers can also keep track of students’ progress as they work through the activity.

All in all, an excellent conference. It is fascinating to see how social media is transforming teaching, and how much teachers are trying to incorporate active learning into their lessons. I figure a conference is worthwhile if I can take home at least three good ideas/lessons/activities from it. After this one, I have more than dozen to try out with my students!

 

A Balancing Act

In Geometry, we are learning about circumcenters, orthocenters, and centroids of triangles. Geogebra is a nice tool to use to explore how the medians, perpendicular bisectors, and altitudes are concurrent respectively, regardless of the shape of the triangle.

Here is a screenshot of the centroid, which is the intersection of the medians of a triangle:

Centroid1

Here is the circumcenter (the intersection of the perpendicular bisectors), with a circumscribed circle:

Circumcenter

And here’s the orthocenter, which is the intersection of the triangle’s altitudes:

Orthocenter

A very nice property of the centroid is that it is the exact center of gravity of a triangle. Here’s a brief video of one balancing on a paper clip: