Lesson 0-5: Mandalas (Day 5)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

Find x.

As in many Geometry problems, all the givens are in the diagram -- and as so often happens with diagrams, none of the vertices are labeled. So let me label them:

Two lines A-C-E and B-C-D-F intersect at C.
Angle BAC = 80, EDF = 100.
DE = 3, EC = 7, AB = 9, BC = x.

We first notice that EDF is an exterior angle -- the interior angle EDC that forms a linear pair with it must therefore measure 80 degrees, the same as BAC. And of course, angles ACB and DCE are vertical angles, hence also congruent. This is enough for us to declare that Triangle ABC and DEC are similar by AA~. It now remains to set up and solve a proportion:

BC/EC = AB/DE
x/7 = 9/3
3x = 63
x = 21

Therefore the missing length is x = 21 -- and of course, today's date is the 21st. Fortunately, there are no errors in this problem -- but I did catch a minor error over the weekend. That problem asks, "Find the area of this triangle to the nearest foot." Of course, Pappas meant "to the nearest square foot."

This is what I wrote last year about today's lesson:

Lesson 0.5 of Michael Serra's Discovering Geometry is called "Mandalas." This is the first of two sections included in the old Second Edition yet omitted from the modern editions.

But what, exactly, is a mandala? Serra explains:

"A mandala is a circular design arranged in layers radiating from the center. The word mandala comes from Hindu Sanskrit, the classical language of India, and means 'circle' or 'center.'"

As Serra points out, other cultures had mandalas, not just the Hindus. The Aztec calendar, for example, was constructed as a mandala.

Many mandalas exhibit threefold or sixfold symmetry. They are related to the regular hexagon, and so the compass and straightedge can be used to construct them. At any rate, the compass should at least be used to draw the circle that is the base of any mandala.

All the mandalas on these pages come from a Google image search. There is no project in this section, but of course "draw your own mandala" is a natural question for this section.

Here is the Blaugust prompt for today:

Tell us about a favorite activity/lesson that makes you jump for joy when you get to use it.

Well, I guess that today's mandala activity might make me jump for joy. It's an opening activity that the students can enjoy as it allows them to exhibit their creativity.

Of course, that's probably not what Shelli meant when she posed this question. She obviously wants us to mention an activity or lesson that we actually taught in the classroom. This requires me to think back two years to my charter middle school classroom.

One day in November that year, I wrote about Benchmark Testing Week -- a very stress-filled week, since there were both print and online Benchmarks to take in both math and English, and they took hours to complete. But I was able to complete some activities during that week. I suppose that these activities made the students and me jump for joy -- if only because anything that wasn't a Benchmark Test made us jump for joy that week. (This week at my old school is probably the early-year Benchmarks, but here I'm referring to the end-of-first-trimester Benchmarks in November.)

Of the sixth grade class, here's what I wrote about a lesson I gave that week:

Yesterday I had a Bruin Corps member present during the sixth grade block. So here's what I did -- I began the class with a Warm-Up division problem from Illinois State -- 2400 / 51. Then I divided the class into two groups -- those who got the right answer and those who didn't. I gave the higher group to my Bruin Corps member, so he could help them move on, just as he and his fellow Bruin have helped with the higher grades this week.

But this led to problems. First of all, some students decided not to answer the Warm-Up. I remember one boy who chose not to answer, so I seated him with the students who didn't know division. Then when I assigned a division question, he answered it quickly -- meaning that he already knew all the steps and was just too lazy to do the Warm-Up. He complained that he had to sit with the students who didn't know division when he already knew it.

There were probably also some students who cheated and started copying the answers when they saw who was getting them right. I reckon that some teachers get around this by simply handing out colored cards rather than telling them where to sit. Then the students can't tell as easily who's getting the right answer, making it harder to cheat. Many of these problems persisted into the homework, where some students either skipped completely or wrote in nonsensical answers, such as "55 divided 8 is 55 remainder 3." And remember -- this is all despite the history teacher giving them time to do the math assignment in his class!

I tried to give the inspiration example I've mentioned earlier -- the Cubs. The North Siders have failed to win the World Series for 108 years before they won it all this week. They didn't let their past failure hinder their present success. Yet the students who can't divide aren't thinking like Cubs -- that is, like champions. They fail to divide, and so they keep coming up ways to avoid division rather than think, I don't know to divide now, but if I work hard (like the Cubs), I will.

Today during the IXL time, the struggling sixth graders continued to struggle. Indeed, I can't say that I've taught a single student to divide -- the stronger students already knowing division and the weaker students coming up with excuses. It doesn't help that I'm trying to crack down on students who don't remember their IXL passwords by telling them they can't use the computers. Weaker students who wish to avoid division just claim that they don't know their passwords!

As I reflect on this day two years later, this was probably one of the few times that I was able to implement Learning Centers that year. For about half the class, this was a good day, but for others, it didn't work. There were too many opportunities for students to avoid the work -- from the one boy who knew how to divide but didn't attempt the Warm-Up to the kids who tried to cheat during the Warm-Up to the ones who claim their computers didn't work. The whole idea of Learning Centers would have worked much better if I'd had tighter classroom management to make sure that students didn't attempt those tricks.

The seventh graders probably has the most enjoyable lesson of the week:

In fact, I realize that the seventh grade lesson is so easy that today, I actually go back and have the students work on the Orienteering STEM project from the Illinois State text. This decision is easy to make, since my support staff member and Bruin Corps member are both present during the seventh grade math block. So I take one group outside, give them compasses, and have them create a map to hide the "treasure" (the textbook), and then another group takes the compasses and map and uses them to find the treasure.

While all of this is going on, my Bruin Corps member watches the rest of the class. She sees that they've already mastered the concept of opposites, so she has them do some general addition of integers that are not opposites. She has them play a game for points similar to the "Who Am I?" games that I played as a sub last year, and buys pizza for the winning group. The winners turn out to be the same group that hides the first treasure -- that is, they are outside for part of the game, yet they still come back to win. And this is more amazing because the smartest student in that group is in fact absent today, so it's not as if he's doing all the work.

Yes, this project from two years ago came from the Illinois State text. But since only some of the students are working on the project, it ended up turning into Learning Centers again.

As for the eighth graders, I finally attempted to teach them some science that week:

Well, I tried to start online Benchmarks yesterday during the math intervention time usually devoted to the other online program IXL. But the problem was that there was a power outage! I'd already charged the laptops before the blackout, but it was impossible to access WiFi during the outage, so the students couldn't access the online Benchmarks. The blackout began at the start of lunch and ended right at -- you guessed it -- 2:25 (that is, P.E. time).

The English teacher suggested that I have the eighth graders finish the written Benchmarks for her own class, since the students couldn't finish them during English class. Well, the students refused to work on them, and when I told them that they were supposed to be working on the essay, one girl called out, "Well, you're supposed to teach us science and you're not doing that!"

"Okay, then," I replied, "let's start the science assignment now."

It's a good thing that I purchased that Common Core Science book last month, since yesterday was my first opportunity to use it -- after all, science was the last thing on my mind with all of this Benchmark stuff. (Notice that there is no science Benchmark, even though eighth graders are supposed to take the NGSS science test.) So I just jumped into Chapter 7 of that book, which as I wrote in my October 10th post, is on Matter and Its Interactions. I just had the students start writing the definitions of some vocabulary from that chapter, but they only got through the first four terms after all the arguing about the essay and writing. Still, at least I got some science in at a time when I thought I'd have very little time for science.

Yes, I admit two years later that this science lesson didn't work as well as I wanted. But then again, this was a spur-of-the-moment science lesson caused when a power outage prevented me from giving the online Benchmarks. The class didn't have a science text because copies of the Illinois State science texts were still a few weeks away from arriving. In the meantime, we had to use the online science text -- which we couldn't because of the blackout.

I could have made that lesson more enjoyable by changing it into a game, similar to the impromptu game played by the seventh graders the following day. But as I wrote earlier, even writing words and definitions was enjoyable compared to taking the Benchmarks.

If you prefer that I answer Shelli's prompt with a Geometry lesson based on the U of Chicago text (outside of my one year of teaching), then one of my favorites is on the area of a circle, which I first posted in my January 16th post this year:

If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.

I wrote about several activities related to the circle area in that January post -- which includes activities created by other teachers. Unfortunately, I've never actually taught this lesson in the classroom -- I was planning on giving my seventh graders a pi lesson for Pi Day, but I didn't quite make it to March 14th that year.

One Blaugust participant this year is Ilana Cyna:

https://www.teachingtothenthdegree.com/uncategorized/cleave-books-resource-pages/

Cyna has no problem coming up with her favorite activities and lessons, since nearly every single page is a link to an activity.

I'm not quite sure where and what Cyna teaches. The Cleave Books link above is a British website, but she also implies that she may be Canadian in the following post:

https://www.teachingtothenthdegree.com/uncategorized/a-fracking-we-will-go-or-not/

Cyna writes:

Science time.

I am going to highlight a project I have done in grade 7 science, but I am sure that it could be modified for other grades. I was looking online for an assignment on the environmental impacts of using oil or natural gas and I came across this website.

Both sources describe a town hall style meeting to discuss allowing oil companies to lease land for fracking natural gas, with the students being given roles within the scenario (oil and gas company representatives, government representatives, First Nations representatives, resource workers, community members, environmentalists).

In this post, Cyna hints that she's a middle school math and science teacher -- in other words, just as I was myself two years ago! In other words, I wonder whether her site could have helped me teach science that year. I mean in general, of course -- no website would have helped me on blackout day.

Then again, she posted several times in July 2016 and then not again until August 2017. Thus my entire first year of teaching fits inside the twelve months when she didn't post. Once again, I don't know whether there were any active math/science bloggers during the August 2016-July 2017 period.