Lesson 0-3: Daisy Designs (Day 3)

Today on her Mathematics Calendar 2019, Theoni Pappas writes:

Find x.

Once again, all the given information is in an unlabeled diagram, so let me label it:

Triangles ABC and ADE are drawn with B between A and D, as well as C between A and E. The following distances are given: AB = 9, AC = 3, BC = 6, BD = 15, CE = 5, DE = x.

First of all, we notice that Triangles ABC and ADE are similar by SAS~. The two triangles have Angle A in common, and so we must check that the sides adjacent to the angle are in proportion:

AB/AD = AC/AE
9/24 = 3/8 (since 15 and 5 are not sides of a triangle)
72 = 72 (after cross multiplying)

 Thus by SAS~, Triangle ABC ~ ADE. So we set up another proportion to find DE:

AC/BC = AE/BE
3/6 = 8/x
3x = 48
x = 16

Therefore the desired length is 16 -- and of course, today's date is the sixteenth. This problem fits well with Lesson 12-9 of the U of Chicago text, when SAS~ is taught.

Even though the proportion is correct, there's an error in this problem. In Triangle ABC, we see that the sides are AB = 9, AC = 3, and BC = 6. So AB = AC + BC, thus violating Triangle Inequality.

It's tricky for Pappas to come up with side lengths that are proportional, yet avoid fractions. The easiest way to avoid fractions is for the sides to be in the ratios 1:2:3, but then these sides would no longer form a triangle.

How can we fix this so that the sides actually form a triangle? Well, as Triangles ABC and ADE are almost drawn as right triangles (with right angles at C and E), we might try the ratio 3:4:5 instead:

AB = 5, AC = 3, BC = 4, BD = 15, CE = 9, DE = x.

These triangles are similar since 5/20 = 3/12, and solving 3/12 = 4/x gives x = 16 as before.

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

Lesson 0.3 of Michael Serra's Discovering Geometry is called "Daisy Designs." Yesterday the students used rulers or straightedges to make designs, and today they'll use a compass.

Serra begins:

"The compass is a geometric tool used to construct circles. You can make some very nice designs with only a compass."

The students are then directed to create a "daisy" using a single compass setting. Serra points out that this construction is related to that of a regular hexagon -- which is required under the Common Core.

On today's worksheet, I take one question from the exercises  -- and the other, as usual, is taken from the project. I found a website that teaches students how to form the daisy designs:

http://geometry2014.weebly.com/daisy-design.html

(The creator of this website is Monica Zimmers, a New York high school teacher.)

The second question directs the students to use a graphing calculator to graph r = a sin (n theta) in polar coordinates, for various values of a and n. The graph of these are, well, "daisies." Of course, I don't expect all students to have graphing calculators (though I do expect them to have compasses).

Here is the Blaugust prompt for today:

How I used something unexpected in my classroom to…

In a way, today's lesson fits this prompt. The unexpected object is a compass, and it is used to make some amazing designs as part of an opening week project.

But in my Blaugust post last year, I wrote about another unexpected resource -- a UCLA college student -- whom I'd been able to use the previous year at the charter middle school:

Cell biology, of course, counts as life science. Therefore, if I had taught science the way I was supposed to, this topic would have been covered in my seventh grade class. Instead, I attempted to teach it to my eighth graders under the NGSS standards.

I've admitted before that life science isn't my strong suit. And so I tried to use my Bruin Corps student, a molecular biology major from UCLA, to explain the details of this topic. In reality, I should have come up with a stronger science program from the start of the year -- well before the arrival of my Bruin Corps student -- and then had him supplement that curriculum, rather than try to have him be the curriculum.

Of course, I couldn't make the students enthusiastic about science if I am not enthusiastic about it. To increase my interest in life science, let's try to tie it to my best subject, math -- enter the mathematical concepts of patterns, sequences, relations, one-to-one correspondence, all playing a role in unraveling the codes and mysteries of the living cell.

Again, this purpose of this is to engage myself with the lesson by tying it to my stronger subjects. But the next step is to engage the students with the material.

In 1994, the first Jurassic Park film had just been released. Since then, it has expanded into a full franchise, with Jurassic World having been released two years ago and the fifth movie, Fallen Kingdom, which was released around the start of this summer. So surely it wouldn't have been that difficult to tie the movies to the science lesson. "Have any of you watched the newest Jurassic Park movie? Do you know how the scientists in the movie brought the dinosaurs back to life? It's called genetic engineering...." This might have sparked an interest in learning the science and convince the students to let me teach them even though I wasn't a true science teacher.

I owed it to my students to teach them life science -- after all, I'd never know how many future doctors I had in my classes. One seventh grade girl once told me that she wanted to become a veterinarian when she grew up. Obviously, vets need to be familiar with life science -- and this includes genetic engineering (what is dog breeding after all).

One year ago today we mourned the passing of a famous singer -- Aretha Franklin, the Queen of Soul. (It's also the 61st birthday of another famous singer -- Madonna.)

It's perhaps only fitting that Aretha would leave us around the first day of school. After all, her most famous song is probably "Respect," and that's a word that is commonly heard in classrooms during the first week of school, when teachers are telling their students the rules. Three years ago, I posted some proposed classroom rules in preparing for my first year of teaching, and naturally many of these rules began with the word "Respect."

One of these proposed rules was terrible -- "Respect order." What in the world does it mean to "respect order" anyway? To me, a classroom with "order" is one in which the students choose to sit down, do their work, and behave. In short, the teacher is in charge of the class. Therefore, the rule "respect order" really means respect the teacher -- and that's how I should have stated the rule.

By the time school actually started, I never actually stated "respect order" as a rule -- but neither did I ever state "respect the teacher." But the idea that "respect order" was a rule pervaded my thinking -- and today I believe that this negatively affected my classroom management. In the end, "respect order" isn't as strong as "respect the teacher."

For example, one girl -- our future vet -- regularly questioned my authority. She often asked why she had to sit down in her assigned seat and why I couldn't give her a calculator to answer her basic arithmetic problems. My answer was something like, "because it preserves order in class" -- that is, because the class would run more smoothly if the students weren't running around the room or doing whatever they wanted. This only led to arguments -- basically, she insisted that order could be maintained even if I let her sit anywhere she wanted and gave her a calculator.

What I should have told her was "Because I said so." After all, students who act orderly in class don't do so because they "respect order" -- they do so because they respect the teacher. And so "respect the teacher" should have been my rule, and "Because I said so" my most commonly response to a student who questions the teacher's -- my -- authority.

Last year on the blog, I wrote a new song -- a parody of her song "Respect," both to honor the Queen of Soul and to introduce the rules to classes.

Here are the lyrics to such a parody:

Respect THE TEACHER
(A parody of "Respect" by Aretha Franklin, RIP 1942-2018)

What you want
CLASS, I got it
What you need
Do you know I got it
All I'm askin'
Is for a little respect when you get TO SCHOOL (just a little bit)
Hey CLASS (just a little bit) when you get TO SCHOOL
(Just a little bit) STUDENTS (just a little bit)
I ain't gonna do you wrong while you're gone
Ain't gonna do you wrong cause I don't wanna
All I'm askin'
Is for a little respect when you come TO SCHOOL (just a little bit)
CLASS (just a little bit) when you get TO SCHOOL (just a little bit)
Yeah (just a little bit)

I'm about to give you all of my LESSONS
And all I'm askin' in return, CLASS
Is to give me my propers
When you get TO SCHOOL (just a, just a, just a, just a)
Yeah CLASS (just a, just a, just a, just a)
When you get TO SCHOOL (just a little bit)
Yeah (just a little bit)
Ooo, YOU KIDS
Sweeter than honey
And guess what?
So ARE my LESSONS
All I want you to do for me
Is give it to me when you get TO SCHOOL (re, re, re ,re)
Yeah CLASS (re, re, re ,re)
LISTEN to me (respect, just a little bit)
When you get TO SCHOOL, now (just a little bit)

R-E-S-P-E-C-T
Find out what it means to me
R-E-S-P-E-C-T
Take care, TCB
Oh (LISTEN to me, LISTEN to me, LISTEN to me, LISTEN to me)
A little respect (LISTEN to me, LISTEN to me, LISTEN to me, LISTEN to me)
Whoa, CLASS (just a little bit)
A little respect (just a little bit)
I get tired (just a little bit)
Keep on tryin' (just a little bit)
You're runnin' out of fools (just a little bit)
And I ain't lyin' (just a little bit)
(Re, re, re, re) when you come TO SCHOOL
(Re, re, re ,re) 'spect
Or you might walk in (respect, just a little bit)
And find out YOU'RE IN TROUBLE (just a little bit)
I got to have (just a little bit)
A little respect (just a little bit)

In this parody, the capital letters represent the changes from Aretha's original song. The main changes are "baby" to "CLASS" and "home" to "SCHOOL." Oh, and of course, students should "LISTEN" to me (the teacher), not "sock it" to me. The third-to-last line is "YOU'RE IN TROUBLE" to show what will happen if you don't respect the teacher. I wanted to preserve the rhyme "honey" and "money," but I couldn't think of any rhyming words that fit the song (and besides, the rest of the song doesn't rhyme, so why should these), so instead I changed "money" to "LESSONS." Meanwhile, I preserved "TCB," meaning "taking care of business," since that's indeed what needs to happen in class.

Here is a YouTube video -- with the original lyrics, of course:

Throughout this month I will link to real Blaugust participants. In fact, today Shelli herself blogs an example of something she struggles with -- keeping up with her "My Favorite Friday" series:

http://statteacher.blogspot.com/2019/08/mtbosblaugust-its-friday.html

For the past few years, I've done a "My Favorite Friday" post as a way to reflect on the week and to focus on the positive reasons that create our "why".  Last year, I fell off the blogging wagon around mid-October and as a result, there was much of the year that I didn't spend time in reflection and celebrating victories.  I definitely want to bring back #MyFavFriday this year though!

Her favorites this week include classroom decorations, gifts, and "hack of a hack" (referring to her Rocketbook hack from earlier this week). Actually, I've been thinking about Rocketbook -- when I was at the old charter, there were arguments about students' inability to see what I was writing on the front board in markers (that often faded). With Rocketbook, I might have avoided such arguments.

If you think that linking to the Blaugust leader Shelli doesn't count as linking to a participant of Blaugust, let me link to Benjamin Leis instead:

http://mymathclub.blogspot.com/2019/08/summer-19-cool-math.html

I wrote last year that Leis runs a middle school math club. He often introduces high school topics to his bright middle school students:

Six  years ago I viewed many Mathematical topics as cut and dry.  "How hard can it be to learn everything there is to know about say Algebra I?"  Is there really anything new to find out about parabolas?  I don't think that exactly anymore as I continue to find further depths to even the most well trod subjects. These are not usually obvious though and discovering them  takes some work.  Towards that goal, I constantly read new material that comes my way.  Its not hard to find a constant stream of new mathematical writing but a lot of what I see tends to be a repeat of something I've read before.  Sometimes a few months go by without learning anything. This summer, however, was surprisingly rich in terms of personal math discoveries.

He also gives an example of an advanced Geometry topic -- Morley's triangle. (Click the above link to see it, since it's visual.)

Here are the worksheets (on a non-advanced topic) for today: