Thus there won't be much "A Day in the Life" coming up. All five classes I sub today has either an aide, or are co-teaching. The first class of the day is study skills, and the rest are English (eighth grade with the aide, seventh grade co-teaching).
Even though I;m not in charge of managing many classes this week or next week, there are still a few things I can say about the subbing. The seventh graders are reading a novel (more like a graphic novel) called Lambert. (And what's this novel about -- quadrilaterals with three right angles, also known as Lambert quadrilaterals?) But the main lesson is about annotation -- and even though the lesson is online, it contains key parts of a traditional lesson: "define," "model," and "your turn."
I think back to yesterday's class and how I failed to teach all parts of the rounding lesson well. If I want to return as a regular teacher in the classroom someday, I must a habit out of teaching all parts of a traditional lesson effectively.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
ABCD is a parallelogram. What's the area of BCDE?
(Here is the given info from the diagram: E is the foot of the perpendicular from B to
Of course, as soon as we see those radical 2's, we suspect that this problem has something to do with 45-45-90 triangles, which commonly appear on the Pappas calendar. (Sorry, 15-75-90 fans, but your favorite special right triangle doesn't appear in the problem, nor any other on the Pappas calendar.)
And indeed, we see that in right triangle BCF, leg BF = 5/sqrt(2) and hypotenuse BC = 5. This is enough to conclude that BCF is a 45-45-90 triangle, so that Angle C = 45. (Technically, there's no such thing as "Converse of the 45-45-90 Theorem." Officially, we're using forward 45-45-90 along with a congruence theorem for uniqueness -- since we're given a leg and a hypotenuse, we use HL.)
Opposite angles of a pgram are congruent, and so Angle A must also be 45 degrees. Thus Triangle ABE is also 45-45-90. Its hypotenuse is 4sqrt(2), and so its legs AE and BE are both 4.
Opposite sides of a pgram are congruent, and so AD = BC = 5. Since E is between A and D and we know both AE = 4 and AD = 5, we conclude that DE = 1.
This is enough for us to find the area of quadrilateral BCDE -- a trapezoid. We know its height and both of its bases, so we just use the formula:
A = (1/2)h(b_1 + b_2)
A = (1/2)BE(BC + DE)
A = (1/2)4(5 + 1)
A = 2(6)
A = 12
Therefore the desired area is 12 square units -- and of course, today's date is the twelfth. Even though the trapezoid area formula appears in Lesson 8-6, we must wait until Lesson 14-1 to teach this problem since it involves 45-45-90 triangles.
By the way, notice how I write the trapezoid area formula as (1/2)h(b_1 + b_2) -- and the triangle area formula as (1/2)hb, even though it doesn't contain parentheses in the text. This is because in ASCII -- without the parentheses -- we would have the "8 / 2 (2 + 2) = 1 or 16?" problem. (Hmm, I notice that by the formula, 8 / 2 (2 + 2) would be the area of a trapezoid with height 8 and both bases 2, but a trapezoid with equal bases would actually be a pgram, so we'd just use the pgram formula.)
Lesson 2-1 of the U of Chicago text is called "The Need for Definitions." This is what I wrote last year about today's lesson:
The part about the definitions of words from outside of mathematics -- such as terrorist -- is even more timely today as the blog calendar has placed this lesson closer to the anniversary of 9/11.
The second chapter of nearly any high school geometry text discusses the logical structure of geometry -- to prepare students for proofs. This includes the U of Chicago text, as well as Dr. Franklin Mason's text, and many others.
Lesson 2-1 of the U of Chicago text deals with definitions. But the introduction to the chapter mentions a 1986 USA Today article concerning a non-mathematical definition: cookie. Normally, as teachers we'd ignore this page and skip directly to the first lesson, except that this article is mentioned all throughout 2-1, even including the questions!
Now, of course, a teacher could have the students discuss the article as an introduction to the importance of precise definitions. Such an introduction is often called an anticipatory set -- a concept that apparently goes back to the education theorist Madeline Hunter.
A teacher could present the article as an anticipatory set, but I should point out that the article is over a quarter of a century old -- after all, my text itself is nearly that old. The article points out that the word terrorist was controversial even back then. As we already know, a decade after the book was written, the 9/11 attacks occurred -- and since then, that word terrorist has been thrown around so much more, with very strong political implications.
And, of course, there was another definition that led to a politically charged debate -- one that occurred just a few years after the publishing of the text. During the investigation during the impeachment of Bill Clinton, the former president questioned the definition of the word is. So we see that there are two fields where precise definitions matter greatly: law and mathematics.
To me, it might be fun to discuss these examples in class. But it may be tough for the teacher to remain politically neutral during such a discussion, so we must proceed with caution.
The images at the end of this post do not mention the article -- I threw out any part of the section that refers to the article. I preserve the discussion about what a rectangle is, and the one definition given in the lesson -- that of convex set.
When approaching the questions, I first threw out Questions 1 through 3, since these questions go back to the article. I kept all of the questions about convex sets, since that's the term defined in the lesson, then kept the question where students guess the definition of midpoint -- a preview of Lesson 2-5.
Now I want to consider including the review questions as well. As any teacher knows, students have trouble retaining what they've learned, so we give review questions to make them remember. I avoided review questions during Chapter 1 since most of them were review of the Lessons 1-1 through 1-5. But most of the review questions in this section are labeled Previous course. I must be careful about these, since it all depends on which previous course is being mentioned here.
Question 20 in the text discusses the definition of words like pentagon and octagon. Like midpoint, this book will define these terms later in the chapter (Lesson 2-7), but this is labeled Previous course. I assume that the intended previous course is probably a middle school course. But -- remembering that this is a Common Core blog -- I decided to look up the Common Core Standards. The only standard mentioning the word pentagon is a 2nd grade standard!
CCSS.MATH.CONTENT.2.G.A.1
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
So in theory, it might have been nearly a decade since the students saw the word pentagon. (The word octagon doesn't appear in the standards at all.) But I figure that upon seeing the question, the students will remember vaguely that these words all refer to shapes with different numbers of sides -- and at least know that a triangle has three sides, even if they must guess on all the rest. This is a good preview of Lesson 2-7.
In the other questions marked Previous course, the course referred to is clearly Algebra I. Once again, I don't want to intimidate the students with Algebra I questions in a Geometry class. Of course, we can see how Questions 21 and 22 came about -- they are clearly translations of the word problems "23 degrees less than the measure of an angle is the measure of its supplement" and "the measure of an angle is six times the the measure of its complement," respectively. I'm torn whether to include such problems. One thing that I definitely want to avoid is algebra problems masquerading as geometry problems -- for example, we take a linear equation from algebra and write its two sides as the measures of vertical angles (provided the two sides equal valid angle measures). The geometry in such a question is trivial -- just set the two sides equal to each other since vertical angles are congruent, then the rest is all algebra. The geometry in a question about complementary and supplementary angles is less trivial, but then -- so is the algebra, since a typical question will often have variables on both sides, and many students struggle with these.
In the end, I decided to keep Questions 21 and 22 but at least give the students a break by making the solutions whole numbers -- notice that as written, the solutions to both contain fractions. Question 23 seems to serve no geometric purpose at all. I decided to drop the second variable z and change the number 225 to 360, since students will often need to divide 360 degrees by various numbers -- for example, when finding the exterior angle measures of a 15-gon. This is the most difficult algebra/arithmetic that I want appearing in the first semester of a geometry course -- nothing beyond this is acceptable.
Finally, we reach Question 24. This is an Exploration question, asking the students to define the words cookie and terrorist. Once again, this makes a lot more sense if the article is mentioned in class. I decided that I'll include this and other Exploration questions, but label them as Bonus questions to emphasize that these questions are optional for the students. Of course, it can be thrown out completely if a teacher wants to avoid politically charged debates over the word terrorist.
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