Today is the second of five days that I'm subbing in an Algebra II and Integrated Math II class. It is also the midpoint of the fourth quarter of the year -- the end of the seventh quaver.
In Algebra II, the students begin working on a Pizzazz packet for basic trig. Today's worksheet is page 165, "Greek Decoder." Students are given two sides of a right triangle and they must use the Pythagorean Theorem to find the length of the third side. Then they match up the letter for each problem with the Greek letters in the answer key to decode the message. (I assume that the Greek part is because Pythagoras was Greek.) The letters don't actually correspond to the real alphabet -- for example, "nu" is S and "pi" is O. The final message is, "Pythagoras was a famous Greek who knew all the right angles!" I believe most students figured out how to complete the worksheet.
During my third period conference, I am assigned to an AP Spanish class. Apparently, it is a class tradition to watch a movie the week after the AP Spanish exam. And the movie they watch today is Stand and Deliver, about the world's most famous math teacher -- Jaime Escalante, who successfully taught AP Calculus AB to students at a school right here in Southern California. (Well, maybe he's the most famous math teacher other than Pythagoras, who taught the members of his brotherhood.)
In Integrated Math I, students continue working on the Geometric Constructions Project. Here are a few things I want to say about the project:
-- First of all, the teacher suggests that the students watch some YouTube videos in order to learn how to perform the constructions. Here are the first two videos mentioned on this project worksheet:
The teacher provides some more links, but I don't post them here to the blog. This is because three of them show how to construct the regular octagon and pentagon, but constructing these challenging figures is not necessary for the project. The last link (Julie Nowak) is actually to a playlist of all of the same videos listed under David Nicholls, so there's no need for me to link to the playlist.
-- On this blog, I'm focusing on the Math I classes because Math I is almost identical to Common Core Math 8 -- a class I will be teaching next year. But as we found out last week, constructions don't appear in Math 8 (to the dismay of the traditionalists who want to see them in middle school), but they do appear in Integrated Math I -- well, Honors, at least. Actually, I spoke to one of the other teachers, and she told me that this is definitely an Honors-only assignment -- the regular students were having trouble just trying to bisect a segment! Again, we apologize to the traditionalists who want to push this down into middle school.
-- I recommend to the students that they start by constructing a hexagon -- even if the final figure contains an equilateral triangle. Let's say the students are drawing Figure #1 (from the project pages I linked to on Friday), which is two interlocking triangles (Star of David). It's much better to draw the hexagon first and connect every other vertex than to draw an equilateral triangle. It's also easier to remember how to construct the hexagon -- I might be tempted to use the hexagon to draw Figure #28, even though it's just an equilateral triangle with no hexagon necessary.
-- I decide to create a completed project of my own to show the students. Since students are to choose one design of their own, I also include my own design. I choose two interlocking circles and two equilateral triangles, as constructed in Lesson 4-4 of the U of Chicago text (Euclid's first theorem) or Level 1 of Euclid the Game.
That's right -- Euclid the Game! By now, many readers were probably wondering how long it would take me to mention Euclid the Game in this post, since every time I mentioned constructions this year on the blog, I inevitably bring up Euclid the Game.
Well, since the students are looking at the two YouTube videos listed above anyway, I want to mention the Euclid the Game website. But as it turns out, Euclid the Game is blocked on the district computers, as are all game websites! You'd think that the district would make an exception for an obviously educational game like Euclid, but oh well!
Actually, the author of the Euclid the Game website is in the process of changing his website and adding five new levels, to bring the total to 25. I tell the students that I've completed about 7-9 levels of Euclid so far, but of course this means nothing to them unless I can show them the game.
-- A major source of confusion is whether the students can complete this assignment in groups, or must they do it as individuals. Despite the teacher directing the class to divide into groups for the assignment, he wants the students to complete the assignment as individuals -- which he clarifies by sending a text.
-- So far, Figures #16 and #17 are the most popular -- probably because they are easy. To me, it's much better to do an easy figure correctly than a more complex figure incorrectly. Recall that constructions are all about exactness!
-- Many of the students insist on using a protractor as a straightedge, instead of a ruler. I don't know why they refuse to use a ruler for a straightedge. We know that protractors are forbidden in classical constructions, and the teacher specifically tells them to avoid protractors on the worksheet. I must be careful to make sure that they're not trying to sneak angle measure into their constructions.
-- Finally, I think I know why I'm unable to draw a circle with the teacher's whiteboard compass. My problem is that, as I hold down my finger to represent the center, the ribbon is wrapping around my finger, so that the radius is shorter after I wrap it around. Again, constructions are all about exactness, so it's not good for me to draw a whiteboard circle that's visibly imperfect and then insist that the students' circles be drawn correctly (even if they have the advantage of using a real compass).
I'll continue to write about the Math I students' progress as they work their way through the Geometric Constructions Project.
Chapter 14 of Morris Kline's Mathematics and the Physical World is "The Motion of Projectiles." In this chapter, Kline shows us the paths of objects that have been launched.
"I value the discovery of a single even insignificant truth more highly than all the argumentation on the highest questions which fails to reach a truth." -- Galileo
So we're still looking at Galileo's research on mechanics. Kline writes:
"By analyzing motion along a straight line, whether it be of bodies falling straight down or sliding down inclined planes, Galileo discovered some fundamental physical principles, namely, the first two laws of motion. These laws are in themselves quantitative statements about velocity, force, mass, and acceleration."
Now Kline considers an object (possibly a bomb) dropped from an airplane flying horizontally at 100 feet per second. Once again, the key to this idea goes back to my high school physics teacher and his line that "vectors operating at right angles are independent." So he writes equations for horizontal:
x = 100t
y = 16t^2
components of the motion. Then Kline uses substitution to eliminate t:
y = x^2 / 625
And this is the equation of a parabola. Later on in the chapter, Kline considers a projectile launched from a gun pointing at an angle A above the ground at velocity V. In order to consider the horizontal and vertical components of the velocity, he must use trig -- just as my Algebra II students are getting ready to learn:
v_x = V cos A
x = (V cos A)t = Vt cos A
v_y = V sin A - 32t
y = Vt sin A - 16t^2
Again, Kline eliminates t to obtain:
y = (tan A)x - 16 / (V^2 cos^2 A)x^2
which is yet another parabola. So Algebra students who wonder why they have to study parabolas only need to look at the equations of projectile motion to learn of their application.
An interesting question Kline asks is, what value of A maximizes the horizontal distance the projectile can travel? He points out that when A is 0, the projectile doesn't move from the ground, and when A is 90 degrees, it just goes straight up and down, not horizontally. So the maximum likely lies somewhere in between 0 and 90 degrees. To determine the answer, Kline sets the vertical height to 0 (that is, when the projectile lands on the ground), solves for t, then plugs this value into the equation for the horizontal distance, to obtain:
x = (V^2/16) sin A cos A
It's possible to use Calculus to find the maximum value of x. It's also possible to use trig to find the maximum -- consider the double-angle formula:
sin 2A = 2 sin A cos A
x = (V^2/32) sin 2A
Then the maximum occurs when sin 2A = 1 -- that is, when 2A = 90, so A = 45 degrees. But Kline uses neither Calculus nor the double-angle formula -- instead, he uses a clever trick.
He reminds us of the definitions of sine and cosine -- in a right triangle with legs a and b and hypotenuse c, sin A is just a/c and cos A is just b/c. So he writes:
sin A cos A = (a/c)(b/c) = ab/(c^2)
Kline then decides to fix c^2, and to maximize ab, we can maximize its square a^2 b^2 instead. So his goal is to maximize a^2 b^2 under the constraint that c^2 (which is a^2 + b^2) is fixed. And now he has reduced this to a previously solved problem:
"We can regard a^2 and b^2 as sides of a rectangle of semiperimeter c^2, which is fixed. The quantity a^2 b^2 is the area of such a rectangle. Of all such rectangles, the area is largest when the rectangle is a square, that is, when a^2 = b^2. Hence the product ab is also a maximum when a = b. In this case, the right triangle above is isosceles and angle A is 45 degrees. That is, for any given muzzle velocity, the maximum range is obtained by firing at an angle of 45 degrees. This result, a famous one, is Galileo's."
So we can see how powerful this trig stuff and the Pythagorean Theorem -- on which my Algebra II students are working today -- can be. It also shows us how remarkable Galileo was to have discovered so many properties of moving objects. But Kline's coverage of the famous Italian scientist is at an end, for Kline closes Chapter 14 with the following:
"But Galileo had already done more than one man's share and in 1642 his infirm body refused to function any longer."
By the way, I've posted the Geometric Constructions Project as if it were a one-day lesson, for last Friday's activity day. But we see that in the actual class, the students have a full week. I don't post week-long projects on the blog and treat everything like a one-day lesson or activity. And so I will post PARCC problems here on the blog as scheduled. But for teachers who are reading this blog, keep in mind that it's more realistic for the students to take a full week to finish the activity.
Question 14 of the PARCC Practice Exam is on rotations and reflections on a coordinate plane:
14. The right triangle in the coordinate plane is rotated 270 degrees clockwise about the point (2, 1) and then reflected across the y-axis to form triangle A'B'C'.
Drag and drop the correct orientation for triangle A'B'C' into the correct orientation on the coordinate plane.
(Here are the coordinates of triangle ABC: A(2, 1), B(5, 6), C(5, 1).)
We've seen before when rotating triangles that when the center of rotation is not the origin, it's always placed at one of the vertices of the triangle -- in this case A. So the rotation image of A is again A. At this point, we could try to find the rotation images of B and C -- but since the question asks us to choose the correct orientation and place it on the plane, it's far better to keep track of the orientation of a single vertex and the orientation of the triangle. Notice that 270 degrees clockwise is equivalent to 90 degrees counterclockwise, so our image would be blue triangle in the upper-left corner, placed so that the image of A is itself.
But that's just the rotation image. Our transformation is the composite of a rotation and a reflection, so we must still perform the reflection across the y-axis. So the final image A'B'C' looks like the blue triangle in the upper-right corner, and the mirror image of A(2, 1) is A'(-2, 1), so this is where we place the image A'.
PARCC Practice EOY Question 14
U of Chicago Correspondence: Lesson 6-3, Rotations
Key Theorem: Two Reflection Theorem for Rotations
The rotation r_m o r_l, where m intersects l, "turns" figures twice the non-obtuse angle between l and m, measured from l to m, about the point of intersection of the lines.
Common Core Standard:
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Commentary: Even though the Integrated Math I classes are studying constructions now, in the classroom I found several old worksheets scattered around -- including these Kuta worksheets that I saw. Notice that Kuta's rotations are always centered at the origin, unlike the PARCC's which are often centered at a vertex. On the other hand, Kuta's reflections sometimes have mirrors that are parallel to the axes, not just at the axes themselves.