Thursday, January 18, 2018

Lesson 9-1: Points, Lines, and Planes in Space (Day 91)

Chapter 9 of the U of Chicago text is called "Three-Dimensional Figures." In past years, we skipped over this chapter and jumped directly into Chapter 10. After all, most questions relating to 3D figures on standardized tests are asking about their surface areas or volumes -- the purview of Chapter 10. As we are following the digit pattern this year, we will cover all of Chapter 9 starting today, Day 91.

I think back to David Joyce, who criticized a certain Geometry text. He writes:

Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.

In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered.

And so Chapter 6 of the Prentice-Hall text is just like Chapter 10 of the U of Chicago text. Joyce laments that students don't learn "the basics of solid geometry" before surface area and volume. But we can't fault Prentice-Hall for this. Even before the Common Core, most states' standards expected students to learn the 3D measurement formulas and hardly anything else about 3D solids.

We can't quite be sure what Joyce means by "the basics of solid geometry." But it's possible that some of what he wants to see actually appears in Chapter 9 of the U of Chicago text. Thus, by teaching Chapter 9, we are slightly closer to Joyce's ideal Geometry course.

And incidentally, there is one Common Core Standard in which 3D solids are mentioned, but not surface area of volume. We'll look at this standard in more detail next week, in Lesson 9-4.

Lesson 9-1 of the U of Chicago text is called "Points, Lines, and Planes in Space." The first three sections of Chapter 9 are the same in both the old Second and modern Third Editions. (As it turns out, the new Third Edition squeezes in surface area in Chapter 9, saving only volume for Chapter 10.)

The heart of this lesson is the Point-Line-Plane Postulate. We first see this postulate in Lesson 1-7, but now it includes parts e-g:

Point-Line-Plane Postulate:
a. Given a line in a plane, there exists a point in the plane not on the line. Given a plane in space, there exists a point in space not on the plane.
b. Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.
c. Through any two points, there is exactly one line.
d. On a number line, there is a unique distance between two points.
e. If two points lie in a plane, the line containing them lies in the plane.
f. Through three noncollinear points, there is exactly one plane.
g. If two different planes have a point in common, then their intersection is a line.

There are several terms defined in this lesson -- intersecting planes, parallel planes, perpendicular planes, and a line perpendicular to a plane.

Actually, I'm still thinking about Joyce's "basics of solid geometry." I know that his website also links to Euclid's Elements. So Book XI of Euclid is a reasonable guess as to what Joyce wants to see taught in class:

https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html

Let's look at some of the definitions and propositions (theorems) here and compare them to the contents of Lesson 9-1. We'll start with Definition 3, since Definitions 1 and 2 will actually appear in tomorrow's Lesson 9-2.



Definition 3.
A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane.
Definition 4.
A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane.
Definition 5.
The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

Definition 3 appears in Lesson 9-1 as a line perpendicular to a plane. Definition 4 appears in this lesson as perpendicular planes. But Definition 5, the angle between a line and a plane, is only briefly mentioned in the U of Chicago text.



Definition 6.
The inclination of a plane to a plane is the acute angle contained by the straight lines drawn at right angles to the intersection at the same point, one in each of the planes.
Definition 7.
A plane is said to be similarly inclined to a plane as another is to another when the said angles of the inclinations equal one another.
Definition 8.
Parallel planes are those which do not meet.

Again, Definitions 6 and 7 are about the angle between two planes, which is not discussed in our text at all. Definition 8, of course, appears in today's lesson -- but just as with lines, the U of Chicago uses an "inclusive" definition of parallel where a line or plane can be parallel to itself. Intersecting planes (our remaining term) are implied in Definition 8 as planes that are not parallel.

Let's look at the propositions (theorems) now:



Proposition 1.
A part of a straight line cannot be in the plane of reference and a part in plane more elevated.

This is essentially part e of our Point-Line-Plane Postulate. Euclid calls it a proposition (or theorem) and even provides a proof, but Joyce argues that the proof is unclear. Thus we might as well consider it to be a postulate.



Proposition 2.
If two straight lines cut one another, then they lie in one plane; and every triangle lies in one plane.

This is essentially part f of our Point-Line-Plane Postulate. If A, B, and C are the three noncollinear points mentioned in part f, then we can take lines AB and AC to be the two intersecting lines that appear in Proposition 2, and triangle ABC to be the triangle mentioned in this proposition.



Proposition 3.
If two planes cut one another, then their intersection is a straight line.

This is very obviously part g of the Point-Line-Plane Postulate. Joyce points out that this is yet another postulate, and that it holds only in 3D, not 4D and above.



Proposition 4.
If a straight line is set up at right angles to two straight lines which cut one another at their common point of section, then it is also at right angles to the plane passing through them.

According to Joyce, this is the first true theorem in Book XI. It asserts that if a line intersects a plane and is perpendicular to two lines in the plane, then the line is perpendicular to the whole plane. Joyce points out that the proof is a bit long, but it works. Theoretically, our students can prove it using the new Point-Line-Postulate and theorems from the first semester of the U of Chicago text.

https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/propXI4.html

Here is a modern rendering of this proof. The idea is that line l is perpendicular to each of two lines m, n, in plane P, with all lines concurrent at point E. Our goal is to prove that line l is perpendicular to the entire plane P by showing that, if o is any other line in plane P with E on o, then l must be perpendicular to o as well.

Given: l perp. m, l perp. n with l, m, n all intersecting at E
Prove: Line l is perpendicular to plane that contains m and n.

Proof:
Statements                                    Reasons
1. bla, bla, bla                              1. Given
2. Choose A, B on m and              2. Point-Line-Plane part b (Ruler Postulate)
    C, D on n so that
    AE = EB = CE = ED
3. Exists plane P containing mn 3. Point-Line-Plane part f (3 noncollinear A, C, E)
4. Choose F on l,                           4. Planes contain lines and lines contain points.
    and o in plane P s.t. E on o
5. Lines AD, o intersect at G,         5. Line Intersection Theorem
    Lines BC, o intersect at H
6. Angle AED = Angle CEB          6. Vertical Angle Theorem
7. Triangle AED = Triangle CEB   7. SAS Congruence Theorem [steps 2,6,2]
8. AD = CB, Angle DAE = EBC     8. CPCTC
9. Angle AEG = Angle BEH           9. Vertical Angle Theorem
10. Triangle AEG = Triangle BEH 10. ASA Congruence Theorem [steps 8,2,9]
11. GE = EH, AG = BH                  11. CPCTC
12. FE = FE                                    12. Reflexive Property of Congruence
13. Triangle AEF = Triangle BEF  13. SAS Congruence Theorem [steps 2,1,12]
14. FA = FB                                     14. CPCTC
15. Triangle CEF = Triangle DEF  15. SAS Congruence Theorem [steps 2,1,12]
16. FC = FD                                    16. CPCTC
17. Triangle FAD = Triangle FBC   17. SSS Congruence Theorem [steps 8,14,16]
18. Angle FAD = Angle FBC          18. CPCTC
19. Triangle FAG = Triangle FBH  19. SAS Congruence Theorem [steps 11,18,14]
20. FG = FH                                    20. CPCTC
21. Triangle GEF = Triangle HEF  21. SSS Congruence Theorem [steps 11,12,20]
22. Angle GEF = Angle HEF         22. CPCTC
23. EF perp. GH (i.e., l perp. o)      23. GEF, HEF are congruent and a Linear Pair                              24. Line l perpendicular to plane P 24. Definition of line perpendicular to plane

So this is probably what Joyce wants to see more of. Propositions 5 through 19 aren't very much different from this one. But as I wrote above, our students will find such proofs difficult -- we had to prove seven different pairs of triangles congruent above, in three dimensions to boot. No modern text teaches such theorems, since no state standards -- pre- or post-Core -- require them.

The proof works -- the definition in Step 24 is satisfied because o is arbitrary. But notice that in the drawing at the above link, Euclid assumes that G, the point where lines AD and o intersect, is between A and D. But this is irrelevant for the proof -- all the congruence theorems used in the proof still work even if G isn't between A and D.

What's worse, of course, is if o is parallel to AD. Notice that AD | | BC (since DAE and EBC, the angles proved congruent in Step 8, are alternate interior angles), so o could be parallel to both. But that's no problem -- just switch points C and D in that rare case, and the proof still works.

Since this is a brand-new chapter for the blog, I have no worksheets for it. And so here is a newly created worksheet for Lesson 9-1.



Wednesday, January 17, 2018

Chapter 8 Test (Day 90)

Today is the Chapter 8 Test. It is also Day 90, the mathematical midpoint of the year. As we already know, most Early Start schools don't actually begin early enough in August to have a true semester of 90 days before winter break.

Last year, the mathematical second semester was when SBAC Prep began. Indeed, the bell schedule was changed so that SBAC Prep would replace most of P.E. time. Since we'd lacked a real conference period anyway, things truly became tough for me once SBAC Prep began.

This is what I wrote two years ago about today's test:

Let's worry about the Chapter 8 Test that I'm posting today on the blog. Here are the answers:
1. This is a triangle tessellation. The triangle is isosceles, so it shouldn't be too hard to tessellate.
2. One can find the area by drawing a square grid and estimating how many squares are taken up on the grid. Since the shape happens to be an ellipse, one can also find its area using the formula for the area of an ellipse -- pi * a * b, where a and b are the major and minor axes. That's right -- I had to slip in a reference to pi this week.
3. 5/2 or 2.5.
4. 12 square units.
5. 40 square units.
6. 6 mm. This is a trick if one forgets the 1/2 -- especially with 3-4-5 right triangles featuring in the last few questions, but here the legs are 6 and 4, not 3 and 4.
7. 10 feet.
8. 4.5s.
9. One can find the area by drawing a diagonal to triangulate the trapezoid. Then one adds up the area of the two triangles.
10. Answers may vary. The simplest such rectangle is long and skinny -- 1 foot by 49 feet.
11. Choice (a). The triangles have the same base and height, therefore the same area.
12. 3s^2.
13. 13 feet.
14. 6 minutes.
15. 780,000 square feet.
16. 133 square feet.
17. 1/4 or .25. Probability is a tricky topic -- the U of Chicago assumes that the students already know something about probability. Then again, it should be obvious that the smaller square is 1/4 of the larger square.
18. 4,000 square units.
19. 13.5 square units.
20. a(b - c) or ab - ac square units.

But here's a 2018 update: again, the test I wrote that year was based on Lessons 8-1 to 8-6 (which is what we covered that year), rather than Lessons 8-4 to 8-9. It's awkward to make the students answer so many questions about the first three lessons of the chapter but not the last three. And so if teachers wish, they may replace the first two questions on the test with the following:

1. Give the circumference and area of a circle with radius 10: a. exactly; b. estimated to the nearest hundredth.
2. Could the numbers 1, 2, sqrt(3) be the lengths of sides of a right triangle?

Answers:
1. a. 20pi units, 100pi square units
    b. 62.83 units, 314.16 square units
2. Yes. (Notice that c here needs to be 2, not sqrt(3).)

Today is a test day, and so as usual I'm labeling this post "traditionalists." Let's see what our favorite traditionalist, Barry Garelick, has to say:

https://traditionalmath.wordpress.com/2018/01/16/everyones-happy-in-happy-land-dept-2/

Chicago Tribune carries this story about a controversial approach to education being tried at various high schools in Illinois :
They’re abandoning most aspects of traditional classroom instruction and reshaping the way kids learn.

Of course, we already know the answer to Garelick's question. No, it doesn't sound helpful -- at least not to traditionalists like him.

So far, this post has drawn five comments. Surprisingly, only one of these was posted by our favorite commenter, SteveH:

“[“I’m scared,” Hardy admitted.
He said the program is not a perfect solution and not every student will graduate prepared for college, work and life after competency-based learning.
Even so, there’s no turning back.
“We had a school on the brink of failure,” he said.]”
Perhaps not prepared for “… work or life after competency-based learning”?
So let’s just lower expectations and make it seem official. Never mind what the requirements are for college, vocational school, work, or even life.
OK, so evidently SteveH's first problem with this "competency-based learning" program is that he sees it as lowering expectations.

Actually, let's back up to Garelick's description of the program itself. We see that it's being taught in Chicago and several other schools throughout Illinois. We already know that Chicago (as in the U of Chicago text and its elementary counterpart Everyday Math) and Illinois (as in the Illinois State text) are hotbeds of anti-traditionalist pedagogy. I wouldn't be surprised if Garelick's opposition to this program is due at least in part to the city and state in which it is being piloted.

The original Chicago Tribune article describes a high school Geometry class, and since this is a Geometry blog, let me post this part in full:

At Ridgewood High School, math teacher Tristan Kumor started off geometry class on a recent day by asking his students, “How do you want to learn today?”
His 9th and 10th graders sat in groups, with one student using Popsicle sticks to build a bridge. The lesson related to triangles. Kumor traversed the room guiding students individually on their progress, which is part of the way competency-based education plays out.
Garelick notes that so far, this sounds just like Project-Based Learning or PBL. Indeed, there might have been a project similar to this in the Illinois State text, but we never reached it last year. Of course, we already know what he would prefer these Geometry students to be doing in class -- answering individual problem sets (or p-sets).

Notice the title of Garelick's post -- "Everyone's Happy in HappyLand." We know that students in a traditional Geometry class might give complaints such as "Why do we have to do proofs?" and "When will we ever use these proofs?" But students in Kumor's class are less likely to complain, because they enjoy building a bridge more than building a proof. The question "When will we ever use this?" never arises. In other words, Kumor's students are happy -- and Garelick acknowledges this by referring to this classroom as "HappyLand."

But Garelick's concern, of course, is that these happy students aren't learning anything. He'd much prefer that students unhappily learn things than happily learn nothing. As usual, there's something that Garelick leaves out -- students who aren't happy doing an assignment are likely to leave the assignment blank.

Presumably, by building a bridge, students learn that triangles are "stable" in that a triangle with three Popsicle sticks doesn't collapse. The reason for this is SSS Congruence -- there is essentially one triangle given three side lengths. Garelick would argue that students would learn this better by writing a traditional proof rather than building a bridge. But I would counter that students learn more by building a bridge than leaving a proof blank -- which is what many students would end up doing.

There are other parts of "competency-based learning" (CBL) that Garelick criticizes. He writes about a "three-before-me" approach, where students are to ask each other for help, not the teacher:

We don’t want teachers teaching  handing it to the student, now do we?

First of all, consider the Latin phrase docendo discimus -- "We learn by teaching." Traditionalists ought to appreciate that this phrase goes back 2000 years, as it's attributed to Seneca the Younger. So you can't get much more traditionalist than that.

Meanwhile, we must also consider the fact that many students, especially teenagers, avoid listening to the teacher at all costs -- especially when the teacher is telling them that they're wrong. Teens are more likely to accept corrections from other students than from the teacher. Garelick just assumes that students will accept the teacher's corrections just because he or she is the teacher.

The next two Garelick comments are about the grading scale. First, CBL doesn't use traditional letter grades -- in particular, there are no F's:

Right, the Jo Boaler approach which holds that mistakes grow your brain. And if mistakes are that powerful, then failing is even better!

No, the idea is that some students think that getting an F is the end of the world. They believe that they are no good at math and simply leave all subsequent assignments blank. Remember, students learn nothing from the assignments they leave blank, no matter how traditional they are. Under the "Jo Boaler approach," students are less likely to leave assignments blank.

Apparently, CBL grades are on a scale of 1 to 4. Garelick responds:

And let me guess; no one gets a 4. At least that’s how it’s played out in other schools that have tried similar things.

Ah -- so now he criticizes the lack of high grades, not the lack of low grades. This strongly reminds me of the French grading system (see my November 15th post), where 20/20 is impossible. But I suppose he goes have a point here -- the impossibility of a 4/4 grade is much different from the impossibility of 20/20, where there are still so many good grades (19/20, 16/20, etc.) possible.

I'm not quite sure what exactly Garelick believes about the lack of 4/4. I do now that by limiting both high and low grades, the grade gap between the top and bottom student is reduced. This is the opposite of tracking, where differences between students are emphasized. Many traditionalists favor tracking, but it is generally avoided for political reasons.

Let's return now to SteveH. He writes:

It appears to be an opt-in pilot program, and I see that Proviso West still offers the traditional AP Calculus track.

This is a response to the first two commenters in the thread. They are concerned that this program is an "experiment" on children -- and indeed, previous traditionalists have accused progressive programs of treating students like "guinea pigs." But that's the problem -- it's possible that there could be a curriculum that is truly more effective at teaching students, but logically, some cohort must be the first to use it, and the parents of that first cohort may think that their kids are being experimented on like guinea pigs.

Of course, traditionalists have a solution -- traditionalism is the most effective pedagogy possible, and so there's no need ever to try anything else. Traditionalism should remain until the end of time. Hence there are no experiments and no guinea pigs needed.

Recall that the U of Chicago has a "Lab School." (If I recall, a former president sent his children to this school.) Parents who send their children there are already aware that new curricula are being tested there, so they won't feel that the kids are being mistreated like guinea pigs. This is another solution to the new curriculum problem -- test the new curriculum only at schools where the parents choose to send their children.

And indeed, this is what SteveH writes. Apparently, at Proviso East High School the new CBL curriculum is being tested, but at Proviso West High traditionalism remains. (There are freshmen in the East High Geometry class and so officially those students are on an AP Calculus track, but SteveH does specify traditional AP Calc track, and CBL isn't traditional.) He allays the fears of the first two commenters that students are being forced to take CBL Geometry.

You end up with a chasm that you can’t Pre-AP away and you can’t claim a program of self-competency.

SteveH explains the problem with "Pre-AP" classes in an earlier post. According to him, the best way to prepare students for AP classes is to teach them traditional math beginning in kindergarten, rather than teach progressive math up through middle school and then place them in "Pre-AP" classes.

That’s the fundamental fuzzies flaw – lower expectations are always less, not more. Engagement plus student-driven PBL is not even a proper vocational education let alone a path to STEM.

I don't know what SteveH does consider to be a proper vocational education if not PBL. I hope it's not just assigning p-sets and assuming that teens will actually complete them simply because the teacher says so.

There is one thing that I agree with Garelick in this post. At the end, we see:

Educators say the grades and transcripts have been a source of concern, with high schools reluctant to change because that data is used in college admissions.
Well yeah, there’s that, but let’s not let that stop such a promising program.
Like Garelick, I'm also concerned that a new grading system can confuse colleges. In past posts, I've written that almost no high school uses a trimester calendar, since college admissions officers expect to see semester grades on the transcripts. Indeed, it also explains why even though schools are split regarding whether sixth grade (or even fifth) is an elementary or a middle grade, nearly all agree that high school starts with ninth grade -- because colleges want to see four years of high school English.

So what should high schools do if they want to pilot a new grading system? This is tricky. Perhaps college applicants -- and only college applicants -- could have their 1-4 scores converted to traditional letter grades. But how should we do this? Should we use the traditional GPA conversion where 4 is an A, 3 is a B, and so on? But then if Garelick is right that there are no 4's, then there are no A's -- and those students would be at a disadvantage compared to seniors from schools that still give out traditional A grades.

Furthermore, such conversion defeats the purpose of having alternate grades if "3" really just means "B" and "2" just means "C," and so on. The conversion between 1-4 grades and letter grades probably shouldn't be a one-to-one conversion.

Otherwise, perhaps 1-4 scores should be reserved for middle school and below only. (Elementary schools often already use 1-4 or other grading systems.) Then when the students reach high school, the expected letter grades can appear on the transcript.

OK, here is the Chapter 8 Test:



Tuesday, January 16, 2018

Lesson 8-9: The Area of a Circle (Day 89)

Lesson 8-9 of the U of Chicago text is called "The Area of a Circle." We all know the famous formula that appears in this lesson.

Two years ago, my Pi Day activity was more geared towards the area. Therefore, I'm posting that Pi Day worksheet today for Lesson 8-9.

Meanwhile, many chapters in the second half of the book are longer than those in the first half -- and this causes a problem in setting up the chapter review and chapter test. Thursday is Day 91, which is when Lesson 9-1 will be taught, and tomorrow is the Chapter 8 Test. This means that today needs to be the Chapter 8 Review as well as Lesson 8-9. Get used to this, since there are several more long chapters coming up in the text.

This is what I wrote two years ago about Lesson 8-9:

I visited several other teacher blogs for ideas on lessons. One of these blogs has a lesson that's perfect for Pi Day:

https://theinfinitelee.wordpress.com/2016/02/08/lesson-area-of-a-circle-or-how-i-got-students-hungry-for-the-formula/

Laura Lee is a middle school math teacher from Minnesota. Here is how she teaches her seventh graders about pi:

I teach CMP. I love discovering pi in Investigation 3.1 of Filling and Wrapping. It’s hands-on and engaging. Students love discovering that pi is a real thing not just a random number and excuse to eat pie on 3/14. If the lesson goes really well, I’ll even get a kid to ask, “Isn’t this a proportional relationship? Isn’t pi a constant of proportionality?” To which I want to use every happy emoji ever created in large, bold, capitalized letters with lots of exclamation points!!!
But then Investigation 3.2 comes along, where students discover the formula for area of a circle as pi groups of radius squares. I’ve never been able to get students to make sense of this lesson. There seem to be lots of understandings (what a radius square is, how to calculate how many times one number goes into another, understanding multiplication as groups of…) needed in order to get valid data that would allow students to conclude pi groups fit inside the circle. It just gets messy and doesn’t seem to make any sense to students, so this year I decided to take a different approach.
I decided to go with composing and decomposing. I would show them how to decompose the circle into a rectangle that is πr by r. Here’s how the lesson went (spoiler alert- a Domino’s pizza makes an appearance):
Notice that last year, I posted a lesson that actually covered area before circumference. Lee's lesson restores the order from the U of Chicago text, with circumference (Lesson 8-8) before area (8-9).

Let's just skip to the part where, as Lee writes, a pizza makes an appearance:

Method 2 (I had been talking about pizza all morning with my first 2 classes, so last hour I decided I should just order a pizza and use that to derive the area formula)
  • Order a pizza (Domino’s large cheese worked great!)
  • Reveal pizza to class, watch them go insane!
  • Have students gather around your front table
  • Slice pizza into 16 slices,
  • talk about circumference of 8 of the slices or half of the pizza: πr, record this on the pizza box
  • then start arranging pizza slices into a rectangle, listen to student “Ahas!” and “No ways!” when they see it is clearly starting to form a rectangle
  • Talk about dimensions of rectangle and then the area

The U of Chicago text does something similar in its Lesson 8-9. The difference, of course, is that the text doesn't use an actual pizza.

Lee writes that for her, the key is proportionality. This fits perfectly with the Common Core:

CCSS.MATH.CONTENT.HSG.C.A.1
Prove that all circles are similar.

Then again, notice that Common Core seems to expect a proof here. How does Common Core expect students to prove the similarity of all circles without Calculus?

Unfortunately, none of our sources actually prove that all circles are similar. What I'm expecting is something like this -- to prove that two circles are similar, we prove that there exists a dilation mapping one to the other. For simplicity, let's assume the circles are concentric, and the radii of the two circles are r and s. So we let D be the dilation of scale factor s/r whose center is -- where else -- the common center O of the two circles. If R is a point on the circle of radius r, then OR = r, and so its image R' must be a point whose distance from O is r * s/r = s, and so it must lie on the other circle of radius s. Likewise, if R' is a point on the circle of radius s, its preimage must be a point whose distance from O is s / (s/r) = r, and so it must like on the circle of radius r. Therefore the image of the circle of radius r is exactly the circle of radius s.

Of course, this only works if the circles are concentric. If the circles aren't concentric, then it's probably easiest just to compose the dilation with an isometry -- here a translation is easiest -- mapping the center of one circle to that of the other. Therefore there exists a similarity transformation mapping any circle to any other circle. Therefore all circles are similar. QED

To get from the area of the unit disk (pi) to the area of any disk (pi * r^2), we are basically using the Fundamental Theorem of Similarity from Section 12-6 of the U of Chicago. This time, though, we are using part (b) of that theorem:

Fundamental Theorem of Similarity:
If G ~ G' and k is the ratio of similitude [the scale factor -- dw], then
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2.

We skipped this formula back when we covered Lesson 12-6 because at the time, we hadn't learned about area yet. [2018 update: Actually, this year we reverse the order -- so far we've covered area but not similarity at all.] Although Wu attempts to prove a special case of the Fundamental Theorem of Similarity using triangles, it's much easier to do it using squares, as the U of Chicago does. 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.

Now here are some Pi Day videos:




OK, I didn't need to reblog the Pi Day video. But I posted a pi video for Lesson 8-8, and so I might as well post a video for Lesson 8-9.

And this is what I wrote two years ago about Review for Chapter 8 Test:

As I mentioned earlier, the Chapter 8 Test is on Wednesday, which means that the review for the Chapter 8 Test must be today.

In earlier posts, I mentioned the problems that occur when a teacher blindly assigns a worksheet that doesn't correspond to what the student just learned in the text. Since I'm posting a review worksheet today, we should ask ourselves whether the students really learned the material that is to be assessed in this worksheet.

For example, most students learn about area at some point in their geometry texts, but only the U of Chicago text includes tessellations in the area chapter. Yet the very first question on this area test is about -- tessellations. So a teacher who assigns this worksheet to the class will then have the students confused on the very first question!

Let's review the purpose of this blog and the reason why I post worksheets here. The purpose of this blog is to inform teachers about the transformations (isometries, similarity transformations) and other ideas that are unique to Common Core method of teaching geometry. The worksheets don't make up a complete course, but instead are intended to be used with a non-Common Core text -- the one that teachers already use in the classroom, in order to supplement the non-Common Core text with Common Core ideas. Another intent is for those teachers who do have Common Core texts, but are unfamiliar with Common Core, to understand what Common Core Geometry is all about. My worksheets are based mainly on the U of Chicago text because both this old text and the Common Core Standards were influenced by NCTM, National Council of Teachers of Mathematics.

So this means that a teacher interested in Common Core Geometry may read this blog, see this worksheet, decide to assign it to the class, and then have all the students complain after seeing the first question because their own text doesn't mention tessellations at all.

I decided to include the tessellation question because it appear in the U of Chicago text. But as of now, it's uncertain that tessellations even appear on the PARCC or SBAC exams. So it would be OK, and preferable, for a teacher to cross out the question or even change it. There's a quadrilateral, a kite, that's already given in the question, so the question could be changed to, say, find the area of the kite, especially if the school's text highlights, instead of tessellations, the formula for the area of a kite.

I admit that it's tricky to accommodate all the various texts on a single worksheet. I included tessellations since this is a drawing assignment that is fun, and I'd try to include them if I were teaching a class of my own. But I also want to include questions that may be similar to those that may appear on the PARCC or SBAC exams.

[2018 update: The decision to include tessellations on this worksheet is even more controversial than it was two years ago, since this year we began with Lesson 8-4 and didn't even cover the earlier Lesson 8-2 on tessellations!]

For example, Questions 2 and 9 are exactly the type of "explain how the..." questions that many people say will appear on those Common Core exams. And so it was an easy decision for me to include those questions.

Another issue that came up is the Pythagorean Theorem. Such questions appear in the SPUR section of Chapter 8 in the U of Chicago, but we didn't cover Lesson 8-7. [2018 update: This year, of course, we covered the Pythagorean Theorem.] I dropped the questions that were purely on the Pythagorean Theorem, but I kept two trapezoid questions where the Pythagorean Theorem is needed to find the length of a side or the height. But these questions will confuse a student who has reached the area chapter, but not the Pythagorean Theorem chapter that may be several chapters away.

Then there is a question where students derive the area of a parallelogram from that of a trapezoid. I point out that in other texts -- especially those where trapezoid is defined inclusively -- this isn't how one derives the area of a parallelogram. In the U of Chicago, the chain of area derivations is:

rectangle --> triangle --> trapezoid --> parallelogram

But in other texts, it may be different, such as:

square --> rectangle --> parallelogram --> triangle --> trapezoid





Friday, January 12, 2018

Lesson 8-8: Arc Measure and Arc Length (Day 88)

Lesson 8-8 of the U of Chicago text is called "Arc Measure and Arc Length." This and the next section are the same in the old Second and modern Third Editions -- except that arc measure appears much earlier in the new version (in Chapter 3).

This is, of course, the lesson when students learn about the number pi. Two of my favorite lessons to teach each year are the Pythagorean Theorem and pi. Both two and three years ago, I rearranged the lessons so that Pythagoras appears near the Distance Formula and pi is taught on Pi Day. But since we're following the order of the text this year, Pythagoras and pi are taught here in the same chapter!

Indeed, since following the digit pattern means that our pi lesson isn't on Pi Day, you might ask, what lesson will I post on Pi Day instead. According to the blog calendar, March 14th works out to be the 128th day of school. Lesson 12-8 of the U of Chicago text is on SSS Similarity -- unfortunately a lesson about triangles, not circles or pi. Luckily, I have two months to figure out how I'm going to celebrate Pi Day on the blog this year.

The seventh grade U of Chicago text, called Transition Mathematics, is much more convenient for setting up the pi lesson near Pi Day. Today's lesson on the circumference of a circle is Lesson 12-4, and Lesson 12-8 is on spheres -- whose surface area and volume formulas definitely use pi. Keep in mind that I'm referring to my old Second Edition, not the new Third Edition -- the Third Edition of Transition Math teaches pi in Chapter 7 and stats in Chapter 12.

Much of my chapter rearrangement in past years was driven by my desire to celebrate Pi Day by teaching the famous constant. Thus I began the second semester with Chapter 12, so that we would be in Chapter 8 on measurement. The chapters following 12 are also related to similarity (such as trig) while the chapters following 8 are also related to measurement (such as volume), and so the net result was that we covered Chapters 12 through 14, and then back to Chapters 8 through 10. This year I wanted to follow the book order, at the cost of severing the link between pi lessons and Pi Day.

Two years ago I tried to combine Lessons 8-8 (circumference) and 8-9 (area), but the worksheet I posted on Pi Day leaned more towards area. A month later, I subbed in a seventh grade classroom where students were learning about circumference. And so the April worksheet I found in that classroom is what I'm actually going to post again today.

This is what I wrote two years ago about today's lesson:

CCSS.MATH.CONTENT.7.G.B.4
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

And we all know what this means -- today was the day the students begin learning about pi!

Of course I am posting today's worksheet on the blog. For this activity, the students are given four round objects and a tape measure, and they are to measure the circumference and diameter of each of the objects. For example, one of the objects is a heart tin -- its circumference is about 47 cm and its diameter is about 15 cm.

You may notice that there's room to measure five objects, not just four. Well, the fifth object is the circle painted on the outdoor basketball court. This is convenient because its diameter is already marked (the free-throw line). But the students, instead of bringing the tape measure outside, use a nonstandard unit to measure the circle -- their own feet. With basketball on the mind of so many Californians today -- here in the south we celebrate Kobe Bryant's final game, while those in the north hope the Warriors win their 73rd game today -- it's great to incorporate the sport into today's lesson.

Notice that students are not to fill out the column "What relationship do you see?" yet. But some students try to come up with a relationship anyway. One student tries subtracting the diameter from the circumference, to write something like, "The circumference is 32 cm more than the diameter." I argue that this student is actually on the right track, if you think about it.

Meanwhile, a few students have already heard of pi, so they already know the relationship. One student cheats by measuring the diameters and simply multiplying each one by 3.14. The regular teacher will probably reveal the relationship between the circumference and diameter tomorrow.

Most of the students enjoy the lesson, but a few wonder why we are doing this activity. But most likely, these students are upset because they finish measuring the basketball court before any other group and is hoping for a reward. Instead, they are caught by another teacher for attempting to return to the classroom and fool around while I'm still out watching the other students.

Let's think about where this lesson fits in the seventh grade curriculum. Last week I wrote that if I were teaching the class, I'd try to reach Chapter 8 by Pi Day. As we see, this class came close -- certainly much closer than last week's Chapter 2 class.

But it can be argued that today is actually a "Pi Day" of sorts. You see, instead of 3/14, today is April 13th, which is 4/13. As the digits of pi appear in reverse, we can think of this as "Opposite Pi Day."

And now you're thinking -- here we go grasping at straws to come up with another math holiday. We already have Pi Day on March 14th, Pi Approximation Day on July 22nd, and Pumpkin Pi Day on the 314th day of the year in November. We had Square Root Day of the Decade on 4/4/16, Square Root Day of the Century on 4/5/2025, and several Square Root Days of the Month -- including yesterday, April 12th, which can serve as sqrt(17) day. And now I insist on adding yet another Pi Day on April 13th just because 3/14 reversed is 4/13! Do I really think that anyone is actually going to celebrate any of these extra so-called "Pi Days"?

Well, actually I didn't invent "Opposite Pi Day" -- the creator of the following video did:


It's yet another parody of Rebecca Black's "Friday" -- as I mentioned back on the original Pi Day, the fact that Friday and Pi Day rhyme is too irresistible for many math parodists to avoid. The poster of this song, who goes by the username "AsianGlow," probably just missed uploading the song on the original Pi Day, so rather than wait a whole year to post it, he just reversed the digits. He uploaded the video exactly five years ago today -- April 13th, 2011.

AsianGlow writes:

Of course by now you know of Rebecca Black making her infamous debut with Friday, but what about Pi Day?

How come Fridays get so much more love? Friday usually gets four times a month to party, but Pi Day only gets one... March 14th...

Does this mean I can only eat pies once a year?! OMG NO! Any date containing 1, 3, or 4 should be a piece of Pi Day! hahaha! See what I did there? You don't think it's funny? Well don't be so bitter and get some sweets in yo life! :D

This is the first video in a while that I did allll by myself! Well the filming anyways...
FilmRebelRoby - http://youtube.com/filmrebelroby - helped me with mastering my vocals.

April 13 is like opposite Pi Day! OMG it's still April fools! That's right, that's Pi!

Notice that AsianGlow's proposal that any date containing 1, 3, and 4 should be Pi Day only applies to March 14th and April 13th. We see that January 34th, April 31st, and 13/4 all just barely avoid being valid Gregorian dates, but they could exist in certain versions of Calendar Reform. In particular, all three date exist in a Leap Week Calendar where every third month (including January and April) have 35 days, 28 days in all the rest, and Leap Week labeled as Month 13. [2018 update -- I wrote about Calendar Reform on the blog two weeks ago, so you can read more about these calendars in those posts.]

I wouldn't have mentioned Opposite Pi Day here on the blog at all had I not subbed in a class learning about pi today. In some ways, I'm celebrating Opposite Pi Day today for the exact same reason as AsianGlow -- we both already missed the real Pi Day, yet we want to celebrate pi today. Both Pi Day and Square Root Day can be easily manipulated so that they fall during the unit on pi or square roots.

Though officially I covered pi on the blog on the original Pi Day -- and that includes both the circumference and area formulas -- the actual lesson focused more on area. This follows the lesson plans on Dr. Hung-Hsi Wu. Yes, I gave circumference first, but the area activity is Wu's. Today I finally post a lesson where students experiment with circumference. Of course, with these two lessons a week apart, the last part of that Common Core Standard:

CCSS.MATH.CONTENT.7.G.B.4
...give an informal derivation of the relationship between the circumference and area of a circle.

isn't fully emphasized here on the blog.

Before we return to 2018, let's think about last year, 2017. Yesterday I wrote about how I should taught the Pythagorean to my eighth graders last year, and so today I'll do the same regarding pi and my seventh graders last year.

Actually, I never reached the lesson on pi last year. That's because I was waiting, as usual, for Pi Day to teach the lesson, but I was out of the classroom before March 14th. In fact, I wrote in a post dated later that month what had actually happened on Pi Day. I decided to give my students one last surprise by delivering a pizza to my old classroom. But the bell schedule was mixed up that day, with school out early the entire week for second trimester Parent Conferences. It turned out that sixth grade was in the classroom at the time I delivered the pizza. Thus the sixth graders got to celebrate Pi Day with a pizza, even though seventh grade is the year that pi appears in the Common Core.

So had I made it to Pi Day, how would I have taught the lesson? Pi Day fell on a Tuesday last year, and at the time, Tuesdays were for projects. I assume that the Illinois State text had some sort of project where students had to measure the diameters and circumferences of various round objects -- in other words, an activity not much different from the one I'm posting today.

On the other hand, I posted that I should have made Tuesdays the traditional lesson day. Still, I see no problem with a brief measurement activity before the traditional lesson -- just as I'd given the eighth graders a brief Pythagorean Theorem activity before the traditional lesson two months earlier.

I had no control over Parent Conferences or the bell schedule. Again, I don't know when seventh grade had class that day -- only that sixth grade was the last class. If I were teaching, I wouldn't have been able to get the pizza -- but I could have sent my support staff aide to purchase it instead. After all, she'd bought a pizza for our eighth grade class four months earlier. (I mention this pizza in my Epiphany post from last weekend, in the "October 2016" section.) As a bonus, I could have had her get an extra pizza to share with my fellow teachers as they waited for Parent Conferences to begin.

So that seventh grade isn't left out of the party, I could bring some other round foods -- such as cookies -- for the students to measure. They only get to eat what they measure, so this is an incentive to do the activity correctly. Meanwhile, sixth grade gets a party but isn't learning about pi. Actually, I remember that there was a pi activity near the end of the Illinois State sixth grade STEM text page as a preview of seventh grade. The Pi Day pizza party would have been a great excuse to do this -- provided, of course, that I was given more than a day's notice as to what the bell schedule would be that day (which, as you may recall, wasn't always guaranteed on shortened days).

Suppose now that I had returned to the school and was planning a Pi Day lesson and party this year, then how would I do it? Again March 14th will be a shortened day -- not because of Parent Conferences (which I believe are a week later this year), but because it's Wednesday. The Common Planning meeting means that once again, kids will go home before 1:59. And so again, I'd be at the mercy of which class of students meets between the time that pizza parlors open and the time that school is dismissed.

If seventh grade gets the party -- which would make the most sense, since they're the ones learning about pi this year -- it would be the second straight Pi Day pizza party for this cohort, since last year's sixth graders are this year's seventh graders! But once again, I have no control over what the bell schedule is.

And as Pi Day is a Wednesday, this is a Learning Centers day. Of course, that day the centers are set up so that students are measuring circles and learning about pi.

For the second straight day, we're avoiding the elephant in the room -- classroom management. When I gave this activity as a sub, there weren't too many behavior issues. After all, if students misbehaved that day, I'd leave their names for the regular teacher. The problem is when I am the regular teacher and there's no one for me to leave names for. And my support aide -- the only adult my students respect in my classroom -- might not be present if she's out buying the pizza!

Notice that in the post from two years ago, I did mention one behavior issue. Let me repeat it:

Most of the students enjoy the lesson, but a few wonder why we are doing this activity. But most likely, these students are upset because they finish measuring the basketball court before any other group and is hoping for a reward. Instead, they are caught by another teacher for attempting to return to the classroom and fool around while I'm still out watching the other students.

Is there anything I could have done to handle this problem better? Perhaps I should have told the students to stay near the court until all students had finished measuring it. This is something I should have anticipated at the time, because it was a sign of things to come. Do the students know what they should do at each step of the activity? And do they know what to do when they are done?

I suspect that if I had given this worksheet to my seventh graders last year, they would have handled it better than most of the other projects they did that year. This is because there's a worksheet, so at least they would know what they are supposed to do. No students, for example, could claim that they had finished in ten minutes or less if most of the boxes on their sheet were empty!

Yes, I know that many of these recent posts have turned into spilled milk and discussion of my class from last year again. But it's important to reflect on my past failures in order to set myself up for future success if I ever return to the classroom. The sky is the limit!

Since today is Friday, this is my first official activity day for the second semester. But of course, Lessons 8-4 and 8-7 already have activity components. This is a tough week for many students as it marks both the return from winter break and the first heavy set of formulas to remember. And so it's nice to have many activities going on during the week.

And on these Friday activity days, I was going to create a worksheet containing the Exploration questions from the U of Chicago text. If you wish to include them in your class, here they are:

a. Measure the circumference of your neck with a tape measure to the nearest half inch or centimeter.
b. Assuming your neck is circular, use your measurement to estimate your neck's radius.
c. What would be another way to get its radius? (Cutting is not allowed, of course!)

These questions almost fit this worksheet anyway. Students can write "my neck" in the "Object" column and its circumference in the second column. The only difference is that now they must calculate its diameter, not its radius, to match the third column.

Monday is the holiday for Martin Luther King, Jr. Day. And so my next post will be Tuesday.