Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Find x.
Almost all of the given information for this problem is in the diagram, and so I have no choice but to go into a long description. Furthermore, in this diagram the vertices aren't labeled, but I'll label them for the sake of you blog readers who can't see the diagram.
In Triangle ABC, D is between A and B.
Angle A = ACD = DCB = a
AD = 21, DB = 7, CB = x
To solve this, we notice that Angles A and DCB are congruent, since each has measure a. And of course, Angle B is congruent to itself. This is enough to conclude that Triangles ABC and CBD are similar by AA~. Thus we may now set up a proportion:
CB/DB = AB/CB
x/7 = 28/x
x^2 = 196
x = 14
Therefore x = 14 -- and of course, yesterday's date was the fourteenth. So there seems to be no problem with this question -- yet I now claim that it contains an error.
Here's the problem -- let's find CD. That's easy -- in Triangle ACD, Angles A and ACD each have measure a, so by Converse Isosceles, CD = AD = 21.
But in Triangle BCD, we have DB = 7, CB = 14, and CD = 21. So the longest side of this triangle is exactly equal to the sum of the shorter two sides. This contradicts Triangle Inequality -- oops!
What gives here? Let's try to calculate the measure of a using trig. According to the Law of Sines:
AB/sin C = CB/sin A
28/sin 2a = 14/sin a
We use the Means Exchange Property (Lesson 12-4) so that all the a's are on one side of the equation:
28/14 = sin 2a/sin a
2 = sin 2a/sin a
2 = 2 sin a cos a/sin a
1 = cos a
0 = a
But of course, an angle of a triangle can't be zero degrees -- oops!
The mistake that Pappas makes here is a fairly common one made by textbook writers and teachers in setting up similarity problems. For example, we might write:
Angles FGH = JKL = 80, FG = 3, JK = 6, GH = 4, KL = 8, FH = 5, JL = ?
The givens provide us with two pairs of proportional sides and a pair of included congruent angles, and so the triangles are similar by SAS~, and therefore JL = 10. But by Converse Pythagoras, Angle FGH is clearly 90, not 80 as given. (Did you notice 3-4-5 there?) The problem is that the question writer is only thinking about providing enough info to conclude that the triangles are similar -- not whether the info is inconsistent -- especially if it contradicts theorems from chapters other than the similarity chapter (such as Triangle Inequality in the Pappas question).
If we had given Angles FHG = JLK = 37 instead of FGH and JKL, we can safely assume that 37 is a rounded value, since the true value (arccos 4/5) is irrational. But it's not accurate to claim that 80 is a rounded value of 90.
And suppose we had instead given JL = 10 (triangles similar by SSS~) and JKL = 80 and asked the student for FGH. Should the student use similar triangles to conclude FGH = 80, or should he use Converse Pythagoras to conclude FGH = 90?
So let's fix the Pappas question. If we keep the angles and try to fix the sides, we have:
AB/CB = 2cos a
k = 2cos a
where k is the constant of proportionality (for the similarity ABC ~ CBD). We see that since a > 0, we conclude that cos a < 1 and thus k < 2. It's also possible to place a lower bound on k -- if a = 60, then Angle ACB = 2a = 120, leaving 180 - 60 - 120 = 0 degrees for Angle B. So a < 60, and since cos 60 is 1/2, we conclude that k > 1.
Pappas probably wanted the constant k in her problem to be a simple natural number. Yet in this problem, we found that k must be strictly between 1 and 2. So k could be 1.0001 or 1.9999, but it can't be exactly 1, 2, or any other integer. The best she could do is let k = 3/2, the simplest rational in the interval (1, 2) -- for example, let AD = 10 and DC = 8. Then x = 12 without any contradiction.
The other idea is to keep the sides as given and just throw out Angle ACD = a -- after all, we didn't need to know ACD to conclude that the triangles are similar. Pappas probably included ACD = a just to add another level of difficulty (so that it wouldn't be as obvious which triangles are similar), but she unwittingly added a contradictions instead.
In fact, we can now prove rigorously that Angle ACD > a. By Triangle Inequality from above, we conclude that CD < 21. So now we can use Unequal Sides -- since AD = 21 and CD < 21, we have AD > CD, and hence Angle ACD > A = a. QED
I'd argue that we shouldn't need to think about trig to create (or solve) a similarity problem. Even though the Law of Sines appears above, we didn't need it -- Triangle Inequality and Unequal Angles are sufficient to conclude that k must be between 1 and 2. The next time you create a similarity problem, please make sure that your givens don't contradict any pre-trig theorem (Triangle Inequality, Triangle Sum, Isosceles Triangle, Unequal Sides/Angles, 45-45-90, 30-60-90).
Today is the first day of school according to the blog calendar. The blog calendar is based on one of the districts from which I hope to receive subbing calls. This district is not LAUSD, but in fact is the district whose calendar I used last year. I first followed this calendar four years ago -- the first year of this blog.
I could have used another calendar -- the other district where I currently sub. This calendar has a near Labor Day start. Actually, I propose another name for this calendar -- the Labor Thanksgiving Calendar. In other words, years ago school started after Labor Day, but in order to allow for school to be closed the entire week of Thanksgiving, the start was moved up in recent years to the week before Labor Day. The reason I favor the other district calendar (the Early Start or Pre-Christmas Finals Calendar) is that it's more convenient for my purposes.
Recall that I'm following the digit pattern for days in the U of Chicago text. Chapter 7 is the last chapter of the first semester, and it contains eight sections. Following the district calendar puts Lesson 7-8 ("The SAS Inequality" or Hinge Theorem) on Day 78. Then Days 79-80 can be review days for the final. But this year, the finals will be given on Days 83-85. This means that Lessons 8-1 and 8-2 are taught before the final.
Notice that the district calendar starts the second semester with Day 86, which would be the day for Lesson 8-6. Lesson 8-1, "Perimeter Formulas," can be summed up in a single sentence ("Just add up all the sides!") and Lesson 8-2, "Tiling the Plane," is also easy since it's on tessellations, so I don't mind squeezing these in before the final. Lesson 8-3, "Fundamental Properties of Area," does give the formula for the area of a rectangle, but then this can easily be sneaked into some of the later lessons. Lesson 8-4, "Areas of Irregular Regions," can safely be skipped, but Lesson 8-5 ("Areas of Triangles") is important. Instead, this year I'll do something special for Lesson 8-6 ("Areas of Trapezoids") on the day after winter break.
On the other hand, the Labor Thanksgiving Calendar has a true 90 days in the first semester, which would force all of Chapter 8 to be in the first semester (and on the first final). I'd rather keep most of Chapter 8 in the second semester, and so I choose the Early Start Calendar to base the blog on (even though I'm much more likely to get subbing calls from the Labor Thanksgiving Calendar district).
For future reference, here is a pacing guide for the entire year:
Chapter 0: August 15th-28th
Chapter 1: August 29th-September 12th
Chapter 2: September 13th-26th
Chapter 3: September 27th-October 10th
Chapter 4: October 11th-25th
Chapter 5: October 26th-November 8th
Chapter 6: November 9th-30th
Chapter 7: December 3rd-14th
Lessons 8-1 to 8-2: December 17th-18th
Semester 1 Finals: December 19th-21st
Lessons 8-6 to 8-9: January 8th-14th
Chapter 9: January 15th-29th
Chapter 10: January 30th-February 13th
Chapter 11: February 14th-28th
Chapter 12: March 1st-14th
Chapter 13: March 15th-April 4th
Chapter 14: April 5th-18th
Chapter 15: April 23rd-May 6th
State Testing Window: May 7th-31st
Semester 2 Finals: June 3rd-5th
Notice that Day 14 is the day after Labor Day. This means that teachers at schools that have a Labor Day Start, but wish to follow my pacing guide, can simply pick it up at Lesson 1-4, "Points in Networks" -- which was my traditional first day of school lesson for the first three years of this blog.
But at the district whose calendar I'm following on the blog, today is the first day of school. Actually, there was a "Freshman First Day" yesterday, so ninth graders attend 181 days of school. We could thus count yesterday as "Day 0" -- but if I were a regular Geometry teacher in this district, I wouldn't bother to teach any math on Day 0 anyway. Some of the Geometry students might be freshmen, but many would be sophomores, and so the first real lesson would be today, Day 1.
Even though there is no Day 0 on the blog, there is a Chapter 0, covering Days 1-10. Since the U of Chicago text doesn't have a Chapter 0, we use Michael Serra's Discovering Geometry instead. Don't forget that my copy of Discovering Geometry is an old text, dated 1997 (Second Edition). My old version goes up to Chapter 16. The modern (3rd-5th) editions only go up to Chapter 13. But Chapter 0 is essentially the same in all editions -- the only difference is that Lessons 0.5 and 0.8 no longer exist in any edition later than my Second Edition.
This is what I wrote last year about today's lesson:
Lesson 0.1 of the Discovering Geometry text is called "Geometry in Nature and Art." In this lesson, students learn that Geometry is all around them.
The main theme of this lesson is symmetry. Serra writes about two types of symmetry -- reflectional symmetry and rotational symmetry. And so we are introduced to two of the main Common Core transformations, reflections and rotations, in the very first lesson.
By the way, I've noticed that other teachers use the Serra text as well. For example, here is a link to Lucy Logsdon -- a Michigan Geometry teacher and Blaugust participant:
https://lsquared76.wordpress.com/2018/08/14/honors-geometry-unit-5-polygon-properties/
Even though Logsdon never mentions the name Serra or title Discovering Geometry, she's clearly referring to this text. Notice that she begins by discussing "Polygon Angle Conjectures" -- and calling these "conjectures" instead of "theorems" is a giveaway that this is Serra's text. I see that the lessons follow the same order as in Serra -- polygon angle conjectures are in Lessons 6.1 and 6.2, kite and trapezoid properties are in Lesson 6.3, midsegment properties are in Lesson 6.4, and parallelogram properties are in Lessons 6.5 and 6.6. The only reason she calls this "Unit 5" instead of 6 is that in his modern editions, Serra combines his old Chapters 1 and 2. Thus the modern chapter numbers tend to be one less than the Second Edition chapter numbers.
The reason that Serra calls these "conjectures" instead of "theorems" is that officially, nothing is proved until the final chapter. But he does sneak a few coordinate proofs and flow proofs in earlier chapters, which is why these proofs appear on Logsdon's blog as well.
Oops -- I'm not an official Blaugust participant, yet I can't help myself from looking at Shelli's list of Blaugust prompts and participants. I still won't actually add this blog to her list (since only real teachers belong there, and I'm not a real teacher). But I still want to look at her prompts anyway. As today is the fifteenth (my reference to yesterday's Pappas question notwithstanding), let's look at the 15th prompt on Shelli's list:
Something I struggle with as a teacher/in the classroom.
And that's an easy one to answer. Two years ago, I was hired as a math and science teacher, yet I struggled to teach science. Last year, I wrote about this in my first day of school post -- and since I'm cutting and pasting that post anyway, I might as well cut and paste this story in.
When I first wrote that post, I referred to two particular students in my old class -- the "special scholar" and her cousin, the "special cousin." The previous summer I'd written a post explaining who those two girls are, but those posts were lost in the Great Post Purge of 2017. I then repeated who they are in my January 6th post. (I won't cut and paste from that post, but it's another post that's all about my struggles in the classroom.)
Now I'll retell this story, now that you know exactly whom I'm talking about:
In the NGSS, however, some of the chemistry-related topics are now considered seventh grade standards. I've written before that I should have stuck to the old California Science Standards in Grades 7-8 and introduced the NGSS only to the sixth graders. This meant that the eighth graders should have received the chemistry lessons.
Of course, there was that Edible Molecule Project -- the first project in the Illinois State science project text. As you may recall, a huge argument between me and the "special cousin" -- the new girl transferring in from another school -- surrounded that project. First, she stated that the previous year, her science teacher had assigned an Edible Cell Project, which she'd enjoyed. She also told me that I was wrong to teach anything other than physical science -- since the previous year had been life science -- and ignored me when I tried to explain the NGSS. Finally, when I did finally assign the Edible Molecule Project, she continued to complain and didn't want to complete the project.
Although the new girl was just being oppositional, she did make some valid points. Chances are that if I'd taught physical science strongly from the start of the year -- as well as treated her cousin (the "special scholar") with more respect -- she'd have been willing to learn science from me. Notice that if I really had taught science from the start of the year, I probably would have been past the chemistry projects in the Illinois State text and been ready to work on those that leaned towards physics.
During the Edible Molecule Project, I didn't really say much about the shapes of the molecules. I just told the students that for a molecule like H2O, each hydrogen atom is bonded to the oxygen atom, not to the other hydrogen. I didn't say, for example, that the bond angle in H2O is 105 degrees.
It's a shame that I wasn't able to teach chemistry to my eighth graders better. After all, I wrote before that I prefer physical science to life science -- and in fact, I have a special affinity for chemistry. Back in February 2016, I explained why:
One last thing I want to mention in this post -- when I was young, I remember being fascinated by an old college chemistry text. Some of the problems in this text required algebra, and one of them was a slope question which used the notation slope = delta-y / delta-x -- where the symbol "delta" stands for something like "difference" or "the change in." So when my Algebra I teacher assigned some slope problems, I started writing "delta" in all of my answers. I still remember her response: "Delta is one of my favorite Greek letters." Believe it or not, I've since noticed that nowadays, some Algebra I teachers actually use "delta" when teaching slope!
And then eleven months later, I wrote about how well I did in my chemistry classes:
I think back to my own days as a science student -- which contain many highs and lows. As a high school freshman, my general science teacher saw some promise in me and wanted to promote me to an Applied Bio/Chem class, but I moved to another school before the end of the first quarter. Two years later, my Integrated Science III teacher at that new school similarly thought I was gifted and not only recommended me for Chemistry, but convinced the magnet at our school to accept me!
But unfortunately, I didn't extend my chemistry fascination to my eighth graders. I could have had fun telling them about tris, a molecule described as having the shape of a Mobius strip.
Returning to Serra:
"Nature displays an infinite array of geometric shapes, from the small atom to the greatest of the spiral galaxies. Crystalline solids..."
Atoms -- but then again, Serra focuses more on macroscopic shapes rather than microscopic shapes. His other examples include honeycombs, snowflakes, and pine cones.
I created the first worksheet of the year from questions from Serra's text. Some of these are labeled as Exercises, while others come from his first "Project." And some of these questions ask students to bring objects in to class. This seems awkward for the first day of school -- but then again, this is the very first lesson in Serra (so it's intended for early in the year). It might be good for teachers to find photos on the internet and show them to the class. Or better yet, the students might be able to come up with pictures of their own to draw -- especially the questions about art from various cultures (which include the students' own cultures).
Yet there are two questions that students might enjoy as an opening day activity. One of them asks students to find the line of symmetry in a work by British artist Andy Goldsworthy, who indeed is still alive (hint: H2O). The other asks students to name playing cards with point symmetry. They might want to draw these -- the three of diamonds has point symmetry, but not the three of clubs. Yes, they might want to try drawing the three of clubs with point symmetry to see why it is impossible.
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