8:00 -- Second period (in high schools in this district, "first period" = zero period) is the first of two Algebra II classes.
Notice that while today is Day 93 in the old district (the one the blog calendar follows), it's only Day 85 in the old district. Just knowing that it's Day 85 should be a huge giveaway what's happening in this class and all of the other classes today. We're approaching the end of the first semester, and so everyone is preparing for finals. All classes have review worksheets for the big test.
I don't sub for Algebra II often in this class, so I'm not familiar with the text in this district. We already know that Geometry uses the Glencoe text, but Algebra II actually uses Big Ideas -- the same publisher as the middle school math texts.
Here is the table of contents for the Big Ideas Algebra II text:
1. Linear Functions
2. Quadratic Functions
3. Quadratic Equations and Complex Numbers
4. Polynomial Functions
5. Rational Exponents and Radical Functions
6. Exponential and Logarithmic Functions
7. Rational Functions
8. Sequences and Series
9. Trigonometric Ratios and Functions
10. Probability
11. Data Analysis and Statistics
The first semester final apparently covers the first five chapters of the text. Indeed, the lesson plan for this week so far is written on the board:
Monday: Graphing Square Root and Cube Root Functions
Tuesday: Review for Chapter 5 Quiz
Wednesday: Chapter 5 Quiz and Final Review Part I
Thursday: Final Review Part II
Notice that the second semester thus begins with exponents and logarithms. This is timed perfectly for e Day coming up on February 7th.
8:50 -- Second period leaves and tutorial begins.
9:23 -- Two minutes before tutorial is scheduled to end, the fire alarm suddenly goes off. Of course, it's only a false alarm -- especially considering that it's raining today!
9:30 -- Third period arrives. This is the first of three sections of our favorite class -- Geometry! The students work on the first semester review worksheet.
Here is the lesson plan for Geometry this week:
Monday: Review for Chapter 6 Quiz
Tuesday: Chapter 6 Quiz and Final Review Part I
Wednesday: Final Review Part II
Thursday: Final Review Part III
The Chapter 6 Quiz only contains the first half of the chapter (Lessons 6-1 to 6-3). Recall that Chapter 6 of the Glencoe text is on quadrilaterals -- this corresponds roughly to Chapters 7 (which is parallelograms) and 5 (rectangles, rhombi, trapezoids, and kites) of the U of Chicago text. The parallelograms of our Chapter 7 were on the quiz, while the quadrilaterals of our Chapter 5 haven't been covered yet. This is reflected in the final review packet, where questions on other quadrilaterals are labeled "SKIP."
10:30 -- Third period leaves for snack.
10:50 -- Fourth period arrives. This is the second of three sections of Geometry.
11:40 -- Fourth period leaves. Fifth is the teacher's conference, which leads directly into lunch.
1:20 -- Sixth period arrives. This is the third of three sections of Geometry.
2:10 -- Sixth period leaves and seventh period arrives. This is the second Algebra II class.
2:40 -- The fire alarm goes off again.
3:05 -- Seventh period leaves, thus ending my day.
By the way, let's compare the days leading up to the finals in both Algebra II and Geometry today to the days leading up to finals in our Geometry course on the blog. We see that in both classes, the teacher gives his last quiz of the semester, then several days of review worksheets. In both classes, the final review packet started on the day of the quiz itself. In neither class did we finish the packet, so we know that tomorrow will be the third day of review in Algebra II and the fourth in Geometry.
But in our course on the blog, after we finished Lesson 7-8 on a Wednesday, I posted two days of finals review on Thursday and Friday. Then we went to Lessons 8-1 and 8-2 on Monday and Tuesday (with a few review questions on the worksheets) before finals started on Wednesday.
Perhaps this teacher's method of finals review is better than mine. We could have skipped 8-1 and 8-2 and spread out our review pages over four days instead of two. I also could have covered 8-1 and 8-2 anyway along with a finals review page. It's also possible for me to post a Chapter 7 Quiz just as this teacher did -- quiz review on Thursday, quiz on Friday plus the first finals review page. Then there would be three days of review (Friday, Monday, Tuesday).
Let's return to our New Year's Resolutions. Here is the focus resolution for today:
2. Keep a calm voice instead of yelling at students.
I definitely felt like yelling at the third period class. At one point, only a handful of students are actually paying attention to my finals review. Other students start treating the entire period as a nonacademic free day, with talking and phones galore.
This time, I try to avoid yelling. I never mentioned this on the blog, but back at the old charter, some students wanted to help me with my yelling problem and gave me a stress ball to squeeze. It's doubtful that I ever used the ball -- while I was aware at the time that I yelled often, I didn't yet understand just how deep my yelling problem was. Sometimes something minor would tick me off (such as my alma mater UCLA losing to USC the previous night) and I'd start yelling in class.
So now I use the stress ball whenever I feel like yelling. At the end of third period, I squeeze the ball and start thinking about how to improve for fourth and sixth periods.
In fourth period, I tell the students that there is to be no talking or phones while they are completing the packets. No one is to act as if he or she doesn't care about the final -- instead, they should act as if their lives depended on it. Indeed, their lives do depend on it -- students who earn higher grades earn money over their lives. I mentioned recently how I must become wary of talking about grades as a form of classroom management, but it's about impossible to discuss finals without mentioning grades.
Indeed, fourth period is an improvement over third period. But still, there are times when it appears only one smart student is answering my questions. In the end, I place one student on my bad list for phone use and not paying attention, and I name fourth period as the best Geometry class of the day (while second period is best overall). But then again, there's an administrator in the room during much of this period. I can't help but think he's sent there after the loud third period -- and so much of the improvement during fourth is due to his presence, not anything I did.
But still, I'm worried that not everyone is well-prepared for the final. I'd hate to think that a student who could have passed next week's final failed because he or she played around on the day I'm here instead of reviewing for the final. The regular teacher might return for tomorrow's review day, or there might be another sub -- one who isn't as strong in math. Thus I feel it's my imperative to make sure that the students are preparing well for the final on the day that I'm in the classroom.
I haven't played the Conjectures/"Who Am I?" game since my birthday. But I don't want to wait until my next birthday (or all the way to 2020, since it's on the weekend this year) to play it again. I wish to play it on special or important occasions -- and what's more important than the final?
And so in sixth period -- the largest of the three classes (35 students -- ironically, third period is the smallest with only 25) I start the game, as usual with the age and weight questions. As usual, this class is noisy since the game encourages them to speak out loud in groups. Fortunately, the game works its magic, as students are definitely more engaged here than in third period.
In each class I cover different questions from the assigned section of the packet that the students are working on (Questions 28-41). Here are some of the questions:
28. Name the longest segment in Triangle ABD.
31. Give the reason that justifies that p | | q:
a) If Angle 2 = Angle 6
b) If Angle 4 = Angle 6
35. Given AC is a median of Triangle ABD, what can you determine?
In the first two classes, some extraneous information is given. In Question 28, the diagram shows Triangle ABC along with ABD. (This is an Unequal Angles question -- the inequality theorems appear in Chapter 5 of Glencoe yet not until Chapter 13 in the U of Chicago.) And in Question 31, there are two parallel lines with two transversals in the diagram. In each case, I only drew the part of the diagram that's relevant on the board.
Question 35 is simple -- it just asks for the definition of median. Yet third period was so loud and so many weren't paying attention that they couldn't figure it out! This was the last straw that caused me to modify the class for fourth and sixth periods.
In third and fourth periods I choose random problems to do on the board, but in sixth period for the game I just go in order. I'm able to get through ten problems in sixth period, including a problem on slope that many students struggled on (yet I don't cover in the other classes). I end up spending a little extra time on slope and decide not to do the parallel and perpendicular parts of the question, so I can answer more questions instead.
Because of this, there are two questions that I don't reach in sixth yet I covered in the other classes -- two problems on finding angle measure (one parallelogram, one triangle). The problem is that some of the angle measures are fading on the worksheet. We're able to figure out the missing numbers in the other classes but I run out of time before I could tell sixth period.
Last but not least, the finals review packet contains an answer key. For certain questions, I ask the students to set up the problem (give the equation, etc.) rather than just read out the answer in order to be placed on the good list (periods 3-4) or receive a group point (period 6).
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
...nothing, actually. All the givens are in the diagram, with none of the points labeled. So once again, I must label the points myself.
In Circle O, AB,
This is another Inscribed Angle Theorem problem. We have that Angle ABD, the inscribed angle, is half of the intercepted arc AD = 112, and so ABD = 56. Now notice that ABD is the exterior angle of Triangle BDE with remote interior Angles E = 39 and BDE = x. So x + 39 = 56, and x = 17. Therefore the desired angle is 17 degrees -- and of course, today's date is the seventeenth.
We also could have used the Angle-Secant Theorem (half the difference of the arcs) to solve this problem as well. Indeed, it suggests how to prove the Angle-Secant Theorem.
Notice that exterior angles of a triangle appear in Chapter 5 of Glencoe (hence on the final in the class I subbed for today) and Chapter 13 of the U of Chicago text. But circles theorems like the Inscribed Angle Theorem appear in the second half of both texts.
Lesson 9-3 of the U of Chicago text is called "Pyramids and Cones." Our text refers to both pyramids and cones as "conic surfaces."
This is what I wrote last year about today's lesson:
This lesson is very similar to Lesson 9-2 in that the focus is on vocabulary. The difference is that today's lesson is not an activity, but a traditional worksheet.
Indeed, students are asked to calculate slant height in two of the problems on this worksheet. As we already know, this requires the Pythagorean Theorem. A right triangle can be formed with the slant height as the hypotenuse and the altitude height as one leg. The other leg is unnamed in the U of Chicago text, but notice that it's actually the apothem of the regular polygon base.
In fact, we discussed this in previous years -- many Geometry texts define "apothem," but not the U of Chicago text. Indeed, Lesson 8-6 explains the trapezoid area formula as follows:
"There is no known general formula for the area of a polygon even if you know all the lengths of its sides and the measures of its angles. But if a polygon can be split into triangles with altitudes or sides of the same length, then there can be a formula. One kind of polygon that can be split in this way is the trapezoid."
Well, another polygon that can be split in this way is the regular polygon. Indeed, all the altitudes and sides are of the same length, because the triangles are congruent. The common altitude of these triangles is the apothem of the regular polygon. (It has come to my attention that apothems do appear in the modern Third Edition of the text. This is in the new Lesson 8-7 on Special Right Triangles, since 30-60-90 and 45-45-90 triangles are used to find the apothems of equilateral triangles, squares, and regular hexagons.)
But let's get back on track with pyramids and cones. Again, we look at Euclid's definitions:
Notice that Euclid's definition of "pyramid" isn't that much different from the U of Chicago's. But Euclid's cones, like his cylinders, are solids of revolution. Therefore they are limited to right cones and right cylinders. The U of Chicago generalizes this with a definition of cone not unlike the definition of pyramid. This allows cones to be oblique as well as right, and pyramids and cones are examples of conic surfaces (the surfaces of conic solids).
Oh, and I also notice one more difference between the old Second and new Third Editions. The point at the top of the pyramid is called the vertex in the Second Edition and the apex in the Third. Euclid, meanwhile, never names the top point of the pyramid.
Let's look at the next proposition for us to prove:
This is basically the Two Perpendiculars Theorem of Lesson 3-5, except that the two given lines are perpendicular to the same plane, not merely the same line. The line version of Two Perpendiculars appears as the last step of the proof.
Let's modernize Euclid's proof:
Given: Line AB perp. plane P, line CD perp. plane P (B, D in plane P)
Prove: AB | | CD
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Draw DE in plane P such that 2. Point-Line-Plane, part b (Ruler/Protractor Postulates)
DE perp BD, DE = AB
3. AB perp. BD, AB perp. BE, 3. Definition of line perpendicular to plane
CD perp. BD, CD perp. DE
4. BD = BD 4. Reflexive Property of Congruence
5. Triangle ABD = Triangle EDB 5. SAS Congruence Theorem [steps 2,3,4]
6. AD = BE, Angle ABD = EDB 6. CPCTC
7. AE = AE 7, Reflexive Property of Congruence
8. Triangle ABE = Triangle EDA 8. SSS Congruence Theorem [steps 2,6,7]
9. Angle ABE = Angle EDA 9. CPCTC
10. DE perp. DB, DE perp. DA 10. Definition of perpendicular lines [steps 6,9]
11. Lines BD, DA, DC coplanar 11. Proposition 5 from Friday (DE perp. to all, steps 2,10)
12. Lines BD, AB, DC coplanar 12. Point-Line-Plane, part e (points A, B)
13. AB | | CD 13. Two Perpendiculars Theorem [step 2]
Some steps look strange here, such as Step 12. Here, Euclid uses the "proposition" (actually a postulate) that all triangles lie in a plane, so the plane containing two sides of Triangle ABD (namely BD and DA) must contain the third (AB). Here, we use our Point-Line-Plane Postulate part e. This is because Step 11 establishes that a single plane (not plane P, by the way -- we can call it plane Q if we want) contains the lines BD, DA, and DC -- that is, points A and B lie in plane Q. Thus by part e, the entire line AB must also lie in plane Q.
The reason the proof is so convoluted goes back to the original Two Perpendiculars Theorem:
If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other. [emphasis mine]
Notice that key word coplanar. Most of the time we use the Two Perpendiculars Theorem, we take it for granted that the lines in question are coplanar. But in this proof, the bulk of the proof is just to establish that AB and CD are coplanar! After all, the two perpendicular statements (AB perp. BD, CD perp. BD) were established all the way back in Step 2, so we could skip directly from Step 2 to Step 13 if we didn't have to prove that the lines are coplanar.
On the other hand, it is truly necessary to prove that all three lines (the two lines and the transversal) are perpendicular. Otherwise, we could prove, for example, that DB | | DA following Step 10, which is absurd. But since three lines (AB, CD, and the transversal BD) were proved coplanar in Step 12 (they all lie in plane Q), we complete the proof that AB | | CD.
David Joyce points out that Euclid omitted a step here. It could be the case that the two lines meet the plane at the same point (i.e., B and D are the same point). But Euclid proves that this is impossible in a subsequent proposition.
Here are the worksheets:
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