Wednesday, March 22, 2017

Lesson 12-4: Proportions (Day 124)

Lesson 12-4 of the U of Chicago text is on proportions. It's easy to figure out why there would a lesson on proportions in a chapter on dilations and similarity.

But I don't have any worksheet from last year to post. This is because I was always pressed for time whenever we reached this lesson, so I always omitted Lesson 12-4 in order to reach the lessons containing actual geometry.

It isn't too difficult to find proportion worksheets online, of course. A Google search for a proportions worksheet gives the following Kuta page as the fourth result:

Notice that Kuta considers this to be a Pre-Algebra topic, and in fact I did cover proportions earlier in some of my middle school classes. So I could give the above worksheet in my class.

But when I taught proportions to my sixth graders, I didn't teach cross-multiplication. Instead, I used tape diagrams and double number lines in accordance with the following Common Core standard:

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Also notice that if we follow the standards in naive order, as the Illinois State Student Journal does, then it would be awkward to show the sixth graders equations for Standard RP3 before they see equations for the very first time in Standard EE5.

By the way, I don't necessarily have time to get into a big traditionalist debate today, but the traditionalist Barry Garelick, quoting the late Ralph Raimi, has something to say about teaching proportions using methods other than equations:

There are some topics that even the most bleeding-heart progressive will use equations to teach -- for example, no one is going to avoid a^2 + b^2 = c^2 when teaching the Pythagorean Theorem. But proportions is one topic where there is a strong progressive resistance to equations, especially in the early middle school years.

Still, Raimi, like most traditionalists, preferred using equations and wished that his teachers used equations from the very beginning:

What was proportional to what no longer needed
to concern me; the relations dictated by the problem led to airtight equations
about the meaning of which I no longer had to think. It was, to me, a

Of course, for high school Geometry lesson, the students obviously should have seen proportion equations by now, so equations are preferred for this lesson.

Tuesday, March 21, 2017

Lesson 12-3: Properties of Size Changes (Day 123)

This is what I wrote last year about today's lesson. Admittedly it isn't much, but I don't have time to write more. Today my middle school students took a 50-question quiz!

Finally, here are the Geometry worksheets for today. They are based on Lesson 12-3, with an extra page for the proof of the Dilation Distance Theorem -- this proof comes directly from PARCC:

Monday, March 20, 2017

Lesson 12-2: Size Changes Without Coordinates (Day 122)

This is what I wrote last year about today's lesson:

Let's get back to Geometry. Lesson 12-2 of the U of Chicago text is called "Size Changes Without Coordinates" -- and recall that "size changes" are what the text calls dilations.

Aha -- so that's where we've heard that word before. In a way, Lorentz dilations really are similar to Common Core dilations -- Lorentz scales all intervals of time by the same factor (depending on the speed of the object) while Common Core scales all distances by the same factor.

Much of today's lesson doesn't need to change from last year's lesson. But there's one activity about dilations that I found on the MTBoS. [2017 update: I posted this activity last Friday.]

Here's what the author Olivia writes about her activity:

At the beginning of the week, I assigned a dilation project in geometry.  Students were to pick a picture from the internet, draw a grid over top of it, then redraw the picture following the grid on a larger piece of paper.  My district does not have an art program, so many of the students are definitely not comfortable when it comes to art.  I heard a lot of negative comments that day from students saying they sucked at drawing, it was going to turn out horrible, and many pleas of students asking me to “please not hang them up!”  They were especially adamant that they WOULD NOT be putting their names on their pictures.  I told them it would be okay and they would turn out great.  I said that if I could do it, anyone could do it!  Well I gave them 2 full days of class time to work on their posters.  I hung up a couple of the posters after the first day, because two of my students finished theirs by working on it in study hall.  The next day, my other classes were all asking who drew what.  I said, sorry guys this class wanted to remain anonymous.  Well, my geometry class came in later in the day.  I told them all that people kept asking about who drew what but I did not rat anyone out about who drew what.  Then, a very exciting thing happened! Students started saying, “Mine looks so good, I’m definitely putting my name on mine” or “I want everyone to know who drew mine.”  They told me that it wasn’t as terrible as they thought it would be and it was actually fun.  I loved seeing students so excited and proud of their own work. It always makes my day brighter when students realize just what they can accomplish.  Everyone ended up putting their names on their finished products, and I was left as one happy math teacher.

Well, there's nothing stopping us from assigning this activity today. I obtained the pictures simply by performing a Google image search for "cartoon character" -- those just happened to be the ones that came up.

There's one thing about activity -- it works better on a coordinate plane. But note that Lesson 12-2 of the U of Chicago text uses Slope and Distance Formulas just to prove that the mapping from (xy) to (kxky) is a dilation with scale factor k.

Friday, March 17, 2017

Lesson 12-1: Size Changes on a Coordinate Plane (Day 121)

Here is today's Pappas question of the day:

How many innings did Team Mexico pitch in this year's World Baseball Classic?

The answer is 17 -- and of course, today's date is the seventeenth.

Yes, I plan to address the controversy regarding the World Baseball Classic in this post -- after all, Mexico was eliminated from the tourney due to a mathematical formula, and this is a math blog. But first, I wish to write about the status of my blog.

It is still the Big March, and times are so tough that I've lost the will to write about my class. And so I'm hereby dropping out of the MTBoS, for now. In particular, I am no longer a part of Tina Cardone's "Day in the Life" challenge -- yet another MTBoS challenge I fail to complete. Even though tomorrow is the 18th, my monthly posting date, I won't be submitting any post.

By the way, many of the other "Day in the Life" posters have already dropped out (and that ironically includes Cardone herself). The few remaining participants only make one post a month, so of course it's much easier for them. I thought that I could handle three posts a week during the school year, but in the end I couldn't.

My plan now is for this blog to return to its roots. The blog name is "Common Core Geometry," and so I'm returning to posting Geometry lessons from the U of Chicago text. Before I was hired as a teacher, my plan had been to convert the day count into the lesson number to cover -- so since today is Day 121 on the blog calendar, we'll start with Lesson 12-1. The good thing about this is that I can cut and paste most of these lessons from the first two years of this blog, so it doesn't require as much effort on my part. Ironically, Lesson 12-1 is one of the few lessons I didn't post on the blog either last year or the year before -- more on that later.

Before we begin though, let me wrap up a few loose ends. I will still write about my class from time to time on the blog. In particular, you may be wondering about Pi Day. Well, I'd wanted to surprise my students by getting them a pizza on 3/14 at 1:59, as is traditional since pi is 3.14159.... The problem is that, as you can tell by the day count, we're at the start of the third trimester. Therefore, this week is the second Parent Conferences Week, with students dismissed at 1:15 everyday. As much as students like pizza, they wouldn't want to stay 44 minutes after dismissal to get it.

So instead, I served pizza to the latest class that met on Pi Day. With the schedule mixed up to conferences, that class turned out to be sixth grade. (See my November 17th post, another Parent Conferences day, to see how the schedule works on days like this.) I would have preferred seventh grade to get the pizza since that's the grade when they learn about pi, but oh well.

I've written the relationship between Pi Day and the ends of trimesters/quarters before (for example, in my November 9th post). At schools that start after Labor Day, the end of the second trimester often falls close to Pi Day. But at Early Start schools such as mine, the second trimester ought to end at least a week before Pi Day -- in fact, it's the third quarter, not the second trimester, that should end closer to the math holiday. But with my school's extra holidays (most notably the longer winter break), the second trimester is extended towards Pi Day. So with Parent Conferences, the school will usually be dismissed before 1:59 on Pi Day.

Okay, let's get to that World Baseball Classic. Its structure is similar to that of the World Cup, the international soccer tournament. All the teams are divided into groups of four teams, and in the first round, each team plays each of the other three teams in the group. This is called round robin. The top two teams in each group advance to the second round.

As it turns out, there are four possible outcomes for the standings after all the games in a group have been played. The most straightforward is for the four records to be 3-0, 2-1, 1-2, and 0-3. Then the two teams that advance are obviously the 3-0 and 2-1 teams. The second possibility is for there to be two 2-1 teams and two 1-2 teams. This is also easy -- the two 2-1 teams advance.

In this year's classic, three of the four groups followed the 3-0, 2-1, 1-2, 0-3 patterns, so there were clear first and second place finishers in those groups. But the two remaining possibilities involve three-way ties -- either three 2-1 teams and one winless team, or one undefeated team and three teams with 1-2 records.

This last case occurred with the group containing Italy, Mexico, Puerto Rico, and Venezuela. As it happened, Puerto Rico was the 3-0 team, so they obviously advanced. The problem is how to determine the second team to advance. The rules state that there should be a tiebreaker game -- but only two teams can play in that game. So the question is, which two teams among Italy, Mexico, and Venezuela should play in the tiebreaker?

It seems fair that the tiebreaker should be determined by runs -- since after all, the objective of baseball is to score more runs than the other team. The Classic actually uses a defensive tiebreaker rule -- it's the two teams that allowed the fewest number of runs. Since Puerto Rico isn't involved in the tiebreaker, only the runs Italy, Mexico, and Venezuela allow to the other two tied teams count. As it turns out, Mexico allowed 19 runs, Italy 20 runs, and Venezuela 21 runs in those games. Yet Mexico wasn't allowed to play in the tiebreaker game.

That's because there's one more component to the tiebreaker rule -- innings. It's not which team gives up the fewest runs, but the fewest runs per inning pitched. In a way, this makes sense in light of the existence of mercy rules -- if a team is leading by ten runs after seven innings, the game ends. So a team that loses 10-0 after seven is penalized more than a team that loses 10-0 after nine. The innings rule also rewards teams that play a scoreless tie through nine and lose 1-0 in extra innings.

The innings rule ends up hurting Mexico. This is because Italy and Venezuela played a 10-inning game, so each team is credited with 19 innings (ten against each other and nine against Mexico.) But Mexico is only credited with 17 innings. Their game against Italy is only considered to last eight innings, because the Italians won it on a walk-off in the ninth with nobody out. Had there been one out, it would be considered 8 1/3 innings, and with two outs, it would be 8 2/3 innings.

And so we now calculate the defensive runs per inning:

Italy: 21/19 = 1.05 runs per inning
Venezuela: 20/19 = 1.11 runs per inning
Mexico: 19/17 = 1.12 runs per inning

Therefore Venezuela and Italy played in the tiebreaker, and Mexico was left out.

Here's the question -- suppose you are on Team Mexico, and you are about to play Venezuela in the final game of the group. You already know the scores of the previous five games, so you know that Puerto Rico is 3-0, and already have two losses, so your only hope is the tiebreaker. So what exactly must you do in the game against Venezuela (besides beat them of course) in order to play in the tiebreaker game?

This is an interesting algebra problem. We already know the scores of the other games, so in particular Italy beat Mexico 10-9 (with none out in ninth), and Venezuela beat Italy 11-10. So let's set up some variables -- in the final game, Mexico scores m runs and Venezuela scores v runs. We now calculate the number of runs per inning:

Italy: 20/19 = 1.05 runs per inning
Venezuela: (m + 10)/19 runs per inning
Mexico: (v + 10)/17 runs per inning

Since two teams advance to the tiebreaker, Mexico only needs to have a better average than just one of the other teams, either Italy or Venezuela. Let's look at Italy first. In order for Mexico to play on, we need Mexico's run-to-inning ratio to be less than Italy's. So we write an inequality:

(v + 10)/17 < 20/19

If this were an equation, an Algebra I student could solve this by cross-multiplying. This is awkward when it's an inequality, though. So instead, we can clear fractions by multiplying by 17(19):

19(v + 10) < 17(20)
19v + 190 < 340
19v < 150
v < 7.89

Notice that we don't get to round 7.89 up to eight. This is an inequality that says that v must be less than 7.89, and 8 is not less than 7.89. So the inequality states that if Mexico beats Venezuela and holds them to seven or fewer runs, Mexico is in the tiebreaker game.

As soon as Venezuela scores its eighth run, the inequality no longer holds. Mexico can still advance, though, if they beat Venezuela by a large enough margin. To figure this out, we set up another inequality, this time to show that Venezuela's ratio is greater than Mexico's:

(m + 10)/19 > (v + 10)/17
17(m + 10) > 19(v + 10)
17m + 170 > 19v + 190
17m > 19v + 20
m > 1.12v + 1.18

This inequality is tricky because the coefficient of v isn't 1 -- it's not as simple as say m > v + 1 (which would state merely that Mexico would have to win by two). Since this inequality is irrelevant when v is less than 8 (as Mexico would already be in the tiebreaker game with a victory), we start out by plugging in v = 8:

m > 1.12(8) + 1.18
m > 10.14

Again m must be greater than 10.14, so Mexico needs at least 11 runs. In other words, if Venezuela scores 8, Mexico must win by three runs, not two. If we plug in v = 9, we obtain:

m > 1.12(9) + 1.18
m > 11.26

So again Mexico would need a three-run victory. Plugging in 10, 11, and so on for v continues to show that Mexico needs to be up by three. Once we reach 16, it changes:

m > 1.12(16) + 1.18
m > 19.1

Then Mexico would need to win by four -- but this is baseball. Scores like 20-16 are uncommon, so for all intents and purposes, only a three-run victory is needed. We summarize this as follows:

In order to advance to the tiebreaker game, Mexico must beat Venezuela and:
-- If Venezuela scores seven runs or fewer, then any margin of victory is sufficient.
-- If Venezuela scores between 8 and 15 runs, then Mexico must win by at least three runs.

All of this presupposes a nine-inning game, by the way. If the game extends into extra innings, Mexico benefits because they can put up zeros, but then Venezuela benefits as well. Then again, an extra inning would help Mexico have a better ratio than Italy at least -- for example, if the game lasts ten innings, then we can replace "seven" in the above with "eight," so that a 9-8 victory in ten innings (but not nine) is enough to make the tiebreaker.

In the actual game, Mexico held an 11-6 lead at the seventh inning stretch. But then Venezuela scored three in the bottom of the seventh, When the eighth run crossed the plate, we moved from the first case above to the second, but Mexico was still in a good position with a three-run lead. As soon as the ninth run scored, Mexico needed to score another run to advance.

In the bottom of the ninth, team captain Adrian Gonzalez believed that a two-run victory was sufficient to advance. He stated that if Venezuela scored a run in the ninth, the team would have intentionally walked or balked in another run to tie the score, then hope to win by two in extras. I wonder what Gonzalez would have said if he knew that the margin had to be three -- walking and balking in two runs makes the game into a farce -- and it's hard to win by three in extras anyway. (In the 11th inning Mexico would start with two runners on base, so a homer would be enough for that three-run lead, but then Venezuela would start the bottom half with runners on base too.) The best way for Mexico to advance was not to give up more than seven runs in the first place.

All of this reminds me of another sport that has a round robin format -- Quidditch. In J.K. Rowling's Harry Potter series, the four teams at Hogwarts play a round robin tournament. In three of the books, Harry's Gryffindor team wins the Quidditch Cup -- but oddly, it's never as simple as just going 3-0 in the three matches. Instead, Gryffindor is one of two 2-1 teams (with the lone loss always to, of all teams, Hufflepuff), and a tiebreaker of total points in the three matches is used.

A few Potter fans have criticized this tiebreaker system, and have written fan fiction in which teams take advantage of the system. Here is such a story:

The author, Crys, has Gryffindor and Hufflepuff agree before the game that each team should let the other score over a hundred unguarded goals in order to win the points tiebreaker. In the end, the two teams that pull off this stunt are tied in the standings -- and rather than have a second tiebreaker, the teams are declared co-champions.

I can imagine something similar happening in the WBC -- except that since the tiebreaker is based on defensive runs per inning, teams must rack up outs, not runs. Say instead of  10-9, 11-10, and 11-9 (which are high scoring by baseball standards), we change these to the lowest possible scores, 1-0:

Italy beats Mexico 1-0 (none out in ninth)
Venezuela beats Italy 1-0 (but in nine innings, not ten)

Also, let's assume that instead of losing to Puerto Rico, all three teams beat the island team. Now if Mexico beats Venezuela, there are three 2-1 teams and Puerto Rico at 0-3. In this case, the rules state that the first place team (as determined by the same run-to-inning ratio) advances to the next round outright, while the other two teams play the tiebreaker game. So the goal now is to avoid the tiebreaker game and advance directly to the second round.

So what must Mexico do now? If they lose they are eliminated, but if they win, they could even advance to the second round outright! We calculate as follows:

Italy: 1/18 = 0.06 runs per inning
Venezuela: m/18 runs per inning
Mexico: (v + 1)/17 runs per inning

In order for Mexico to advance outright, it must have a better ratio than Italy. So we have:

(v + 1)/17 < 1/18
18(v + 1) < 17
18v + 18 < 17
18v < -1
v < -1/18

Oops -- Mexico can't give up a negative score!That's This means that it's impossible for Mexico to avoid the tiebreaker game -- unless the game goes extras. It's easy to see that if Mexico pitches 10 shutout innings, both teams will have given up one run in 18 innings, and if Mexico pitches 11 shutout innings, it will have a better ratio than Italy. A 1-0 victory over Venezuela in 11 innings will give Mexico the best ratio of the three tied teams and allows them to avoid the tiebreaker.

So Mexico comes up with the following plan -- for the first nine innings, both teams simply agree not to swing at the ball. The pitchers on both teams make every pitch directly into the strike zone, and so every single batter strikes out. After both teams put up nine innings of zeros, they agree to play the game straight in the tenth. As you can see, this is the World Baseball Classic equivalent of the Quidditch match in the above link -- instead of unguarded goals, we have uncontested outs.

But why, you may ask, would Venezuela agree to this charade? Notice that Venezuela at this point is already 2-0 and is guaranteed at least a tiebreaker -- so all it has to play for is avoidance of the tiebreaker to advance outright.  To do so, Venezuela needs to have a better ratio than Italy -- and here comes Mexico agreeing to put up nine zeros to improve Venezuela's ratio! And so Venezuela agrees to the farce!

There's one caveat, though. Venezuela has no incentive to strike out in the bottom of the ninth. At this point Venezuela can just swing for the fences and a 1-0 walk-off win -- this would put them in the second round outright, as their record would be 3-0. The agreement then would be to play strikeout ball for 8 1/2 innings and then play real baseball from the bottom of the ninth on. Mexico would be at a disadvantage since Venezuela has one more active at-bat -- it may be safer just to play real baseball all along and go for the tiebreaker game.

But just imagine what would happen if such a game were to be played. The fans in the stands would be bored watching every player take three straight pitches -- but stats geeks would love a game in which one team pitches a no-hitter (indeed a perfect game) for nine innings while the other team takes a perfecto into the ninth. And Italy would be furious -- they think they're guaranteed a spot in the second round with their 0.06 run ratio, and suddenly they're forced to play a tiebreaker because two teams choose not to swing at the ball. At the very least, it would force the WBC to consider changing the tiebreaker rule to prevent such a stunt from being played again.

Well, as long as the tiebreaker rule exists, we have some interesting Algebra I problems. Who would think that we could ever see (v + 1)/17 < 1/18 (or even 18(v + 1) < 17) in a real-life scenario? This question may be a bit tricky for my eighth graders though, as I never did cover inequalities at all, or even equations which require clearing fractions.

That's funny -- I said that today I want to return to Geometry, but instead I end up writing much more about an Algebra I problem. Well, let's start the Geometry right away.

Lesson 12-1 of the U of Chicago text is on Size Changes on a Coordinate Plane. If we remember from last year, "size change" is what the U of Chicago calls a "dilation." This is a good place to make my grand return to the U of Chicago, since I like emphasizing the Common Core transformations, which include dilations.

In the past, I skipped over Lesson 12-1. This is because I was mainly concerned with circularity -- dilations are used to prove some of the properties of coordinates, but right in this lesson, coordinates are used to prove the properties of dilations.

But last year, I was fed up with juggling the order of the U of Chicago text (and this year, I got in trouble trying to juggle the Illinois State text as well). This year I want to stick to the order as intended by the authors of the U of Chicago text. And furthermore, we've seen that the actual dilation problems on the PARCC and SBAC involve performing dilations on a coordinate plane -- not using dilations to prove properties of coordinates! So Lesson 12-1 is more in line with PARCC and SBAC.

Here is the main theorem of Lesson 12-1 along with its coordinate proof:

Let S_k be the transformation mapping (x, y) onto (kx, ky).
Let P' = S_k(P) and Q' = S_k(Q). Then
(1) Line P'Q' | | line PQ, and
(2) P'Q' = k * PQ.

Let P = (a, b) and Q = (c, d) be the preimages.
Then P' = (ka, kb) and Q' = (kc, kd).

(1) Line P'Q' is parallel to line PQ if the slopes are the same.
slope of line P'Q' = (kd - kb) / (kc - ka) = k(d - b) / k(c - a) = (d - b) / (c - a)
slope of line PQ = (d - b) / (c - a)
Thus line PQ | | line P'Q'.

(2) The goal is to show that P'Q' = k * PQ.
From the Distance Formula,
PQ = sqrt((c - a)^2 + (d - b)^2).
Also from the Distance Formula,
P'Q' = sqrt((kc - ka)^2 + (kd - kb)^2)
        = sqrt((k(c - a))^2 + (k(d - b))^2)      (Distributive Property)
        = sqrt(k^2(c - a)^2 + k^2(d - b)^2)    (Power of a Product)
        = sqrt(k^2((c - a)^2 + (d - b)^2))       (Distributive Property)
        = sqrt(k^2)sqrt((c - a)^2 + (d - b)^2) (Square Root of a Product)
        = ksqrt((c - a)^2 + (d - b)^2)              (Since k > 0, sqrt(k^2) = k)
        = k * PQ                                             (Substitution) QED

At the end of this post, it's back to posting worksheets based on the U of Chicago text. This time, I post an activity from last year where students dilate cartoon characters. This activity makes more sense this year than last year since it requires using coordinates.

This lesson could actually help my eighth graders as well. We were supposed to cover dilations earlier but we ran out of time. For that matter, slope and the Distance Formula are also part of the eighth grade curriculum. I wouldn't make eighth graders perform the two preceding proofs with so many variables, but specific numerical examples are within the reach of eighth graders.

Next week is Days 122-126, so I will post Lessons 12-2 through 12-6 then. Fortunately, most of these will be cut-and-paste worksheets from last year -- and I definitely need a break next week! This is because I actually have tickets to the three games of the World Baseball Classic that are being played right here in Los Angeles, at Dodger Stadium. So cutting and pasting is all I'll have time to do!

Tuesday, February 28, 2017

Big March Hiatus (Days 109-120)

The pressures of the Big March can get the best of us -- how much more easily, then, can a first-year like me succumb to the pressures of the Big March?

The pressure on me right now is so great that I will not be posting to the blog at all in March. I hope that the next post here will be in April.

Monday, February 27, 2017

Science Week (Days 107-108)

Here is today's Pappas question of the day:

A = {perfect cubes < 32}, B = {multiples of 3}, A intersect B = ?

See that A is the finite set {1, 8, 27} and B is the infinite set {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...} -- there's no point in listing elements of B greater than 32 since they can't possibly be in A. So the intersection of these sets is {27} -- and of course, today's date is the 27th.

Even though the intersection of two arbitrary sets can be very large (even infinite), the intersection of two sets in a Pappas question is always a singleton (a one-element set) -- since that lone element must be the date. Thus Pappas questions almost never ask for the union of two sets, only the intersection.

Some readers might point out that the intersection isn't actually {27} but {27, 0, -27, -216, -729, ...}, since both cubes (but not squares) and multiples of three can be zero or negative. Usually, in a Pappas problem, it's assumed that we're talking about natural numbers, since only these can be the date.

It might be possible to give this problem in a middle school classroom. Set theory isn't usually taught in most schools these days, except for special situations (such as N(S) from the U of Chicago Geometry text from last year). It might be possible to ask a middle school student this question:

Name a natural number less than 32 that is both a perfect cube and a multiple of 3.

But I won't ask my students anything of the sort this week -- the second week of the Big March. In many ways, even though I consider the Big March to begin the Tuesday after President's Day, at least that's a four-day week. We truly feel the grind of the Big March today, the Monday after Prez Day, since it's the first of several five-day weeks.

The idea that the Big March doesn't really start until today is definitely true in school districts that have something called a "February break." I've already mentioned how some districts combine Lincoln's Birthday with President's Day to form a four-day weekend. Well, some other districts end up taking the entire week of President's Day off instead.

This idea of a "February break" or "President's Day vacation" isn't common in California. The only time I've seen it here back when there were furlough days due to budget cuts. Recall that when furlough days were common a few years ago, they were immediately inserted on the Monday, Tuesday, and Wednesday before Thanksgiving. (These days proved to be so popular that now we regularly have Thanksgiving week off even after we stopped needing furlough days.) Anyway, in the old year-round schools at LAUSD (see my June 19th post for more info), some tracks were already off all of November. So these tracks inserted the furlough days after President's Day instead. Other than that, I've never heard of a California school observing a week off in February.

As it turns out, one of the districts with a full week off just happens to be New York. There are so many New York teachers writing for "Day in the Life," and so I keep reading about the February break in their posts. Since New York schools have such a short winter break, President's Day vacation is greeted with open arms. For them, today marks the start of the Big March, with no more days off until spring break. (Wow, from "Regents" to "February break," I'm learning more about Big Apple schools than ever I wanted to this year!)

Okay, back to what I was saying about this week in my classroom. I'm not going to give any math questions like the Pappas question this because I'm not teaching much math this week. I warned you about this in my last post, and near the start of eighth grade block, I received the email message that makes it official -- I must indeed establish separate math and science grades this trimester. And so I declare this week to be Science Week, and I'll have to teach mostly science this week. Notice that the announcement about science doesn't affect today's lesson, as it's just coding Monday anyway.

Moreover, near the end of eighth grade block I received another tough message -- this one about the visit from Illinois State coming up on Wednesday. I was wondering whether I'd have a hour-long observation of one of my classes that day. Well, actually I'll be giving an hour-long presentation on how I plan to teach the Illinois State curriculum the rest of the year. That's going to be hard -- of course I'd rather have the hour-long observation, since it's not as if I wouldn't have to teach during that hour anyway. Now I have to come up with things to say for a full hour -- mostly likely a detailed lesson plan of what I'll be doing, week by week, until the SBAC. Yes, from the sudden need to plan science lessons to the presentation, I'm definitely feeling the pain of the Big March this week!

First things first -- since this is a two-day post, I might as well write about Science Week. Notice that this actually isn't my first Science Week of the year. The week between Rosh Hashanah and Yom Kippur, back in October, was my first Science Week. I used the High Holidays as an excuse to tell the students about lunar calendars and hence the motions of the earth, moon, and sun. But this time, I want to make sure that the students are learning the correct material for their respective grade levels.

Exactly one month ago, I wrote about how I should have taught science this year:

Of course, I now know what I should have done about science. I already mentioned how I'm fond of three-week cycles -- well, I should have used the original Study Island period by cycling among Foldable notes, a project, and a quiz on Study Island. The science period should occur every week, even after music tinkered with the schedule. This would have led quite nicely to the current test prep block for science, where I could continue the same three-week cycle. Avoid all pre-tests until the students are confident that I'll actually teach the material.

But what should have I done about the whole state standards/NGSS mess? (Right now, I try to explain NGSS to the students, but they don't listen. Based on my bad lessons, they think that I know nothing about science, so why should I know what NGSS is?) Perhaps I could have taught eighth grade physical science and seventh grade life science, but NGSS for sixth grade, as the transition to NGSS will be complete by the time they reach eighth grade. For life science, I don't feel comfortable with an animal dissection lab, but there are some microscopes in the room, so I could have used these in a lesson.

So this is exactly what I want to do this week. I want to give a physical science lesson to eighth grade, a life science lesson to seventh, and an NGSS wild card lesson (which could be on any of the three strands -- earth, life, or physical) to sixth grade. To determine which lesson to teach, I look at the list of standards under Study Island. I use Study Island as a guide because this is the website that purports to give the California version of the Next Generation Science Standards.

Let's start with sixth grade first. The first topic (well, the second topic -- the first topic is called "Pretest," which I'm skipping as per above) is "Structure, Function, and Information Processing." It consists of two subtopics -- "Cells" and "Living Systems." So it's clearly a life science lesson.

Now I can't actually use the three-week cycle that I mentioned last month, but I can make it into a three-day plan, as follows:

Tuesday: Foldable notes
Thursday: Science project
Friday: Quiz

On Wednesday, I'll teach the math topic that I originally scheduled for this week. After all, this is my Student Journal day.

So what sort of project should I give my sixth graders on Thursday? Well, we know that the only way to see cells is with a microscope, and I mentioned last month that we indeed have microscopes.

But where exactly am I going to get a cell to show the students? Well, cells are easy to find, as every single organism is made up of cells, but that doesn't tell me how to get a cell slide that my students can put under the microscope. This is the problem I have with teaching science -- I don't know where exactly science teachers get things. All I remember from my own days as a science students is that the teacher gave us the slides and we put them under the microscope. Of course I made no thought as to where the slides come from.

If I can't get the slides, the project could be to made a model of a cell. This doesn't necessarily mean that it will be the Edible Cell Model that I mentioned in that same post from a month ago.

In seventh grade, the second topic (after the first topic pretest) is "Matter and Energy in Otganisms and Ecosystems," with "Energy in Ecosystems" as its first subtopic. I want to focus on life science with my seventh graders, and it's clearly a seventh grade topic.

Here is the plan for seventh grade:

Tuesday: Foldable notes
Friday: Quiz

Recall that seventh graders don't meet on Wednesdays, so Thursday is the day that I must reserve for the math Student Journal. There's no time for a science project this week, but Energy in Ecosystems isn't a topic that naturally lends itself to a project anyway, except for something like cutting out pictures of different organisms and use them to create a food web. But this is more easily drawn in the Foldable rather than cut out.

Notice that so far, I have yet to mention either Green Team or Bruin Corps. Remember what I wrote in my post from a month ago -- one reason I've dug such a hole with science is that I was too dependent on Green Team and Bruin Corps to provide science lessons.

In particular, there's no sign of the Green Team project starting soon. And it might be possible for me to ask the Bruin Corps member who comes on Thursdays -- a biology major -- to provide me with some cell slides. But I won't, because I wish to avoid being dependent on a college student to provide me with science lessons.

We now move on to the eighth grade lesson. The second topic is "Genetics & Biotechnology" -- but I actually covered this back in November and December, back when I was still asking my Bruin Corps member to help me with the lessons. The third topic is "Natural Selection and Adaptations" -- but this is a life science topic, and I prefer sticking with physical science in eighth grade. (Of course, genetics is also life science, but at the time I was still milking lessons from my Bruin Corps member.)

There's a good reason to teach the fourth topic this week, "Space Systems." Not only is it a physical science lesson, but in a way I've already started this topic. First of all, the "Earth, Sun, and Moon" lesson from October is the second subtopic. The first is "The Universe" -- and I begin this subtopic today, some of the Monday Five questions I give today are on the solar system! The workbook from which I get the Monday Five often gives themed questions during the week, and the theme this week happens to be outer space. There are questions on the length of Jupiter's and Saturn's year, the distance of Earth and Neptune from the sun, and shooting starts.

By the way, the seventh grade Monday Five is also themed. This week's theme is spiders -- and hey, I can fit spiders into this week's lesson. Here's an example of a food chain that contains spiders:

Producer: Pea Plant
Primary Consumer: Aphid
Secondary Consumer: Beetle
Tertiary Consumer: Spider
Quaternary Consumer: Bird

On Friday in all classes will be a quiz. I've already scheduled a 50-point quiz for this Friday, and I might as well keep it, but make it into a science quiz. After all, the whole point of Science Week is to establish a science grade. So let's take a look at how my grading should work.

Recall that our online grading program automatically weights the assignments by categories:

40% Formal Assessment and Projects
20% Quizzes
15% Participation
15% Homework
10% Classwork

First of all, the points are weighted so that they fit these categories. So if I were to give 100 points in each category, they are weighted so that each Formal Assessment point is worth four times as much as a Classwork point. I think this is deceptive to the students, and so I circumvent this weighting by giving 400 Formal Assessment points, 200 Quiz points, 150 each for Participation and Homework, and finally 100 Classwork points, for a total of 1000 points in math.

Second, even with the points weighted properly, many students still think that 1% is one point -- so students with a grade of 79% say that they are "one point away from a B." In reality, 79% can represent a score anywhere from 790 to 799 out of 1000, so they may be anywhere from one to ten points away from a B.

Third, notice that I won't have time to give all five components during science week. Notice that if I were to give that one quiz and no other grade, that quiz would comprise 100% of the grade, not 20%, since four categories are missing. If I were to give only that quiz and Classwork, the quiz would make up 2/3 of the grade and the classwork 1/3, since 20% is double 10%. On the other hand, if I gave only that quiz and a project, then the quiz would be only 1/3 of the grade and the project 2/3, since 40% is double 20%.

All of this seems to present an appealing solution. Since there's only one week of science, it's reasonable to let the total number of points be 100 -- then one point really is 1%. There's already going to be a 50-point quiz, and Quizzes are 20%. So all we have to do is find another component that's also worth 20% and assign 50 points to that category. Then each of the 50-point assignments is half the grade, as desired. The problem, of course, is that there is no other category worth exactly 20%, nor are there two categories that add up to 20%.

The easiest way to make the math work out is to include Quizzes, Participation, and Homework. We see that these categories add up to 20% + 15% + 15% = 50%, so the percents end up doubling to 40, 30, and 30 respectively. So I could make the quiz worth 40 points instead of 50, and let the participation and homework be worth 30 points each.

But this is also problematic. With only one week available, there would most likely be only time for one homework assignment worth 30 points. If a student fails to turn it in, that student must be perfect on everything else just to get the lowest possible C -- so if that student then gets just one problem wrong on the quiz, the grade earned is F (since there are no D grades). Notice that there will be parent conferences at the end of the the trimester, where the report cards are given. And this isn't going to play well at parent conferences -- a student gets one question wrong on the quiz (which is still an A on the quiz) and a perfect score in participation (another A), yet due to not turning in one homework assignment, the grade that appears on the report card is F.

This is why I'm suddenly so obsessed with science grades -- I'm thinking about the parents. Any parent in that situation would cuss me out over giving an F for missing one homework assignment, then cuss me out for not teaching science until just before grades are due and then basing the grade on just one week of work. And this is with some parents already upset with me over the math homework that I'm required by Illinois State to give online, even though not everyone has Internet access!

So I'm still trying to figure things out. Perhaps I should use the 40-30-30 grading, but counting the Foldable as "homework." That way the points are weighted properly, yet since the Foldable is done in class, students who don't do the homework aren't severely penalized. (Or I could give the homework but use the controversial 0=50% grading system for that assignment only, so the student who misses the assignment but excels on everything else earns a B rather than an F. Or I could give three homework assignments worth 10 points each, but that means I have to come up with three different assignments on the fly.)

Well, let's see how my science lessons go tomorrow. This is actually an exciting time to teach science, as scientists have made two major discoveries in recent weeks:

-- Astronomers have discovered seven new exoplanets -- that is, planets outside our solar system. It's possible that at least one of these planets is Earthlike, or habitable.
-- Here on Earth, geologists have discovered two new continents.

One of these new continents is Zealandia -- named after the only part  of it that's above sea level -- New Zealand. But I'm actually more interested in the other new continent -- Mauritia.

Recall from some of my 2014 and 2015 spherical geometry posts that the antipodes -- or point directly opposite -- of California is in the Indian Ocean. I think it would be fun if the antipodes of California turned out to be a continent after all -- and it's Mauritia that's nearer the antipodes rather than Zealandia.

But since my coordinates are approximately Latitude 34N, 118W, the antipodes would have to be near Latitude 34S. Mauritius, for which Mauritia is named, lies at around Latitude 20S, and all maps of Mauritia I've seen so far have the new continent extend northward from Mauritius towards the Equator, and not southward towards 34S. So unfortunately, there appears to be no new continent at my antipodes after all -- oh well.

This is a two-day post. My next post will be Wednesday, March 1st -- and I'm glad that tomorrow's a no-post day, since I need the extra time to create that one-hour presentation for Illinois State!

Friday, February 24, 2017

Test #6 (Day 106)

Here is today's Pappas question of the day:

A person is jogging around a 1/4-mile track at 6 mph. If they jogged for an hour, how many times did they go around the track?

Well, if the speed is 6 mph, then in one hour six miles would have been jogged. Since there are four laps to a mile, six miles is 24 laps. So the answer is 24 -- and of course, today's date is the 24th. This is the sort of problem I could give in either my sixth or seventh grade class.

Today is the day of a major test. The eighth graders tested on statistics and substitution, the seventh graders on integer operations and equivalent expressions, and the sixth graders on percents and the simplest equations.

The eighth grade test is fairly easy. This is mainly because the 100-point test structure covers the last two standards, with more emphasis on the penultimate standard as they've been exposed to it for a much longer time. And that standard was statistics -- obviously it's easier to determine whether a graph shows positive or negative correlation that it is to solve a system of equations. Next week's general quiz, worth 50 points, should be completely on solving systems.

By the way, I plan on introducing the elimination method next week. The next standard, 8,EE 8c, is on applying systems of equations. Just as this week's homework was set up for substitution, next week's is set up for elimination -- the questions are set up as "The sum of two numbers is ... and their difference is ... What are the two numbers?" A system with x + y = something as one of the equations and x - y = something as the other is just begging to be solved by elimination.

Meanwhile, the seventh graders struggled the most on the test. Integer operations form the bulk of their test, and with this class not meeting on Wednesdays, there's less practice time. Since there aren't many topics that are much more important than negative numbers, I may consider having most of my upcoming Warm-Ups be on negatives, even as we move on to other standards. This extends my predecessor's idea of having more eighth grade systems practice well beyond the current unit.

I've said it before and I'll say it again -- I call the current lesson "integer operations," but notice that the word integer doesn't appear in the Common Core Standards. Instead, the current standards are technically about rational numbers. The field Q of rational numbers includes fractions and their negatives -- since the students already learned about fraction operations in fifth and sixth grades, learning about signed numbers extends their knowledge to the entire rational field.

More sixth graders pass the test than seventh graders. The percent problems are tricky, of course, especially when they must rewrite the percents as fractions -- but at least they don't have to add (or subtract) any of the fractions, which is always harder than multiplying them.

The simple equations section is a little tricky only because inequalities are included. Many students are tricked when the variables are on the right side of an inequality -- we must remember that in an inequality like 4 < x, x is actually greater than 4. Students should either plug in various values such as x = 5 (so 4 < 5 is true), or just reverse the inequality as x > 4.

Today's song for music break is all about solving equations, as all three grades are working on them:


When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
A letter alone on the left side,
A number alone on the right side.
That's all you have to do,
To solve it!

Alternate 7th grade verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Whatever you do to the left side,
The same done to the right side.
That's all you have to do,
To solve it!

Alternate 8th grade verse:
When you see an equation,
Or problems that involve it,
All you have to do,
Is solve it!
Move the variables to the left side,
Move the numbers to the right side.
That's all you have to do,
To solve it!

There are a few other things going on today. First of all, my grading software suddenly changed all of my classes. For some strange reason, instead of the STEM classes that I've had since the first day of school, I'm now considered to have separate math and science classes! This change applies to sixth, seventh, and eighth grades. And all the grades that I've entered so far this trimester suddenly appear under the science classes, even though they're for the most part math grades!

I'm still trying to figure out what this means for my grades. It could mean that now I'm required to give separate grades in math and science. Changing all of those science grades to math could be a major inconvenience -- but more importantly is the fact that I haven't done much science, especially in sixth and seventh grades.

I'm still waiting to hear from the administrators regarding this issue. If it turns out that I need to establish science grades, then I'll declare next week to be a science week. All lessons next week will be on science, except for the usual math Student Journal day (Wednesday for sixth and eighth grades, Thursday for seventh grade). In particular, the 50-point general quiz scheduled for next Friday will become a science quiz.

So what sort of science would I teach each grade next week? Well, I'll cross that bridge when I get to it -- notice that Monday is going to be coding no matter what, so hopefully I'll find out whether I must establish a science grade by Monday night. I'm not counting on the Green Team curriculum being ready by next week, but it would provide a great project for the third trimester science grade.

What I don't want, though, is for me not to find out I need to give a science grade until the day that grades are due. This would result in a epic embarrassment -- I have to give a science grade yet didn't teach much science, and the students and parents would see my big science failure on the report card!

Second, it turns out that this upcoming Wednesday is the day that the curriculum developers from England will be coming in town (and not earlier as I thought). I'm assuming that this is part of the Common Planning meeting, but sometimes they come in to observe as well. In addition to the science situation, such an observation would also affect my lesson plans for next week. Recall that the Illinois State text also has a science component, so I could do a science assignment from the text that day.

Meanwhile, during SBAC Prep time for sixth and seventh grades, I was actually able to administer an online practice test. This was not without problems though. It takes a long time to make sure that each student has the correct ID number, and the Wi-Fi connection required for the students to access the site fails for 10-15 minutes during the sixth grade period. But at least the students can begin to familiarize themselves with the SBAC platform.

By the way, here's something interesting about the SBAC platform that I found out -- the official online calculator for the California version of the SBAC is Desmos. You might remember Desmos from the past MTBoS challenges -- the topic was My Favorite Lesson, and teachers kept mentioning Desmos in their lessons. I didn't have access to Desmos, and so I never used it in my lessons (especially not as a substitute teacher).

Well, I may be using Desmos sooner than I thought, now that it's a part of the SBAC. There have been some eighth grade Illinois State STEM projects that require the use of graphing calculators -- and my classroom has only one graphing calculator, my own. So maybe I could try using Desmos on the laptops the next time there's a graphing calculator project -- and then I justify this to the administrators (who may wonder why I'm using software other than Illinois State, IXL, or Study Island) by informing them that Desmos will help the students prepare for SBAC. I'm not sure whether I want to create Desmos accounts for the students, but it appears that students could log on to Desmos using a Google account -- and the coding teacher has already provided them with Google accounts.

Speaking of the MTBoS, here's a link to high school teacher Brian Palacios, the "Day in the Life" participant whose monthly posting date is the 24th. He hasn't made his February 24th post yet, but I do see something interesting for January 24th (and no, there's no mention of Desmos):

Palacios writes:

7:35am | I arrive at school. Today’s the first day of Regents Exams, a.k.a. state exams. They last four days. I enter the main office to move my time card and look for the proctoring schedule for the day.

I've read the blogs of so many New York teachers (including those who participate in "Day in the Life") that I know what "Regents" are. I never knew, though, that they're given at the end of the first semester (as well as, I presume, the end of the year).

In fact, reading Palacios here makes me understand fellow New Yorker Wendy Menard's "Day in the Life" post better. Her posting date is the 21st, but she actually writes about January 25th. This would be the second day of the four-day Regents. It explains why Menard doesn't mention finals, as her students would be taking Regents instead.

Speaking of Menard, Palacios mentions actually seeing Menard at a meeting:

The PLT begins and the theme is Next Steps. The facilitators are Wendy Menard and Jose Luis Vilson. They’re awesome. The discussion gets fairly off topic after some time, but no one seems to mind.

Surprising, Menard doesn't mention this  meeting in her January 25th post, although she does bring it up in a February post. Notice that the main topic of this meeting is race. I try to avoid mentioning race in my school day posts, but nearly half of the Palacios post is about race or politics. This includes a certain book (not movie, but book) that Palacios is reading:

7:40pm | On the train home. I complete The Mathematician’s Shiva and begin, excitedly, Hidden Figures.

Palacios wraps up his post just like Menard's -- by speculating about a new class he'll teach soon:

I’m also super excited about getting closer to teaching a legit mathematics elective course. Mathematics was one of the founding principles of my school and, sadly, there is a glaring lack of mathematics-based initiatives that exist right now. I want to try and change that. What’s great is that I got word from leadership at the end of the day today that there are plans for me to teach a Discrete Mathematics course in the near future.

Well, I may be teaching three "new" science courses in the future -- as in next week. It all depends on what I hear from my school's leadership about the grades.