Wednesday, May 11, 2016

PARCC Practice Test Question 16 (Day 159)

Today is the fourth of five days that I'm subbing in an Algebra II and Integrated Math I class.

Before we get to our Morris Kline chapter and our PARCC question, let me point out that this is a traditionalists-labeled post. This is the special traditionalists post that I've been planning to write -- and it's about a topic that affects both the school I'm subbing at today and the school I'll be working at next year.

The school where I'll be working is on a block schedule. I've alluded to block schedules in the past here on the blog, but I've never written about them extensively. This is despite my having attended block schedule schools as a student, as well as having subbed at such schools. But now that I'm headed for a full-time position at a block schedule school, I'd like to discuss what a block schedule is, as well as my opinion of the block schedule, in more detail.

In short, a block schedule is a schedule on which fewer than six classes each day. Typically, on one day, the odd-numbered periods 1, 3, and 5 meet, and the next day, the even-numbered periods 2, 4, 6 hold classes. When fewer classes meet, the result is that each period is longer than they would be on a more traditional six-period day.

In some ways, block schedules are closer to college schedules -- after all, not all college classes meet everyday, and some classes are longer than an hour. Block schedules are also closer to what students follow in other countries. Again, much of my knowledge of the British school system comes from the Harry Potter series. J.K. Rowling often has Harry attend "Double Potions" with Professor Snape. She means that Harry's Potions class is twice as long as usual -- just like a block schedule. And notice that the wizard doesn't attend his least favorite class every school day -- again, just like a block schedule.

There are many different kinds of block schedules. We now consider two components of any schedule:

-- which classes meet on which days
-- how long each class meets

We will look at each of these components separately:

The Daily Schedule

It's possible to classify most block schedules as being one of three types:

-- Pure Block Schedule
-- Common Block Schedule
-- Hybrid Block Schedule

The Pure Block Schedule

On the Pure Block Schedule, every single school day (with the possible exception of the first or last day of a semester, etc.) is a block day. So on Monday odd classes meet, then even classes on Tuesday, odd classes on Wednesday, even classes on Thursday, and odd classes on Friday. But this means that even classes meet the following Monday -- a week after odd classes met on Monday!

That is, there's no simple relationship between the day of the week and which classes meet, due to the unfortunate fact that the school week has an odd number of days. Schools must print out calendars at the beginning of the year to indicate which days are odd days and which are even days. Often these will be called "A" and "B" days, or given two different colors (usually the school colors).

The Common Block Schedule

I call the following plan the "Common Block Schedule" because it's the most prevalent. Basically, schools avoid the "week of odd length" problem by designating one day of the week to have a traditional schedule where all six classes meet. Then the other four days are block days. This allows the school to have the day of the week line up with the day's schedule.

Often, Monday is chosen as the all-classes day. Then Tuesday is an odd day, Wednesday is an even day, Thursday is an odd day, and Friday is an even day. One advantage of choosing Monday as the day for all classes is that most holidays are on Mondays (MLK Day, President's Day, etc.). When Monday is a holiday, Tuesday through Friday can remain block days without any problems. But some schools choose Friday as the all-classes day instead. If Monday is a holiday, Friday becomes a block day -- the classes that would have met on Monday are the only ones that meet on Friday, so that the week remains balanced.

Many schools have a weekly Common Planning Day. School is out an hour earlier -- or possibly it starts an hour later -- and teachers attend meetings during that extra hour. Many schools that have block schedules have all classes meet on the Common Planning Day and the other four days be the block days. This may seem strange -- classes are already shorter when all six periods meet, so why make them even shorter by having all classes on Common Planning Day? The reason for this is so that all classes are affected equally by the meeting day. If Common Planning Day had been, say, an odd day, then students would end up spending more minutes in the even periods for the week.

I first heard about block schedules when I was a seventh grader, attending a junior-senior high school that ranged from Grades 7-12. That year, our school had a traditional six-period day, but there was a proposal to have a block schedule the following year, when I'd be an eighth grader. Our school already had late start days on occasional Wednesdays (about once every three weeks). Under the block schedule, the late start days would fall every single Monday instead. Then Tuesday through Friday would be the block days.

The block schedule was tested out in May that year, during standardized testing. Back then we didn't have Common Core or the SBAC -- instead California had the CLAS test. CLAS was for eighth and tenth graders (I don't remember which elementary grades tested), but we seventh graders had a mock CLAS test, as did the freshmen. Many schools still do this today during SBAC time -- they have a traditional six-period day most of the year, but switch to a block schedule at testing time.

In the end, the block schedule proved to be so unpopular that it was rejected -- except for the part about having every Monday be a late start day, of course. (And not only that, but CLAS itself was rejected by the state, so I ended up never taking the real CLAS in eighth grade!)

During my freshman year, I moved to a different district -- and my new high school actually had a block schedule. It was a Pure Block Schedule, so I was given a calendar to determine which days were odd days and which days were even days.

The following year, we switched to a Common Block Schedule, with all classes on Mondays. As it turned out, that only lasted for three years -- the year after I graduated, the school returned to the Pure Block Schedule. A few years ago, my alma mater returned to the traditional six-period day.

I mentioned block schedules back during my Calendar Reform posts. Some versions of calendar reform change the number of days in a week. If there are an even number of days (or at least an even number of school days) per week, then schools can have a Pure Block Schedule and yet have the blocks tied to days of the week.

The Hybrid Block Schedule

Some schools only have two block days per week, rather than four. At the school where I've been subbing all week, odd classes meet on Wednesdays and even classes meet on Thursdays. On the other three weekdays, all classes meet.

This explains why I'm posting this today -- I'm actually following a block schedule today as a sub! I see 1st and 5th periods today, which are both Algebra II, as well as 3rd period conference. And tomorrow I will see the other Algebra II class 4th period and both Integrated Math classes, which are 2nd and 6th periods.

Some of the students I tutored last year attend a school with even fewer block days. Every two or three weeks, "lab days" (block days) are announced. As usual, these lab days appeared in pairs, with odd classes meeting one day and even classes on the other.

The Hourly Schedule

Now that we've seen how schools determine which classes meet each day, we'll now look at how long each block period lasts.

Consider a simple traditional six-period day -- let's say we have first and second period before snack, third and fourth period before lunch, then fifth and sixth period. We can naively convert this to a block schedule simply by dropping the passing periods between Periods 1/2, 3/4, and 5/6. So now we have three blocks instead of six periods. Each traditional period is just shy of an hour, so each block ends up just shy of two hours. Taking into account that for each block, an unnecessary passing period is converted into class time, each of the three blocks ends up being exactly two hours.

But here's the problem -- many students and teachers alike will agree that two hours is just too long for a single class! The reason that there are so many different versions of block schedules around is that different schools add different things to the schedule, just to prevent the blocks from being two full hours. Hardly any school has a block schedule as simple as having three two-hour blocks.

Here are some of the different things added to a block schedule to shorten the blocks:


When a block schedule was proposed at the school I attended as a seventh grader, a thirty-minute advisory period was added to the day. This allowed the school to make each block about ten minutes shorter, so each period was about 1:50 in length.

Actually, the advisory period served a second purpose. Recall that my school covered Grades 7-12, so there were two separate lunches, one for middle school and one for high school. Notice that having a second lunch is much easier under the traditional six-period day than a block schedule -- some schools are forced to have one of the lunches split one of the blocks. At my school, the proposal was that while one grade span was at lunch, the other span would be in advisory.

A Second Day of Common Planning

Recall that the school I'm subbing at today has a hybrid schedule, with two block days per week. This school doesn't have a weekly day of Common Planning when school is out an hour early. Instead, school is out 30 minutes earlier each of the two block days. This has the same effect as an advisory period -- it shortens each block by ten minutes, so each is about 1:50 in length.

By the way, next week the AP ends and the SBAC begins. Just as my old middle school did during CLAS week, next week the school will switch to a Common Block Schedule with an all-classes day on Monday. School will not be out early any day that week, so blocks will be the full two hours.

A Seventh Period

Some schools add, instead of a 30-minute advisory period, a full 7th period to the school day. This 7th period is about an hour long and meets everyday, just like a traditional period. Such a schedule has the effect of shortening each class by 20 minutes each, down to around 1:40 in length.

This is what the school I attended for Grades 9-12 did: my school had a magnet program where students are encouraged to take seven classes rather than six. Neighborhood (i.e., non-magnet) students were supposed to attend tutorial during 7th period. They could choose any of their classes to attend that period. If students are receiving a low grade in a class, teachers can require them to attend their tutorial on a particular day.

One advantage of having a 7th period was that some classes -- for example, athletic team sports -- want their students to practice everyday, not every other day. So sports were scheduled for 7th period.

Back when I was in high school, I was on the track team. Instead of 7th period tutorial, I was placed in a special sports tutorial class for athletes, which was basically study hall. For me, sports tutorial was 4th period. During my junior year, I was admitted to the magnet program, so I was given an extra class instead of sports tutorial.

Over the years, many students took advantage of tutorial and simply hid somewhere on campus rather than go to one of their classes. Those who were caught were given a two-hour detention -- the first hour was the remainder of 7th period, and the second hour was after school.

After I graduated, the school allowed at first juniors and seniors, and later on all grades, to leave home an hour early if they had parental permission -- the thought being that it's better for them just to go home than loiter around campus and disturb the students who needed help. (I believe that sports tutorial was eliminated -- instead athletes were unscheduled first or second period and so got to arrive late every other day.)

Now the school has returned to a six-period day, so no one gets to go home early or arrive late. The magnet students still have seven classes -- this is accomplished by having two 1:30 blocks occur while the neighborhood students attend three traditional classes (so the magnet students end up fitting four classes in three periods). This schedule is still Pure in that the two blocks attended each day are still not tied to the days of the week, but to a special calendar.

I've seen other schools get around the tutorial problem by simply requiring students to go to a certain period during tutorial. Notice that a school may have a Common Block Schedule (so that the day of the week is sufficient to determine whether it's an odd day or even day) and yet still require a special calendar for tutorial (since it must rotate among six classes).

The seventh period was convenient for students in my magnet program. I've seen some schools that are the opposite of magnets implement a 7th period -- at some inner-city schools, so many students are failing and deficient in credits that they need a seventh class in order to catch up.

At other schools, the biggest problem is tardiness. So I've seen some schools schedule the extra class to the beginning of the day, before the blocks. When it's at the beginning of the day, the class is often numbered zero period rather than 7th period. Electives were often scheduled for zero period -- a class that the students enjoy, so that they'd be motivated to arrive on time.

One local school that has a unique block schedule is the King Drew Medical Magnet. It is officially part of the LAUSD, but I believe that students in several local districts are eligible to attend:

This school has a Hybrid Block Schedule in that only Tuesdays and Wednesdays are block days. To prevent the blocks from being two hours long, an extra class meets on block days -- but this extra class is not a 7th period per se. (Notice that this schedule also has a 20-minute homeroom at the beginning of the day, so each block is shortened to just shy of 1:30.)

Instead, this extra class, which meets after snack, is one of the three classes that doesn't have a block period that day. On Tuesdays, the block periods are 1-3-5. Since the extra class meets between 1st and 3rd periods, 2nd period logically fits in the schedule at this point. On Wednesdays, the block periods are 2-4-6, and so the extra class, meeting between 2nd and 4th periods, is 3rd period.

This arrangement affects the non-block days, too. Since 2nd period has an extra meeting on Tuesdays, on Mondays five classes meet -- all except 2nd period. And 3rd period has an extra class on Wednesdays, so on Thursdays every class happens except 3rd. Only on Fridays do students have a traditional six-period day.

Notice that on this schedule, all six periods follow the same pattern -- the block is the second meeting of a particular period in every case. This means that a teacher can schedule, say, a pre-lab lesson for the first meeting, the lab during the second (block) meeting, a post-lab lesson for the third meeting, and then a quiz on Friday, the fourth meeting.

The big disadvantage of this schedule is evident when we see the "Professional Development Week Bell Schedule." PD days occur on some (but not every) Tuesday. This means that the classes that meet on Tuesdays (1-2-3-5) always have less time than the ones that don't (4th and 6th). Indeed, we see that the so-called "blocks" for 1-3-5 on a PD Tuesday are only seven minutes longer than the non-block meetings of those same classes on Monday!

That PD occurs on Tuesdays is an LAUSD policy. So it's not as if the magnet can change PD day to Fridays, which would fit better with this Hybrid Block Schedule. The alternative would be to make Tuesday the all-classes day, but then some classes would have the block as the third meeting of the week and others would have it as the second meeting.

Embedded Support Time

One of the districts I sub at has several schools on a block schedule. At these schools, blocks are officially the full two hours -- but the last 20 minutes were "embedded support," where only students who need to catch up need to stay. This has the same effect as 7th period tutorial in that the real classes are 1:40 in length -- but since the students are already in class, the teachers can just tell their failing students to stay, rather than hope that they actually show up for 7th period.

A Fourth Block

If we are going to have a 7th period to shorten all the blocks to 1:40, the next natural step is to add an extra block instead. With four blocks per day, each block can be about 1:30 long.

It's possible to fit six classes into four blocks -- each class meets two out of every three days. So we can have a Pure Block Schedule with "A," "B," and "C" days. As is usual for a Pure Block Schedule, these letters aren't tied to any particular day of the week.

It's also possible to fit six classes into four blocks using a Hybrid Block Schedule. Instead of "A," "B," and "C," one local district has block days on Tuesday, Wednesday, and Thursday. Then both Monday and Friday are all-classes days -- and both days have school starting about 30 minutes later, similar to the Hybrid Block Schedule in my own district.

Of course, a four-block schedule also admits an "A"/"B" schedule with eight periods. If having seven periods allows students deficient in credits to make them up, then having eight periods makes credit recovery even easier. And students who pass all their classes might be able to graduate in three years, rather than four, from a high school with eight-period schedule.

I've seen block schedules that combine some of the ideas that I've listed here. For example, one local district has an interesting block schedule for its middle schools. These schools have a Hybrid Schedule with block days on Wednesday and Thursday -- just like the high school where I'm subbing today, but with a twist. You see, the students are enrolled in seven periods rather than six, but one of the periods (3rd) appears to be more like an Advisory or Homeroom. Also, lunch is also one of the numbered periods (5th or 6th, depending on the grade level). So students actually have eight periods on three days of the week.

On Wednesdays four of the classes meet for a block period. As we've seen above, when there are four blocks, each one is about 1:30 in length. It's not as simple as having odd blocks on one day and even blocks on the other, since two periods (Advisory and lunch) have special meanings -- instead, four classes plus lunch meet on Wednesdays. So on Thursdays there are only three block periods plus lunch -- but Thursday is the Common Planning Day. Because one of the "blocks" is actually 3rd period Advisory, this period isn't much longer than it is on Monday, Tuesday, or Friday. So the net result is that school is out more than two hours earlier on Thursdays.

The 4 x 4 Plan

The 4 x 4 Plan is a special version of the Four Block Schedule, but with one big difference -- instead of alternating between "A" and "B" days, all the "A" classes meet everyday the first semester, and all the "B" classes meet the second semester.

I first heard of the 4 x 4 plan back when I was in high school. Our school opened up a satellite campus for students who were deficient in credits. I found out that students were able to catch up because the satellite campus had a 4 x 4 Block Schedule.

One possible advantage to the 4 x 4 plan over simply alternating "A" and "B," even with four blocks and eight periods, is that students may forget material during the day the class doesn't meet (especially in a class like math). Of course, they could do the homework the night they don't have the class -- but if these students did homework, they wouldn't be deficient in credits in the first place.

Under the 4 x 4 plan, a whole year's material is completed in a semester. This means that each semester on a traditional schedule is like a quarter under 4 x 4. Students must take finals at the end of every quarter under 4 x 4, just as they take semester finals on a traditional schedule. And each quarter on a traditional schedule is like a quaver under 4 x 4. Sometimes teachers only mark grades of "D" or "F" on the quaver progress report on the traditional schedule, but under 4 x 4, every student must get a grade every quaver.

I've also seen versions of the 4 x 4 plan where the basic unit is the trimester. Here's a link to a website promoting the 3 x 5 block schedule:

Just multiply it out -- 4 x 4 is greater than 3 x 5, so more credits are possible under the 4 x 4 plan than the 3 x 5 plan. According to the link, each of the five periods is 68-75 minutes -- raising the question whether they are even long enough to be called "blocks" anymore. Trimesters of any sort are rare in California high schools.

The Block Schedule and My Plans for This Blog

I've described so many types block schedules in the post, so which of this will my middle school for next year be following? As it turns out, it follows the 30-minute advisory model -- just as my own middle school tried to implement so many years ago. This means that each class will end up being about 1:50 in length. But at this school, advisory occurs at the beginning of the day.

My new school has a Common Planning Day every Wednesday. Therefore it follows the Common Block Schedule with all classes meeting on Wednesdays. All-classes days on Wednesdays are much rarer than those on Mondays or Fridays, but we must admit that it's the next most logical day to place the all-classes day after Monday and Friday. This means that odd classes meet on Mondays and Thursdays, and even classes meet on Tuesdays and Fridays.

I will be teaching three preps next year -- sixth, seventh, and eighth grade. With three preps, I'll definitely have less time available for blogging. So my plan is to take advantage of the block schedule and post only three days per week. Since my plan is for this blog to focus on the Common Core Math 8 class, I'll post only on days when the class meets -- it may be either Monday-Wednesday-Thursday or Tuesday-Wednesday-Friday. Of course, it's possible that I could be assigned two Math 8 sections, with one odd and one even block -- then what? Well, I'll cross that bridge when I come to it.

My focus for this week on the blog is the Integrated Math I class -- and that class doesn't meet at all today since it's 2nd and 6th periods only. So I'll follow my blog plan for next year's block schedule today -- I won't write about my subbing at all today. Of course next year, I won't even post on days when Math 8 doesn't meet, but today I'm writing this special post to explain what a block schedule is.

Traditionalists and the Block Schedule

Speaking of today's special post, I said that I'm labeling it as traditionalists -- but so far, I haven't said anything about traditionalists at all.

The following link belongs to a website created by Jeff Lindsay. It is one of the oldest sites on the web -- Lindsay first created the website in 1999, back when I was still a senior in high school (and following a block schedule as a student), and it's most recent update was in 2008. Despite the age of this website, it nonetheless appears in the top ten Google search results for "block schedule":

It's definitely accurate to call Lindsay a traditionalist -- after all, in the following link, Lindsay promotes Direct Instruction, a strong traditionalist pedagogy:

Lindsay begins:

Block scheduling is a restructuring of the school day into classes much longer than the traditional 50-minute period. In one common form, students have four long class periods per day instead of seven or eight. A course that normally covered the entire school year could then be compressed into an intense half-year course. Fewer, longer classes are said to allow new styles of teaching rather than old fashioned direct instruction. Education is supposed to become less stressful, more relaxed, and more enjoyable, bringing a long list of educational advantages. Many educators and parents see it as a recycled form of the Modular Scheduling concept which was tried and abandoned in the 70s and 80s.
The case for block scheduling in high schools is being pushed from a variety of sources. Some of these sources offer long lists of euphoric virtues of block scheduling without any hard data and without any serious consideration of the down side. Regrettably, school boards across the country seem to be adopting an experimental program on the basis of its purported advantages without demanding or considering hard data. While there may be some interesting success stories at several schools, the case for block scheduling has not been established through serious, long-term scientific studies. At best, the case for block scheduling is tenuous and, in some cases, is contradicted by scientific studies. It seems to have fared no better than Modular Scheduling. Pending clear evidence in favor of a change, I feel that we should not rush to jump on the bandwagon.

Lindsay devotes most of his block schedule pages to the argument that students don't learn as much on the block schedule as they would on a traditional bell schedule. As we can see, much of his argument directly attacks the 4 x 4 Plan -- which is not the schedule adopted at either the school where I'm subbing today or the school where I'll be teaching in the fall. But in the following paragraph, Lindsay mentions the "alternating A/B block" which is more common:

We should also consider some experience in Washington. The Washington School Research Center has published a study, "Schedule Matters: The Relationship between High School Schedules and Student Academic Achievement," by Duane Baker, Jeff Joireman, Joan Clay, and Martin Abbott, Oct. 2006. This study of schools in the State of Washington looks at student achievement data in math, reading, and writing for Washington High Schools as a function of the schedule type of the schools. Schedule types included traditional seven-period days (21.6% of schools), traditional six-period days (41.2%), 4x4 block (14.2%), alternating A/B block (7.1%), and block (15.9%, some traditional periods and block periods). The authors write that "the results of the study led to two fundamental findings. First, the seven-period and Modified Block schedules were, overall, the highest performing schedules correlated with reading, writing, and math WASL. Second, the 4x4 and A/B Alternating Block schedules were, overall, the lowest performing schedules correlated with reading, writing, and math WASL." While they try to argue that the data don't necessarily prove that the block is a problem, there certainly wasn't dramatic support for any academic benefits from the block. As always, it remains appropriate for parents and teachers to ask tough questions of those touting the block. "What will the effect on academic performance be? Do you have evidence to show that there won't be any harm to our kids?"

It's hard to tell exactly what Lindsay means by a "Modified Block" schedule. It could refer to the school where I sub (which I call a Hybrid Schedule -- every class has three traditional and one block period per week), or it could be more like the current schedule at the magnet program I graduated from (three regular periods and two block periods per day). It does seem interesting though that this "Modified Block," whatever it is, was the second-highest performing schedule. So it fared better than the traditional six (but not seven, which was the highest) periods.

We see that to Lindsay, like most traditionalists, academic performance and results have priority over all other considerations. A schedule, or a pedagogy, is best if it produces the best results. But my problem is that the traditionalists assume that the students will just meekly go along with their ideas "just because." They say that individual homework sets are the best way for the students to practice with the material, and I agree -- provided the students actually do the homework.

One rationale for the block schedule is that with fewer passing periods, there will be much fewer opportunities for student mischief in the halls. Traditionalists who prefer the six-period day apparently assume that students will behave just because there's a traditional schedule, just as they assume that students will do the individual homework just because it's assigned to them or listen to the teacher just because he/she is "the sage on the stage."

I believe that progressive pedagogy (especially in the higher grades) and the block schedule take students who are already inclined to break rules and dodge hard work, and put them in a situation where they are less inclined to do so.

But Lindsay does have a valid point about "tutorial" as it once existed at my high school (and he describes a similar situation on his website). Lindsay would be happy to know that my high school has abandoned tutorial and mostly returned to the six-period day (except for magnet students).

Oh, and many students like the block schedule because they don't have to carry as many heavy books to every class. We already know the traditionalist solution to this problem -- lighter textbooks.

What to Teach During the Longer Block Periods

On Part 3 of his website, Lindsay summarizes the problems with block scheduling. Again, many of his concerns are geared towards the 4 x 4 Plan, but one problem that affects all classes is what to teach during the classes that are 1:30-1:50 in length:

1. Problems due directly or indirectly to attention span limitations:
  • Longer classes are incompatible with the attention spans of most students (20-50 minute attention spans are commonly cited).
  • Instead of trying to cover twice as much material in a longer class period, the natural tendency is to water down the material to maintain interest, resorting to movies, games, doing homework in class.
  • Either due to attention span limitations or to the watering down of material, learning is likely to be less effective, especially in courses such as math and science (as demonstrated in Bateson's study).
  • Learning Disabled children may be especially disadvantaged by the longer classes of block scheduling.

And of course traditionalists oppose the idea of watching movies or playing games in class -- and they believe that homework in class encourages laziness.

What exactly did I teach in the block classes I covered today? I'm sorry, but that will have to wait until tomorrow's post, when I'll describe the block classes I subbed for in more detail.

Meanwhile, it seems interesting that Lindsay would mention science as a class that would be watered down by the block schedule, considering that more time for science labs is frequently mentioned as a benefit of the block schedule! Then again, I can easily see a traditionalist like Lindsay turn one of my pro-block arguments around to an anti-block argument -- more time for science labs is great, provided the students are actually working on the lab the entire 1:30+ block. If they are wasting time after the first hour, then the students might as well have a six-period day. The real question, and one not addressed on Lindsay's website, is which are actual students more likely to do -- a science lab for the entire block period, or an individual problem set for science homework?

Speaking of science, this is a good time to segue into our reading. Chapter 16 of Morris Kline's Mathematics and the Physical World is called "Deductions from the Law of Gravitation." In this chapter, Kline tells us knowledge of gravity leads to a further understanding of the universe.

"Who by a vigor almost divine, the motions and figures of planets, the paths of comets, and the tides of the seas, first demonstrated." -- Newton's Epitaph

Kline begins:

"In scientific work the proof of the pudding is in the eating. By applying his mathematical law of gravitation to the motion of objects near the surface of the earth and to the motion of the moon Newton found that the deductions agreed with known facts."

So we see that in this chapter, Kline is still describing Newton's work. Now we get to see how Newton's laws are used to solve problems. Here, I will describe problem he worked on -- calculating the mass of the earth. It's not as if we can put the whole planet on a scale and weigh it!

Kline writes:

"To do this we shall first make an auxiliary calculation. Suppose that an object of mass m is suspended at C on a string AC. Of course gravity pulls straight down on this mass. Nearby a one-ton mass of lead is placed and it pulls the mass m to B." (By "nearby" he means one yard away.)

Many readers may be confused at this point -- but the idea is that gravity means that all objects are attracted to each other -- not just large objects like the earth, moon, or sun. We only notice gravity when one of the objects is large. But in this case, Newton is trying to see how much a one-ton mass of lead can attract m in order to determine how much bigger the earth is.

Kline uses the parallelogram rule to add the force vectors on the mass m -- PQ is the downward force of gravity attracting the mass to the ground, and PR is the lateral force of gravity attracting the mass to the ton of lead. Then PS is the diagonal of a rectangle. Kline writes:

tan SPQ = QS/PQ = PR/PQ

Now SPQ is a tiny angle since the force of gravity attracting the mass to the lead is small, but it's still measurable -- Kline states it to be 0.00000045 degrees. The tangent of this angle is 8 * 10^-9.

Afterward, Kline returns to Newton's original equation for the law of gravitation:

F = GmM / (r^2)

to obtain:

PQ = GmM / (4000 * 1760)^2

where 4000 miles is the distance to the center of mass of the earth, and 1760 is the conversion factor to yards (since the lead is one yard away). He then writes that PR = Gm since the lead is one yard away and has a mass of one ton. So he writes:

PR/PQ = (4000 * 1760)^2 / M

where M is the mass of the earth. He then plugs in the value of PR/PQ already calculated as 8 * 10^-9 to obtain:

8 * 10^-9 = (4000 * 1760)^2 / M
M = 6.5(10)^21

So the mass of the earth is about 6.5 sextillion tons. Since a ton (avoirdupois) is almost 1000 kg (also known as a metric ton), we see that the mass of the earth is about 6(10)^24 kg.

Nowadays, we find the mass of the earth simply by typing in "mass of earth" into a Google search, but now we can appreciate how much work was put into that number that magically appears on the computer screen.

Question 16 of the PARCC Practice Exam is on dilations and similar triangles:

16. Triangle APQ is the image of triangle ABC under a dilation centered at vertex A with scale factor 1/2. Triangle RBT is the image of triangle ABC under a dilation centered at vertex B with scale factor 3/4. Which statement about triangles ABC, APQ, and RBT is correct?

A. All three triangles are similar.
B. None of the triangles are similar.
C. Triangles APQ and RBT are not similar because they were dilated with different scale factors.
D. Triangles APQ and RBT are not similar because they were dilated with different centers of dilation.

To put it simply, the answer is (A). This is because any figure is similar to its dilation image -- after all, the whole point behind using dilations is to study similarity. But as we can see with the answer choices, just because ABC is similar to both APQ and RBT, it isn't obvious that APQ and RBT must therefore be similar to each other.

Let's recall the definition of similar as given in Lesson 12-5 of the U of Chicago text:

-- Two figures are similar if there exists a similarity transformation mapping one to the other.

If you remember from back in February, I had a huge problem with the definition of similarity -- this was all because of a problem on the old PARCC where we had to know what similar triangles were in order to prove the properties of dilations! So it's circular to use dilations to define similarity, then use similarity to prove that dilations work. The way out of this circularity is just to assume a Dilation Postulate which states the properties of dilations.

Anyway, this definition of similarity given in the text actually works better for this problem. We see that two figures are similar if there exists a similarity transformation -- not necessarily a dilation, but any "similarity transformation" -- mapping one to the other.

But what exactly is a "similarity transformation," anyway? Many texts define it as the composite of a dilation and an isometry -- and this definition is often used in the proof of AA Similarity, where a third triangle is produced that is similar to one of the triangles and congruent to the other. But the definition given in the U of Chicago text is:

A transformation is a similarity transformation if and only if it is the composite of size changes and reflections.

Recall that a "size change" is just a dilation. But I wish to emphasize the plural here -- size changeS -- which implies that the composite of two dilations is a similarity transformation. In particular, the composite of the dilation centered at A of scale factor 2/1 (mapping APQ to ABC) and the dilation centered at B of scale factor 3/4 (mapping ABC to RBT) is a similarity transformation. Indeed, it is itself a dilation of scale factor 3/2 (though its center is hard to find). The composite of two dilations is always either another dilation or else a translation.

PARCC Practice EOY Question 16
U of Chicago Correspondence: Lesson 12-5, Similar Figures
Key Theorem: Definition of similar

Two figures are similar, written F~G, if and only if there is a similarity transformation mapping one to the other.

Common Core Standard:
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Commentary: Again, dilations and similarity appear in Integrated Math II, not Math I, so I have no worksheet from the classroom to post here. So I came up with two questions that are in the same spirit as the PARCC question.

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