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.

Wednesday, February 22, 2017

Student Journal: Systems of Equations (Days 104-105)

Here is today's Pappas question of the day:

An equilateral triangle circumscribes a circle of radius sqrt(22/(3sqrt(3))). Find the area of triangle.

As I wrote earlier, I'll be switching to mostly algebra questions soon on my Pappas questions soon, as I'll be giving actual eighth grade systems of equations. But here's a geometry question to tide us over, since this is technically a geometry blog.

This question contains a common Pappas trick -- the apothem is given as sqrt(22/(3sqrt(3))), a very strange-looking number. But this is necessary to make the answer a whole number (the date). This happens often with equilateral triangle problems -- the area of an equilateral triangle with a integer side, radius, or apothem can never be an integer, but must contain a sqrt(3) factor. Even if the side, radius, or apothem itself contains a sqrt(3) factor, the area still has a sqrt(3) factor. In order to make the area a whole number, the side, radius, or apothem must contain the fourth root of 3.

When I see this sort of problem, it's actually easier just to use a variable, such as a for apothem, rather than the unwieldy sqrt(22/(3sqrt(3))). Then we substitute it back in later on.

The one thing I know about equilateral triangles is that they contain several 30-60-90 triangles. The actual proof of this is somewhat tricky -- the key is that angle bisectors of a triangle are concurrent -- they meet at the incenter of circle. Since the triangle is equilateral, it's equiangular -- the angle bisectors cut the 60 degree angles in to 30 degree halves. Along with the fact that angle bisectors of an isosceles (or equilateral) triangles are also altitudes, we conclude that the angle bisectors divide the whole triangle into six congruent 30-60-90 triangles. The apothem a is the short leg of these triangles, with the long leg a sqrt(3) and hypotenuse 2a. The original equilateral triangle has a base twice the length as the long leg, or 2a sqrt(3), and a height that of the short leg and hypotenuse combined, or 3a. So the area of the large triangle works out to be 3sqrt(3) a^2.

It only remains to plug in the given value of a:

a = sqrt(22/(3sqrt(3)))
a^2 = 22/3sqrt(3)
3sqrt(3) a^2 = 22

So the area is 22 -- and today's date is the 22nd.

Today is Day 104 here at the middle school, but it's only Day 100 in kindergarten. It's the first of three party days to occur at our K-8 school in the next few weeks.

Eighth grade, meanwhile, isn't anything like a party today. The students begin solving systems -- perhaps one of the most difficult topics in Common Core 8. On one hand, it's surprising that systems would even be included in the eighth grade standards -- but then again, I did say that every first semester Algebra I topic seems to appear in the Common Core 8 standards, and that would include systems of equations.

I remember back when I was student teaching an Algebra I class. We fell behind, and we had to rush through the systems of equations. We only had time to teach one method -- and it was graphing! A few years later, I told this story to an Algebra I teacher at another school, and she was surprised that we didn't choose elimination as our one method to teach, rather than graphing!

In some ways, I'm in the same predicament now. I'm rushing to cover all of the major content, or MC, standards this year, so today I only have time to teach one method of solving systems. This time, we haven't even reached graphing of lines yet, so instead of graphing, I choose substitution.

Why am I favoring substitution over elimination? Well, as it turns out, the Illinois State online homework that I'm required to give favors substitution. Indeed, the students are asked to solve four systems, and three of them already have x or y isolated in one of the equations. The fourth system is actually graphing -- but no equations are given. The lines are already graphed, and the students only need to identify the point of intersection. As it turns out, this is a lousy question -- the correct answer is given as (13, 2q), which is clearly a typo. I suspect that the q is supposed to be 9, and that the intended solution is (1.3, 2.9), but there are no decimals given in the computer solution.

Today's substitution lesson took a while to get through -- I suspect that part of it is that some students still struggle with multi-step equations. If you think about it, a substitution problem nearly always leads to a multi-step equation -- after all, if one equation is already solved for y, then we're to substitute an expression containing x into another equation also containing x. Students must know what to do next based on whether the two x-terms are on the same or opposite sides of the equation.

Of course, this means that equations not already solved for x or y are out of the question! We know that some systems are easy to solve with elimination (such as x + y = 4, x - y = 2) while others require multiplication before elimination can take place. Those students who struggle with multi-step equations might find elimination easier in some cases.

By the way, I often refer to elimination as "the method of three names." That's because I've seen three different names for this method -- the other two are "addition-subtraction" and "linear combinations."

I may introduce elimination as part of tomorrow's Learning Centers. This will be the first centers day with something on the line -- a major test is coming up on Friday. At least I've done centers once before, so hopefully they'll work out well tomorrow.

There isn't much of a science lesson today. It's the last Wednesday of the month, so there's already an Aspiration Assembly taking up much of the science period. (See my December 14th post for more info here.) Then afterward, the dean comes in to warn the students about gentrification yet again. (See my January 6th post for more info here.) There ends up being only 20 minutes for science. But, already anticipating less time for science, I have the students log on to Study Island and start answering questions related to their recent science projects. After all, in order to pass the California Science Test they'll need to be able to answer questions.

The sixth graders today are solving equations for the first time. But the equations given in the Illinois State text are trivial to solve -- the equations already have x or y isolated! The first equation they solve today is 5 + 7 = q. Yes -- there's that letter q as a variable again! One of my students sees q and thinks it's actually a 9 -- until I asked her whether 5 + 7 = 9. As usual, the seventh graders don't meet on Wednesdays.

This is a two-day post. My next post will be on Friday.

Tuesday, February 21, 2017

Say It With Words, Pictures, Tables. and Symbols (Day 103)

Here is today's Pappas question of the day:

y = -4x + 57
(1/2)x - (1/3)y = 3

Find the sum of the x & y coordinates of the point of intersection.

Hey -- this is one of those systems questions I was discussing last week. To solve this problem, we notice that this is well-suited for substitution since one of the equations is already solved for y. But it may be better to clear the other equation of fractions before the substitution:

(1/2)x - (1/3)y = 3
3x - 2y = 18
3x - 2(-4x + 57) = 18
3x + 8x - 114 = 18
11x = 132
x = 12

y = -4x + 57
y = -4(12) + 57
y = 9

So the solution is (12, 9) -- but the question asks for x + y. We add 12 + 9 = 21, so it's 21 -- and of course, today's date is the 21st.

Let's consider the suitability of this problem in my eighth grade class. Even though I'm discussing this problem today on the blog, I didn't give my eighth graders this question, because they don't actually start learning systems until tomorrow.

First of all, we notice that strange final step -- find the sum of the coordinates. This is merely to shoehorn the natural solution (x, y) into a Pappas problem whose answer is the date 21. Theoni Pappas herself does this quite frequently for problems whose solution is an ordered pair -- she often asks for x + y or xy.

The bigger problem, though, are the fractional coefficients in the second problem. Even though I taught several techniques for solving equations, clearing fractions wasn't one of them. We could attempt the substitution without clearing fractions first:

(1/2)x - (1/3)y = 3
(1/2)x - (1/3)(-4x + 57) = 3
(1/2)x + (4/3)x - 19 = 3

but this is error-prone -- students are more likely to make a mistake with the fractions than actually find a solution for x. Most likely, I'd pose the second equation with the fractions already cleared:

3x - 2y = 18

As I wrote earlier, I can give systems of equations as Warm-Up problems well past this week's lessons -- and I'll run out of Illinois State problems, so Pappas questions will work fine. I may indeed consider asking the students for x + y or xy, with the answer as the date. Then since the students already know the date, they can use this information to check their work.

OK, that's enough about the Pappas question -- let's get down to business. Today marks the first day of that time of year I call the "Big March." This is how I explained the Big March last year:

Consider the three major breaks of the school year: winter break, spring break, and summer break. I suppose that now I should actually say four major vacations, since Thanksgiving break is now the fourth major break of the year. Of these four breaks, which one do you look the most forward to?

To me, the toughest stretch of the school year is that from President's Day to spring break. Think about it -- the first school holiday of the year is often Veteran's Day. Then, starting with Vets Day, there is another holiday every 2-4 weeks. A few weeks after Vets Day is Thanksgiving. A few weeks after Turkey Day is winter break. A few weeks after coming back from the holidays is Martin Luther King Day. A few weeks after King Day are the holidays for Lincoln and Washington. But once we reach President's Day, there are no more holidays until spring break, which may be a month or two after Prez Day, often depending on the Easter date.

Therefore, I consider the current stretch, from President's Day to Easter, to be the toughest stretch of the year. I've named this difficult period of time the Big March -- this name evokes the military where soldiers often have to travel long distances on foot, and it also refers to the month of March, the month that constitutes the majority of this period.

Hold on a minute, you may be saying. Last year, I referred to this tough stretch of the year as the "Long March," but now I'm calling it the Big March. Let me explain why.

I wanted to post some pictures around my room today to mark the beginning of this stretch. But the phrase "Long March" is often used to refer to events involving Communist China (or sometimes even Germany) prior to World War II -- and I don't want to use those pictures. It's distasteful, bordering on outright offensive, to compare suffering six straight weeks of school without a holiday to the actual suffering of those who lived under totalitarian regimes. (Imagine calling this time of year the "Long Holocaust" or something like that!)

And so this year, I decided to change "Long March" to "Big March." A Google search for "Big March" does lead several big marches that occurred this year around the country, such as the Women's March or the March for Life -- but nothing like the 1930's and 1940's events. In order to avoid those political marches, I put up pictures of two "Big Marches that you can find right now on Google" -- one a march to save a library, the other an anti-bullying campaign (and both of those marches occurred in Europe).

In addition to the pictures, my music break featured a song warning the students about the upcoming Big March. It's set to the tune of another song about a march -- "The Ants Go Marching."


We go big marching in Week 1, hurrah, hurrah
We go big marching in Week 1, hurrah, hurrah
We go big marching in Week 1,
We can't even stop to suck our thumb
And we all go marching down, counting down
'Til we get out of school, BOOM! BOOM! BOOM!

We go big marching in Week 2, hurrah, hurrah
We go big marching in Week  2, hurrah, hurrah
We go big marching in Week 2,
We can't even stop to tie our shoe
And we all go marching down, counting down
'Til we get out of school, BOOM! BOOM! BOOM!

We go big marching in Week 3, hurrah, hurrah
We go big marching in Week 3, hurrah, hurrah
We go big marching in Week 3,
We can't even stop to climb a tree
And we all go marching down, counting down
'Til we get out of school, BOOM! BOOM! BOOM!

We go big marching in Week 4, hurrah, hurrah
We go big marching in Week 4, hurrah, hurrah
We go big marching in Week 4,
We can't even stop to shut the door
And we all go marching down, counting down
'Til we get out of school, BOOM! BOOM! BOOM!

We go big marching in Week 5, hurrah, hurrah
We go big marching in Week 5, hurrah, hurrah
We go big marching in Week 5,
We can't even stop to take a dive
And we all go marching down, counting down
'Til we get out of school, BOOM! BOOM! BOOM!

We go big marching in Week 6, hurrah, hurrah
We go big marching in Week 6, hurrah, hurrah
We go big marching in Week 6,
We can't even stop to pick up sticks
And we all go marching down, counting down
'Til we get out of school, BOOM! BOOM! BOOM!

Here's a short bridge that I added to the song:

It's the Big March!
The toughest time of school.
There's no day off,
It isn't very cool.
We got to work hard,
As a general rule.
It's the very Big March! BOOM! BOOM! BOOM!

The six weeks mentioned in the song refer to the length of this year's Big March. As it turns out, a PD day was announced for the end of March, so our Big March ends before Easter. Oh, and it happens that this PD day messes up the blog's day count again, so that one of the upcoming eighteenths of the month (my "Day of the Life" posting day) lands on a multiple of three (which is my scheduled day off from posting). So here we go again!

As I've written before, there are some bright spots during the Big March. Tomorrow, the primary grades will celebrate Day 100, based on the kindergarten day count. Next week is the big party day for the intermediate grades -- Dr. Seuss Day, observed on the late author's birthday, March 2nd. And of course, the big party day for middle school will be Pi Day. March 14th -- that is, if I have anything to say about it!

I wrote in my last post about how to a non-teacher, all this about Big March sounds like whining. We know that many workers don't even get President's Day off, so they aren't sympathetic to teachers who have no days off from Prez Day to the end of March. Indeed, yesterday there was a worker inspecting the roof in preparation for installing a new one, and there was also an appointment with a repairman (actually a woman) to fix the dishwasher. Those workers go much more than six weeks without a day off, so to them teachers shouldn't complain about the Big March.

So far, this post is quite lengthy, yet I've said nothing about today's mathematical content. Well, it's a Tuesday, so it's time for another STEM project. Learning Module 12 of the Illinois State text for eighth grade is called "Say It With Words, Pictures, Tables, and Symbols." In this project, students learn how to convert patterns into equations.

The first pattern is a simple grid -- 1 by 1, 2 by 2 (and no, I'm not still talking about those ants in the Big March song). Students are to draw the 5 by 5 grid and write an equation to predict how many squares are in the 6 by 6 and 10 by 10 grids. Finally, they have the opportunity to graph their equations using a graphing calculator.

I know why the Illinois State text asks students to explain what they are doing -- but I think that too much time was spent on explaining and not enough on exploring more patterns. In fact, Dan Meyer, the King of the MTBoS, is quite outspoken on this sort of problem. In his old "Makeover Mondays" series from 2013, he wrote about how to improve this project:


  • Motivate the move to generalization. Why should we move from counting by hand in part A (easy!) to generalizing to a formula in part B (hard!)? The problem just asserts that the contractor wants to. I’d rather put the student in a position to say to herself, “My word. I’d rather eat chalk than count up all these tiles. Does anybody have a faster way?”

In other words, the project ought to have asked the student to find how many squares are in a grid of 6 by 6, 7 by 7, and so on, until the student wants to find a formula.

Here are a few other things that happen in my eighth grade class. After showing my top student -- the girl who transferred in from Algebra I -- the graph of y = x^2, I told her about parabolas and how all of them have similar graphs. The hope, of course, is that I'll be able to teach her a full unit on quadratic equations soon so I can recommend her for Geometry next year.

Meanwhile, some of the other girls in the class remain unmotivated. My support provider tells the students that boys aren't really interested in dating girls who aren't smart. This is the perfect opportunity to bring out Danica McKellar's Kiss My Math -- a book I purchased for the sole purpose of motivating the girls in my classes. McKellar gives a survey in which she asks teen boys what they are looking for in a girlfriend. The results -- 51% want the total package (looks and smarts) while 41% don't care either way. Only 8% of teen boys desire to be smarter than their girlfriends -- and McKellar recommends that girls not even waste time with that 8%.

Oh, and there was a quick observation during the eighth grade block. A visitor from the charter board comes in just as the eighth graders are about to begin the project. I believe that the observation goes well for the most part.

In the other graders, sixth graders begin work on a project where they pour water from a smaller container into a larger container. This way, they compare the relative sizes of a cup, pint, quart, gallon, bucket, and dishpan. One group has the misfortune of having to determine how many cups fill a large dishpan. They end up using 17 cups to fill half of the dishpan, and conclude that there are about 34 cups in the dishpan. This is slightly more than two gallons.

I was actually trying to time the sixth grade water projects so that they line up with the Green Team science unit. The Green Team curriculum has been delayed -- but fortunately, it's the next project that's more directly related to water conservation.

The seventh grade project is called "I Have...Who Has?" This is a classic -- students are given cards in which each card connects to the next. In this case, the teacher begins with an expression such as "Who has 2(3x + 4)?" The student who has the card "I have 6x + 8," replies by reading this card, which links to the next question "Who has 5(7x + 9)?" The student who has the answer reads out the next question, and so on.

I assume that there are "I Have...Who Has?" cards packed away in one of my Illinois State boxes, but it takes forever to find anything in them! So instead, I found an old "I Have...Who Has?" worksheet from my student teaching days! This was from a high school class of students who failed the first semester of Algebra I and ended up in Pre-Algebra, so this is appropriate for seventh grade. But instead of the intended equivalent expressions, these cards contain equivalent fractions. Oh well -- it's not as these students can't use extra help with fractions. (My old student teaching kit also contains a similar activity -- though not expressly called "I Have...Who Has?" -- for Geometry terms.)

This project doesn't go well at all, due to student behavior. I end up kicking several students out of the class for excessive talking and disruption.

The "Day in the Life" poster for the 21st is New York high school teacher Wendy Menard. She hasn't posted her February 21st post. Here's a link to her January 21st post -- well, "sort of":


January 21st fell on a Saturday. Menard was hoping to write about her participation in the Women's March that day. (What was I saying about big marches earlier again?) But as it turns out, an injury prevents her from participating.

So instead, she writes about the last day of the semester, January 25th -- which she's calling a "key day" in the DITL series (even though Cardone doesn't mention this as such). It's interesting that's no mention of finals (except for "final grades," not final exams) anywhere in the post. In fact, she teaches new content to her Algebra II class -- other classes get to play with Lego bricks.

Menard concludes her post by mentioning future teaching assignments -- Geometry in the second semester, and AP stats in the fall. Half of a year of Geometry is a bit odd -- but I'm one to talk, as I must deal with the repercussions of having just half a year of science tomorrow.

Saturday, February 18, 2017

MTBoS A Day in the Life Post: February Reflection

 This post fulfills my monthly requirement for Tina Cardone's "Day in the Life" project. My monthly posting day is today, the 18th. Not only is today a Saturday, but since the length of February is 28 days (a multiple of the seven-day week), March 18th must fall on the same day of the week -- yet another Saturday. Let's look at my recent posting schedule:

November 18th -- PD Day (leading up to Thanksgiving break)
December 18th -- Sunday (leading up to winter break)
January 18th -- Field Trip Day
February 18th -- Saturday (President's Day weekend)
March 18th -- Saturday

The "Day in the Life" posters for the 4th (18-14), 11th (18-7), and 25th (18+7) have also commented that their posting days seem to fall on the weekend or other non-school days more often than not.

As usual, "Day in the Life" on the weekend means it's time for Cardone's special reflection questions:

1) Teachers make a lot of decisions throughout the day.  Sometimes we make so many it feels overwhelming.  When you think about today, what is a decision/teacher move you made that you are proud of?  What is one you are worried wasn’t ideal?

Well, I like the activities I came up this week for sixth and eighth grades this week. I took advantage of the three-day week (due to PD on Thursday and Friday) to give special activities on Tuesday. I found these activities -- where else? -- on the MTBoS.

In sixth grade, we played Tax Collector, which I found on the website of Denise Gaskins:


And in eighth grade, I provided some Mathematical Dating Advice (as it was Valentine's Day, after all), which I found on the website of Sarah Carter:


As for a decision I'm less proud of, well, the worst decision I made this year involves science. In my December 18th "Day in the Life" reflection post, I wrote about the problems I had when giving a Green Team pretest. And many of my other November and December DITL posts mention how I'm officially a science teacher, yet I haven't taught much science.

This week, my science struggles continue. Lately I've been trying to give science projects to my eighth graders, but not all of them have turned out well. The project I gave this week required the students to cut out pictures of physical changes (such as melting) and chemical changes (such as burning, for example). But instead, I had the students draw the pictures because I don't have access to a printable copy of the pictures. The problem is that I don't have copies of the Illinois State science text -- I could access the projects online, but not print them.

Last week, the project had the students identifying six "mystery substances." The problem is that these substances were a mystery even to me, the teacher! The Illinois State teacher's edition doesn't specify what the substances are supposed to be! So I just cobbled together some Ajax. salt, alcohol, and other household items. One of the items was supposed to be magnetic, so I used steel wool.

During the PD meetings on Thursday and Friday, I spoke to my counterpart -- the middle school math and science teacher at our sister charter school. She told me that her school, unlike mine, actually has copies of the Illinois State science text. But she just skipped the "mystery substance" project because she couldn't figure out what to use as the substances either.

And with all of this, I can't say whether my science efforts will lead to my eighth graders getting even one question right on the California Science Test. There is too much confusion with regards to the content of the test -- whether it includes only physical science (the old state test) or elements of earth and life science as well (the Next Generation Science Standards).

2) Every person’s life is full of highs and lows.  Share with us some of what that is like for a teacher.  What are you looking forward to?  What has been a challenge for you lately?

Well, I'm looking forward to the rest of my President's Day weekend, because the challenge is definitely what lies ahead. It seems interesting that Tina Cardone would associate the months of February and March with Rejuvenation -- probably because we are now finally closer to the upcoming summer than the previous summer. But I've always considered this time to be the toughest time of the school year.

Let's think about it for a moment. From the first day of school on, we have never gone more than a month without a day off from school. First there's Labor Day, then a few weeks later are the Jewish high holidays (or Columbus Day at some schools.) A few weeks after that is Veteran's Day. A few weeks after that is Thanksgiving, A few weeks after that is winter break. A few weeks after that is Martin Luther King Day. A few weeks after that is President's Day.

But then we enter a stretch, from President's Day to Easter, devoid of holidays. Depending on the particular school district and when Easter is in a given year, there may be no holidays at all in the entire month of March. Furthermore, this is often the time when math classes reach some of the more difficult units in the year (such as factoring quadratics in Algebra I), big projects become due in English and other classes, and so on.

I've noticed how tough it is from Prez Day to Easter ever since I was a young students myself. And so I now refer to this toughest time of the year as the "Big March" -- named after the month that largely makes up this period, but also brings to mind a hardworking soldier.

Non-teachers who read this may feel little sympathy for our "Big March." So what, they may say, that we have to work from President's Day to spring break without a day off? In the private sector, not everyone even gets Prez Day off, and they certainly don't get a week off for spring break! They may suggest that we teachers try a real worker's "Big March" -- after New Year's Day, they might not have another off day until Memorial Day!

Here's how I respond to such naysayers -- we teachers work with students, who are too young to appreciate what a private sector work schedule is. All they know is that every few weeks, they get a day off from school until the Big March, when there's no day off. They get antsy, and we teachers have to deal with that. That's why the Big March is a tough time for us teachers.

At my school, students have a slight reprieve from the Big March. It was just announced yesterday that there would be another PD day coming up right at the end of March. Thus the Big March doesn't extend all the way until Easter (which is late this year, April 16th), but instead covers the rest of February and most of March.

For Cardone this may be a time of "Rejuvenation," but for me, I must prepare for the first Big March of my fledgling teaching career. As a substitute teacher the last few years, the Big March was tough, but of course I wasn't called in to sub every single day. But now I really do have to work six weeks between off days.

3) We are reminded constantly of how relational teaching is.  As teachers we work to build relationships with our coworkers and students.  Describe a relational moment you had with someone recently.

This week was Valentine's Day -- a day strongly associated with relationships. My students gave me candy that entire day -- and in fact I ended up eating nothing but sweets until dinner! After the holiday, I had to give some of my candy away -- in fact, during the PD, one of my fellow teachers felt drowsy and wanted some candy to boost her energy, and so I handed her some of mine.

Lately I fear that I've been criticizing certain students too often. Some of my sixth graders are smart but talk too much, and some of my brightest seventh graders have the opposite problem -- they fall asleep in class. I tried pulling them off to the side and let them know that they're smart, but they need to change their attitude to go along with their intelligence. I told one of my sleepy seventh graders about the time when I was myself in seventh grade and nearly failed art class. These pep talks may have helped -- but not if I turn around and start yelling at them again the next day.

In fact, one eighth grader has had enough of all of this. I got upset at her this week when I passed out Valentine's pencils and she decided to take two pencils for no reason. She eventually apologized, but she rejected my own apology for lashing out at her -- probably because she correctly notes that I'll end up yelling at her again at some point. At least I acted like the adult I should be by accepting the girl's apology.

But her actions show that it's too late to repair my relationship with this student. It's far better for me to maintain good relationships with my sixth and seventh graders, as these are the students I'll need to reach not just the rest of this year, but into next year as well.

4) Teachers are always working on improving, and often have specific goals for things to work on throughout a year. What have you been doing to work toward your goal?  How do you feel you are doing?

My goal was to become the ideal classroom manager. Well, let's see how my classroom management went on Wednesday, the last day I had students:

-- Some of my eighth graders took out their cell phones. They put them away only after I threatened to inform the history teacher that they had cell phones out.
-- Some of my sixth graders are loud in class. They become quiet only after I threatened to inform my support staff member that they are loud in class.

So in both cases, the only way to compel the students to be complaint is to invoke the name of some other adult on campus. This doesn't sound like anything the ideal classroom manager must do.

The reason for the cell phone incident is simple -- both Tuesday and Wednesday, the history teacher collected cell phones from all the middle school classrooms. Students had to turn in the phones even if they weren't using them. When no eighth graders surrendered their phones, the history teacher specifically asked me to inform him if any student took out a phone.

On the other hand, the sixth grade situation is more alarming. These students continually talk in class and don't become quiet until my support staff member threatens to take away their nutrition or lunch, or punishes them during P.E. class (which she's in charge of). They've decided at the start of the year that she's the only adult in my room that they need to listen to and respect -- and there's nothing I can do to gain that respect.

In both Thursday's and Friday's PD meetings, we continued Responsive Classroom training. The topic for Thursday was Academic Choice, and Friday's was Morning Meeting. But discipline continued to be a running theme throughout both day's meetings.

All teachers received a copy of the book The First Six Weeks of School, which is published by Responsive Classroom. Of course, we're well past the first six weeks (unless by "first six weeks" we mean "Big March"). But it reiterates the ideas that I mentioned on this blog -- according to the education writers Harry and Rosemary Wong, the first part of the school year is critical.

I refer to the first part of the school year as the "Wong Unit." I think of the "Wong Unit" as the time it takes for me to learn all of the students' names -- which should be shorter than six weeks. Then again, six weeks is a "hexter" (half of a trimester). In a way, this is how long it takes for the students to understand all the academic expectations -- they can't fully appreciate what these expectations are until they receive grades (progress reports) for the first time.

Whether we call it "first six weeks," "first hexter," or "Wong unit," I believe that my problems with the sixth grade extend even further back than this! That one day I "subbed" in November (and which I mentioned in my November DITL posts) revealed flaws in my teaching style -- when a student challenges me ("I wasn't talking!"), my first instinct is to yell.

My New Year's Resolution in 2016 (not 2017, but 2016) was to think about whether my instincts in the classroom made sense -- and if not, to do something about it. By yelling, I was breaking that resolution, since yelling does nothing to instill good behavior in students. And so I should have been improving my classroom instincts as a sub in January 2016 -- seven months before I stepped in the classroom the first time. My support staff member has better instincts than I do -- and that's why she commands the respect of my students more effectively than I can. I have renewed that Resolution for 2017, in the hopes that I may fulfill it this year.

If there's anything I could have done during the actual first six weeks of the school year, it may have been to develop better relationships with my students -- not just learn their names (as I wrote above), but understand their learning styles. That one eighth grader I had a problem with this week is one of the lowest students in the class. I should have realized that math would be a struggle for her and tried to be more understanding when giving her help from the very start of the year. Perhaps if I had done this, she wouldn't have been compelled to steal an extra pencil from me in the first place.

5) What else happened this month that you would like to share?

Earlier this month, a famous mathematician passed away -- Raymond Smullyan. I wrote about him on my personal blog back on New Year's Day 2016 (yes, the same day I wrote that ill-fated New Year's Resolution for 2016). He was famous for coming up with puzzles -- in fact, this is what I wrote last year about him:

David Kung wraps up the lecture with some classic Knights and Knaves problems, which he attributes to the American mathematician Raymond Smullyan (who apparently is still alive at 96 years of age). Here is a link to some of these problems so you can try them out for themselves:


Well, he passed away last week at 97 years of age.

At the time I wrote about Smullyan, I was not yet a full-time teacher. I was watching David Kung's lectures for pure recreation and not anything to do with an actual classroom.

But notice that it's possible for teachers to give Smullyan's puzzles to their students. For example, the aforementioned Denise Gaskins writes that she has her math club students work on his puzzles:


This isn't quite a Smullyan puzzle, but it's today's Pappas question of the day:

1/3 of the people at a party were women, 1/9 were girls, 1/6 were men and 7 were boys. How many at this party?

Notice that 1/3 + 1/9 + 1/6 = 11/18, leaving 7/18 of the party as boys. Therefore there must be eighteen guests at the party -- and of course, today's date is the eighteenth.

This concludes my "Day in the Life" post for February. My next DITL post will be on -- yes, Saturday -- March 18th, and my next personal post will be on Tuesday, the start of the Big March.

Tuesday, February 14, 2017

Activity: The Mathematics of Love (Days 101-102)

Here is today's Pappas question of the day:

(2 + 3i)(8 - 5i)

When in standard complex form, the imaginary part of the above = _____i.

To answer this, we can use FOIL -- actually just OI as only these contribute to the imaginary part. So we have -10i + 24i = 14i. The imaginary part is 14 -- and of course, today's date is the fourteenth.

This is obviously not a middle school problem -- it's Algebra II at the earliest. This means that once again, I give Illinois State questions for Warm-Up, not Pappas questions.

Today is the special activity day that I have planned, at least for sixth and eighth grades. As I wrote before, I have to do Student Journals in seventh grade because the class doesn't meet tomorrow. This class is learning about integer operations. This isn't exactly the topic I want to rush -- that this topic is already getting an extra day, but it really needs much more time. As you would expect, the students are confused when I had to jump from addition/subtraction to multiplication and explain why 7(-2) isn't 14 since the positive number is bigger -- or even worse, 5.

The sixth grade activity comes from Denise Gaskins -- one of the other teachers I met during an MTBoS challenge last month. I promised her that I'd do one of her activities in class and tell her about the experience, so here it is:


The particular activity I choose for today is called "Tax Collector":


This is what I wrote in my comment to Gaskins:

Hello! I commented here during last month's MTBoS challenge. We agreed that if I used one of your games in my classroom, I tell you about it, so here goes!

Anyway, I played Tax Collector with my sixth graders today, and they seemed to enjoy it very much! I began with me as the tax collector and the class as the taxpayers. As it turns out, the first number chosen was 11 -- and the end, the class just narrowly lost to the tax collector, 110-100. If they had started with 19 (or even 17) and made all the same subsequent choices, they'd have won the game!

Afterwards, I divided the class into pairs, with one student as the tax payer and the other as the tax collector. No taxpayer wins, or comes quite as close as our initial game with the whole class, even though by now they knew that it was best to start with a large prime like 19. A few taxpayers believed they had won, but often it was because the tax collector didn't take all the factors correctly -- for example, a taxpayer started with 20, and the collector took only 10 instead of 1, 2, 4, 5 as well.

All in all, it was the most fun we had in class in a while. Thanks for the activity!

In fact, you can see what makes the game so difficult for the taxpayer. If the payer takes a prime, the collector can take only 1. But the payer can never take a prime on any move but the first, since no matter what number the payer takes on the first move, the collector takes 1. So the payer can only take at most one prime the whole game -- the collector is guaranteed all but one of the primes. This is why it behooves the payer to take the largest possible prime on the first move. I'm not sure that my students recognize 19 as a large prime number, but they do learn to start with a large odd number.

The tricky part of the game is when to take an abundant number like 12. If the taxpayer takes such a number too soon, the collector gets many of its factors and takes the lead. But if the payer waits too long to take the number, all of its factors are gone and the collector keeps the number itself. The girl who mistaken thinks that she beats the collector does learn that it's good to take 10 before 20 -- if done correctly, after the payer takes 20, the collector gets only 4.

As I wrote to Gaskins, this is a great activity to get the students thinking about factors -- which is great during the current lessons on fractions and percents.

I want to discuss the eighth grade project -- but first let's look at the song for music break. Since today is Valentine's Day, I just had to sing the Square One TV song "The Mathematics of Love."

By the way, speaking of Valentine's Day, last year I wrote that one group that receives many V-Day goodies is teachers. The students give me candy all day, and one seventh grader even offers me a special holiday doughnut. Meanwhile, the K-1 teacher gives me some red velvet cupcakes to thank me for moving a large toy oven out of her classroom. Along with the fact that I wat waffles with syrup for breakfast and the school cafeteria serves yogurt for lunch, it means that the first thing I eat today that isn't sweet is dinner!

Barry Carter provides us with the lyrics:

The Mathematics Of Love

Lead vocals by Larry Cedar

Backup vocals by Reg E. Cathey, Cris Franco, Luisa Leschin, and Beverly Mickins

Featured vocals by Arthur Howard

A five, six, seven, eight!
One night one night the stars were glowing
Two hearts two hearts were overflowing
Three words hit like a bolt from above
Bum bum bum
Four arms four arms were hugging tightly
Five times five times I kissed you lightly
So goes the mathematics of love
The mathematics of love
One two three forever
I’ll keep on counting the ways
One thousand nights I’ll hold you
And love you all of my days (and love you all of my days)
One night one night the moon was shining
Two hearts two hearts were intertwining
So goes the mathematics of love
The mathematics of love
Seven eight nine tenderly
I’ll hold the memory of
The one night two hearts thundered
The mathematics of love
Great, Tony! You got it!
One two three forever
The mathematics of love
One more time!
The mathematics of love
Alright! Take five!
In this song, the gag is that the lead vocalist reads all the numbers in the song as Roman numerals, so he sings "eye night," "eye eye hearts," "eye eye eye words," and so on. I admit that of all the songs from Square One TV, this one is my favorite.

Now as it turns out, my eighth grade project is based on this song. Part of the project has the students convert from Arabic to Roman numerals and vice versa. This is actually inspired by a project mentioned in the Illinois State text. Recall that two weeks ago was "Input, Process, Output," where the students are introduced to functions. There's a follow-up to this project -- but it's printed only in the teacher's edition, not the student texts. Anyway, the students learn that conversion to Roman numerals is also a function, with an input, process, and output.

The Illinois State teacher's text actually has the students focus on attempting to add, subtract, multiply, and divide Roman numerals. Instead, I mainly have the students convert the numerals and provide only one example for each operation. Naturally, the students find it easier to convert back to Arabic numerals before doing any calculations. The most common mistakes are just as you would expect -- confusing IV = 4 with VI = 6, and IX = 9 with XI = 11, and so on.

The text also asks whether the set of Roman numerals is closed under addition (or any of the other three operations). Notice that as soon as an additively closed set contains 1 (or Roman numeral I), it automatically contains infinitely many natural numbers, so we must ask whether one can write arbitrarily large natural numbers in Roman. At first glance it would appear not, since M is the largest number that has its own letter, and it only represents 1000. But sometimes we place a bar (which is called vinculum in Latin) over a Roman numeral to multiply it by 1000. So V-bar is 5000, M-bar is a million, and so on. So we can write Roman numerals as large as we please, if we use arbitrarily many vincula when writing them. Only then can the set of Roman numerals be closed under addition.

This is only part of today's activity -- the "mathematics" part. Since the activity is called "The Mathematics of Love," I need to represent the "love" part of the lesson. To accomplish this, I take a page from one of my usual fallback blogs -- Sarah Carter:


Even back when I was the age my students are now, I often found myself thinking about the boyfriend and girlfriend functions. As we see from Carter's blog, boyfriend and girlfriend are supposed to be functions, but in practice, many people cheat and so they're no longer functions, as one input (boy) is paired with more than one output (girl, etc.).

This example fits perfectly today, since it fits both the holiday (V-Day) and the content (functions). I begin with some examples of functions (such as father) and non-functions (such as brother) before moving on to boyfriend. This is actually part one of my activity -- part two is Roman numerals.

As it turns out, this week I was supposed to sign my sixth graders up for a Playworks activity, but I don't find out until it's too late. There was an open spot this morning, but I missed it. In a way, I don't mind that much -- the sixth grade math activity goes well. In fact, the students behave reasonably well during the activities -- of course they want to misbehave during traditional lessons. I'd much rather give up the SBAC Prep time to Playworks -- with all the talking that's going on, precious little test prep is accomplished anyway.

This is a two-day post, so we look ahead to tomorrow's lesson, particularly the science lesson. The next project in the Illinois State science text is on physical and chemical changes. Notice that the seventh graders at my sister school already had this lesson back in November -- the day my car broke down and I subbed for my counterpart.

I don't know what tomorrow's Common Planning meeting is about ye. But I do remember hearing that there would be another meeting with the Illinois State developers on the third Wednesday in February -- which works out to be tomorrow, the 15th. Right now, the administrators aren't checking to make sure that we're implementing the Illinois State curriculum (hence today's non-Illinois State sixth grade activity). Instead, the concern is with the SBAC Prep time and making sure that the students are taking the practice SBAC test online. But our student laptops are old, and they can't be upgraded enough to download a browser powerful enough to access the website. (That's another reason why I could have given up "SBAC Prep" time to Playworks!)

Again, this is a two-day post, covering today and tomorrow. So the next school day post won't be until after the five-day weekend, on Tuesday the 21st. But recall that my monthly posting date, for "Day in the Life" is the 18th, a Saturday. So not only is it a weekend, but it's smack-dab in the middle of a long holiday weekend!

Oh well -- my next post will be "Day in the Life" on Saturday.

Monday, February 13, 2017

Coding: Google Calendar (Day 100)

Here is the today's Pappas question of the day:

ABCD is a four-digit number with the following properties:

1. C is the only even digit, with all the other digits odd.
2. The digits 5, 6, 7, and 8 do not appear.
3. Each digit is greater than the previous digit.

Question: What number is AB?

To solve this problem, we notice that by Rule #3, our number probably starts with a low digit so that it can end with a high digit. So let A be the smallest possible thousands digit, 1. This works by Rule #1, which tells us that A must be odd. We know that B must also be odd, so it can't be 2 -- so let's try 3 for B. Now C can be 4, since by Rule #1 it must be even. As for D, it can't be 5, 6, 7, or 8 by Rule #2, so we must go all the way to 9 for D. As this is the largest digit and we chose the smallest possible value for A, B, and C, there are no other possibilities. The number ABCD is 1349 -- but the question asked for AB. So the answer is 13 -- and of course, today's date is the thirteenth.

I call this a Pappas question, and her calendar does occasionally give digit questions, although none of them are exactly like this. This is actually more like one of the Monday Five questions that I gave my seventh graders today. It's not exactly the question from the worksheet, though -- I manipulated it so that the answer would come out as thirteen. Thus both Pappas and the worksheet influenced today's question.

Monday Five -- that's right, today must be a coding Monday. And unfortunately, this is a one-day post, so I'll be writing about very little math today outside of the Monday Five question.

The sixth graders are creating a survey, and they are able to answer each other's surveys. The seventh graders are continuing with spreadsheets -- today they research data on whatever topic they like (such as sports) so that they can create a spreadsheet with that data. The coding teacher decided that Unity is too tricky for the eighth graders. So instead he shows them some Google apps -- today's app is Google Calendar.

Since today's a coding Monday, and I have nothing else to write about today, let me discuss some issues related to the school calendar. After all, the eighth grade coding lesson is about Google Calendar, so my calendar discussion fits perfectly!

Lincoln's Birthday. As I wrote in my last post, today is the Lincoln's Birthday holiday at many schools, but not the LAUSD and hence not my charter. The districts for which I subbed last year did observe Lincoln's birthday, which is why I observed the holiday on the blog.

On the other hand, remember that our charter has PD days coming up on Thursday and Friday. Thus the students, in effect, will be able to enjoy a five-day President's Day weekend.

Some other districts observe Lincoln's Birthday differently. Instead of celebrating Honest Abe on the Monday before President's Day, they do so on the Friday before President's Day. The result is the observance of a four-day Prez weekend. When I was a young sixth grader, my school did this -- plus there was a PD day on the following Tuesday, so we had a five-day weekend. This only lasted for one year -- we returned to having consecutive Mondays off the following February.

But our upcoming five-day weekend has nothing to do with Lincoln. We teachers must put in a full day of PD on both Thursday and Friday, with Monday as our only day off.

Day 100. Today is the 100th day of school, a day celebrated at elementary schools. I've mentioned this on the blog before as a mere curiosity -- but notice that I now actually work at a K-8 school, which means that Day 100 is finally relevant. Day 100 is often celebrated in kindergarten and first grade classrooms, as these are the years when students learn to count to 100.

Despite this, no one actually celebrates Day 100 at my school today. There are two reasons for this -- first, notice that tomorrow is Valentine's Day. It's awkward to have a big party today for Day 100 and then a big party for V-Day tomorrow -- a three-day week with parties on two of the three days! The official recommendation from our director (principal) is to wait to celebrate Day 100 until after the five-day weekend, on Tuesday the 21st.

The second reason is that, for the kindergartners, the 100th day of school is next week anyway. The first day of school for Grades 1-8 was Tuesday, August 16th, but kindergartners didn't go to school that week. Indeed, I don't think our kindergarten teacher was hired until the following week. The Kinder teacher herself counted Monday, August 22nd as Day 1 (which was our Day 5), and so her Day 100 is our Day 104, Wednesday, February 22nd.

The only relevant Day 100 is counted by the grade that actually celebrates it -- kindergarten, not middle school. So our school will have the Day 100 party one day later than the recommendation by our director, on the Wednesday of next week.

The three-day week. Recently, my weekly plan has been the following:

Monday: Coding
Tuesday: STEM Projects
Wednesday: Student Journals
Thursday: Learning Centers
Friday: Assessment

This week there is no school on Thursday and Friday, so the resulting week looks like this:

Monday: Coding
Tuesday: STEM Projects
Wednesday: Student Journals

This eliminates the assessment at the end of the week. In fact, when I submitted my plans to cover all Major Content (MC) before the SBAC, I left this week open as I knew it would be a short week. So I didn't even bother to list a Learning Module for this week. This allows me to give more time for topics that I know will be difficult for the students.

In eighth grade, I actually gave an extra week for EE7b, on solving multi-step equations. I already started Learning Module 11 -- the corresponding standards of which are actually SP1 and SP2, which are in the statistics strand. Notice that SP1 and SP2 are not MC -- yet they appear on the second trimester benchmark tests that are coming up soon. I still don't like how Illinois State provides both a "Year View" pacing plan and benchmark tests, yet one doesn't correspond to the other!

For sixth and seventh grades, I'll continue the standards that I began last week -- since both of these are huge ones. Sixth graders are learning about percents for the first time. And the seventh grade standard is integer operations -- I covered mostly addition last week, so that leaves subtraction, multiplication, and division.

Seventh grade is especially challenging since that I don't see that grade on Wednesdays! It means that tomorrow is the last time I'll see them for a full week! Naturally, the Student Journals are when the students learn new material, so I'll be doing that with the seventh graders tomorrow.

I also use the SBAC prep time to attempt to sneak in new material, especially for seventh grade. In fact, today I mention a practice SBAC question based on EE1 -- on basic factoring. In a way, this is just GCF factoring, or the distributive property in reverse. But it's tricky when embedded into an SBAC question which directs the students to find which step contains an error.

Other issues. There have been several other things going on in my classroom that I'll mention in this coding Monday post.

The instructional aide has been continuing to work with my classroom management. This time, she tries to enhance my classroom management via desk arrangement. The seats, previously divided into groups of four, are now arranged in order to form a sort of horseshoe shape, which allows the teacher to reach the students more easily. The ideal horseshoe has already been implemented in the English class next door, but due to the location of my front board and projector, a modified horseshoe is in place in my classroom.

I remember back when I was student teaching, and I was provided a copy of Fred Jones and his work Tools for Teaching, which I still own. The horseshoe is one arrangement that Jones recommends.

We have been required to submit a list of high-, middle-, and low-achieving students to our director, along with plans for how we will help each group. The new seating arrangement goes hand-in-hand with these plans.

Given a list of high- and low-achieving students, some teachers prefer to group homogeneously, by placing students at the same level together. The alternative is to group heterogeneously, by seating a high student next to a low student. Traditionalists tend to prefer the former -- indeed, they often advocate going one step further to full-blown tracking (which is controversial -- I explain why in vacation posts and keep the controversy out of school day posts). The English teacher also appears to be grouping the students homogeneously.

There seems to be conflicting directions from the administration. On one hand, coming up with plans for each level implies homogeneous grouping. But we were also asked to list helpful students -- that is, strong students who can assist weaker students in heterogeneous groups. In the end, I seated the student so that stronger students can help the weaker ones out -- a difference from English class. But the seventh grade groups somehow end up more homogeneous. This could be helpful during the shorter seventh grade week, when the traditional lesson could bleed into Learning Centers (so I'll want the higher group to work on the new material independently while I help the lower group).

On the other hand, since the other two grades do meet on Wednesday, I could do projects with sixth and eighth grades tomorrow. Notice that as there is no new Learning Module this week, there is no STEM project either. I want to take advantage of this and provide activities that don't come from Illinois State. Which projects will I choose? Well, you'll have to wait until tomorrow's post to see.