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.

No comments:

Post a Comment