Last night I tutored my geometry student. He is now in Chapter 6 of the Glencoe text, which covers various polygons. Let's see how this chapter is organized:

Section 6-1: Angles of Polygons

Section 6-2: Parallelograms

Section 6-3: Tests for Parallelograms

Section 6-4: Rectangles

Section 6-5: Rhombi and Squares

Section 6-6: Trapezoids and Kites

Clearly, Chapter 6 of the Glencoe text corresponds best to Chapter 5 of the U of Chicago. Angles of polygons (most notably, their

*sum*) is in U of Chicago's Section 5-7. Notice that the parallelogram consequences in Glencoe's 6-2 and 6-3 wait until U of Chicago's 7-6 and 7-7, respectively. The rest of the Glencoe chapter corresponds to U of Chicago's 5-4 and 5-5.

My student had just finished Section 6-4 in Glencoe, on rectangles, and he asked me whether I had a worksheet prepared for rectangles. But there are two main differences between the U of Chicago text and a more traditionalist text such as Glencoe. The first is the U of Chicago's dependence on

*symmetry* (reflected in the Common Core standards), while Glencoe uses triangle congruence to derive the parallelogram properties. The other is the U of Chicago's

*inclusive* definitions of the various quadrilaterals. Recall that in the U of Chicago, a parallelogram is a trapezoid, and a rectangle is an isosceles trapezoid. Glencoe, like most texts of its era, defines

*trapezoid* exclusively.

My worksheets are based on the U of Chicago. In this text, rectangles are covered in Section 5-5, "Properties of Trapezoids," because a rectangle is a trapezoid -- indeed, an isosceles trapezoid. In fact, the properties of the rectangle derive trivially from those of the isosceles trapezoid. This would only confuse my student since he is only in Section 6-4, while trapezoids don't occur until Section 6-6 of his text. So my worksheet shows how the properties of the rectangle and isosceles trapezoid are related, while he doesn't even know what a trapezoid is yet! Glencoe can define

*rectangle* before defining

*trapezoid*, since a rectangle isn't a trapezoid in that text. On the other hand, notice that

*rectangle* is defined after

*parallelogram*, since a rectangle is still a parallelogram in that text.

Now my student wants to see a proof involving rectangles. I decided to prove the least obvious properties of rectangles -- namely that their diagonals are congruent. Notice that my Section 5-5 worksheet contains a proof that the diagonals of

*isosceles trapezoids* are congruent. And so I was forced to show him the Quadrilateral Hierarchy and explain what a trapezoid is. When I finally showed him a proof, I used a traditional one based on SAS -- since that is what he's expected to learn in his Glencoe class -- rather than use the Rectangle Symmetry Theorem.

I like the idea of showing that, since rectangles are isosceles trapezoids and isosceles trapezoids have congruent diagonals, so do rectangles. Rectangles inherit all the properties of isosceles trapezoids -- just as they inherit all the properties of parallelograms. The inclusive Quadrilateral Hierarchy means that there's less to memorize, and less to prove. But the benefits are lost if we suddenly spring the inclusive hierarchy on someone who learned the

*exclusive* hierarchy!

Meanwhile, today I subbed in a math class. The regular teacher has three sections of Calculus (periods 3-5) and two sections of 9th grade Integrated Math (per. 1 and 6). I point out that to opponents of Integrated Math, this is ironic, since it's because of Integrated Math that there might not be any Calculus students in three years -- and this is especially true since "9th grade" Integrated Math is based on

*8th grade* packets.

The "9th" (8th) grade MathLinks packet for today's class is "Slope and Slope-Intercept Form of a Line." I noticed that once again, today's lesson is geared more towards algebra than geometry, even though the 8th grade Common Core Standards has both algebra and geometry. I was hoping that there'd be more geometry now that we're in the second semester, but so far this has not turned out to be the case.

Here are the sections of the current Student Packet 8-8:

8.1 Introduction to Slope

8.2 Input-Output Investigation

8.3 Slope-Intercept Form

8.4 Skill Builders, Vocabulary, and Review

The students are now in Section 8.2, "Input-Output Investigation." Today, the packet directs the students to play a game: the Input-Output Game. Here are the rules of the game, as written by the teacher himself:

*One person thinks of a combination of cups & counters -- the other person then asks two questions (numbers) to determine the equation, the value of y when x = 0, and the slope (also graph the equation). After this, the players switch responsibilities and repeat the process.*
Notice that this defines a function whose slope is the number of cups, and whose intercept is the number of counters. That is, the input

*x* represents the number of counters in each cup.

But this game turned out to be highly problematic. Here are some of the problems that I had when I played this game in first period:

-- Many students couldn't choose a good number of cups and counters.

-- When they did choose a number of cups and counters, they wrote the number of cups and counters under "Input" and "Output." (In other words, they wrote the values of

*m* and

*b* where the values of

*x* and

*y* are supposed to go!)

-- Because this is a partner activity, if the first student doesn't perform his/her task correctly, then the second partner can't perform his/her task at all.

-- When I tried to correct them, many students were talking and not paying attention to me at all.

And all of this is despite the teacher having shown them how to play the game during yesterday's class! (Then again, the teacher showed them a specific number of cups -- three -- and counters -- two -- and input values

*x* = 2 and

*x* = 4. It's easier for the students when they don't have to choose the numbers!)

I remember back when I was student teaching in a traditionalist Algebra I course, there was a similar activity near the beginning of the school year. The students were to divide into pairs. One student would choose a value of

*x* -- say

*x* = 4 -- and then perform the following tasks based on the number they chose:

Multiply both sides by 4: 4

*x* = 16

Add 10 to both sides: 4

*x* + 10 = 26

Divide both sides by 2: (4

*x* + 10)/2 = 13

Subtract 1 from both sides: (4

*x* + 10)/2 - 1 = 12

The second partner now receives the equation (4

*x* + 10)/2 - 1 = 12 and now must solve that equation. The whole point of this activity is for the students to see that they must perform inverse operations in order to solve the equation -- instead of

*subtracting 1*, they should

*add 1* to both sides. Instead of

*dividing by 2*, they should

*multiply by 2* on both sides, and so on.

But once again, the problem was that the first student didn't do his/her task right -- and the second partner's success is contingent on the first student's. Both that day and today, it was often the

*second partner* who was feeling frustrated, since the second partner couldn't do

*anything* if the first student makes a mistake.

Now this is often the point where traditionalists point out the flaws with a more progressive curriculum. The partner activity failed because when one partner couldn't perform the task, the second can't even begin. Instead, the students need to be led not by a fallible student partner, but by the one person who's supposed to know already what he/she is doing -- the "sage on the stage," the

*teacher*. My problem with pure traditionalism at this age is that it presumes that the students are even willing to listen to the teacher. I can easily see a teacher giving a traditionalist lecture on this material and the students simply tuning him/her out -- especially when it's time to graph.

(By the way, just as another word for the progressive philosophy is

*constructivism*, there is a similar-sounding word for the traditionalist philosophy --

*instructivism*.)

To me, I could improve the lesson not simply by tossing it out completely to replace it with a traditionalist lecture. Instead, I intervene at the points where students are likely to make a mistake. Here are the changes that I implemented by the time I reached the other 9th grade class -- 6th period:

-- In both the student teaching class and today's subbing class, the students had trouble choosing numbers. I conclude that it's a bad idea to have students choose anything in this sort of activity, except to choose a partner. (And some teachers oppose having the students choose their partners as well!) Instead of having the students choose the numbers, I handed one partner a post-it note

*telling* them how many cups and counters there were. I prepared the post-it notes during the teacher's conference period (2nd).

-- I told the student to whom I gave each post-it note to

*cross out* the chart showing the input and output. That chart is only for the second partner, so the first student should cross it out rather than put numbers in the wrong places. In theory, the students switch roles after each game, so that the first partner becomes the second. This means that I can only play half as many games as the teacher intended -- but I'd much rather play

*fewer* games

*correctly* than

*more* games

*incorrectly*.

-- After that, instead of having the second partner

*choose* the input values,

*I*, the teacher, make that choice. I chose

*x* = 3 for the first input, and

*x* = 1 for the second. So the second partner writes these values under the "Input" column. The first person multiplies the input by the number of cups, then adds the number of counters, and then reports this number as the "Output" for the second person to write down. The second person's job is still to guess how many cups and counters there are.

-- The second partner gives the equation, the value of

*y* if

*x* = 0 (that is, the

*y*-intercept), and the slope. Finally, both partners work together to graph the line. My job is to make sure that the lines are graphed correctly.

Well, with these changes, 6th period fared somewhat better than 1st. A few students were still confused, especially with the graphing. But at least some were less confused than period 1 was with everything

*before* the graphing.

In case you're curious, the Calculus classes were working on a project of their own -- an "Optimal Can Project." The students were to divide into groups and determine what dimensions maximize the volume of a cylindrical can for a given surface area. The connection to geometry, the topic of this blog, is obvious -- we need to formulas for the volume and surface area of a cylinder. We will reach Chapter 10 of the U of Chicago text, on the volume and surface area formulas, some time in late March or early April.

In some ways, the cylinder optimization project is a bit difficult. One only has to do a few

*rectangle* optimization questions -- that is, give the dimensions of a rectangle to maximize the area given a fixed perimeter -- before one conjectures that the optimal rectangle is the

*square*. The teacher has to modify the question -- such as stating that the rectangle needs fencing on only three sides (because the fourth is either a river or a wall) -- in order to prevent the answer from being a square. But with a cylinder, the answer is less predictable.

Sometimes I like to post on the blog based on what I tutor or teach. I'm considering posting an activity similar to the one that I taught today in class, except that it would be on

*vectors*, since that's what we covered today. I didn't want to create the activity before trying it out in 6th period today, and by then it's too late for me to create the activity since I want to make sure that students don't have to

*choose* the vectors, so I'll have to make a cut-out page with all of the vectors on it. Instead, I post my originally planned lesson for Section 14-6, which contains many of those properties of vector addition from the Common Core Standards that I mentioned yesterday.