Month: February 2015

Two Organizational Things I Do

I don’t know if I’ve blogged about these things before. These aren’t Two Classroom Ideas That Will Completely Change Your Teaching or anything. In fact, I’m willing to bet that many of you have tried or currently do something similar. But for me, these two things have made my life easier and my classroom run more smoothly. So in case this helps…

The First Idea

In Geometry, I want my kids to learn to use multiple tools, and find the tools that are the most useful to them at any given moment. One moment they might need patty paper to trace something. Another moment they might need eraseable (this is key!) colored pencils to emphasize different things. Another moment they might want to pull up Geogebra on their laptops. And another moment, they might need a ruler to draw a straight line. Who knows. So what I did at the start of the year was create geometry buckets, populated with the tools that each group might need at any given time.

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I have a different bucket for each group. I color coded most of the items in the bucket (with the exception of the protractors, because I didn’t want to cover any of the angles!). I store the buckets in the room. At any point, kids are allowed to grab them. Sometimes they have to, because they are asked to measure an angle, or draw a circle. When I have them use the giant whiteboards, they have their dry erase markers and an eraser in the buckets. But most of the time, when a kid needs some patty paper, or a ruler to make a diagram, or colored pencils to organize their ideas or annotate a diagram, they’ll just grab the bucket and bring it to their groups. And at the end of each class period, the kids will just put them back.

I thought things would get lost or mixed up. But it’s been a semester, and I just went through the buckets and have found only a few colored pencils were in the wrong boxes, and only a single compass migrated from one box to another. I love these buckets of geometry tools!

The Second Idea

I do tons of groupwork in my classes. And I try to switch up groups often enough for some spice, but let them work together long enough so they can learn to work together (I try to do it two times a quarter). However, when kids are in groups, passing things out and collecting things can be annoyingly time consuming. And if my kids know one thing about me as a teacher: I don’t waste time, not a second.

So here they are: something I’ve been doing for the past few years. Folders. Specifically, each group gets one folder.

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On the front (not photographed), I have a label with the kids’s names on it. Inside are two pockets. The left hand pocket is for things I normally would hand out. (Mainly: the packets that I make for kids to work collaboratively through.) The right hand side has two purposes: (1) I have kids turn in nightly work sometimes, so they will put it in there, and (2) when I mark up the nightly work, I put it back in there and students collect it the next day.* There are also some “The Dog Ate My Homework” forms for when a kid doesn’t turn in their work. Instead of them calling me over and giving me a story explaining what happened, I just have them fill out that form saying why they didn’t have their work.

One huge benefit for having these folders is that it allows me to mix up where the groups sit each day.** When I walk into the classroom, kids aren’t sitting down usually. They are waiting for me. I throw down each folder on group of desks, and then kids sit at the group of desks with their folder on it. That way: kids are in different locations each day, mixing things up. The group in back won’t always be in back! Sometimes I give a kid the folders to put down, and sometimes the power, the sheer power of who sits where, goes to their head. (“Oh, you’re standing by this group of desks? Too bad, I’m putting your folder waaaay over at that far group of desks.”) Fun times.

Now you might say: each day you have to put in the packets you’re going to hand out the next day? Nope.Well, sometimes. But usually not. In classes I’ve taught before, where I have my ducks in a row, I do a massive photocopying of the papers for the entire unit. I lay them out, and fill up the folders. Then I’m pretty much set for a week or two (or more!). Below is a picture of me doing that today!

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That is all. Go back to your regularly scheduled lives now.

UPDATE: I forgot to say: I color code the folders for each class. So red folders = my geometry class, blue folders = one of my precalc classes, green folders = the other one of my precalc classes. I also use a lot of file folders to organize things for me. And for those, I use the same color folders for each of my classes. So, for example, when I give a geometry test, I bring a red file folder to class. And then I keep the taken geometry tests in that file — and when I’m going home, I just throw that red file in my backpack so I can mark ’em up.

*The fact that there is one folder per group also has the added bonus that when one kid forgets to put their name on their nightly work, you know exactly whose it is, because it is in the folder for that specific group (and usually all the other kids put their name on it).

**Someone, somewhere, told me that there was some ed research that suggested that kids sitting in the same spot every day helped them learn better. I have my doubts about that.

Angle Bisectors of a Triangle and the Incenter (and lots and lots of Salt)

In Geometry, we’re about to embark on the whole triangle congruence bit (SSS, SAS, ASA, etc.).  Below is the plan we have outlined.

The TL;DR version: By learning about angle bisectors, we motivate the need for triangle congruence. We have students figure out when they have enough information to show two triangles are congruent. They use this newfound knowledge of triangle congruence to prove basic things they know are true about quadrilaterals, but have yet to prove that they are true. Finally, we return back to angle bisectors, and show that for any triangle, the three angle bisectors always meet at a point. As a cherry on the top of the cake, we do an activity involving salt to illustrate this point.

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STEP 1: To motivate the need to figure out when we can say two triangles are congruent when we have limited information, we are showing a problem where triangle congruence is necessary to make an obvious conclusion. 

We’re going to see the need for triangle congruence to show that:

any point that is equidistant to an angle (more precisely: the two rays that form an angle) lies on the angle bisector of that angle.

This will be an obvious fact for students once they create a few examples. But when they try to prove it deductively, they’ll hit a snag. They’ll get to the figure on the left, below. But in order to show the congruent angles, they’ll really want to say “the two triangles are congruent.” But they don’t have any rationale to make that conclusion.

Hence: our investigation in what we need in order to conclude two triangles are congruent.

When kids do this, they will also be asked to draw a bunch of circles tangent to the angles. All these centers are on the angle bisector of the angle. This will come up again at the end of our unit.

fig3

Kids will be doing all of this introductory material on this packet (.docx)

STEP 2: Students discover what is necessary to state triangle congruence.

This is a pretty traditional introduction to triangle congruence. Students have to figure out if they can draw only one triangle with given information — or multiple triangles.

We’re going to pull this together as a class, and talk about why ASA and SAS and SSS must yield triangle congruence — and we’ll do this when we talk about how we construct these triangles. When you have ASA, SAS, SSS, you are forced to have only a single triangle.

I anticipate drawing the triangles in groups, and pulling all this information together as a class, is going to be conversation rich.

The pages we’re going to be using are below (.docx) [slight error: in #4, the triangle has lengths of 5 cms, 6 cms, and 7 cms]

STEP 3: Once we have triangle congruence, we’re going to use triangle congruence to prove all sorts of properties of quadrilaterals.

Specifically, they are going to draw in diagonals in various quadrilaterals, which will create lots of different congruent triangles. From this, they will be asked to determine:

(a) Can they say anything about the relationship between one diagonal and the other diagonal (e.g. the diagonals bisect each other; the diagonals always meet at right angles)

(b) Can they say anything about the relationship between the diagonals and the quadrilateral (e.g. one diagonal bisects the two angles)

(c) Can they conclude anything about the quadrilateral itself (e.g. the opposite sides are congruent; opposite angles are congruent)

We have a few ideas percolating about how to have students investigate and present their findings, but nothing ready to share yet. The best idea we have right now is to have students use color to illustrate their conclusions visually, like this:

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STEP 4: This is a throwback to the very start of the unit. Students will prove that in any triangle, the angle bisectors will always meet at a single point.

Here are the guiding questions (.docx)

And we will finish this off by highlighting the circles we drew at the start of the unit. Notice we have a single circle that is tangent to all three sides of the triangle. The center of that circle? Where the angle bisectors meet. Why? You just proved it! That location is equidistant to all three sides of the triangle.

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STEP 5: The reason we’re highlighting these circles is that we’re going to be cutting out various triangles (and other geometric shapes) out of cardboard, elevate them, and then pour salt on them. Ridges will form. These ridges will be angle bisectors. Why? Because each time you pour salt on something flat, it forms a cone. The top of the cone will the the center of the circle. We’re just superimposing a whole bunch of salt cones together to form the ridges.

The other geometry teacher and I both saw this salt activity at the Exeter conference years ago. Here are some images from a short paper from Troy Stein (who is awesome) on this:

The general idea for this activity is going to be: kids take a guess as to where the ridges are going to be, kids pour the salt and see where the ridges are.

As the figures get more complex, they should start thinking more deeply. For example, in the quadrilateral figure above, why do you get that long ridge in the middle?

My hope is that they start to visualize the figures that they are pouring salt on as filled with little cones, like this:

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The material I’ve whipped up for this is here (.docx):