Introducing rigid motions

Today we introduced ourselves to rigid motions for the first time in geometry. Here’s the run-down:

  1. Give our definition of congruence.
  2. Give students one minute of think time and four minutes of write time to brainstorm all their possible answers to “How can I show that these two objects are congruent?” (I should have called them “figures”. I probably should have at least rotated one of them.)
  3. Have every student write their favorite method up on the whiteboard.
  4. Ask students to respond to the various methods on the board. Lead a pretty good discussion.
  5. Settle on, and define, superposition as our method of choice for the time being.
  6. Give students one minute of think time and four minutes of write time to brainstorm all their possible answers to “What moves are valid to use for superposition?”
  7. Have every student write their favorite move on the board.
  8. Ask students to respond to the various moves on the board. Lead another pretty good discussion.

Here’s a pdf of the (unedited) responses my students put on the board. Some thoughts:

  • Students struggled to differentiate between the definition of congruence and the method by which figures could be shown to be congruent. Many students said we could show the two figures congruent by showing they had the same size and shape. This is true, but also just the definition.
  • The Greeks’ obsession with compass and straightedge geometry stands very far afield from my own students’ relationship to geometry. Some form of “just measure it” was common to almost every student.
  • My students understand that figures have properties, but struggle to identify which of those properties are sufficient conditions for determining other things. Last year, I kind of assumed that students came with both abilities.
  • My students are more comfortable with translation and rotation than reflection. Most students are aware of all three concepts, if not the terms.
  • Two of the three classes came up with superposition on their own. Cool.

I’ve been stressing the idea of informal proof and making a case for why a thing is true (or false) this year. Today we went through each item on the board and formulated arguments for the truthiness of them. We came up with counterexamples for things they claimed were false (“draw two figures with the same area that aren’t the same shape”, “draw two figures with six squares that aren’t the same size or shape”). We made arguments to explain why certain pairs couldn’t exist (“why can’t we draw two congruent triangles, one with a right angle and one with no right angles?”). We talked about how it’s important to actually be clear in what you write, so people understand what you mean.

The conversations were good. Not great, but good. I am a much better facilitator of discussion than I was when I first started all this. I still need to be better. But students were engaged and thinking, and, most important when compared to last year, students were developing their own counterexamples and arguments to develop mathematical understanding.

Tomorrow we will practice our rigid transformations by using them to show that various pairs of figures are congruent and begin to think about how the coordinates change under each operation so we can develop algebraic rules. If the future, we will hopefully utilize these experiences to move toward more formal proofs.

This year is going a lot slower than last year, but I think it is paying off in a big way, because my students are actually thinking and justifying their own thoughts.



Last year I introduced geometric sequences via zombies in algebra 1. Because zombies = instant engagement. (I’m only half-joking here, by the way. Zombies really did equal instant engagement for 95% of my algebra 1 class last year.)

Here’s how it went down:

  1. Have the gross zombie on the projector when students are walking in.
  2. Answer “You watch The Walking Dead!?” affirmatively a bajillion times.
  3. Have students take out some paper to keep track of their work on.
  4. Demonstrate the concept of one zombie eating another zombie with fun role play. Explain that there are no zombie hunters for today’s simulation.
  5. Let students get to work answering the question on the board. Wait for someone to ask how many people live in Seattle. Go to slide two. Remember how great Seattle is.
  6. Let students continue working to answer how long until Seattle is completely zombiefied.
  7. Share results. Talk about how the growth rate is different from the linear models we’ve seen in the past.
  8. Switch to slide three. Take bets on how long it will take. Let students get to work.
  9. Share results. Talk about how everyone overestimated the time it would take because they didn’t really realize how quick “exponential growth” is. Show the graph on slide four. Talk about the two tails.
  10. Work through slide five, introducing notation.
  11. Have students suggest their own special zombie diets. Calculate the growth rate for these problems. (Zombies eat twice a day; zombies eat three times a day; zombies eat every other day. Zombies have to contend with zombie hunters.)
  12. Work through the Zombiepocalypse: Day 2 handout during the following class period.

The Day 2 handout might be a little too verbose for your needs. My algebra 1 students do well with lots of repetition and scaffolding. They were able to work through these two days well, and they also seemed to enjoy it.

I got the idea for this from someone out there on the internet, but I have no idea who. So if this looks like what you did, let me know so I can give you some credit.

Curricular Goodies: Walking Dead (ppt) | Zombiepocalypse: Day 2 (docx). They were run through the Pages and Keynote wringer, so hopefully the formatting isn’t too wonky.

Partitioning a line segment (and finding the midpoint)

One of the Common Core standards for geometry is partitioning a directed line segment into a given ratio (G.GPE.6). This falls firmly within the category of Things I Never Learned in School.

Cursory Googling led to a nice little formula, which Shmoop calls the “section formula”: \displaystyle \left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right).

Formulas are nice, but my students don’t do so well with them. As a general rule, they aren’t willing to really commit to memorizing anything. By the end of the year, this results in a crazy mess of half-remembered formulas. I’m trying to avoid that this year, so I’m leaning more toward processes and thinking that are generalizable… even if this means students spend longer solving specific cases than they otherwise would have to.

Thus, the proportions method of finding midpoint. I really like this because last year my students had no idea what the midpoint formula meant… despite the fact that I felt like we drew it out pretty well. On the other hand, my students are really good at proportional reasoning. I also really like this method because it is generalizable: it doesn’t just work for finding the midpoint when given two endpoints, but it also works for finding one endpoint when given the midpoint and another endpoint, or for finding the point that partitions a line segment into a given ratio. It even works when given one end point, the point of division, and the ratio. In other words… it always works. One process to rule them all. Holla.

So here’s what you do: “Find point C on \overline{AB} such that AC:CB=4:3, if A=(2,3) and B=(4,5).”

First, draw a picture. The picture is very important, because when you’re doing these problems, you’re talking about directed line segments, so order matters. I always draw the lines as though they were perfectly horizontal, but you could draw them accurately if you liked. What’s important here is the order of the points.

We use (x,y) for the coordinates of C, because we love algebra and when we don’t know something, we use a variable.

Now you set up a fancy proportion to find the x-coordinate. We need to find distance from A to C considering only the x-coordinates. Hence, x-2. Again, order is important: subtract the more-right coordinate from the more-left coordinate.  Do the same for the distance from C to B and get 4-x. When all is done, you get the proportion \displaystyle \frac{AC}{CB}=\frac{4}{3}=\frac{x-2}{4-x}. Solve for x and you get the x-coordinate of C is 22/7.

Now do it again, but considering the y-coordinates. The proportion is \displaystyle \frac{AC}{CB}=\frac{4}{3}=\frac{y-3}{5-y}. Solve for y and get the y-coordinate of C is 29/7.

So \displaystyle C=(22/7,29/7) and AC:CB=4:3.

If you want to find the midpoint, use a 1:1 ratio. If you want to find an endpoint when you’re given one of the endpoints and the midpoint, just follow the exact same process, and it will all work out in the end. My students grasped this a lot quicker than they did the midpoint shenanigans last year. And they can solve problems they couldn’t have last year. Go team.

If you’d like to derive the midpoint formula, use A=(x_1,y_1), B=(x_2,y_2) and the midpoint C=(x,y). Go through the proportions and out pops \displaystyle C=\left(\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}\right)

Here’s the worksheet I used when I introduced the process. We walked through problem #1 together with me just helping to organize everyone’s thinking. Here’s a solution key that is hopefully legible.

Algebra 2 scope and sequence

At long last, we conclude our tour of course outlines. The biggest problem I had with the way I taught algebra 2 last year was that each topic was taught in isolation. Once we finished quadratics, we never really went back and did anything with them. And then we learned about polynomials, and forgot about them. And then we learned about rational functions, and forgot about them.

So my primary goal for this upcoming year was to come up with a curriculum that—surprise!—flowed and referred back to previous topics. (It turns out that all of my courses last year suffered from the same problems….)

With that in mind, I finally settled on aligning the course around what I’m calling the Five Expectations. They are:

  1. Create and solve equations.
  2. Graph functions.
  3. Transform and combine functions.
  4. Describe and interpret functions and their graphs.
  5. Find and use inverses.

In many ways, the course outline that I came up with looks a lot like the traditional one I used last year. In the end, I decided that the problem wasn’t that the sequencing of units was wrong, but that there weren’t any threads flowing throughout the course to bring these various function families together. My hope is that the Five Expectations will focus my teaching and the students’ work in such a way that each class of functions is seen as a new tool for doing the mathematics that the Expectations describe.

Of note about the algebra 2 standards list is that there is a lot of material listed that is review. I ultimately decided to leave it all in, because (1) the rest of the department teaches those topics in algebra 2 and (2) I noticed last year that my students still really struggled with the review material. Your mileage may vary.

As with the algebra 1 and geometry outlines and standards documents: feel free to use in any way you find useful. If you have suggestions for making the documents better, please share!

Algebra 1 scope and sequence

Many of the same problems that plagued geometry last year also affected algebra 1. The largest of which (to me) was a lack of flow. Just like in geometry, it is easy to teach units in algebra and then only come back to them on a surface level, if at all. So with the new curriculum outline, I wanted to have some sort of logic to the unit order: one unit should hopefully flow to the next somewhat naturally, and later units should build on and use earlier units.

As always, feedback welcomed, and feel free to use and abuse to your heart’s content. If you do make improvements, I’d love to see them!

Geometry Scope and Sequence

Last year was my first year teaching at the high school, and it was definitely an interesting experience. One of the most frustrating things for me was that I didn’t make the move until August, which did not leave much time for planning. Whereas in the past I had come into the year with a list of standards and an idea of how my courses would flow, I went into last year with, essentially, the textbook.

Geometry was the course that suffered the most for my lack of preparation. While our textbook (Holt) isn’t horrible, I never really felt like the year had any sort of flow. And if I felt that way as the teacher, I don’t want to think about how the students felt. In particular, my biggest regret is that I don’t believe my students ever really developed a strong understanding of proof.

So the goal for this year was to write a curriculum that was logical, had a flow to it, and actually developed students’ ability to write and understand proofs. In writing the curriculum, I went back and reviewed the Van Hiele model. In light of this, the course attempts to start at Level 0–1 before slowly moving to Level 3–4 as the year progresses. My hope is that each individual section will do the same.

My second largest failing last year was that too many topics were taught and then dropped. In a course like geometry, where I am asking students to refer back to previous theorems, concepts, and definitions on a daily basis, constant review is critical. To this end, the beginning of the year is filled with some introductory material which will be called upon and extended throughout the year. It is my hope that each section will have many natural opportunities to go back and review, use, and extend earlier sections.

I wanted to focus on the overall flow of the course, so there are no unit lengths, assignments, “I can…” statements, or assessments. Instead, each section has a brief description of the overall goal and then a few points on the specific theorems, constructions, and understandings that students should learn. Of course, that means it’s probably not very useful to anyone, but I’m sharing it anyway. I would love any feedback that anyone wants to share.