The children understand that sin, cos, and tan are side ratios. The children! They understand! They are not making ridiculous mistakes, and they can answer deeper understanding questions like, "Explain why sin(11) = cos(79)." I think right triangle trig is a frequent victim of the "First ya do this, then ya do this" treatment -- where kids can solve problems but have no idea what is going on. There's often not a ton of time for it, and it responds well to memorized procedures (in the short term). So, if your Day One of right triangle trig involves defining sine, cosine, and tangent, read on! I have a better way, and it doesn't take any longer.

First, build on what students have already learned about similar triangles. Ideally, this unit immediately follows that one. On Day One, I assign each pair of students an angle. (You guys have 20 degrees. You all have 25. etc etc, all around the room, so each pair of students is responsible for a different angle.) They work through this document (docx pdf), using Geogebra to do the measuring. They write down the length of the side opposite and adjacent their angle, for triangles of five different sizes. It's important that they write down the lengths, divide them with a calculator, and experience surprise and wonder why they are all exactly the same. (Geogebra made this soooo much better and easier than when I did this with rulers and protractors. So much better. In fact, one of my Matt colleagues basically deserves a medal for all the times he's said "Why don't we just do this with Geogebra?" this year.)

On this day, they just do opposite/adjacent ratio, share the ratio for their angle in a shared spreadsheet, and then everyone has access to the shared spreadsheet (an opp/adj-only trig table) to solve some problems (in that same document). The thing is, they are figuring out how to use what they have learned to solve the problems; they're not just repeating a procedure that was demonstrated. This took one 45-minute period, including checking Chromebooks out and in. I collect the sheets and look for students who had a strategy for #11 (how to solve when the variable is in the bottom of the ratio) so they can share their strategy next class.

Next two classes, I provide them with a table of all three ratios (to tape in their notebook) for angles of 5-degree increments, and they work their way through this page with appropriate help. For example, in the first set of problems, I just had them label the sides first. Then choose the ratio for all the problems, then solve for an answer. This particular document is not terribly pretty, because I had limited time to put it together. In every class, someone wanted to know why they couldn't use like hyp/adj if that equation was easier to solve. For those that asked, I pointed out that we couldn't look up hyp/adj in the table, BUT, they could use the other angle in the triangle. (And yes I'm aware they could use 1-over the ratio in the table, but that seemed like an overly complicated strategy to suggest.) I gave them a few find-sides and find-angles problems (limited to the angles in their table) to practice for homework. They did not all get to the back, but the kids to catch on/work quicker had something to do after the basic problems.

Today I spilled the beans that these ratios have special names, and we could look them up in our calculator. We mostly spent the period getting used to looking stuff up in the calculator including some hot Plickr action, and working on these problems which they are finishing for homework. I told them they only had to do one "Explain why," but they had to complete all the rest.

Tomorrow on our block day, we are going to go outside and figure out the heights of some really tall things (docx pdf). There are lots of "measuring tall things" activities out there, but I heavily adapted this document, so thanks to Christopher Conrad for posting it.

# f(t)

## Monday, March 2, 2015

## Friday, January 23, 2015

### On Making Them Figure Something Out

Often when I don't really know a great way to teach something, I end up defaulting to Making Them Do Something and Making Them Notice Something, and then finally Making Them Practice Something. I think lots of people do, and it's not bad. It's loads better then Telling Them Something, and it's certainly better than Children with Nothing To Do. But lots of times, the learning that comes out of MTDS and MTNS doesn't really stick that great. They can maybe do an exit ticket, but ask them a question that relies on The Thing in a week, and you just get a bunch of blank stares. So in my planning, I'm taking as a signpost Making Them Figure Something Out or MTFSO. I think the best existing curricula

So here's an example: the discriminant in Algebra 2, or said another way "What kind of roots does this quadratic equation have?" We're not particularly concerned in Virginia with them being able to define the word "discriminant," but they should be able to recognize whether the solutions are rational or irrational, real or non-real. Given a quadratic equation, they should be able to figure out what kind of solutions it has, and know how to describe those kinds of numbers. Here is a sample released item (question 50 of 50 in this document). (Which, you may notice, has exactly zero correct answers. Good job, Virginia. (Poor Virginia. Maybe she'll marry well.))

You can find all kinds of MTDS/MTNS lessons about the discriminant. Here and here, for example. But here's what I came up with to turn it into MTFSO. We had already spent a day on simplifying radicals (including with imaginary results), a couple days on solving by undoing and solving with the quadratic formula. Then we made a big map of different kinds of numbers with special attention to recognizing rationals vs irrational and real vs non-real.

For this particular lesson, they first solved four equations using the quadratic formula: x

Then, they sat in groups of 2-3 at big whiteboards and got one of these sets of questions:

I saw different approaches... start with a desired answer and try to work backwards. Write out the quadratic formula with blank spots and repeatedly fill in and try values. Start with an equation very close to one we had just worked with and see how it worked out. And of course the bane of all group work reared its head - one kid grabs a marker and takes over while everyone else is happy to let them.

*depend*on MTFSO, and it must be nice to be working with one of those.So here's an example: the discriminant in Algebra 2, or said another way "What kind of roots does this quadratic equation have?" We're not particularly concerned in Virginia with them being able to define the word "discriminant," but they should be able to recognize whether the solutions are rational or irrational, real or non-real. Given a quadratic equation, they should be able to figure out what kind of solutions it has, and know how to describe those kinds of numbers. Here is a sample released item (question 50 of 50 in this document). (Which, you may notice, has exactly zero correct answers. Good job, Virginia. (Poor Virginia. Maybe she'll marry well.))

You can find all kinds of MTDS/MTNS lessons about the discriminant. Here and here, for example. But here's what I came up with to turn it into MTFSO. We had already spent a day on simplifying radicals (including with imaginary results), a couple days on solving by undoing and solving with the quadratic formula. Then we made a big map of different kinds of numbers with special attention to recognizing rationals vs irrational and real vs non-real.

For this particular lesson, they first solved four equations using the quadratic formula: x

^{2}- 10x + 9 =0, x^{2}- 6x + 9 = 0, x^{2}- 7x + 9 = 0, and x^{2}- 4x + 9 = 0. When we debriefed their solutions, we spent time describing the types of solutions, but we did not belabor the point about why the roots came out each way.Then, they sat in groups of 2-3 at big whiteboards and got one of these sets of questions:

A1) Come up with a new, original quadratic equation whose roots arereal and irrational. Demonstrate that your equation works by using the quadratic formula to solve it.

A2) Come up with a new, original quadratic equation whose roots arereal, rational, and unequal. Demonstrate that your equation works by using the quadratic formula to solve it.

B1) Come up with a new, original quadratic equation whose roots are(In the first class to do this, I gave out the tasks haphazardly. But from that experience, learned that "rational" and "equal" are harder to find than "irrational" and "imaginary." So I adjusted accordingly for the second class. Each set consists of one of the easier ones, and then one of the harder ones.)imaginary. Demonstrate that your equation works by using the quadratic formula to solve it.

B2) Come up with a new, original quadratic equation whose roots arereal, rational, and equal. Demonstrate that your equation works by using the quadratic formula to solve it.

I saw different approaches... start with a desired answer and try to work backwards. Write out the quadratic formula with blank spots and repeatedly fill in and try values. Start with an equation very close to one we had just worked with and see how it worked out. And of course the bane of all group work reared its head - one kid grabs a marker and takes over while everyone else is happy to let them.

After groups had a chance to figure stuff out and explain how they did it (it all came down to paying attention to what kind of number was under the radical, of course), we summarized with some notes:

And that is that. A general understanding of what was going on persisted to the next day... We'll see how Monday goes.

## Sunday, January 11, 2015

### SSA ASA and All the Rest

I finally, Finally, FINALLY have a plan I like for introducing triangle congruence theorems.

For a few years there (2? 3?) I had been trying to make a go of Triangles a la Fettucine as described in this MT article. And I just couldn't work it! The principle of the thing is sound, and it always started out okay, but quickly got tedious and the kids would both lose interest and not get the point. Many students just would stubbornly not get the memo that you had to use the entire length of the colored-in sides, but you could use any length of the uncolored sides. The mechanics of the thing got in the way of seeing the larger picture.

In this new lesson, I deliberately separated the triangle-creating phase from the seeing the larger picture phase.

Before the lesson, I gave students the first page of the pre-assessment from this Shell Center Formative Assessment Task. (Thanks someone on Twitter who suggested that.) I did not use the lesson itself, because I felt my students weren't at the point of being able to understand what it was asking. The pre-assessment was compared to what they could do afterward, of course, but also to kind of get their juices flowing about angles and side lengths and what congruent means. They worked on the pre-assessment for about 15 minutes.

Phase 1 was constructing some triangles out of construction paper based on various given information using straight edge, protractor, and occasionally a compass. Here is the instruction page. (Thanks again, Twitter, for some helpful feedback making it better before it went live to the children.) Students were in groups of 3 or 4, and

Before they started I ran a quick protractor clinic, and they were off. The groups worked on creating triangles for 30-40 minutes. Beforehand, I created a set of reference triangles out of cardstock, so I could quickly assess their products for accuracy when they were done. I just kept the correct ones and discarded the inaccurate ones (where an angle was not measured correctly, for example.) (Instead of making them re-do incorrect ones, but I think that would be a fine thing to do if you can swing the time.) The groups that finished first, I handed off the cardstock triangles and put them in charge of assessing other groups' work. By the time triangle construction was complete in all three of my Geometry sections, I had 10+ correctly-made copies of each triangle.

I suppose you could try to recreate this experience with dynamic geometry software somehow, but I'm dubious that there's a replacement for creating physical triangles with your hands. Students directly encountered having choices for how to finish making the triangle (when they were only given the lengths of two sides, for example) vs being locked into only one possible triangle (when they were given ASA, for example). In the future, I'd like to be more deliberate in assigning each student one of each.

After triangle construction, we ran through a quick lesson on notation and naming conventions for congruent polygons. Then I put up a kind of standard-looking congruent triangle proof, where enough information was given (or able to be inferred) to show that all three pairs of sides and all three pairs of angles were congruent. We wrote out a 9-step proof that proved the triangles congruent by SASASA. "Wasn't that a pain?" I said. "Yes," they said. "Wouldn't it be nice if we could know triangles were congruent to each other with a smaller set of information?" I said. "Yes," they said.

Justin Lanier and Pershan? possibly others? brought up the issue on Twitter that triangle-uniqueness (will this given information only allow you to make one triangle?) is a cognitively different thing from triangle congruence (can I be sure these two triangles are identical?). That was a slippery thing that always poked at the edges of my thoughts this time in the school year, but I'd never thought to explicitly address it. I think this lesson does a nice job of bridging those two related understandings. The triangle-constructing compels one to think whether there are choices to be made in what this triangle looks like... or is it unique? But putting all the triangles together on a poster highlights the question of do all these triangles have to be identical, using certain given information? Pretty seamlessly, I think, and without having to dwell on it.

Using a recording sheet (created by my colleague, Matt), they did a gallery walk, recording whether, in fact, all the triangles were congruent, which parts were the given information, and drawing a sketch. (He provided the triangle outlines in the rightmost column -- I took those off.) Then we had a quick discussion and they made their first foray into identifying which theorem applied based on given information (the back of that sheet).

And that was that! On their quiz Friday (usually we have a half-period quiz on Fridays), I asked another question very similar to one of the pre-assessment questions, and every single student showed growth in their ability to explain why the given information was not enough to guarantee unique triangles.

(I'm not sure if this lesson is useful in Common Core land -- over there, you're supposed to link congruence to rigid transformations. Which I do here in Virginia, informally, but it doesn't rise to the level of students performing transformational proofs.)

For a few years there (2? 3?) I had been trying to make a go of Triangles a la Fettucine as described in this MT article. And I just couldn't work it! The principle of the thing is sound, and it always started out okay, but quickly got tedious and the kids would both lose interest and not get the point. Many students just would stubbornly not get the memo that you had to use the entire length of the colored-in sides, but you could use any length of the uncolored sides. The mechanics of the thing got in the way of seeing the larger picture.

In this new lesson, I deliberately separated the triangle-creating phase from the seeing the larger picture phase.

Before the lesson, I gave students the first page of the pre-assessment from this Shell Center Formative Assessment Task. (Thanks someone on Twitter who suggested that.) I did not use the lesson itself, because I felt my students weren't at the point of being able to understand what it was asking. The pre-assessment was compared to what they could do afterward, of course, but also to kind of get their juices flowing about angles and side lengths and what congruent means. They worked on the pre-assessment for about 15 minutes.

Phase 1 was constructing some triangles out of construction paper based on various given information using straight edge, protractor, and occasionally a compass. Here is the instruction page. (Thanks again, Twitter, for some helpful feedback making it better before it went live to the children.) Students were in groups of 3 or 4, and

*the group*was responsible for creating the nine triangles (There are ten on the sheet, but the last one is impossible.) For thoroughness, I'd love if every student had to create all the triangles, but I was afraid 1) that would take way way too long and 2) many students would tire of it midway through and check out. I think my instincts were right on both counts. The more difficult constructions were marked with a *, which I told the students, which allowed for some self-moderated differentiation (by less-confident students quickly claiming responsibility for non-* triangles).Before they started I ran a quick protractor clinic, and they were off. The groups worked on creating triangles for 30-40 minutes. Beforehand, I created a set of reference triangles out of cardstock, so I could quickly assess their products for accuracy when they were done. I just kept the correct ones and discarded the inaccurate ones (where an angle was not measured correctly, for example.) (Instead of making them re-do incorrect ones, but I think that would be a fine thing to do if you can swing the time.) The groups that finished first, I handed off the cardstock triangles and put them in charge of assessing other groups' work. By the time triangle construction was complete in all three of my Geometry sections, I had 10+ correctly-made copies of each triangle.

I suppose you could try to recreate this experience with dynamic geometry software somehow, but I'm dubious that there's a replacement for creating physical triangles with your hands. Students directly encountered having choices for how to finish making the triangle (when they were only given the lengths of two sides, for example) vs being locked into only one possible triangle (when they were given ASA, for example). In the future, I'd like to be more deliberate in assigning each student one of each.

After triangle construction, we ran through a quick lesson on notation and naming conventions for congruent polygons. Then I put up a kind of standard-looking congruent triangle proof, where enough information was given (or able to be inferred) to show that all three pairs of sides and all three pairs of angles were congruent. We wrote out a 9-step proof that proved the triangles congruent by SASASA. "Wasn't that a pain?" I said. "Yes," they said. "Wouldn't it be nice if we could know triangles were congruent to each other with a smaller set of information?" I said. "Yes," they said.

after school gluestick par-taaaayyyy |

In the meantime (the lesson covered by this post spanned five days, FYI) I had glued all the triangle A's to a poster, all the triangle B's to a poster, etc. Before they looked at them, I had them predict (using the original instruction sheet) whether they thought all the copies of each triangle had to be congruent. Will all the triangle A's be congruent? Will all the triangle B's be congruent? etc. Reminders that for congruence, it's okay if you have to reflect or rotate one to make it look exactly like the other.

Justin Lanier and Pershan? possibly others? brought up the issue on Twitter that triangle-uniqueness (will this given information only allow you to make one triangle?) is a cognitively different thing from triangle congruence (can I be sure these two triangles are identical?). That was a slippery thing that always poked at the edges of my thoughts this time in the school year, but I'd never thought to explicitly address it. I think this lesson does a nice job of bridging those two related understandings. The triangle-constructing compels one to think whether there are choices to be made in what this triangle looks like... or is it unique? But putting all the triangles together on a poster highlights the question of do all these triangles have to be identical, using certain given information? Pretty seamlessly, I think, and without having to dwell on it.

Using a recording sheet (created by my colleague, Matt), they did a gallery walk, recording whether, in fact, all the triangles were congruent, which parts were the given information, and drawing a sketch. (He provided the triangle outlines in the rightmost column -- I took those off.) Then we had a quick discussion and they made their first foray into identifying which theorem applied based on given information (the back of that sheet).

And that was that! On their quiz Friday (usually we have a half-period quiz on Fridays), I asked another question very similar to one of the pre-assessment questions, and every single student showed growth in their ability to explain why the given information was not enough to guarantee unique triangles.

(I'm not sure if this lesson is useful in Common Core land -- over there, you're supposed to link congruence to rigid transformations. Which I do here in Virginia, informally, but it doesn't rise to the level of students performing transformational proofs.)

Labels:
geometry,
lessons,
reflection

## Wednesday, November 19, 2014

### Graphles to Graphles

New game! My Algebra 2 students struggle with stating the domain and range for reasons including: trouble understanding and writing inequalities, and a lack of comfort with the coordinate plane. We spent a day on looking at graphs and identifying their domain and range. We learned to deploy our wonderful domain meters and range meters that I learned about from Sam. But for maybe 25% of the students, the cluebird was stubbornly refusing to land.

So, I thought asking the question backward might be a good way to attack it. Instead of here's the graph, what's the D and R? ask, here's a D and/or R, draw a graph. I mean, I know this is pretty standard fare. The thing is, I didn't want to do examples and a worksheet, or hold-up-your-whiteboard so I could somehow assess 22 graphs in a split second. It seemed like there should be a better way.

So I did what I do, which is ask on Twitter. And I got lots of helpful ideas, but this was the one that I latched onto and ran with:

To prep: Make game cards. I printed each page on a different color card stock. Student play in groups of 4-ish, so plan accordingly. I printed 6 sets. (John suggested having students submit constraints, but, for this crew, I decided to unload that part and create cards with the constraints.) You'll also need a mini-whiteboard, marker, and eraser for each student. Check your dry erase markers, because nothing kills a math game buzz like a weaksauce marker. (I'll admit to a minor teacher temper tantrum where I uttered the words "I'M NOT THE MARKER FAIRY! I DON'T POOP MARKERS!" Teacher of the year, right here, folks.) Also, you'll need some kind of token that players can collect when they win a turn. I use these plastic counting chips that I use for everything, but anything would work, candy, whatever.

Doing a demo round with a few kids playing and everyone watching will pay off, in the more-kids-will-know-what-is-up sense.

Here's how the game plays:

So, I thought asking the question backward might be a good way to attack it. Instead of here's the graph, what's the D and R? ask, here's a D and/or R, draw a graph. I mean, I know this is pretty standard fare. The thing is, I didn't want to do examples and a worksheet, or hold-up-your-whiteboard so I could somehow assess 22 graphs in a split second. It seemed like there should be a better way.

So I did what I do, which is ask on Twitter. And I got lots of helpful ideas, but this was the one that I latched onto and ran with:

```
@k8nowak make it a game? Charades-ish: Each student puts in 2 requirements. Team draws two and has a minute to make a function that fits.
```

— John Golden (@mathhombre) November 17, 2014

The end result is, I'd argue, more like Apples to Apples than Charades (hence the title). To prep: Make game cards. I printed each page on a different color card stock. Student play in groups of 4-ish, so plan accordingly. I printed 6 sets. (John suggested having students submit constraints, but, for this crew, I decided to unload that part and create cards with the constraints.) You'll also need a mini-whiteboard, marker, and eraser for each student. Check your dry erase markers, because nothing kills a math game buzz like a weaksauce marker. (I'll admit to a minor teacher temper tantrum where I uttered the words "I'M NOT THE MARKER FAIRY! I DON'T POOP MARKERS!" Teacher of the year, right here, folks.) Also, you'll need some kind of token that players can collect when they win a turn. I use these plastic counting chips that I use for everything, but anything would work, candy, whatever.

Doing a demo round with a few kids playing and everyone watching will pay off, in the more-kids-will-know-what-is-up sense.

Here's how the game plays:

- Someone is the referee.
- To begin the turn, the referee turns over two (or one, or three) different-colored cards,
**and reads them out loud**. (I feel the reading aloud is important practice for interpreting inequalities.) You could do, like, first round is one card, second round is two cards, third round is three cards. Whatever suits your needs. - The other players have one minute to sketch a graph meeting the constraints on the cards. The referee is responsible for timing one minute.
- The players hold up their mini-whiteboards so the referee can see. The referee should disqualify any graphs that don't match the cards.
- Of the remaining graphs, the referee picks his favorite. This player wins a token.
- The turn is over, and the player to the referee's right becomes the new referee.
- Repeat.

It was great! Here are things I liked about it:

- 100% participation 100% of the time. At no point should anyone be kicking back.
- Nowhere to hide. There were a couple kids who had to come to me and say, "Miss Nowak, I really don't know what's going on." which I don't think they'd be compelled to do if we were just doing some practice problems.
- Good conversations. Especially reasons for why graphs were disqualified. "You need an arrow there! The domain goes to infinity!" That sort of thing.
- Students were necessarily
*creating*and*evaluating*. Take that, Bloom! - Built-in review of what makes a graph a function vs not a function.
- My chronic doodlers had a venue to express themselves. Especially if the graph didn't have to be a function.
- Authentic game play. You could use your knowledge of what a referee liked to curry favor.

## Thursday, November 13, 2014

### We Got a Problem

We spent practically the whole period (35-ish minutes) on one problem today. This one, that Justin wrote about recently, that he found on Five Triangles:

Since we just spent a few days naming pairs of angles made by parallel lines and proving what's congruent and what's supplementary, I was provisionally hoping we'd get, at the end, 10 or so minutes for students to present various solutions to the class. That did not happen in any of my three class periods. Because I didn't have the heart to interrupt them. At the 10-minutes-left mark, too many were still making passionate arguments to their small groups about why they thought their solution worked.

Here's what we did: The big whiteboards were on the tables. Groups of 3 or 4. I stated the problem while showing this diagram. Made a big deal of starting with a regular old piece of copier paper and making a single fold. 3-5 minutes silent, individual think time. No class made it to 5 minutes without a buzz starting. I wrote a time on the board and said, by this time, everyone in your group needs to be prepared to present a solution to the class. Also, the answer is not as important as the reasoning that got you there. If you are saying something like "this angle has this many degrees," you have to explain the reason why that must be true.

Then I started listening and circulating. There was almost 100% engagement, and I have to think it's due, to a high degree, to the problem itself. This problem just felt do-able, but not obvious, to every learner in the room -- the sweet spot.

When a group would start crowing, in their 9th grade way, that "MISS NOWAK. WE GOT IT," I refused to confirm or deny that their answer was right, and asked a randomly selected group member (everyone was supposed to be able to explain the solution) to walk me through the reasoning. I played the role of highly annoying and dense skeptic. "Wait, how did you know that angle was 90?" "Because it's a RIGHT ANGLE." "Wait, how do you know it's a right angle?" "... ... ...BECAUSE A SHEET OF PAPER IS A RECTANGLE." "Oohhh, right." And then, when they got to a part that was not justified (an assumed bisector, an assumed isosceles triangle, trying to use two sets of parallel lines to leap to a conclusion about congruent angles), I wasn't shy about saying I wasn't convinced. In their groups, they had already harvested the low hanging reasoning fruit. I figured my experienced eye was valuable for training a spotlight on flaws in their arguments. And they responded well, in a back-to-the-drawing-board kind of way.

I'm in the habit of slagging myself on here, but I'm going to take a moment and describe a few times I witnessed and celebrated some great, inspired ideas with at least one learner today:

This seemed evident: the importance of being able to articulate how you know things are true. I think that was one of the purposes of proof that Pershan hit on last summer... knowing why a right answer is right. In a world (of school Geometry) where if I'm careless, I'm too often asking kids to "prove" things that they think are already obvious, I want to make as much room as possible for problems like this where something is not obvious and needs justification.

I'm not entirely comfortable with leaving 72 kids hanging about what the correct solution was, and why. What I

They should have sent a poet. I should have done this on a block day.

Since we just spent a few days naming pairs of angles made by parallel lines and proving what's congruent and what's supplementary, I was provisionally hoping we'd get, at the end, 10 or so minutes for students to present various solutions to the class. That did not happen in any of my three class periods. Because I didn't have the heart to interrupt them. At the 10-minutes-left mark, too many were still making passionate arguments to their small groups about why they thought their solution worked.

Here's what we did: The big whiteboards were on the tables. Groups of 3 or 4. I stated the problem while showing this diagram. Made a big deal of starting with a regular old piece of copier paper and making a single fold. 3-5 minutes silent, individual think time. No class made it to 5 minutes without a buzz starting. I wrote a time on the board and said, by this time, everyone in your group needs to be prepared to present a solution to the class. Also, the answer is not as important as the reasoning that got you there. If you are saying something like "this angle has this many degrees," you have to explain the reason why that must be true.

Then I started listening and circulating. There was almost 100% engagement, and I have to think it's due, to a high degree, to the problem itself. This problem just felt do-able, but not obvious, to every learner in the room -- the sweet spot.

When a group would start crowing, in their 9th grade way, that "MISS NOWAK. WE GOT IT," I refused to confirm or deny that their answer was right, and asked a randomly selected group member (everyone was supposed to be able to explain the solution) to walk me through the reasoning. I played the role of highly annoying and dense skeptic. "Wait, how did you know that angle was 90?" "Because it's a RIGHT ANGLE." "Wait, how do you know it's a right angle?" "... ... ...BECAUSE A SHEET OF PAPER IS A RECTANGLE." "Oohhh, right." And then, when they got to a part that was not justified (an assumed bisector, an assumed isosceles triangle, trying to use two sets of parallel lines to leap to a conclusion about congruent angles), I wasn't shy about saying I wasn't convinced. In their groups, they had already harvested the low hanging reasoning fruit. I figured my experienced eye was valuable for training a spotlight on flaws in their arguments. And they responded well, in a back-to-the-drawing-board kind of way.

**Props**I'm in the habit of slagging myself on here, but I'm going to take a moment and describe a few times I witnessed and celebrated some great, inspired ideas with at least one learner today:

- You extended that line to make it intersect another line!
- You marked those lines as parallel with some arrows! And those other ones too!
- You made an estimate of a reasonable answer!
- You grabbed a piece of scrap copier paper and made a physical model to look at and play with!
- You suggested your group start over and draw a clearer diagram, so more people would know what you were talking about!

**The Value**This seemed evident: the importance of being able to articulate how you know things are true. I think that was one of the purposes of proof that Pershan hit on last summer... knowing why a right answer is right. In a world (of school Geometry) where if I'm careless, I'm too often asking kids to "prove" things that they think are already obvious, I want to make as much room as possible for problems like this where something is not obvious and needs justification.

**Questions**I'm not entirely comfortable with leaving 72 kids hanging about what the correct solution was, and why. What I

*want*is for this to keep bugging them, and for them to make little doodles and sketches of it in their spare time, and for them to not be able to leave it alone. I did not want to ruin anyone's fun. At the same time, I think many kids could have benefited and learned from seeing a few different correct arguments for why the angle had to measure 140 degrees. This is still an open question for me -- what's the best way to handle kids/groups that don't arrive at the correct answer? Do you let the question hang, or do you interrupt everybody so groups who made it down a valid path can have time to show what they did? I'd love to pick it up tomorrow, but I'm going to be out (whaddup, #NCTMRichmond!) so we'll have to see if anyone has any memory of what happened today on Monday.**Regrets**They should have sent a poet. I should have done this on a block day.

## Monday, November 10, 2014

### DDT, Y'all

Today in Geometry we tried Dance, Dance Transversal as popularized by Jessica and Julie. The kids dug it, and nailed an exit ticket identifying names of pairs of angles. I followed Julie's plan pretty closely. I loved that the kids were up and moving around for a good 20 minutes of class. (Was anyone else traumatized by that Grant Wiggins article? I'm very on the lookout for ways to make kids move.)

I just want to add one more resource to the arsenal: a powerpoint I made to show while playing. The slides auto-play the different moves. There are some initial slides that demonstrate where to put your feet for each cue. Then, the first two game slides are timed with a 1.5 second delay and worked well with Problem, and the second two game slides are timed with a 1 second delay and worked well with Dynamite.

Here's video. In case you're wondering, I did also play along, every period. Because their dancing did not have enough FLAVOR, and I had to demonstrate. Try to ignore the one kid doing some kind of demented hopscotch:

I just want to add one more resource to the arsenal: a powerpoint I made to show while playing. The slides auto-play the different moves. There are some initial slides that demonstrate where to put your feet for each cue. Then, the first two game slides are timed with a 1.5 second delay and worked well with Problem, and the second two game slides are timed with a 1 second delay and worked well with Dynamite.

Here's video. In case you're wondering, I did also play along, every period. Because their dancing did not have enough FLAVOR, and I had to demonstrate. Try to ignore the one kid doing some kind of demented hopscotch:

## Saturday, November 8, 2014

### Gallery Walk for Noticing Features of Inverse Functions

I put a call out on Twitter last week for good things for inverse functions. I got a few helpful responses but nothing that was really

*the thing*. So here's what I made.
The day before, we had worked with inverse functions as doing and undoing equations. I started with ciphers. Students walked in and the board said, IQQF OQTPKPI, DGCWVKHWN DTCKPU! with no explanation from me. I just greeted them and took attendance and acted nonchalant. One kid sidles up and goes all sotto voce, "Miss Nowak, does the first part say

I had one of them explain how the encoding was done with this example. Then, they wrote secret messages using their own shift

*Good Morning*?" Since the good morning part was a pretty easy crack using context, after a minute or two someone notices that all the letters are shifted over by two, and can't keep from blurting it out, and we're off.I had one of them explain how the encoding was done with this example. Then, they wrote secret messages using their own shift

*n*cipher, traded, decoded. I babbled a little bit about Caesar and Enigma (I really want to show them this Numberphile video, thanks for the tip Mike Lawler). The encoding and trading and decoding only took about ten minutes, we went through one from beginning to end: what was your message, how did you encode it, how did you decode it. The alphabet was written on the board along with a counting number under each letter, the idea being that if your encoding added 5, the decoding would be subtract 5.
We spent the rest of that day couching inverses in terms of equation rules.

The next day, I wanted them to notice all the nice things that are true for functions and their inverses: the symmetry over

Each pair of students got one of these (the first page). They tacked their cards to the paper, completed the tables, graphed each function in a different color, and computed

As the mini-posters were completed, they were hung up around the room. I said, hey, you all had different functions and now they're up there with their inverses. There are some neat things that are always true about a function and its inverse. Walk around and look at them all, and write down at least two things you notice. If you look at page 2 of this same document, the first question has space for them to write down observations.

They sat back down, and they shared their noticings with the class. I had Desmos up on the projector with some pre-loaded functions, so we had a concrete thing to point to as they were sharing.

Then they got to work on the rest of that page 2, which is lifted directly from lesson 6 of this eMathInstruction textbook (thanks Sam for pointing me to this resource). Some of them were able to just do those problems, some needed help restating the given information and what the problems were asking.

So there you have it. I especially liked this lesson for the social, discussion, get-up-and-move-around aspects. These Algebra 2 classes have not responded positively to problem-posing when they haven't been "shown how" to do a problem first, but, we have been successful with lessons like this where we break the questions into clear chunks while still requiring that they do some thinking and figuring out. It's a bit of a tightrope walk but that's how you get down a tightrope, right? One tiny step at a time?

*x*+ 5 and*x*- 5 is fine and pretty obvious, but what about more complicated rules. Kids had mini whiteboards, I'd throw a function on the board and they'd try to write the inverse. Each time, they wrote down operations done in the original function, inverse operations in reverse order, then do that to an x. So for example if the given function is 3*x*^{2}- 5, they write down "square, multiply by 3, subtract 5" and then write down "add five, divide by 3, square root." Plop down an*x*and do those things to it. The biggest hurdles were order of operations (so they might write down "multiply by 3, square, subtract 5"). Also, always undoing the whole of what came before. So in this example, they'd be likely to write*x*+ 5/3 instead of (*x*+ 5)/3. But we just kept going and honestly, they didn't want to stop until they were getting them right. (I just got an idea about how to make this part better. Compute like f(3) (or something) and run the result through their inverse to see if a 3 comes out. (Instead of just you're right or wrong because I say so.) Have to figure out how to make that manageable.)The next day, I wanted them to notice all the nice things that are true for functions and their inverses: the symmetry over

*y*=*x*, that the inputs and outputs trade places, that*f*^{-1}(*f*(*x*)) =*x*. So, each student got one of these cards. They figured out the inverse of that function using the technique from the day before. There was another student in the room with the inverse of their function, so they had to get up, talk to people, and then sit with their partner.Each pair of students got one of these (the first page). They tacked their cards to the paper, completed the tables, graphed each function in a different color, and computed

*f*^{-1}(*f*(0)) and*f*^{-1}(*f*(1)). They needed various levels of support interpreting instructions, but it helped to have them working in pairs on the same piece of paper - there was a natural reason for them to talk to each other to figure it out. My colleague Lois is teaching the same course, and got a coach to come in for one of her sections, which was a great move.As the mini-posters were completed, they were hung up around the room. I said, hey, you all had different functions and now they're up there with their inverses. There are some neat things that are always true about a function and its inverse. Walk around and look at them all, and write down at least two things you notice. If you look at page 2 of this same document, the first question has space for them to write down observations.

Then they got to work on the rest of that page 2, which is lifted directly from lesson 6 of this eMathInstruction textbook (thanks Sam for pointing me to this resource). Some of them were able to just do those problems, some needed help restating the given information and what the problems were asking.

So there you have it. I especially liked this lesson for the social, discussion, get-up-and-move-around aspects. These Algebra 2 classes have not responded positively to problem-posing when they haven't been "shown how" to do a problem first, but, we have been successful with lessons like this where we break the questions into clear chunks while still requiring that they do some thinking and figuring out. It's a bit of a tightrope walk but that's how you get down a tightrope, right? One tiny step at a time?

## Friday, November 7, 2014

### Fire Up Blogging Machine

Yo. This year is hard. New building, blah blah. I'll pause a minute for no one to be surprised.

I feel, very often, like I'm not that good at this. I know that everyone does sometimes. I know, I know. I think it has very much to do with attending to formative assessment every day. (Every. Damn. Day.) Measurement: making it hard to lie to yourself since... measuring was invented.

The children are charming and testy and pathetic and confident and devious and brave, all in the same day, all in the same 45 minutes. There are 70 ninth graders that move through my room, and the thing is that a ninth grader is like the weather in Buffalo in April -- if you don't like it (or if you do), just wait five minutes.

I have lessons that I want to write up for this blog, the problem being my artifacts (documents, pictures, student work, etc) are all over the place. The file system on the school network is unreliable, so teachers all use either Dropbox or Drive or flash drives to store and/or keep a backup of everything (even though Dropbox isn't installed at school -- it's web interface only, 2005-style). Colleagues have been very generous sharing (bewildering Virginia-standards-based) materials with me. So all the stuff I've modified or created and used is on... the school file system and Dropbox and Drive and a flash drive. That puts just enough of an annoying-barrier in the way of assembling blog posts. I have got to get my computer file organizing act together.

So, those are some lame excuses for the radio silence. More coming. I'll figure this out.

I feel, very often, like I'm not that good at this. I know that everyone does sometimes. I know, I know. I think it has very much to do with attending to formative assessment every day. (Every. Damn. Day.) Measurement: making it hard to lie to yourself since... measuring was invented.

The children are charming and testy and pathetic and confident and devious and brave, all in the same day, all in the same 45 minutes. There are 70 ninth graders that move through my room, and the thing is that a ninth grader is like the weather in Buffalo in April -- if you don't like it (or if you do), just wait five minutes.

I have lessons that I want to write up for this blog, the problem being my artifacts (documents, pictures, student work, etc) are all over the place. The file system on the school network is unreliable, so teachers all use either Dropbox or Drive or flash drives to store and/or keep a backup of everything (even though Dropbox isn't installed at school -- it's web interface only, 2005-style). Colleagues have been very generous sharing (bewildering Virginia-standards-based) materials with me. So all the stuff I've modified or created and used is on... the school file system and Dropbox and Drive and a flash drive. That puts just enough of an annoying-barrier in the way of assembling blog posts. I have got to get my computer file organizing act together.

So, those are some lame excuses for the radio silence. More coming. I'll figure this out.

## Friday, September 12, 2014

### Not One Mention of Karl Gauss

There are things I'm supposed to teach and they're strung along into a curriculum and I often get to some of these things and I'm like, "wtf? why?" So when I get to those things I think, okay, why might this ever be useful or interesting.

Why might someone want to sum an arithmetic sequence? That's a list of numbers that keep increasing or decreasing by the same amount. So for example, 3, 5, 7, 9, ... is an arithmetic sequence and so is 8, 3, -2, -7, -12, -17, ...

The only times this has ever been useful in my life is when I needed a shortcut for adding a bunch of things really fast. Which doesn't come up that much, but it does come up once in a blue moon.

So I decided to impress the children with my lightning addition capability. "Impress" is maybe a tad ambitious as words go because they are 16 but their interest can be piqued.

When they walked in I gave them a slip of paper with 20 boxes and asked them to come up with an arithmetic sequence and write down the first 20 terms. They could use their favorite numbers, or their least favorite numbers, or numbers they were indifferent toward. I made a little form special for this purpose because if one wants humans to take one seriously, one must make it obvious that one has prepared for their arrival. (And plus I'm already best friends with the guy in my department with the good paper cutter.)

The little form also had a spot for the

Then I asked them to trade slips with a partner and check each other's math before handing it in, because if my answers didn't match what was on their paper, there was going to be hell to pay. From first period in particular, the class that is still resisting engaging and accusing me of "teaching weird." They weren't going to abide errors in arithmetic.

They wrote their names on the list of 20 things. They cut off the sum and handed me the list of twenty things. They kept the sum. The anticipation was building. And by that I mean I said, "Children. Prepare to be amazed," and the children made me try again because I was too monotone.

So then I shuffled up the little slips of sequences and started saying, B, your sum is 210. C, your sum is 384. D, your sum is 2440. E, your sum is -24.

They were astonished! I only made two mental arithmetic errors in two class periods, which was convincing enough that they wanted to know my secret. We wrote some sequences on the board and we stared at them for a while. No one figured it out. Then I was the worst magician in the world and spilled the trick under the mildest of pressures, the eventual Virginia SoL's. We don't have unlimited time here, people. The commonwealth says so.

Did they learn anything from this little stunt? No. But they were

Why might someone want to sum an arithmetic sequence? That's a list of numbers that keep increasing or decreasing by the same amount. So for example, 3, 5, 7, 9, ... is an arithmetic sequence and so is 8, 3, -2, -7, -12, -17, ...

The only times this has ever been useful in my life is when I needed a shortcut for adding a bunch of things really fast. Which doesn't come up that much, but it does come up once in a blue moon.

So I decided to impress the children with my lightning addition capability. "Impress" is maybe a tad ambitious as words go because they are 16 but their interest can be piqued.

When they walked in I gave them a slip of paper with 20 boxes and asked them to come up with an arithmetic sequence and write down the first 20 terms. They could use their favorite numbers, or their least favorite numbers, or numbers they were indifferent toward. I made a little form special for this purpose because if one wants humans to take one seriously, one must make it obvious that one has prepared for their arrival. (And plus I'm already best friends with the guy in my department with the good paper cutter.)

The little form also had a spot for the

*sum*of the 20 terms. They were to add them up on a calculator and write it down.Then I asked them to trade slips with a partner and check each other's math before handing it in, because if my answers didn't match what was on their paper, there was going to be hell to pay. From first period in particular, the class that is still resisting engaging and accusing me of "teaching weird." They weren't going to abide errors in arithmetic.

They wrote their names on the list of 20 things. They cut off the sum and handed me the list of twenty things. They kept the sum. The anticipation was building. And by that I mean I said, "Children. Prepare to be amazed," and the children made me try again because I was too monotone.

So then I shuffled up the little slips of sequences and started saying, B, your sum is 210. C, your sum is 384. D, your sum is 2440. E, your sum is -24.

They were astonished! I only made two mental arithmetic errors in two class periods, which was convincing enough that they wanted to know my secret. We wrote some sequences on the board and we stared at them for a while. No one figured it out. Then I was the worst magician in the world and spilled the trick under the mildest of pressures, the eventual Virginia SoL's. We don't have unlimited time here, people. The commonwealth says so.

Did they learn anything from this little stunt? No. But they were

*ready*to learn something, which is saying something.## Saturday, August 23, 2014

### Arguing about Shapes

Spirited discussion kicked off my Geometry courses this year. I used a task that is in the IM task bank but not published yet. (Authored by Victoria Peacock and Yenche Tioanda with some revisions by me.)

Update! This task is now published. Here's a link.

Here it is:

Each group was given a printout of the six set of shapes. (I printed one set per half-sheet, and chopped them into 6 half-sheets before class.) Tools provided for their use (in small bins casually opened on each table without explanation of how they were supposed to use them) were patty paper and scissors.

They had five minutes of individual thinking time, and then they were asked to come to a consensus, for each set, in their groups of three or four. (I should mention that at the beginning of the class, we did ten minutes of Talking Points, described by Elizabeth here. So the expectation was set up that everyone participates, and they already had an opportunity to negotiate the awkwardness of talking to each other.)

The mathematical purpose of the task was to begin to develop a definition of "congruent" based on transformations. That a figure is congruent to another if every point on it can be matched up with every point on the other through a series of reflections, translations, and rotations.

The pedagogical purpose was to begin to illustrate the importance of justification and mutual agreement on definitions in their mathematical conversations. I intentionally used the ambiguous phrases "the same" and "not the same." The idea was that part of the work was for the group to come to a consensus about what "the same" meant in their group. In practice, this went like this:

One group: "We said set C is

Me: "Great."

Other group: "Wait a minute, we said set C

Me: "Also great."

Yet another group: "So which is it? We said they are the same."

Me: "... ... ... because... ?"

(Set C is the ribbon-y looking figures.)

Other news: my poker face is amazing.

To bring it all together, I wrote on the board "Our definition of the same" and wrote down all the things they hashed out that were okay and not okay. There was disagreement. Some particularly spirited disagreement over sets B (the lightning bolts) and D (the four equilateral triangles... or is it really only two figures?), in particular. If you saw my tweet about the kid thinking like a topologist, he had a passionate defense of his assertion that set B was the same. This was a 100-minute block period, so we had the luxury of letting the discussion happen.

Then I said, okay, so here's a little secret: what we think of as mathematics is just the result of what everyone has agreed on. We could take

Next up: how to describe transformations precisely.

Update! This task is now published. Here's a link.

Here it is:

**For each set of shapes, write down whether you think they are***the same*or*not the same*, and explain why you think so.Each group was given a printout of the six set of shapes. (I printed one set per half-sheet, and chopped them into 6 half-sheets before class.) Tools provided for their use (in small bins casually opened on each table without explanation of how they were supposed to use them) were patty paper and scissors.

They had five minutes of individual thinking time, and then they were asked to come to a consensus, for each set, in their groups of three or four. (I should mention that at the beginning of the class, we did ten minutes of Talking Points, described by Elizabeth here. So the expectation was set up that everyone participates, and they already had an opportunity to negotiate the awkwardness of talking to each other.)

The mathematical purpose of the task was to begin to develop a definition of "congruent" based on transformations. That a figure is congruent to another if every point on it can be matched up with every point on the other through a series of reflections, translations, and rotations.

The pedagogical purpose was to begin to illustrate the importance of justification and mutual agreement on definitions in their mathematical conversations. I intentionally used the ambiguous phrases "the same" and "not the same." The idea was that part of the work was for the group to come to a consensus about what "the same" meant in their group. In practice, this went like this:

One group: "We said set C is

*not*the same because you have to flip it."Me: "Great."

Other group: "Wait a minute, we said set C

*is*the same because we thought flipping was okay."Me: "Also great."

Yet another group: "So which is it? We said they are the same."

Me: "... ... ... because... ?"

(Set C is the ribbon-y looking figures.)

Other news: my poker face is amazing.

To bring it all together, I wrote on the board "Our definition of the same" and wrote down all the things they hashed out that were okay and not okay. There was disagreement. Some particularly spirited disagreement over sets B (the lightning bolts) and D (the four equilateral triangles... or is it really only two figures?), in particular. If you saw my tweet about the kid thinking like a topologist, he had a passionate defense of his assertion that set B was the same. This was a 100-minute block period, so we had the luxury of letting the discussion happen.

Then I said, okay, so here's a little secret: what we think of as mathematics is just the result of what everyone has agreed on. We could take

*our*definition of "the same" and run with it. In geometry there's a special word "congruent" where specific things, that everyone agrees to like a secret pact, are okay and not okay. Then, I erased "the same" and replaced it with "congruent," and made any adjustments to the definition to make it correct. They had heard the word congruent before, and had the perfectly reasonable middle school understanding that congruent means "same size and shape." I said that that was great in middle school, but in high school geometry we're going to be more precise and formal in our language.Next up: how to describe transformations precisely.

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