Why Perpendicular?

I realized part of the reason I haven’t blogged much this year (despite saying it was one of my main goals!) is because my team is so awesome. There are four regular ed teachers who teach double period Algebra 1, and we’ve done it in various combinations for several years now. This year, we’re newly combined and changing up the way we teach some things.

My contributions are often sharing awesome things I found (but that others wrote)–great for my students but not really anything to blog about. I’ve been loving Math Equals Love INB pages. I’ve implemented way more Desmos activities. I’m not standing still–just not very original I guess.

Anyhow, last week we did parallel and perpendicular lines. We started with parallel and a fairly routine look at parallel lines, focusing on identifying slopes and writing equations of lines that were parallel through a given point (largely to practice writing equations). I (badly) started introducing perpendicular at the end of the period, and homework focused on parallel lines. That day, C (one of my awesome team members) came in to talk through what he had just tried in his class: he gave his kids dot grid paper and had them determine where perpendicular slopes came from.

He tried writing up an activity sheet for me (which he is graciously letting me share below) BUT I didn’t use it either. I tried out what he discussed the next day in my class–and after using his sheet myself to practice decided to instead give dot grids in dry erase pockets. That way, students aren’t limited to a certain number of points (which may mean you have to start at only one of a couple places to be able to complete a square.

When students entered, the Do Now was posted on the board: Make as many squares as you can on a 3 by 3 dot grid.


And as the bell rang, each student got a dry erase dot grid, marker and eraser. My morning class got the idea; my afternoon class (which OF COURSE C was observing so we could talk through this) did not listen at all. Some kids were trying to make squares on an entire sheet of dot paper (whhhhhhy???). Finally, we got the idea and I had students submit in a poll how many squares they got. I should have been clearer that I would have students come up to show their squares, which hopefully would have made them count a bit better! For a 3 by 3 dot grid, the max is 6, but none of my students got more than 5. I showed them the 6th and then told them to use 4 by 4 dots to make as many squares as they could. Again, the maximum a student found was 18, but you can get as many as 20.

Warm up over, we took out our notebooks as well. I had students draw a segment with a slope of 1/3 on their dot grid–and then complete a square. Some students could not figure out how to do this, so I was glad for the dry erase. When we all had it, I had them copy the square into their notebook, and then make a square with a slope of 3/4 (which was tricky so I would maybe switch that for something easier?) on the dot paper. We followed the same process for -1/4 and -2/3 (I may have one of these be an integer slope for next year).


Then I asked them what they noticed about the slopes. After they discussed with partners, we shared out and used those ideas to build our definition of opposite reciprocal slopes. Previously, I’ve usually just said that perpendicular slopes are opposite reciprocals and practiced finding them–but I didn’t feel like students understood WHY that happened. This took a little more time, although not as much as I would have thought, and I think students have a much clearer idea of what makes two slopes perpendicular to each other. I still did Quick Poll them on the calculators to practice finding the slopes, and I want to add more in about differentiating between the two–when I just asked for perpendicular students were generally fine, but asking for one OR the other led to way more wrong answers. Maybe something where they have to circle or underline or write the symbol would help? I thought about having them find both the parallel and perpendicular, but the problem seems to be deciding WHICH one to find.

From there, we practiced more writing equations with Math Mats.

Let me know if you try this out–it’s still very much a work in progress, so we would love any feedback you may have.



This has become one of our favorite quick activities in my Algebra 1 classes this year–and credit for this one goes out to my resident once again.

Remember at birthday parties, where you would whisper something to the person next to you, and then they would whisper it, and so on? And it changed from “Lesley has a cute shirt with rainbows” into something totally inappropriate.

Kinda like that.

The materials are super simple–get a bunch of paper (maybe from your enormous stack of leftover mental math?) and cut it into fourths. Then get problems, cut on to slips of paper.

You could cycle one problem through the whole group (I’ve done similar before with one whiteboard and called it a drill), but we prefer to have all the students working, so one problem per student will do.

  1. 1. Students in groups of 4
  2. Each student gets 4 blank slips of paper and labels them ABCD
  3. Each group gets four slips of paper, numbered 1-4.
  4. Each student picks up a problem, does it [distributes 2(3x-5)] and then passes it to the next person. They hold on to the original problem.
  5.  Pass once to the right, factor out the common factor (or a different step), put the piece you were given on the bottom and pass it on with a new top.
  6. Repeat a total of four times, until you get your original problem back.
  7. Have the group check how well they did keeping their problem the same.

We’ll be doing it with multiplying binomials/factoring quadratics, and we also tried it with translating between words and algebra. It fits a lot of places and can be a quick single round activity or something a little longer–lots of fun and a new favorite. I highly recommend you try it out!

What’s your favorite quick practice structure?

Final Answers

Last week, my AP Statistics class took their final exam: 20 multiple choice questions and 4 free response, basically a half-AP.

I wanted to review the answers with them, especially noting some common mistakes, and also give them a chance to reflect on how it went. I’m really excited with how it went down.

Students went into their (new!) groups of 4 (new seating chart today) with a copy of each question, distributed one per person. My second class had half groups of 3 and half 4, which didn’t matter here (the ghost just got the fourth question and they passed through the ghost).

They got 2 minutes to silently answer as much as they could on the question in front of them, then passed and had two more minutes, and so on. Then they had 10 minutes as a group to come up with their best answers to all four questions. There were a ton of good discussions, both in writing before they were allowed to talk (“I got that too!” “I think its center not spread because…”) and verbally.

Once that wrapped up, I reviewed solutions and they scored their group solution. It also gave me a chance to remind them how rigorous the AP exam is–for students who are used to getting perfect scores fairly easily, the difficulty of the AP exam comes as a shock, and they have a hard time wrapping their heads around 75% correct on multiple choice being GOOD.

It was one of those periods in which I kind of felt like I wasn’t doing much, but I think it was so valuable for my students. This would even work well for a word problem (each person does one step) or explaining several different kinds of problems for a review. By the time they were allowed to talk, several students were really invested in justifying their perspective.

Have you ever tried a write-around strategy like this? Any tips for how it could go even better next time?

Fun With Fractions Day 4

Catch up here on earlier adventures, including Fraction Strips, Comparing Fractions and Equivalency.

Today we worked with ordering and locating fractions on the number line. We started off with a half sheet with 0-1 number lines on one side, and 0-10, 0-100 and 0-1000 on the other. We worked through five sets (we skipped the last one because it was taking forever), having students place the numbers on the line and then putting them on the board & debreifing their results. It went fairly well, although it felt a little draggy. It might be helpful to do one number, then another, then another to help keep the class on a more similar pace.

From there we moved on to a fraction line up. Nothing really revolutionary there, but we tried to be very intentional in our set up. Each group got a poster paper and ten fractions, so two for each student plus two left over.

  • First, look at your fractions and decide which one is bigger and which is smaller.
  • Then decide who has the biggest fraction in the group.
  • Draw your number line and decide what number to go up to.
  • Take one minute to decide where you’ll place your first fraction.
  • Go around and place on fraction at a time. First put your fraction down, then mark its location, then explain to your group why you placed it there.
  • Your group should give you any feedback (bonus: try to do it in the form of a question, instead of saying “wrong, it should go here.”
  • Continue until your first eight fractions are placed then place the last two as a group.

After they were finished, we had them do a gallery walk and write their comments on the other groups papers–I even got part of this on video and some of it was great! One thing I would do a bit differently is to have them move TWO groups away from their own–I saw a lot of looking over shoulders trying to see their paper to compare it.

One group was a little disappointed to see that all the groups had the same fractions–I think next time I might do 8 identical fractions and then change up the last two just a little (so use all fractions with a similar idea behind them like 13/25 and 7/15 and 9/19) so they can especially look at something new on the gallery walk, in addition to confirming their own thinking.

We’re moving on to number lines tomorrow, but with large numbers and then decimals, but I’ll continue this next week Tuesday when we pick back up with mixed numbers and adding fractions.

Powers of Ten and Place Value

My resident and I are pulling our hair out over our Honors Algebra 1 class. They’re happy to work, but they don’t want to think. If they don’t know what comes next, the majority of the class just shuts down. Seriously?!? I’m guessing part of this comes from their backgrounds–they got into the Honors program here in high school, but they’re in Algebra–the top kids passed the Chicago Algebra Exam and are taking Geometry as freshmen instead. That means they have a complex and aren’t usually the very top–just close to it. So they really, really don’t want to be wrong–they’d rather just not try it. I’ll win this one eventually but man am I frustrated right now. Particularly since I think the best way to get what I want is to give a little right now to get them working for me (even if it isn’t really thinking for me) so that they get in the habit of trying things my way for the future.


We wanted to do some early work around estimation, especially in terms of using powers of ten (both for estimating with operations and for place value/number sense understanding). They can all name their place values and identify numbers, but don’t seem to have a strong concept of equivalence–its all procedural.

So we came up with pyramids.

The basic idea is pretty straightforward–find the product of the numbers on the left side and fill in the pyramid (this is the part they found easy). Down the right side, we gave them similar products, but changed the number on the left as well, so for instance 3.2*10000 or 320*10. The back side has only one number each and they had to find all of their own products using different powers of 10.

We ended up also making a matching chain to practice the powers of 10 for in class, which gave us a good idea of where they struggle (anything with a decimal). If anyone is interested, I’d be happy to share it.

(This document was made by my awesome resident Mrs. D after we brainstormed the idea together.)

Cows (Translating Expressions)

Between moving into a new “room”* and three preps, its been pretty crazy/busy. I actually have a team this year (for only one of my three preps, but beats last year when I was an island x 3) and we’ve been trying to get a little ahead on our planning.

My team plans on-level Algebra 1 (I also teach the Honors section), which meets for two periods every day–almost two hours. This past week we moved into writing basic expressions and translating expressions between words and Algebra.

After making a vocabulary chart, we did some basic practice and moved on to the sheet I affectionately (and somewhat nonsensically) call Cows.** It’s the first real group activity of the year, and my scared freshmen still don’t really want to talk to each other (although its nice to figure out who will talk no matter what and who only speaks when spoken to for future group composition).

It starts off easy, with Part I on both sides moving through some basic translation. We did a mix of having students write their answers on the board (selected by a marker left on their desk as we circulated), writing a few things we’d seen on the board and having someone read their answer. Some expressions had several interesting ways of being represented–like “half a number”–and students weren’t always sure which were equivalent. Part II is also pretty straightforward, and then Part III happens, usually on day two of the work (it took me the better part of the double period to get through Parts I & II).

On the Words to Algebra side, the problems get at some basic misconceptions (Larry is four times as old as Bobby) and unit conversion versus equation errors (If three feet equal one yard is the equation 3 f = 1 y correct?). I’m a stickler for defining a variable, and I start in on it here. What is the difference between a variable and a label (or unit)? Is “apples” a variable? Does it represent a quantity? I correct a few “f = Fred” issues and we start to get the idea that f needs to equal a quantity about Fred–his age, the number of plaid shirts he owns, how many fish he caught (apparently I have specific ideas about who Fred is, too.)

And then Algebra to Words happens. 3m + 2b = 60. No big deal, right? “Three of a variable plus….” Nope. Not cutting it here. Define a variable as a quantity with some kind of context, and roll from there. The students really want to turn this in to something like “3 monkeys plus 2 bananas equals 60”. 60 whats, no one knows, including them, and some of them take awhile to catch the issue. I usually do some counting on my fingers; looks like 5 to me.

In class, I did a practice problem first, and modeled how I would think it through to develop a problem, bouncing ideas off of my resident across the room:

Alright, m and b. So I could use monkeys and bananas. What about them? I could do…weights. So 3 monkeys times the weight of a monkey plus 2 bananas….bunches of bananas? Yeah, bunches of bananas times the weight of a bunch of bananas is the weight of the monkeys and bananas…in a van. Going to a new zoo. They’re very small monkeys. Or maybe big bananas. (I realize I could get into specifics around using the same units, like pounds, for both, but that isn’t my focus so much. I had a group catch that on their own in their problem, and handled it then. Since its not my focus, and what we are trying to do is pretty tricky, I try not to overwhelm with details.)

My 1st period class especially worked super hard on this, and came up with some great examples, such as “3 lbs of food for each monkey times the number of monkeys plus 2 pounds of food for each buffalo times the number of buffalos equals 60 pounds of food.” I realize the buffalo are probably starving to death, but the construction is pretty decent.

I know this isn’t perfect, and I’d love to refine it more, so please let me know any feedback you may have. I’ve never done a real revision, and I do like the results I’ve gotten from this version in the past, but I’m positive it could be even more valuable.

What do you think? What would you change? Are there questions you would add? Remove? Tweak?

*Its…basically what you would think teaching in a storage room is like. One of my students told me Friday she had a dream about me. She was telling them to give me my old room back because I’m a good teacher. If only it worked…

**After resistance, my whole team now calls it Cows, even after Conor swore he wouldn’t because it makes no sense. Ha! Also, you can’t even see why I call it Cows on this uploaded one. I’ll try to update with a photo of my beautifully illustrated version.

Rate of Change Cards


This is a classic–five years ago, at my past school, Drew & I sat around (with a few other colleagues, I don’t recall which ones) at our monthly meeting and talked about graphing, coming up soon in our Algebra 1 classes.

“They just can’t calculate slope from a table.”

“Well, my kids don’t seem to connect what the numbers mean to the rate of change. They don’t get what it means.”

“I wish they could just explain it in a sentence.”

And so on. We had some common concerns, but other things were more specific/personal pet peeves. We focused in on our students understanding of slope as a rate of change and some of the things we wanted them to be able to do:

  • Calculate slope from a table
  • Assign units to slope
  • Describe the rate of change in a sentence

In retrospect, this lesson was probably also the start of my obsession with word problems and understanding what a problem actually means.

We decided that we wanted our students to be able to put together those pieces and then scaffolded the problems. There are three sets of cards, and we did each set on a different color (so we could easily see where each group was). To make sure students were finding what we were asking for, each group got an answer blank. With the students in groups of 4, we handed out the answer blank and then the first set of cards. Each student was told to take 1 (they are numbered 1-4) so that they could take a little more ownership–this would probably be a great place for each group member to have a role, but I’m not that good.

1. Ms. Cutter lives 75 blocks from Solorio and bikes to school.

Use the table to find her biking speed and

time required to ride to school.

























Here’s the first problem. Stop it. Now.

I don’t care what the answer is, and I didn’t ask you for it. (This is actually worse with teachers than with students–teachers have no interest in following my directions because they don’t want to think about rate of change this way either, or at least not the ones I’ve presented to!) We wrote it like this because what the problem is asking shouldn’t always change our approach–jumping straight to an answer can take away understanding what you’re actually doing.

We start with determining which variable is independent/dependent and x/y, and then label the top and bottom of the fraction–units only, no numbers. Then we find the rate of change from the table (yes, my bike is jet-propelled–it makes a funny joke so I’ve never changed it.) Set 1 all has an intercept of 0, so these aren’t too tough. As soon as most of the class has the rate, fill that in on your fraction as well. The last thing we need to do is state the rate of change as a sentence. Nothing fancy, “She bikes five blocks in one minute” “Every minute she goes five blocks” or whatever. But say what it means.

Finish up Set 1, and start on Set 2.

5. Ms. Cutter bikes to Marquette Park to see the

tennis team and then bikes home.

Use the table to model her trip.

minutes after tennis

total blocks traveled























Same as the first, but now we’ve got an intercept. We’re still focused on the rate of change, so we start with our units again, but then discuss (first in groups then as a class) what has changed and how to handle it. You can use starting point or intercept here, but we want to get across that we can’t un-ride or un-carry or un-paint. It’s happened already, so it won’t be affected by anything else. The only change here is to fill in that starting point as well, although it doesn’t need to carry over to the sentence.

It’s getting a little bit harder now, and if you have a single period class, you’re also out of time now. Hand out some tables and have students find the rate of change and state it in a sentence for homework to reinforce what you’ve done today. On to Set 3–the students are realizing this was not nearly as easy as we thought but are also starting to understand what the rate is, how to find it and what it means.

The only new part of Set 3 is writing a function for the situation, which you can also skip and come back to later (I would still do the rest of the card).

I think this is a good framework. But I haven’t had the opportunity to discuss it with anyone in the five years since we wrote it, and I haven’t revised it either. I’d love to hear what changes you would make or how you would tweak it to make it even better. I would appreciate any feedback or suggestions you have in the comments. And if anyone is interested in the follow up lesson after this, I’d be happy to post about that later.

I’m going to be out of the country for a week, so I’m not ignoring your comments if I haven’t replied. I really would love to hear what people would do with this lesson.

Function Family Picture Project

This was the final piece in our Function Families unit, after we sorted, compared, and folded.

FuncFam Hands

The goal: Write a bunch of equations, including at least one of each of the eight types we studied this year.

My resident kicked it off with a picture he made, using at least one of each type of function.


Students worked with their partner to write an equation for each function, name the type and define domain and range. Whatever they didn’t finish in class was homework–there were 20 functions total, so most had some work to do.

The next day, the project was introduced and students had the next two nights plus three days of class time to do it.

No source 😦 Please help if you know!

Students were given two minutes to draw, and he narrated the whole time:

  • S is starting with a periodic function.
  • You can’t steal, but you can borrow someone else’s idea and make it your own.
  • Just start drawing–you don’t have to keep everything. I had a rocket ship on mine when I started, and then the boat showed up.
  • H has a logarithmic and exponential already.

The goal was to create a sense of urgency here to prevent paralysis–the student staring at their paper for 15 minutes while they wait for inspiration to strike. It mostly worked–there were very few students with a blank paper after that.

Here’s the sketch the student above started with:

Function Families Hands Sketch

They were given the rest of the period to start getting their picture together, and near the end we reviewed domain and range*. They were told to start writing their functions, but that we would review the harder ones in class the next day.



For the rest of the two days of class time, we started out with a Do Now of writing a function that was trickier and then they had the rest of the period to work. The overall assignment was a function list, including domain & range, a reflection, an artistic rendering and a sketch with each function labelled.


This project was a ton of fun, and a great review going into finals. It was also so gratifying to see how well my resident put all of this together and led the class. He has been hired for my school and next year, and will be taking my place in Honors Advanced Algebra (I’m moving to Algebra 1). He’s going to be great, and I’m so proud of how well he has done.

*This is a HUGE HUGE win for domain and range, which has been a source of trauma and stress ever since Piecewise Functions. No drama this time. Not even commentary in the reflections. Silence. They get it. They totally get it. Awesome!

Comparison Matrix: Function Families

One of the teachers at our school leads a PLC as well as some whole-school PDs around literacy skills. She tries really hard to share strategies you can actually use in ALL content areas (you know the drill, they show you something, tell you it can totally be used in math, but they aren’t sure how and have no examples. Sigh.)

Our latest strategy was the comparison matrix, and it seemed like a great fit for Families of Functions.


We weren’t sure if our students had seen this structure before, and function families are a huge and wide-reaching topic, so my resident wisely took half a period to intro the concept.

Criteria Option 1 Option 2 Option 3 Comparison

This is the basic concept (files below), and we started off with cell phones. First students were given two minutes to brainstorm some criteria they could compare the phones on, and fill those in down the left side. Then they had about five minutes to talk through filling in the columns for each type of phone.

We called on a couple of students, several of whom were hilariously into it. “Well, this network is 4G, and this is only 3G, so I think the wifi is important, but it really depends on what you want it for.” ….Um, an example? Although next time I’d totally have them look at something I did need compared–they did a good job!

Then we introduced the “Comparison” column and talked through good ways to make a comparison, and how its more than a list.

Once we were set on the overall concept, we got to work. Each pair of students received one of four versions of the comparison matrix, comparing three types of functions.


They had two days in class to work on the comparison matrix and got some amazing work back.



We both loved overhearing student conversations during this project–so interesting to listen in on what they’ve learned and how it all fits together.

I don’t know why the first page is blank, but I couldn’t seem to fix it. Just scroll down.

Have you ever tried a comparison matrix before? I had never heard of it.

Function Family Foldable

…say that five times fast!


I don’t have a file to share for this one, because we went low-key. Its handwritten. My resident put this together, and I encouraged him to do it by hand because you’re much less likely to want to shoot yourself than when you make one electronically.

They turned out great though, and the students are referring to them left and right as they work on their projects. A lot of students made gorgeous and detailed ones, but this student sits fairly near my desk in my first period (and gets there early) so I took pictures of her cover and interior pages.

LinearAVFF QuadPolyFF ExpLogFF RadicalPeriodFF

(I have no clue why the last one is smaller).

Finals start on Wednesday–almost there!