Tag Archives: algebra I

Make Your Own Best Mistake

One of the fun activities we bandied about in the Talk Less session at TMC16 was to ask students to create the best mistake for a particular problem. Each year in my Algebra I classes, my students (9th graders) have had some portion of Algebra I the year before. Many have had courses that didn’t take them all the way through, some experienced all of it but didn’t do so well, and some had PreAlgebra. We are a private 9-12 school, so we are in charge of placing them after an assessment. This is all to say that while my students need a year of Algebra I, many of them also need a fresh look at the subject–especially the intro topics that they most definitely hit last year. This activity is perfect for this situation.

Introducing this activity was as simple as can be. I gave them one equation to solve and asked them to create the best mistake they could. I had each table (3 or 4 students per table) put their mistake worked out on the board. After they were all done, a representative needed to explain their mistake. Here were all the results from one period.

I’ll tell you what, as happy as I was with these results, the next period wasn’t so great. As much as I don’t want to degrade or judge a mistake that a student thinks is relevant, they started going down a road of simply dropping numbers or introducing new ones. Responses weren’t as thoughtful. A lot of that is on me. Maybe the setup required more work than I indicated before. Maybe I need to respond to the specific combination of students in that room. Regardless, I’ll be coming back to this activity.


Guess and Check and early abstraction

I did Central Park in class today. I’d never really done it because I wasn’t sure how it would go over with our short classes1, and I was afraid that my classes would struggle with that kind of abstraction this early. But I went for it. Holy heck, it was awesome. They loved it. A few needed a little help, but even my weakest students did very well with it and were really encouraged. Many satisfied looks, many arms thrown into the air when all 16 of those parking lots were filled.

For the past 2 years, I have tried teaching Exeter’s Guess and Check method for problem solving. Glenn Waddell gives an excellent summary of it here.  For a summary of the summary, students use a table to organize guesses for a problem solution. After a few iterations, a student can look at the rows of the table to formulate an equation and then solve.

I see it as a very reliable strategy that scaffolds beautifully, but for 2 years I have not been able to sell it. They don’t see the point, they refuse to believe it’s better than what they would otherwise do, etc, etc. But I saw moments in the Central Park experience that made me think merging Guess and Check with a scaffolded Activity Builder may be a better path to teach it.

I’m not 100% sure what that looks like just yet, but I’d love to hear suggestions if you have them. I wonder if a table in the grapher could help, though that is not very easy to see on the dashboard.

I will admit that I made little to no attempt at describing my students’ struggle with Guess and Check. I will try that in a follow up. I just wanted to get this down now.

140 minutes — I know, right?!?

What is a best mistake?

We have been playing around with the prompt “What is the best mistake?” in Algebra I lately. Today we did another similar exercise where I gave them two flawed solutions to a homework problem and the prompt “Which is the best mistake?”

Like in past iterations, students buzzed about what that means, and I was predictably quiet save for my warning that they have 60 seconds to formulate an argument.

Anyone in this field will tell you that a good answer is one with a solid reasoning, a good warrant. I heard a few that I liked. The best argument all day–all year so far–was when a student said “#1 is the best mistake because it is a mistake I would make.”

Is that against the rules?

In Algebra I, we’re having some fun with order of operations and also still getting to know one another (via some nice problems). As students finished a quiz the other day, I put the numbers 1 to 10 on the board and told students to see if they can use four 4’s and any operation they like to get each value. Sure enough, as kids finished the quiz, they would saunter up to the board and write down a solution. A few values started to gather multiple approaches.

We saw some like this   or this .

Then one student wrote this

An audible “ohhhhh” could be heard around the room. Then,

Student: Is he allowed to do that? Is using 44 against the rules?

Teacher: I don’t know. It’s pretty cool, right?

Student: Yea.

Teacher: Yea, I’m not sure. I didn’t make up this game. But, it feels like a good solution. So I see no reason to deny it.

Sometimes the students are too hung up on rules (or unstated potential rules) to get creative and think outside the box. I say, take a shot. Worst thing that can happen is that you are wrong and have determined an extreme case that doesn’t work.

How do kids learn to play video games?

During a discussion about (-3)2 vs -32, I remind my 9th graders that Desmos and TOC (the other calculator) agree with what we said in class.

Student: We were never allowed to use calculators in 8th grade.1

Me: That may be, and there is a time and place for being allowed to use calculators, but I guess my point in saying that was if you’re at home working on a problem and you don’t remember how it all works, your calculator is a good tool to use to play around and test stuff out.

Students: <thoughtful look>

Me: Let me ask you something. When you get a new video game, the first thing you do before you play is crack open that manual and begin reading from the start, right?

Students: <snickers>

Me: OH COURSE NOT! What do you do?

Student: I just play around, see what happens.

Me: Exactly. You hit some buttons, explore territory, see how it works. Trial and error. Maybe explore a tutorial they made for the game, or work through a level challenge. Now, from time to time, you may need to ask a friend who has more experience, or go online and look at a forum on the game, or watch a video of some other person playing, and that’s a good thing.

Students: <nodding>

Me: So why can’t math be just like that? I’m here to tell you it can be.

My apologies that this transcript is mostly just a monologue. But the students really liked the analogy. I am certain of two things:

  1. There is no way this is the first time someone compared learning a video game to learning math.
  2. This has legs.

Anyone? Research, blog posts, rumors?

1 I have my own thoughts on this, which range from: good way to reinforce fundamentals, to why not give them every tool, to there’s a good chance this 9th grader is an unreliable narrater.

Claim and Warrant all over the place

At TMC16, I attended a 3 day class led by @stoodle and @Plspeak. For reference, here are their materials on the wiki and a collection of the tweets from the class.

Chris started the class by demonstrating how he uses Claim/Warrant statements to structure class discussion. He models formal debate by forcing students to present a solution or opinion in terms of “My claim is ______. My warrant is ______.” The warrant is the justification.

Another strategy they demoed was in presenting two solutions (and sometimes two errors) and having students pick which they liked best.

So today, on the 3rd class day with my Algebra I students, I gave it a whirl.
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First Day: Speed Demon and Talking Points

Today was Day 1, folks. All students come for 10 minute mini-classes earlier in the week, so today they came in with some minor stuff (seats, etc) out of the way and we were ready to math. And we mathed.

I have two preps this year: AP Calculus AB (mostly 12th grade, 3 sections) and Algebra I (mostly 9th grade, 2 sections). My general strategy on first days is to get as much math in as I can. Especially in a day where other classes may be reading syllabi and passing out books, students will remember classes that dig in and get to it.

Each year, I start with Nathan Kraft‘s Speed Demons 3 act in order to set up average rate of change and begin a great discussion about shrinking that time interval for greater accuracy for rate of change. In the past I would play it, hold a good discussion with students about how on earth we can measure things like distance on this video, and then send them off to measure, calculate, and make conclusions at home.

This year I saw it as a good opportunity to have my table groups work on this problem together to start the year. In each section, the conversation went similarly: we acknowledge the need to measure a distance on the road, they suggest using cars, people, or telephone poles as a unit and extrapolating, they acknowledge that those units in this context are ok but flawed, and then someone suggests Google Maps. In each class, there was a burst of energy when I suggested they take out their laptops and check out the satellite. From there, paper, whiteboards, chalkboards, and rulers were flying around. Here are a few shots of the action, including a nice use of “ruler on the screen” to get a good measure.

They concluded that all cars were speeding. We had a good chat about accuracy and what could have lead to more reliable speeds. The best line of the day was a student that said which interval we used would have a lot to do with that–as in, an interval near the start, middle, or end of the car’s journey. I have been doing this problem for a few years and I had never considered different pieces of the cars’ journeys, just shrinking one interval down.

For reflection

  • Lots of great discussion at tables once real work started, but in the class discussion there was still way to much student-teacher talk and not enough student-student talk.

In Algebra I, I dipped my toe into the Talking Points pool for the first time. I used the “Being Good at Math” set from this collection Dylan Kane posted from TMC14 and credit Cheesemonkeysf. Let’s hash this out list-style…


  • Lots of great on-topic chats at the tables which is no small feat for 9th graders in a math class on day 1.
  • We had several braves souls willing to step forward and talk about changed minds in the group chat at the end. They did a good job describing their process as well.
  • I was happy with most of the attitudes they had coming in–most tended to growth over fixed mindset.


  • Many groups had no idea what to do when they all agreed or disagreed together after round 1.
  • I worry about making sure this type of activity sticks with them as we move into traditional content.
  • Either few students were changing their minds OR few were willing to admit it. Either way, there is room for improvement.



My big theme this year is reflection, both for me and the kids. I’m most happy that I made some inroads on that today.