# One Problem Lesson Plans

I’ve heard of people who spend an entire class period on one problem.  One problem!  With middle schoolers!  Most days, getting middle schoolers to focus on anything for 40 minutes, let alone a math problem, is an insurmountable task.  (Side note:  I’ve been working on a grad paper recently aka trying to make myself sound formal by using big words like insurmountable that you would likely never hear come out of my mouth if I were to ever have a conversation with you in person.)

I couldn’t wrap my head around finding a problem that would engage middle schoolers for 40+ minutes.   I didn’t have a clue what that type of problem would look like.  I didn’t know where to begin with a lesson like that.

I worried that my students wouldn’t “learn” as much by spending so much time on one problem compared to multiple problems on a worksheet or some other form of practice.

Enter this problem:

I learned I was wrong.  So very wrong.

Several weeks ago, I gave my students the Open Middle problem above.

Oh. my. goodness.

I was not prepared for the awesomeness that took place that day.  I still smile thinking about it.  I anticipated the problem taking 5-10 minutes, maybe.  Some students worked on it for the entire 40 minutes!

The concept of the problem was simple.  Students knew how to write equivalent ratios.  They understood they needed the digits 1-9 and knew they could only use each digit once.  But the answer?  That wasn’t quite as easy to find.  They were hooked.

And so was I.  I wanted to find other problems to re-create that atmosphere in my classroom.  I completely underestimated the rich conversation that could take place from what I considered a simple task.

By the end of that class period, I knew I needed to do more of this in my class, but what sealed the deal for me was listening as students tried to figure out how to continue working together on the task as a group after class.  They asked me if they could do a group chat with each other that night so they could keep working on it.  Then I overheard, “If I figured it out, I’ll email you! And if you figure it out, email me!”

They were excited over solving a problem in a way that I hadn’t seen from them before.

This week I used the problem below in the same class.

From Chris Smith‘s newsletter via Jo Morgan’s blog.

I was actually nervous for this lesson after how well the lesson with the Open Middle problem went.  I tried not to hope for the same results I got the first time but was worried it would flop.  It didn’t, and again, I was amazed at the conversation that resulted from this one task.

This week I used the problem below from 1to9 Puzzle with my other two sections of 6th graders.  I thought it would take about 5 minutes.

I quickly realized it would likely take longer than I anticipated and saw the opportunity for those students to experience what my other class had.  While it didn’t relate to the content we were covering like the other two problems, I decided to deviate from the lesson plan and give students more time on this.  It was well worth it.

In the few times I’ve done problems like this, a couple things stand out to me.

1. I am amazed at how many students don’t know how to guess, check, and adjust their answers on problems like these.   Some students could not wrap their head around the idea of just picking numbers as a starting point and going from there.  It was an eye opener for me, and I realized I need to continue to incorporate more situations where students need to do this.
2. My doubts about whether students would “learn” as much from doing one problem like this rather than another practice activity were erased.  The conversations amongst students while doing problems of this nature still amaze me.
3. One of my absolute favorite parts of doing these are watching students’ reactions when they finally find a solution.  They are SO proud of themselves.  This past summer I had the privilege of spending a lot of time learning from Sara Van Der Werf.  One of the things I heard from her over and over again was how one of her goals in her classroom is to get kids addicted to the cycle of being puzzled and becoming unpuzzled.  I was able to physically see this in my students more while doing these types of problems than possibly anything else I’ve done so far this year.

Do I have the “one problem lesson plan” down pat?  Absolutely not.  Is it even close to great?  No.  So far it’s really been pretty unintentional.  I’ve pretty much just been lucky and stumbled upon problems that have turned into great lessons.  I need to get better at bringing everyone back together to close the lesson after doing a task like this.  I’ve added finding more tasks like these to my summer to-do list.

UPDATES:  I’ll add more of “one problem lesson plans” below as I try them in my classroom.

This problem found here was another winner with students.  After students found a solution, they continued to work to find other solutions with me telling them to.  Sigh.  I needed that little reminder that week that we were in fact making progress.

# Parallel and Perpendicular Lines

One of the main focuses during my first year of grad school was the idea of constructivist teaching.   If you’re unfamiliar with the idea of constructivism, in short, it’s teaching in a way that gets the students to discover (construct) on their own what you want them to learn.  At first, I really struggled to wrap my head around how that would work in a math class.  My goal my first couple years teaching was to explain the math so well that students  didn’t have any questions.  My goals are now very different than those first few years! The more I started using constructivist activities in my teaching, the easier I found it was to implement more of those types of activities into my classroom.

When I started implementing more activities that led students to discover the math last year, I was not very familiar with Desmos activity builder.  I’m slowly becoming more familiar with how to use it and am loving how it makes these discovery-type activities run much smoother for both me and my students.  There was little to no direct instruction when my 8th graders were learning about parallel and perpendicular lines.  For the most part, students discovered it all through a few Desmos activities I created (with the help of some Twitter friends).  🙂

#### Here are the links to the Desmos activities.

Parallel Lines

Perpendicular Lines (Thanks to Ilona for the help with this one!)

Equations of Polygon Sides (Thanks to @GrainBrowth for the help with this one!)

##### As I think back on how the lessons went, a few things stand out to me.

1.  I probably shouldn’t have been, but I was surprised that nearly every student used “guess and check” to find the answer to the question below.  That meant that we had to spend some time talking about ways that students could come up with a solution without using Desmos since I have yet to have students use Desmos for tests.  However, I think because of this I had far fewer students write “y = 4x + 10) for their answer because they saw what happened in Desmos when they tried that.

2.  When I give an exit ticket at the end of class after doing one of these activities, I’m still somewhat amazed at how well students do.  It’s encouraging to see that this type of instruction works.

3.  In the past when I’ve taught this concept, my lessons looked something like this:

• Get students to understand parallel lines have the same slope
• Get students to understand perpendicular lines have opposite reciprocal slopes
• Spend time using those ideas to solve problems

This year I decided to spend about 2 days on just parallel lines and solving problems with parallel lines.  Then about 2 days just on perpendicular lines and solving problems with that.  Then, I combined the two before having students look into the equations of polygon sides.  This seemed to go better for students.  The biggest mistake I saw students make was when they were asked to write a line perpendicular to another line through a specific point.  Many students who made a mistake on this type of problem, knew to change the slope because the lines are perpendicular.  However, when they would find the y-intercept, they would leave the slope the same, and then at the end when they wrote their final equation, they changed the slope.  However, in the past when I’ve taught this, students have made the same mistake.  I don’t have the data to back this up, but it seemed that fewer students made this mistake this year, and I taught this concept to more students this year than in the past.

4.  As students were working through the activities, I wish I had a better way to quickly check which students were on the right track.  The Desmos teacher dashboard is awesome, I’m just not that great at using it efficiently yet.  I mentioned to my co-teacher that I wish I had a better way to know if students were on track or not, but that I also had the same problem when students were doing this types of activities without Desmos activity builder -it’s actually worse without it.