Exponent Unit

This was the first time I’ve taught exponents without explicitly telling students the “rules” at some point within the unit.  Many students still said things like, “Oh, so when you divide, you subtract the exponents.”  I have mixed feelings over this.  Yes, I want my students to notice patterns, but not at the expense of understanding the math they are doing.  This is one of the things I struggle ensuring as a teacher -that after my students have noticed patterns, they still understand what is actually happening.

I started the unit with a modified version of Andrew Stadel’s exponent mistakes worksheet.  (I know I found someone else’s version of this worksheet that I modified, but I can’t remember where I got it.)  This was something we came back to periodically throughout the unit.  On one of the last days of the unit, we went over the correct answers as a class for the first time.  After going over the sheet, I asked my students to think back to their reaction when I first gave them the worksheet.  Many sort of freaked out and several others were convinced that some of the problems were actually correct.  It was fun for me to see them realize they had learned something throughout the unit because they could now correctly do all of the problems.

Screen Shot 2017-05-13 at 2.48.58 PM

The rest of the first day we focused on identifying the base and writing things in expanded form.  The next several days I spent at least one full day on the product rule, power rule, and quotient rule.  The link for the worksheets I used is at the end of this post.  Again, I know I modified those worksheets from ones I found somewhere online at one point, but I can’t remember where I found them.

Screen Shot 2017-05-13 at 2.39.25 PM

I used this Which One Doesn’t Belong? as a warm-up one day.  I’ve really been loving using these as warm-ups this year.  I love how much vocab students use while doing these.

E1

About this point in the unit, I was not in my morning class a few days in a row due to state testing with my 6th graders.  I was looking for self-checking practice for students on exponent problems.  The challenge for me was we hadn’t talked about the zero power yet or negative exponents.  Most everything I was finding online included those types of problems.  Here’s what I came up with.

I modified Kate Nowak’s row game to work for where my students were at.

Screen Shot 2017-05-13 at 2.11.00 PM

I don’t know if “Two Truths and a Lie” is the correct name for the next worksheet I created, but I couldn’t think of another name and was running out of time, so I went with it.  Basically, students were to simplify 3 different problems.  Two of the problems would have the same answer (the two truths) and the other problem had a different answer (the lie).

Screen Shot 2017-05-13 at 2.11.13 PM

I also had a sheet of Yohaku puzzles ready which I LOVED, but I didn’t end up using it then.  I did, however, use it later in a few of my classes.  I love that there are so many different solutions to these puzzles.  I definitely want to look at the other puzzles on that site for future use.

Screen Shot 2017-05-13 at 2.11.38 PM

When I was finally back with all classes after state testing, we reviewed using this Desmos activity I created.

Screen Shot 2017-05-13 at 2.20.59 PM

absolutely LOVE this Desmos activity from Mathy Cathy for an introduction to zero and negative exponents.

Screen Shot 2017-05-13 at 2.22.40 PM

We ended the unit with some more practice combining all different types of problems.


Here is the link to download the worksheets from this unit.

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:

Screen Shot 2017-03-29 at 8.52.30 PM

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.

113 3

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.

C7mDCnCVQAAqhbS.jpg-large

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.

C90r02CV0AAgeWt.jpg-large