Tuesday, April 16, 2013

Mistakes to the Half Power

Today, after completing Day 128 of Estimation 180, we briefly reviewed the first four questions of the handout from Day 1 of exponent mistakes. Read about Day 1 here. We then attacked the next four questions and each of my four Algebra sections had different ideas, explanations, successes, and challenges. It was beautiful.

Question 5 brought the best conversations of the day. Many students thought that 100 to the half power was 5000 because they went 100*50, thinking 100 should be multiplied by half of itself (50). There were two classes with one student in each that thought the answer was 10, but each student had a difficult time articulating their reasoning. Those two students weren't necessarily convinced with their answer either. I took the lead on this one after hearing this from a couple of students:
It can't be 5000, that's too high. It's gotta be somewhere between 1 and 100, but it isn't 50.
I wrote the following on the board:
100^3 =
100^2 =
100^1 =
100^0 =
Me: Do me a favor and evaluate each of those four expressions.
Respectively, students gave me:
100^3 = 1,000,000
100^2 = 10,000
100^1 = 100
100^0 = 1

It helped that we already proved a zero exponent simplifies to one. PHEW! So I stood there with a nondescript look on my face and asked students to make observations. They got my "What do you wonder? What do you notice?" face.
Me: Let's talk number sense here everyone. Where would we place 100 to the one-half power?
Many students quickly placed it between 100^0 and 100^1. So how is that 5000 looking? Does that make sense? Some keen observers (only a few throughout the day) noticed a pattern of decreasing zeros. One million decreases by two zeros to get ten thousand. Ten thousand decreases by two zeros to get one hundred and one hundred decreases by two zeros to get one. No one really said it was because they were dividing by 100 each time, but I let it slide. I didn't want to take anything away from some of the lightbulbs lighting up in class.
Sheena: Since you take away two zeros each time, and 100^1/2 is between 100^0 and 100^1, you only take away one zero to get 10.
Me: (to the class) What do guys think?
Class: Yea! Totally! That really makes sense.
Me: Is it enough to convince you all? Will this work every time? So let's try this:
I asked students for factors of 100. Students gave me:
Me: Which one of these is a perfect square?
Class: 10*10
Me: So let's rewrite 100 as 10^2 and keep the half power.
Class: Woah! It's ten to the first. So it's ten!
Me: So what can we conclude about something to the half power?
Student: It's the square root.
I toss up a couple expressions for confidence: 49^1/2. They shout, "seven!" or 64^1/2. They shout, "eight!" Now, if you want to really mess with some of your kids, throw 27^1/3 up on the board and ask, "So what is something to the one-third power?" You might get lucky with a kid that says 3. Maybe walk them through that one. For my honors kids, I would ask, "What number on the board is both a perfect square and perfect cube?" You know you have a smarty pants when they say 64. That's a gem right there. Don't expect that often in 8th grade. That's my Elijah! (the closest I'll get to Fawn's Gabe.)

Question 5 stole the show and our explanations for questions 6-8 were not as spectacular. That's fine. However, I think I'll switch the order of the questions and put this question #6 last (after current question #8). I found that the flow of explanations from students for current question #8 (quotient) really helped explore/explain negative exponents, making question #6 easier for students.

In one of my classes, Arielle came out of nowhere and gave us this gem. I immediately put it up on the Quotes of the Week board.

That's right! We're exploring these rules and students are defining them through observations and patterns. I think students have a better understanding of these properties and rules when given incorrect solutions (mistakes). In case I don't have time to wrap things up with you about tomorrow, the handouts from this week will be at the bottom of this post. Michael Pershan recommended I tell students that some solutions are incorrect and some are correct. I tried that out for Days 2 & 3. We only had about ten minutes to start today's handout. After their abbreviated individual time, I put these two things up on the board:
  1. Share with your group the one solution you feel most confident about.
  2. Select one question you want to know the answer to most.
I had students stand and vote once (by raising hand) for the question they wanted to know the answer to the most. I kept a quick tally. Most classes picked question #8. Still standing, I told students to either face the hallway if they thought the question was correct or face the windows (opposite the hallway) if they thought the question was incorrect. I take a quick tally. Sit back down and have students share out loud. I'd resume Fish Bowl if I had more time. Try this sometime. Fun. 
Tomorrow: more group work, less teacher. 



  1. Simply put, you are giving students opportunities to make sense of math. Great job.

    1. Thanks Robert. I'm trying. So are my students. I'm proud of them!

  2. I am shamelessly stealing all three lessons to do with my 8th graders beginning on Friday. Soooo much better than me standing and spouting rules while they write things down.

    1. I hear ya! Been there, done that. Finding mistakes was so much more effective.
      BTW, there's no shame in stealing. Please share!

  3. these lessons are going in my folder for next year ... our unit on exponents was mostly a disaster ... can't wait to try a different approach!

    1. Hi Beth,

      Glad you'll find them useful. I'd be curious to see what would happen if you can sneak them in this year with students.

  4. Thank you! As my students prepare for end-of-course exams, your hand-outs will be great tools for revisiting properties of exponents, facilitating communication, and dispelling misconceptions!

    On a side note, it was so interesting this year that, once I showed students how to make sense of negative exponents generating patterns (i.e.. start with 3^4, then 3^3... see why 3^0 = 1, and use patterns to handle negative exponents) some students chose not to "believe" the "rule" we'd discovered - they generated a column of patterns to arrive at a conclusion about a number raised to a negative power instead. While I was happy they had composed their own path of simpler problems, it was SO INTERESTING that they preferred to do that (when feasible of course) over just using a "rule".

    1. I'm glad you can use this. I find it interesting too that your kids preferred their conclusion over using a "rule". More power to them though for taking the time to do so!

  5. I used your day one handout with my 7th graders and it was really successful. The link to your day 2 handout no longer works, however. Would you be willing to update and share it? Thank you!

    1. Hi Jen,

      Glad it went well with your 7th graders. I fixed the links for all the handouts.

  6. We had an awesome time with this! Stay tuned for a blog post later today at http://joyceh1.blogspot.com

  7. I was hoping to use this with my 8th graders, but the links don't work. Would you be willing to update or share? Thanks.

    1. I too am looking for the links to the handouts please.