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:
50*2
10*10
4*25
10*100

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. 

Half-power,

1002

Day 1 handout

Day 2 handout

Day 3 handout

Thank You Math Mistakes

Every year I dread teaching exponents like nothing else. I still do. It's one of those concepts (units) I have a tough time relating to, and if I have a tough time, imagine my students. For me, exponent rules and properties have been reduced to nothing more than a good puzzle. Not sure admitting that here is wise, but I do enjoy puzzles. However, to me, expressions with exponents don't necessarily lend themselves to having applied meaning for middle schoolers, or at least no simple contextual application I can relate with. This is where you jump straight to the comments and say something like,

"Andrew, exponents are seen in [insert awesome idea] and here's a link." or

"Mr. Stadel, students can relate to exponents when they [insert other awesome idea]." or

"Stades, I'm surprised you haven't done a 3 Act lesson on exponents using [insert other fabulous idea]."

"This year will be different." You ever mutter that to yourself? Well this year is and will continue to be. I shamelessly request you to temporarily stop reading this blog and visit Michael Pershan's website Math Mistakes. Seriously, type mathmistakes.org into your url address. The more I research teaching strategies and content, the more I'm starting to see the benefit of students learning math by identifying mistakes, correcting them, and justifying their reasoning while doing so. Inspired by Michael's post on exponents, here's the handout I threw at my 8th grade Algebra students today with the following directions:

The following statements are all INCORRECT.

1. Identify the mistake(s).
2. Correct.
3. Justify (show) your reasoning.

Me: I'll give you guys 3-5 minutes of individual time to work through as many questions as possible. Then you'll share and discuss your ideas with your group followed by a whole class discussion.

Students worked individually on the questions for a few minutes. When most students were at least 75% complete with the handout, I told them they had until the end of today's estimation song Can't Buy Me Love (in its entirety) to share their solutions and reasoning with their group members. I heard some great stuff. Next we did our Fish Bowl where students come up to the front, walk us through their work and reasoning on the board while we listen and watch both intently and quietly waiting to ask questions. Here's the four we got through today.

Students in the Fish Bowl identified the mistake as multiplying two by five, producing ten. Students also correctly explained how two to the fifth power is thirty-two. The fun part (for me) was seeing multiple representations. 

1. Common: 2*2 = 4, 4*2 = 8, 8*2 = 16, and 16*2 = 32.
2. Grouping two's so 2*2 = 4 twice. 4*4 = 16 and 16*2 = 32.

The mistake here was explained a few different ways, but mainly revolved around the student forgetting the negative.  So far, questions 1 and 2 have a lower entry point for today's task, considering these students haven't seen exponents since last year. Multiple representations looked like this:

1. Common: -2*-2*-2 = -8
2. Preferred: (-2)(-2)(-2) = -8

The classic. I always enjoy this example. Students explained that the mistake was made by multiplying -6 and -6, producing positive 36. Many students pointed out that the exponent is "attached" to the closest term (6) and not the entire expression (-6). Multiple representations included:

1. Common: -(6*6) = -36
2. Common: -6*6 = -36
3. Extra: -1*6*6 = -36

This was an extremely fun conversation to have in class. In fact, for some periods, we never established a conclusive answer to the question before the bell rang. Here's what they came up with. Some students simplified it to thirty-seven and many were convinced of this at first. Very few simplified it to one, but couldn't convince anyone why. Some students offered the following:

Joey: My fifth grade teacher told me anything to the zero power is one.

Raquel: There's some rule that says it's one. It's just the rule. 

Me: Who's convinced by their reasoning? No one? 

Here are the multiple student representations we saw:

1. Pattern: 37^2 = 37*37,  37^1 = 37, so 37^0 = 1 (this didn't convince many).
2. "Anything over itself is 1," some said. Therefore, 37^5 over 37^5 is one and you subtract the exponents (5-5) to get 37^0. Therefore, 37^0 = 1 (this convinced many).

I forgot to mention that before we did Fish Bowl, I asked students what was different about what they initially did with the handout. Here are a few things they said:

1. You gave us individual time instead of just going straight to group work.
2. We had to make corrections.
3. We had to try and figure out the questions without you telling us any rules.

The last observation was my favorite. This activity gave students a desire to listen to each other and want to know the answer to these questions. I wasn't at the front of the room blabbing out rules, properties, their names, and examples. I didn't provide students with guided practice. This combination of activities and strategies felt right. More to come tomorrow, but we can't ignore that there's something to this mistake idea. What do you think?

Mistakenly,

1227

*UPDATE: The next day goes like this.

Day 1 handout

Day 2 handout

Day 3 handout

Capturing Time (musically)

Recently, I had the idea to do a theme of "song lengths" over at Estimation 180. Inspired by a recent comment from Fawn, I chose Santana's Oye Como Va. At first, I opened up iTunes and took a screen shot of the music player.  I threw in an album cover and edited it to look like this, asking "How long is Santana's Oye Como Va?":

I can get away with directly asking the question at Estimation 180. How would you make your estimate? I'd make my estimate based on the time played so far (1:26) and the location of the playhead in the timeline. I absolutely love that students have to use time here, specifically 60 seconds in a minute. Furthermore, I'm hoping students use some type of spatial reasoning with the timeline, either as a fraction, percentage, proportion, or something else. But that's it. Can we go anywhere else with this? This task feels constrained. This doesn't capture the medium of music correctly. There's got to be more, right?

The more I thought about it, I was curious of better ways (or the best way) to capture time and music. Let me rephrase that. If I were going for a more perplexing approach and wanted to create a 3 Act task to share at 101qs.com, how would I go about doing that? I remembered that I own the djay app and experimented with a really lengthy Jethro Tull song titled, Thick As A Brick. This is where I need your help. I'd appreciate you checking out Act 1 and letting me know the first question that comes to mind. Or watch it here and leave a comment/question in the comments.

Based on some initial questions, I'm thinking of revising Act 1 where the virtual record player looks more like this. (notice the record?)

The virtual record player opens up many possibilities with this task. There's a white tape marker on the record for precise tracking when playing the track. I feel there's a lot more math opportunities here, but at the same time it feels a little contrived?
Am I over-thinking this?
What do you see here?
What are your thoughts?
I need some help. Thanks in advance.

Spin it,
339