# Month: October 2013

## Slides

Simple slide stealing in three flavors: KeynotePDFPowerPoint.

## Feedback Form

Attended the workshop? Let me know what you thought.

Estimation 180

http://www.estimation180.com/

A great resource for developing students’ number sense, estimation skills, unit sense, and ability to explain their reasoning in concise, specific ways.

Statistics Learning Centre

http://learnandteachstatistics.wordpress.com/

A blog all about teaching and learning statistics from Middle Earth New Zealand.

Illustrative Mathematics

http://www.illustrativemathematics.org/

A free online source of rich tasks illustrating the Common Core mathematics standards.

Progressions (Tools for the Common Core)

General website: http://commoncoretools.me/

Progressions category: http://commoncoretools.me/category/progressions/ or http://ime.math.arizona.edu/progressions/

HS Statistics and Probability document: Here

Emergent Math’s PrBL Curriculum Maps

http://emergentmath.com/my-problem-based-curriculum-maps/

“Geoff Krall Combs The Internet For Lesson Plans So You Don’t Have To”

Join Twitter, follow some of these people, and check out their blogs.

@rachelrosales

# Solved the Student Engagement Problem Yet? Me Neither.

Find yourself thinking about student engagement in your classroom? Me too. If you haven’t read Dan Meyer’s summertime pieces on engagement, you should.

Here they are:

The Unengageables

The Unengageables, Ctd.

# Absolute Value Problem of the Month

Last summer I stumbled across the Problem of the Month corner of the Inside Mathematics website. I love the idea of a schoolwide problem with multiple levels of difficulty, since students of varying abilities at multiple grade levels can join in at the “low floor” of Level A and push themselves as far as they can toward the “high ceiling” of Level E. (If you’ve never seen the Level A to Level E progression, here’s a sample.)

## My Problem of the Month/Week/Whatever

In the spirit of these problems, I’ve created my own sequence of Level A to Level E challenges involving absolute value.

The sequence was originally inspired by a problem I saw in a student’s SAT review book. The SAT problem is more or less the same as Level D, and seems like an appropriate challenge for my Algebra 2 students. We’re about to begin our study of equations involving absolute value in my Algebra 1 class, so I wanted to adapt the problem (by “lowering the front of the of the ramp”) so it would be more accessible to that group of students. I also wanted to push the problem a bit further for my more advanced Algebra 2 students.

Here’s what I was thinking at each level.

## Level A

Find one value of $x$ for which $|3x-17|>x$

Before we worry about efficient methods for solving equations involving absolute value, I want my students to develop their reasoning abilities with equations (or in this case, inequalities) involving absolute value. The expression on the left is a type I’ve seen hundreds of times, but the inequality as a whole (in particular, the inclusion of a variable on the right side of the inequality) is a new twist for me as a teacher.

I anticipate some struggle here as students become familiar with the context, but once they realize what’s going on (and start plugging in various values of x to test them one at a time) I expect they’ll find success fairly quickly. In fact, if students start with integers (as I image they will) there are only four that will not satisfy the inequality.

## Level B

Find one value of $x$ for which $|3x-17|

After I’ve baited them into the problem with an early dose of open-ended success, students will turn to a very similar problem with far fewer solutions. However, the method most students use to attack Levels A and B will likely be the same, so this second stage demands perseverance more than new methods.

## Level C

Find four more values of $x$ for which $|3x-17|

If students have been thinking only in terms of integers, this third level will force them to break out of that. I anticipate comments along the lines of “This is impossible! I’ve already found all four solutions!” and I look forward to hearing the next part of that conversation in each group, as students wonder aloud (or ask me directly), “Are we allowed to use non-integers?” (To which I’ll reply: “Did I say you couldn’t?”)

## Level D

Describe all the values of $x$ for which $|3x-17|

Now we’re pushing toward the idea of an interval, and possibly the use of more efficient techniques to solve for the endpoints. Students may have a head start here based on what they stumbled across in Level C.

## Level E

Describe all the values of $x$ for which $|ax-b|

If any of my students race through Levels A-D, I want to have a challenge that may hold their attention for a bit longer. Maybe they learned something about the relationship between 3, 17, and the interval endpoints in Level D. This fifth challenge will push them to describe that relationship with clarity.

## Disclaimer

I haven’t used this yet with my students, though I will very soon. And I’m not sure I’ll get the whole school on board this round with an untested problem. For all I know, this could be too easy, to difficult, too boring, or too something else. I’ll post an update after we’ve explored the problem. Feel free to drop a comment on the quality of the problem, or ideas for improving it (whether you use the problem with your students or not).

## Credit To Desmos

With that disclaimer aside, I should mention that without Desmos I wouldn’t have created this sequence of absolute value inequality challenges. When I saw this problem I was feeling rather lazy. So instead of breaking out pencil and paper I just graphed the two expressions in Desmos:

Problem solved! Any x-value between 4.25 and 8.5 will satisfy the original inequality ($|3x-17|).

But then I wondered about changing the parameters. Time for sliders!

Click to be transported to a land of dynamic slider action!

Anyway, that’s how Level E was born. I still had to lower the ramp at the front of the problem, so that’s where Levels A-C came from.

I’d love to hear what you think. If you have anything to share, drop a line in the comments!

# Mistakes, Radicals, Rational Exponents, and Partitioning?

A strange thing happened a few days ago. One of my Algebra 1 students stopped by Thursday afternoon to receive extra help on a topic. (That in itself is not the strange part.) He pulled up a chair, we discussed what he had been struggling with, and from that I typed out a review handout for him to work on while I helped another student from another class.

I asked him to complete as many of the problems as he could during the next 10 minutes or so. Then we’d chat about what he got right, what he got wrong, and what he skipped. A little while later he had this:

## Trouble with Radicals, But Not with Rational Exponents?

When we first sat down, I figured he would have trouble with rational exponents. Nearly all of my students (this year and in the past) who struggle with this topic have almost no trouble with square roots in radical form, some trouble with cube roots in radical form, and (if they have any issues at all) massive problems with rational exponents.

This student’s struggle was more or less the exact opposite of what I typically see. He had trouble with radicals (square roots, cube roots, anything in radical form), and almost no trouble with rational exponents. (My conjecture on #21 is that the times table in his brain has a blank spot at 8 times 8.)

## Concept vs Notation

The evidence clearly shows that this student doesn’t have a conceptual deficiency. Instead, his struggle is with…

Remediation for notation is usually fairly simple. We talked for a minute or two about how radicals (unknown and unfamiliar to him) relate back to rational exponents (known and familiar). Several minutes later, he came back with this:

Things are looking up, even if they’re still not perfect.

## Shifting My Approach

This exchange has me rethinking the direction of the conceptual/notational connection I’ve been trying to draw out for years. In working with an expression raised to the 1/2, I’ve always angled our conversations toward (and silently rejoiced inside when a student shouts):

I treated square roots like the native language, the most helpful representation, and rational exponents as this foreign thing that needs to be converted back to familiar territory.

It’s true that students are more familiar with radicals (at least in my experience with middle schoolers), but I’m quickly starting to believe that rational exponents are dramatically more informative when it comes to thinking conceptually (and when it comes to working procedurally).

## August, Every Year

When students enter my classroom, our first discussion about exponents (which invariable happens within the first couple weeks of school) goes more or less like this:

Me: What does $2^{3}$ mean?

Students: Eight!

Me: No, not “What is its value?” What does it mean, what does it represent?

Students: Oh (why didn’t you say so). 2 times itself three times.

Me: What?! You mean…

$2$
(two)

$2\times 2$
(times itself once)

$2\times 2\times 2$
(times itself a second time)

$2\times 2\times 2\times 2$
(times itself a third time)

Me: There. 2 times itself three times. Wait… That’s…

Students: No, you got it wrong. That’s 2 to the fourth!

Me: But you said…

Students: Yeah, but we didn’t mean…

Me: Grrr…

### TWO MINUTE TIME WARP…

Me: And that’s why it’s more useful to say it like that. So, how would you state the meaning of this: $10^{4}$

Students (in unison, with a three-part harmony): Four factors of 10!

Me: Perfect!

I want them to express powers this way for a number of reasons. At the very least, saying it the other way is flat out wrong. But describing $a^{b}$ as “b factors of a” has proven immensely useful in developing properties of exponents (which, for what it’s worth, I don’t hate as much as many in the MTBoS, probably because I’m easily entertained, and maybe also because my simple brain enjoys finding and justifying simple patterns).

## Okay, So… What Exactly Are We Talking About?

By now, of the 12 people who started reading this post, and the three who are still reading, at least two of you are wondering: What does this have to do with rational exponents and your struggling student?

Well, several weeks ago in Algebra 1 (the above student’s class) we had our first discussion of rational exponents. As usual, I was trying to elicit from them the idea that “to the 1/2” can be thought of as “square root,” and so on.

But a few students—bless their little hearts—wondered: Why?

And another student—bless his heart—applied our beloved “b factors of a” phrasing to come up with this:

What would that even mean? I knew, and you know, too, because we’ve seen this movie before (or at least accidentally read a spoiler in some blog comment or Facebook news feed overpopulated by comments you were never interested in reading in the first place; I digress).

But my students didn’t have half a clue what “half a factor of…” would mean, and I was on the edge of my seat to see where they would take this. (Correction: I was standing. But I fully expect I was standing on the edge of wherever it was that I was standing.)

## What They Saw

After a few more minutes of discussion, here’s what they saw and (more or less) how they described it.

• If you need to find “some number” to the 1/2, write the number as two identical factors. Then “take” one of them. That’s your answer.
• If you need to find the value of “some number” to the 1/3, write the number as three factors of the same number. Then “take” one of them. That’s your answer.

## What I Wondered

I’d never have a conversation on rational exponents take that turn, so now I was curious… What would my students do with other rational exponents? The next day, on their Topic 2 assessment, I invited students to attempt two challenge problems on the back:

1. Find the value of $16^{1/4}$
2. Find the value of $100000^{1/5}$

The results were mixed, but a majority of those who attempted the problems were spot on. Here’s a sample:

I guess in some ways this doesn’t differ much from the classic treatment:

$16^{1/4}=2$ because $2^{4}=16$

But it somehow strikes me as different, as offering more potential for extension, at least in the form my student wrote it on the review sheet that inspired this post. And now I’m wondering another thing. Would these same students—without any additional instruction from me—be able to evaluate $8^{2/3}$?

My guess is some of them could, and I expect they’d treat it like this:

Write 8 as three identical factors, take 2

In fact, I’d wager that with a brief class discussion, most of them would be equipped to handle any of these:

Write 16 as four identical factors, take 3

Write 100 as two identical factors, take 3

Write 4 as two identical factors, take 5

Write 100,000 as five identical factors, take 3

## For My Next Trick, I’ll Be Misusing the Word Partition

This idea of partitioning a number into identical factors and selecting a portion of those factors feels an awful lot like multiplying whole numbers by rational numbers:

Write 20 as four identical terms, take one

Write 21 as three identical terms, take two

Write 8 as two identical terms, take five

I don’t know if you can use partition in that sense (the factors sense), but I couldn’t shake the notion that these two problem types have a lot more in common than I ever thought before. (Maybe now that I’ve rambled all over the page I’ll be able to get some sleep at night. Or was it the kids waking up in the middle of the night that was disturbing my sleep… Too tired to remember.)

I’m a little bit nervous about hitting “publish” on this one. I feel like there are four likely responses to the post—for anyone persistent enough to ramble (as in walk) through to the end of these ramblings (as in babble):

1. Thanks for blathering, but I have no idea what you just said.
2. Thanks for nothing, everyone already knew everything you said.
3. Thanks for trying, but I think you need a mathematical intervention to work out some of your own misconceptions.
4. Thanks for sharing, that’s a nifty connection. I might use it one day with my own students.

If you made it this far, let me know which of those reactions best describes your own. Or go off script and drop a more thoughtful comment.

Either way, thanks for playing!

# Summary

Another big evening of sharing resources and challenges created by the teachers, plus further exploration of Dave Youngs’ Fascinating Triangle mathematical microworld.

# Resources

Slides

Two flavors: PDF, Keynote

MATHCOUNTS

Workout 2. Official handbook is here. This evening we’ll spend a little more time than we usually do working on problems and sharing solutions.

Estimation 180

Days 51-60, here we come! In particular, we’ll look at Day 51 (soda capacity) and Day 52 (glass vase capacity). The rest of the challenges from Days 51-60 are here:

Just like last week, after completing these two official challenges, teachers will share the estimation challenges they created for homework.

Course Themes

Arithmetic to algebra (#1) and four representations (#5).

The Fascinating Triangle

Here’s the game plan for today:

Visual Patterns

A third week of sharing our own Visual Patterns, followed by a journey into two (basic) quadratic patterns (which we didn’t actually have time for last week):

Big Ideas in Algebra

Last week we ran out of time (again) and didn’t have an opportunity to discuss the Session 3 reading assignment, or the various comments teachers in the class left on the Session 3 post. We’ll carve out some time for that discussion in Session 6.

# Assignments

We take a break from reading and “create your own” assignments to focus nearly all of our attention on the Fascinating Triangle. In addition to spending another hour or two delving into one or more extensions, teachers will write a 1-2 reflection on the exploration process, plus “appendices” with “scratch work selections” of their explorations and findings.