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Desmos in my Pocket


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A cool thing happened in Algebra 1 today. We’ve been working our way through an “Applications of Quadratics” unit (one that needs a healthy dose of revision in the near future). Today: projectile-motion problems.

The basic flow for each scenario:

  • Set the context (catapult, arrow, ball from roof, etc.)
  • Substitute an input, evaluate, explain the result in context
  • Substitute an output, solve, explain the result in the context

One student was struggling with the two-solution result to bullet #3 above. “How could there be two answers?”

I’ve been asked this question before, and feel like I’ve been able to help students reasonably well with a combination of questioning, hand-waving, sketching, etc.

But today? I reached into my pocket and added one more element to the conversation:

2014-04-24 22.15.01

 

Now, I love using Desmos on a laptop. Nothing beats that graphing experience in my mind. But to have a functional version of Desmos sitting in my pocket, ready to bring into a conversation at a moment’s notice… That’s cool.

Closer to One


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Earlier this evening I was poking around map.mathshell.org looking for bite-sized middle school tasks. I found a limited but great set of questions here and was inspired to ask this:

Just in case Twitter messes up the formatting on whatever device you’re reading on, here’s the question again, as well as my challenge to you and a solution of my own:

The Question

Which is closer to 1? Does it depend on n? Explain.

image

My Challenge

I’m curious to know how others reason through and/or visualize this problem. If you find yourself with 5-10 spare minutes this week, scribble out an approach/solution/representation (or two) on a napkin, take a photo, and drop a link to the image in the comments. I’ll update this little ditty on Friday by embedding images in the post.

Pictures, words, algebra, numbers, proof with words, proof without words… It’s all fair game here.

A Solution

Don’t peek until you’ve given it a go yourself (and hopefully shared your solution in the comments), but here’s my visual argument, sans words. (As in, “I’m not providing any words.” Let me know if you think it needs a comment to clarify.)

Wasting a Day vs Wasting a Year

Today in Algebra 1 I tried a sorting activity designed—in theory, that is—to strengthen student understanding of the vocabulary involved and the methods used in solving quadratics by factoring and by completing the square. We’ve been struggling with this lately, so I thought I would try another tack.

It didn’t go so well, so I feel like I wasted the day. And you know what? I’m okay with that.

Because after a year of tweeting and blogging and interacting with teachers a thousand times better than I am, I have a vision of what my classroom could be like, and (1) it doesn’t look the same as what I’ve been doing and (2) it’s worth pursuing.

What concerns me far more than wasting a day is not trying new things, not experimenting, not seeking opportunities to grow, and instead sticking with my status quo and through that static and stunted approach ending up wasting an entire year.

So I’ll keep tinkering, I’ll continue trying things that may fail. And sometimes I’ll feel like I lost a day (like today). But at the end of the year, we’ll have something better than if I had played it safe all along.

Oh, and next year… pick up where I left off, and I’ll keep tinkering and trying and failing and improving and building. And I think my students and I will be far better off in the long run.

Excuse me while I head off to redesign something.

Complete the Quadrilateral Domino Activity


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I’m teaching a geometry course at FPU this semester. I taught the same course three years ago, but my philosophy and practice have shifted so significantly in that span that I’ve scrapped what I used the first time around and am in full create-and-curate mode.

With that in mind, I’ve been frequenting Geoff Krall’s fantastic curriculum maps rather, um, frequently as of late. My latest joy-filled find: Don Steward’s Complete the Quadrilateral (via Fawn Nguyen).

Inspired by these posts—and a separate domino-style activity shared by Mike Chamberlain at a common core workshop last semester—I spent a good portion of my three-day weekend turning the complete-the-quadrilateral task into 21 dominos of self-checking goodness.

Set Up

Teachers, print this

Screen Shot 2014-02-10 at 6.21.30 PM

…and use these…

Skalm_2

…to get a stack of these…

2014-02-10 18.25.41

…which you’ll shuffle and put in one of these…

2014-02-10 18.27.34

…and hand out to groups of these:

Classroom

How to Play

Working in groups of 2-4, students will:

  • Join dots to complete the indicated quadrilateral on each domino (Multiple possibilities? Maximize the area!)
  • Determine the area and perimeter of each figure
  • Connect the domino chain, from start to finish

Just to clarify the whole “domino chain” bit, each domino contains one answer on the left (to a previous card’s question) and one question on the right (with an answer to follow on the next card). I’m fairly certain that last sentence doesn’t make any sense, so… “Hey-look-a-picture!”

2014-02-10_19_53_45

Solutions

If you need a hand with completing the quadrilaterals, check out the source, or even the original source. (Note that I removed four problems—Fawn’s #9-11 and 14, due to identical figures mucking up the uniqueness of my domino chain—and that my final problem matches Don’s, not Fawn’s.) For help with the area, perimeter, and domino sequence, note that my handouts contain the dominos in order (reading one column at a time, from top to bottom):

Complete_the_Quadrilateral_Domino_Activity__Set_A__png

Variations

The teachers in my geometry class served as guinea pigs for this revamped version of the activity, and after a debriefing discussion, I think it would be wise for me to create a few variations to allow for more students to participate without becoming too overwhelmed/frustrated. (Translation: The task took a very long time, and with the four required steps—complete the quadrilateral, find the area, find the perimeter, match the domino—I’m afraid student interest will fizzle out as frustration grows.)

Here are several I’m wondering about (including one I’ve already added):

  • Split the deck into two groups of 10 dominos (Set A, a bit easier; Set B, more challenging), allowing for three levels of difficulty to differentiate between or within classes (Set A only, Set B only, Sets A and B)
  • Reduce the deck to 10 dominos, remove perimeter from consideration (if the self-checking domino action is to stay in tact, I think I should make sure I have 10 distinct areas)
  • Same as #2, but remove area from consideration (not sure if I would then make the rules “maximize perimeter” or if that’s just too awkward)

Request for Feedback

There’s a lingering fear in my mind that I took a great activity and ruined it. If you think that’s the case (or not the case), I’d love to hear why in the comments. Also, if you have any other suggestions for tweaking the length of the activity without losing its original challenge or appeal, let me know.

[One-Minute Makeover] Algebra 1 • Topic 7 Assessment


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Previously

If you’re just tuning in, check out the first post in the series. Or Topic 5, if you fancy. (Topic 6 doesn’t exist. It’s a long story.)

Algebra 1 • Topic 7 Assessment, Before

There’s a point in the school year—toward the end of the first semester, usually—when my brain and my body get together to discuss whether there’s enough left in the tank to write something decent (be it a lesson, assignment, assessment, or whatever). Apparently, late in November 2011, the exchange must have been something like this:

Brain: “Okay, body, whaddya say? Let’s write a quality assessment for Topic 7, shall we?”

Body: “Must… sleep… so… tired…”

Brain: “What’s that? You think all we can manage is a pile of whatsit?”

Body: “Erghhhh… Where are we?”

Brain: “Okay, then, that’s the plan! Mediocrity, coming right up!”

At any rate, the result of my/our/their efforts was nothing to write home about (except maybe to lodge a complaint). I hereby present to you, two questions worth their weight in zero g:

Screen Shot 2014-02-07 at 10.58.06 PM

Screen Shot 2014-02-07 at 10.58.11 PM

The worst part of it? I based these questions off two I found on the CST, a multiple-choice test the quality and usefulness of which I regularly sneer at. A strange thing it is to despise one’s assessment muse.

Now then… On to happier times!

Algebra 1 • Topic 7 Assessment, After

The first thing that had to die: The multiple-choice-ness of the problems. I’m not opposed to all multiple-choice problems in the world, just most. There are some decent questions here and there. In fact, quite a number of the ones I see in preparing students for the AP Calculus exam strike me as worth their weight in… I don’t know, maybe salt. (Modern day market value, of course.)

But multiple-choice on a graphing linear equations assessment? Not a good fit, in my estimation. For starters, students with no idea of what they’re doing could luck their way into a perfect assessment score, especially when there are only two questions. Next, the format invites students to select an answer without showing much of their thinking. And beyond that, I left myself no room for questions that demand any measure of critical thought. (More on that in a moment.)

With those concerns at least partially in mind, I wrote a new assessment with four questions:

Screen Shot 2014-02-07 at 11.07.47 PM

Screen Shot 2014-02-07 at 11.08.02 PM

Screen Shot 2014-02-07 at 11.08.09 PM

Screen Shot 2014-02-07 at 11.08.15 PM

Here’s what I like: Goodbye multiple-choice format, hello (potential for) students showing a record of their thinking.

The questions aren’t amazing, and they’re quite limited in scope as they’re all really just begging for an equation in slope-intercept form (my beloved point-slope form comes up in Topic 8). However, I think they’re a dramatic improvement over the original.

Here’s what I still can’t stand: The assessment is still overwhelmingly focused on procedural understanding.

I suppose this might not change until I revisit/rewrite my skills and concepts list for Algebra 1 to include a more rich approach to graphing lines, but it’s still disappointing to look at a mid-November assessment and see a total lack of “explain-your-reasoning-this” or “explain-your-reasoning-that.”

Wrap Up

A quick word about #3 before moving on… I chose to display the ordered pairs so that students wouldn’t struggle with miscounting too-little-toner tick marks. When Desmos adds a “grid density” feature for improved (read: bolder) printing, I’ll consider removing those labels. In the meantime, they’re staying.

More than any assessment in this series so far, I’m super-excited to hear suggestions on how to address the weaknesses of the updated assessment. Have an idea? Please share!

Feb 8, 2014 Update

Great idea from @BridgetDunbar:

[One-Minute Makeover] Algebra 1 • Topic 5 Assessment


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Previously

It all started with this post, and the most recent assessment adventure is here.

Algebra 1 • Topic 5 Assessment, Before

When I first made the shift to standards based grading, I threw together a list of skills as best as I could. When I considered equations involving absolute value, I figured, “What else would I want them to be able to do beyond solving a simple equation and solving a more advanced equation?” Fool. Anyway, here’s the pair of questions on the original Topic 5 assessment:

Screen Shot 2014-02-05 at 9.34.19 PM

Screen Shot 2014-02-05 at 9.34.25 PM

I was actually half-proud of the sequence of lessons I taught leading up to that original assessment. We built (or attempted to build) an understanding of equations involving absolute value by launching from a verbal approach with heaps of number line action, to a graphical approach with fantastic lines of intersection (and this was pre-Desmos!), before finally exploring an algebraic approach (the one and only representation I remember seeing as a student).

At any rate, a mediocre-to-decent set of lessons followed by a decidedly weak assessment left much to be desired.

Algebra 1 • Topic 5 Assessment, After

One issue I found in the original assessment that I tried to address in the update is that I had very little sense of who was struggling with the algebra and who was struggle numerically with the concept of absolute value. In other words, pick a kid who failed the assessment. Was his weakness only in solving equations, or was he unable to even evaluate expressions involving absolute value.

With that question in mind, I added three questions to the front end of the assessment:

Screen Shot 2014-02-05 at 9.48.38 PM

Screen Shot 2014-02-05 at 9.48.43 PM

Screen Shot 2014-02-05 at 9.48.49 PM

#1-2 are nothing special, but they did give me a better sense of how deep a particular student’s struggles went. And #3 is standard assessment rewrite fodder for me, as you’ve no doubt seen in previous one-minute makeovers. Nothing like a little “explain the error” and “redo it correctly” to see what’s going on inside a student’s head.

The last two questions on the updated assessment are literally cut-and-pasted from the original assessment:

Screen Shot 2014-02-05 at 9.52.34 PM

Screen Shot 2014-02-05 at 9.52.39 PM

I decided the main point of this assessment was still, “Can you solve an equation involving absolute value or not?” (Is that a worthwhile goal? I’d love to hear your thoughts in the comments.)

Wrap Up

I’m still largely dissatisfied with this assessment, though I don’t really know what to do with it. I feel like our in-class approach to this topic—warts and all—was richer than the assessment would suggest. In the next one-minute makeover I’ll explore some options for incorporating questions that dig into students’ understanding of this topic from a verbal and a graphical approach. And while I’m at it, the quick nod at a numerical approach could be strengthened, possibly by shifting evaluation from out-of-context “Hey, what’s the value of this?” questions to determining whether particular x-values are solutions of a given equation (via substitution).

Interested in the lessons and assignments that preceded this assessment. The brave shall enter through this door.

Anyway, more reflecting and tinkering is on the way. In the meantime, drop your own thoughts and questions in the comments below.

[One-Minute Makeover] Algebra 1 • Topic 4 Assessment


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Previously…

If you’re just tuning in, consider checking out the first post in the series, or the most recent post.

Algebra 1 • Topic 4 Assessment, Before

We’re drifting into a little section of my Algebra 1 curriculum that I’m at least a little bit ashamed of. I have no one to blame but myself since I put the textbook on the shelf and created lessons, practice, and assessments from scratch. Big plans for improvement in the months ahead, but for now, warts and all…

Here are the two-and-only questions from Form A of my old Topic 4 assessment:

Screen Shot 2014-01-21 at 9.20.33 PM

Screen Shot 2014-01-21 at 9.20.38 PM

Oh, the shame! I’ll talk more about the gap between this assessment and all-that-is-decent-in-this-world in a moment. For now I’ll just remark that what these questions actually demand of students is so far below what I originally intended that they are essentially useless as an assessment tool in Algebra 1.

Algebra 1 • Topic 4 Assessment, After

The first major flaw in the original assessment is that the questions appear out of nowhere and drift away from our attention just as suddenly. So I replaced these two unrelated questions with a sequence of five questions related to a single scenario. Here are the questions from the new Form A:

Screen Shot 2014-01-21 at 9.44.42 PM

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Screen Shot 2014-01-21 at 9.45.21 PM

My initial intention was for students to use expressions and/or equations to answer the questions on the assessment. On the original version, almost none of my students approached the problems algebraically. Many were able to answer the questions (and many were not), but nearly everyone who answered correctly did so with nothing more than some numerical tinkering.

While I’m not opposed to numerical tinkering (quite the contrary; I think it’s a fantastic practice for students), in this class and on this assessment I was hoping to see whether they could write an expression to model a situation and use the expression to answer another question or two in an efficient manner.

With the original assessment, this was a lost cause. With the updated version (particularly #3-5) I was able to measure at least part of what I set out to measure.

Oh, But It’s Still So… (aka “Wrap Up”)

While my current assessment is an improvement over the first version, it still strikes me as terribly inadequate. Here we are, at the end of a unit on linear modeling, and there’s a massive void when it comes to two hugely important things: (1) At no point is any connection made between the verbal/numerical/algebraic representations and a graphical one (and it would be so easy to fix this!), and (2) The scenario is decidedly boring and contrived.

I have ideas for how to address #1, but am at a loss for how to remedy #2 in the space of a single-page assessment. More to think about for the next round of revisions.

One additional minor/medium flaw I see in the updated version is this: At no point do I ask students to explain their reasoning, justify their thinking, etc. (And word on the street is that those are cool things to do.)

Until next time…

[One-Minute Makeover] Algebra 1 • Topic 3 Assessment


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Previously…

The first post in the series is here. The previous post (Topic 2, Part 2) is here.

Algebra 1 • Topic 3 Assessment, Before

When I first drew up this assessment, my goals were to evaluate students’ ability at simplifying linear expressions and solving linear equations. Here’s what the two questions of Form A looked like:

Screen Shot 2014-01-19 at 8.50.22 PM

I had the same all-my-eggs-in-one-basket problem with this original Topic 3 assessment as I did in an earlier assessment. If students aced these two questions, I knew they were capable of what they ought to be able to do. However, if they missed one or both, I was stuck without much information. There was no gradation in the all-or-nothing results.

Another issue: the assessment focused entirely on procedural skills and demanded nothing from students in terms of demonstrating deeper conceptual understanding.

Algebra 1 • Topic 3 Assessment, After

As was the case with Topic 2 (detailed in posts here and here), I addressed the above concerns by lengthening the assessment quite a bit. The revised Topic 3 assessment weighs in at two pages and a total of ten questions.

In the first three questions, I try to get a read on whether students understand conceptually what a solution of an equation is. (For the record, what I’m looking for is something along the lines of “this value does/does not satisfy the equation,” along with numerical support—via substitution—of that claim.)

Screen Shot 2014-01-19 at 8.59.26 PM

Screen Shot 2014-01-19 at 8.59.33 PM

Screen Shot 2014-01-19 at 8.59.38 PM

After that, students move through a series of four increasingly difficult linear equations, giving me the leveled progression I was lacking in the original assessment that would help me distinguish the “almost there” from the “completely lost.”

Screen Shot 2014-01-19 at 9.04.09 PM

Screen Shot 2014-01-19 at 9.04.18 PM

Next up, an error-analysis/explain-your-reasoning style question:

Screen Shot 2014-01-19 at 9.05.49 PM

And to close, two more “solve” questions (including one at the same level of difficulty as the original Topic 3 assessment):

Screen Shot 2014-01-19 at 9.06.47 PM

Screen Shot 2014-01-19 at 9.06.52 PM

Wrap Up

The net result of the these changes is a much stronger assessment, with improvements in at least two categories. The new assessment (1) provides me with more specific insight about student strengths and weaknesses, and (2) demands more of students in the way of critical thinking and clear communication.

I fully expect that this new assessment could be improved in half a dozen ways. Part of the beauty of teaching (and writing many of my own lessons and all of my own assessments) is the opportunity for continual improvement over the years. This job will never leave me bored!

Is there anything in particular you liked about the improvements I already made to my Topic 3 assessment? Do you have a few more ideas for making it even better? Share away!

[One-Minute Makeover] Algebra 1 • Topic 2 Assessment (One More Time!)


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Previously…

It all started here. In the last post, I looked at additive and multiplicative inverses. Onward!

Algebra 1 • Topic 2 Assessment, Before

The second half of my original Topic 2 assessment assessed whether students were able to evaluate expressions involving integers and various operations (including radicals, rational exponents, and a few other things). My original approach included a single question, with everything all smashed together:

Screen Shot 2014-01-19 at 8.56.41 AM

For those who were able to evaluate the expression correctly, I got precisely the information I needed (“Johnny can do this, that, and the other thing.”). But for those who answered the question incorrectly… Was it because they were lost on everything? Or because they struggled with one skill in particular? While a close look at their work would often reveal the answer to that latter question, I find that I’ve stripped one of the benefits of SBG (specific insight into specific strengths and weaknesses) right out of the question.

Algebra 1 • Topic 2 Assessment, After

To address that weakness, I bumped this section of the assessment up from a single question to several (three, in fact):

Screen Shot 2014-01-19 at 9.03.15 AM

I lose a minute or two more of class time to administer the assessment, though I gain a quick and clear sense of who’s struggling with exponentiation, rational exponents, and simplifying expressions involving multiple radicals. Note that while grouping symbols are entirely absent from #9 above, they make an appearance in some of the other assessment forms, including this one:

Screen Shot 2014-01-19 at 11.57.51 AM

Something’s Still Missing…

Even with this more discrete-ified set of questions—which I view as an improvement over the original—I still feel like this assessment is short on critical thinking and “explaining your reasoning.” A nice quick addition might be to present students with an expression (similar to #9 above) with two (or three) incorrect step-by-step approaches (each of which has exactly one error). Ask the students to identify the error in each approach and then show their own (100% correct) step-by-step solution. Here’s a quick mockup:

Screen Shot 2014-01-19 at 12.24.16 PM

Screen Shot 2014-01-19 at 12.24.26 PM

Screen Shot 2014-01-19 at 12.24.33 PM

Wrap Up

I’ve now written four of these “better-assessments-in-sixty-seconds” posts. Since I’ve taken two posts to address each topic (the content fell rather naturally into four categories, rather than only two), I might want to consider breaking these apart for the purpose of grade book entries. I might even leave the assessment handout itself unchanged, but the idea of more refined grade book categories for tracking student mastery certainly has its appeal.

Thoughts on that last thought? Comments on something else? You know what to do.

Cheers!

P.S. Disclaimin’ the Naming’

I’m terrible at coming up with imaginary student names for my handouts. So I often use my students’ names or my kids’ names (I have lots to choose from in this second category, now!). Today I borrowed some names from a list of fictional butlers. Oh, I also have a preference for names to follow an A, B, C, etc., pattern.

[One-Minute Makeover] Algebra 1 • Topic 2 Assessment


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Previously…

Last time in this quick-look-at-improving-assessment series (which began here) I shared my attempt at improving the questions related to distribution on an Algebra 1 assessment. As always, you can check out the topic list here (or here, if you want “I can…” statements as well).

Algebra 1 • Topic 2 Assessment, Before

This time we’ll take a look at a series of questions related to operations on numbers. Here’s the rubbish version (from the original Form A):

Screen Shot 2014-01-19 at 7.34.45 AM

I was trying to get a read on whether students understood what additive and multiplicative properties are. For reasons similar to those shared in the first post in the series, this question type wasn’t particularly effective. Also, there’s the issue of “What am I actually trying to accomplish with these questions?” I don’t think I had that settled in my mind when I wrote the original assessment, and that led to the lackluster questions shown above.

Algebra 1 • Topic 2 Assessment, After

If this assessment was going to improve at all, I first needed to nail down what I wanted to accomplish. Then I needed to work on better ways to ask questions (even just spicing up the originals with “explain your reasoning” or “defend your answer” would have been a nice start.

At any rate, I decided on three goals, so I wrote three mini-sections of the assessment. Here they are:

Goal 1: Students will know the colloquial nicknames for additive inverse (opposite) and multiplicative inverse (reciprocal)

And here’s how I attempt to measure that on the new-and-hopefully-improved assessment:

Screen Shot 2014-01-19 at 7.40.02 AM

Simple, but to the point. On to the next one…

Goal 2: Students will be able to use the idea of additive and multiplicative inverse (possibly without even knowing their names) in order to “make one” and “make zero”

Here’s how I tried to assess that skill:

Screen Shot 2014-01-19 at 7.40.06 AM

I decided that this was actually the main reason we were exploring additive and multiplicative inverses in the first place, so a rather direct assessment question seemed appropriate. On to the third goal related to inverses…

Goal 3: Students will justify their reasoning with verbal and numerical support

The content isn’t profound or complex, so I thought it might provide a nice opportunity for students to create their first “mathematical” argument, one with complete sentences and mathematical “evidence.” With these two questions, I’m really trying to pave the way for more complex arguments students will make in Geometry, Algebra 2, Precalculus, and Calculus.

Screen Shot 2014-01-19 at 7.40.13 AM

Wrap Up

Now that I’ve written three of these posts, I’m wondering if I should add student work. I don’t have anything for the original versions, but for some of the revamped assessments I took pictures of strong and weak responses in order to facilitate in-class discussions the following day. If I can dig those images up, would they be worth posting? Share your thoughts (on this last question, or in general) below.

Cheers!

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