# Modeling in Algebra I: An example

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The modeling method I’ve been using to teach physics has really made me think about way I teach Algebra.  Here’s a good example:

The topic:  Exponential Functions

The way I taught it last year:

Step 1: Introduce the topic with some notes and some example problems.  I used a PPT presentation to show the students the equation y = a*b^x and explain what all the parts mean.  Then we looked at some example problems.

Step 2: Explore the topic with an activity.  We did the classic M&M’s activity to explore a “real world” example of exponential growth.

Step 3: Practice.  We used problems from the text, most of which did not have a context.

Step 4: Assess.  We had a test after about a week or so.  Done.

The way I taught it this year (modeling):

Step 1:  Explore the topic starting with this clip from the movie Contagion.  I got the idea from Leslie Macfarlane on 101 Questions.  I used this handout with it.  We started by discussing what factors would effect the spread of a virus.  We came up with a bunch, but most importantly: the type of virus (R-0 in the video clip) and the number of people who currently carry it.  Here’s what their handouts looked like filled in:

Step 2:  The kids figured out on their own (in groups) that the flu virus doubles each cycle and the smallpox virus quadruples each cycle.  They also were able (with a little help) to figure out that the cycle # (x) needed to be the exponent in the function.  Then, it wasn’t hard to get them to write the function like this:

(total # of people infected) = (starting number of people) * (mutiplier)^x

Step 3:  Generalize it!  We then discussed a general form for the equation.  They needed to make a connection between the R-0 number from the video and the multiplier they used to generate their charts.  As soon as I brought up the question, they noticed that it was just (1 + R).  At this point, my head almost exploded from awesomeness.  Now, they understood that the “multiplier” they used was given by one plus the growth rate.  How did that happen?!  I didn’t “teach” it to them!!  So now they’ve got y = a * b^x, they know what the “b” means, and since we graphed each initial scenario, it was simple for them to see that “a” is the y-intercept.

Step 4:  Expand the generalization to new scenarios (exponential decay).  We used this handout to figure out that if you are losing a certain percentage each cycle, then it’s just (1 – R) instead.

Step 5: Practice.  I gave them 3 or 4 scenarios of each type to practice.  For each scenario, they made a chart and a graph (using Desmos or by hand).  The answer to any questions asked was secondary to showing me that they could analyze the situation using the exponential model.

Step 6: Assess.  Two short multiple choice quizzes, some questions with contexts, some without.  Then an open ended task in which they had to explain to me the differences between two functions.  Retakes are available on all three assessments as needed.  They also take the assessments when they are ready.

Why is this new way better?

I have students now who are willing to try figuring things out on their own.  They see that it can be done.  Especially in Step 4, I saw a bunch of students working to find a function that defined an exponential decay scenario.  They didn’t say “Mr. Owen, what’s the formula?” or “I don’t know how to do this because you didn’t teach us this yet.”  Check out the example below.  This girl decided that (hmw * .10) = y  and that the next value in the table was given by (original amount – y).  I just had to suggest using substitution to combine those two expressions.  From there, it wasn’t too hard for her to see that to do that 10 times, you just raise the expression to the tenth power.

Here’s an example of a practice quiz done by a student (this one has no context):

# SBG = I got this! SBAR = Wait, what?!

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SBAR = Standards Based Assessment and Reporting

Same thing?  Maybe.  Depends on who you ask.  But the second one really gets at the heart of this problem I’ve been having.  And I know I’m not the only one with this issue.  Here goes:

As soon as I began implementing SBAR in my classes, I knew I wanted a system in which kids could work at their own pace and assess when they were ready.  Awesome! Let’s do that.  Well, I’ve got the A and I’ve got the R, but putting them together is a mess!  Here’s the setup:

I’ve been using ExamView for my assessments and it’s been working out great.  Kids take quizzes when they are ready.  You can read about that here. Now, once the assessments are taken, I have to get those assessment scores from ExamView to ActiveGrade, my standards based gradebook.  But it’s not as simple as a cut and paste from one spreadsheet to another. Each student is taking a different set of quizzes each day, and even if they were taking the same quiz, one kid may be on attempt 1 and another on attempt 3.  So the plan I’ve worked out goes like this:

1.) Kids take whichever assessments they are ready for that day (all online, but they turn in constructed response items on paper). The questions change each time they take an assessment.

2.) I score the constructed response items right away.

3.) I enter (by hand) all scores from constructed responses and multiple choice quizzes on a spreadsheet. The spreadsheet has three columns for each learning target (attempts 1-3). That way I can tell how many times that student has done that quiz.

4.) I transfer the scores from the spreadsheet to ActiveGrade.  In AG, I enter “group assessments” called “3.1 Take One” “3.1 Take Two” and “3.1 Take Three.” Then I take the handwritten spreadsheet and match it up to those three “group assessments” entering in scores wherever there are blanks.

The problem is: THIS PROCESS TAKES FOREVER!!  What I need is a system that seamlessly integrates the assessment and the reporting.

To be clear, what I am talking about is a system in which I can create question generators (like what I’m doing now with ExamView Test Generator), give assessments using those generators (like what I’m doing now with ExamView Test Player), have them auto graded (like what I’m doing now with ExamView Test Manager), and then have the results reported in a standards based gradebook for students and parents (like what I’m doing now with ActiveGrade).  See where the disconnect happens there?  ExamView does the assessment and ActiveGrade does the reporting.  Why don’t they talk to each other?  Does anyone know of a software that does this?

# ExamView for Re-assessments

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Sorry if I sound like a commercial in this post, but this software is really helping me assess better and more frequently.  With that said…

If you’ve ever used eInstruction’s ExamView software, you know that certain questions will recalculate values and give you a whole new question.  But did you know that you can create these “question generators” yourself to fit your needs?  It’s true!  You can make multiple choice or constructed response items that will calculate new values with just a click.  With this capability, you can create a quiz that students can retake as many times as you’ll let them.  I’m still planning to let students re-assess in other ways, like presenting in class and making short videos (haven’t tried this one yet, but I want to), but the number of students needing re-assessments on each learning target has been too large to keep up with so far.  This is a way to cut down on the time required to re-assess all those students.  Of course you need the software first, but it comes with a bunch of textbooks, so you might already have it.  Here’s a document that I found (from William McIntosh who works for eInstruction)  with some good instructions on how to get started making your own dynamic questions: