A more coherent flow of Algebra topics

You may have heard about the awesome work Jonathan Claydon (@rawrdimus) has done to reorganize topics in Algebra 2 to create a more coherent flow. When I found out about his project, I was ecstatic because my project for the last 14 years has been to do the same to Algebra 1 in a way that makes it more accessible to students with weak preparation, while simultaneously making it more meaningful (and in that sense, more “rigorous”). If you’ve followed my blog for the last 2 years, you’ve seen that I’ve begun to open up about my aspirations and my unit outlines.

Today I’m posting a tidbit of my curriculum outline here for feedback and for your reflection. If you skim it too quickly, it may look like any other curriculum. If you look at it more closely, I think you’ll see some differences between this and most other curricula (I’m curious if I’m wrong about that — please let me know).

By the way, Algebra 1 in Virginia is similar to Math 8 in CCSS states, but with quadratics thrown in at the end.

One feature of my curriculum is that topics include differentiated extensions. These aren’t just harder versions of the same problems the class is already studying. They’re also not the same as the Depth of Knowledge questions that Robert Kaplinsky likes so much. Instead, they’re designed to expose hidden connections between several topics. Students who do the extensions won’t just be more computationally proficient, and they won’t just understand this topic better — they’ll also see math as more of a coherent whole.

Here are the first 2 units. The column called “Connection to a big idea” lists connections that I think Algebra 1 students typically miss. I intend these big ideas to be themes that run throughout the whole year, similar to Jonathan’s theme of “how is the graph of an equation related to the solutions of the equation?” Of course, there are many more big ideas in Algebra, but these represent the hidden themes that I’ll work to surface wherever possible.

Topics Screenshot 1 REV

I don’t know any other teachers who do polynomial addition/subtraction in their beginning-of-year unit on combining like terms, but I think the reasoning at the bottom of the “timing” column is solid.

The differentiated extensions represent the 30% of the curriculum that I want all students exposed to, but don’t think everyone needs to practice extensively. They’re part of Algebra 1, so everyone will see them at least a bit. Students who are up-to-date on their power standards will get them in depth: they may watch a flipped-class video or get some small-group time with me, and then they’ll do independent practice. Meanwhile, students who are still working on the power standards will participate in some whole-class discussion and a few whiteboard practice problems if appropriate, but won’t study them in depth until they meet the power standard goals. (Those goals are assessed as part of my quiz generator project). What’s a power standard and what’s differentiated extension is determined by my state standards and my sense of what’s critical for Algebra 2, and a big part of this curriculum outline project is to clarify the difference.

(All students will eventually get basic exponents rules, but students who started them early via the extension from Combining Like Terms will get into more difficult exponents problems).

Here is the next unit:

Topics Screenshot 2 REV

[Update 5/31/18: I’ve been working on more differentiated extensions for the Grouping Symbols unit. Since the key idea is building up expressions, advanced students can work on modeling situations with expressions. That’s tough to teach, because students always have a zillion questions and I can’t spend the whole period with my extension kiddos — I need to have substantial time to sit with students who are struggling and reteach power standards like combining like terms, evaluating expressions, or building expressions. So I created this Desmos activity on modeling with expressions. It has so many Desmos bells and whistles it’s virtually a steam train — I worked hard on the video embedded in it. With all the embedded supports, students should be able to work toward modeling situations with expressions without too much assistance from me. Then they may be ready for Desmos’ Central Park the next class, probably with a bit of help from me. It may seem crazy to you to teach modeling with expressions as an extension rather than a core, whole-class activity, but the reality is that in Virginia, this skill isn’t really in Algebra 1. Instead, Virginia focuses modeling questions on slope-intercept or standard form, or on totally contrived questions like, “Write an expression that represents two more than the product of 5 and the sum of 3 and half a number.” Which is really more of an English grammar question than a math question, to be honest. Ugh.  /end]

You may be wondering what “building expressions” looks like on a quiz. Here’s a sample formative assessment:

Building Expressions 1

 

Building Expressions 2

I don’t know of any other teachers who even ask questions like these, though maybe I’ve just been living under my rock for too long. I’ve found that a few months of sporadic preparation along these lines means all students can solve literal equations, though. Pretty cool.

By the way, the Building Expressions questions above are part of my quiz generator, which I think will be getting some nice upgrades this summer. Someday the entire item bank will be big enough I can share it publicly without worrying about test security.

Writing up these outlines of my units were a good reflection for me, so I’ll keep it up until I have the whole year outlined.

Is this how you already teach? Do you have strong objections or ideas for improvement? Thank you, math teacher community, for helping us all grow.

Update: Rose Roberts writes in on Twitter to mention throwing radical expressions into the mix, too:

 

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