Algebra with a capital A!

I teach at a self-contained special education high school in SoHo in NYC.  Our math department does a good job of incorporating “algebraic thinking” into every problem we pose or task we assign.  Though, as they are in high school, our students are aware of what they “should be learning.”  In other words, they see what their peers without disabilities are doing in math class and generally it is not what they are doing.

So the desire to learn Algebra comes up quite frequently.  (The capital A is intentional in this case!)  Algebra is like our student’s white whale.  So, I try to be the boat to their Ahab.

My frustration, however, is that the typical approach to Algebra, with a capital A, is heavily language based. Vocabulary such as variable, dependent, independent, inverse, and substitute are very basic to capital A-lgebra, but they are also complex terms (and ones which have alternate meanings in everyday speech) that our students would require most of the year just committing to memory.

So I have been on a search for capital A-lgebra work that bypasses this vocabulary at least at the very beginning.  Cut to Fawn Nguyen’s Visual Patterns and Heinemann’s Transition to Algebra.

I began with a pre-assessment task about toothpicks

We used popsicle sticks.

We used popsicle sticks.

More fake toothpicks.

More fake toothpicks.

Then we investigated Visual Patterns…

From Pattern #11

From Pattern #11

I modified the structure so that the table scaffolded the students though the first 5 steps of the pattern.  Then the scaffold is removed for the 10th step.  We spent a long time talking about what “nth” meant, but I never said, “This is a variable.”  Should I have?

We carried on…

From Pattern #2

From Pattern #2

Extension questions from Pattern #2

Extension questions from Pattern #2

Looking for the patterns.

Looking for patterns.

Not a fully formed reasoning, but the beginnings are there.  We spent a long time at the beginning of the year “noticing and wondering” so you can see some of the remnants of that here!

What I liked most about Visual Patterns is that it was VISUAL!  Shocking, I know.  My students always respond positively when concepts are presented with a hands-on component, visually, or preferably both!

Then we moved on to Transition to Algebra, a new curriculum from Heinemann.

I began with this Van de Walle number trick as a “hook.”  The students had post-its and I “guessed” their numbers.  When they were all left with five and screamed for me to tell them how I did it, my answer was simple.  “It was algebra.”  No capitalization required.

 FullSizeRender (4)

Then I introduced a modified version of the Heinemann number tricks.

Students were "tricked" by me!

Students were “tricked” by me!

Then they tricked each other!

Then they tricked each other!

Today, I attempted to use the Heinemann worksheets as printed.  Usually, I have to modify some part of a worksheet (a printed grade level, extraneous information unrelated to the problem/task, or a cartoon animal meant for the elementary set) however, the Transition to Algebra worksheet was fairly successful as is.

Here’s how it went…


They especially liked the role play, which led to an organic “mini-lesson” about Lena’s strategy.  Most of the students had used Jay’s strategy of working backwards.

This student does mostly mental computation.

This student does mostly mental computation.

This student "shows his work."

This student “shows his work.”

Overall, the work from Visual Patterns and Transition to Algebra has led to a richer and deeper investigation of capital A-lgebra, than a boat load of “front-loaded” vocab would have.  Next, we’ll be moving on to what Transition to Algebra refers to as “mobile puzzles.”

And I can’t wait to pose questions that I’ve run across on twitter, such as this.

Hat tip to David Wees

Hat tip to David Wees

6 thoughts on “Algebra with a capital A!

  1. Andrew,
    I love this post. Your perspective as a special ed HS teacher is enlightening, and has resonance for all struggling learners (and their teachers).
    I don’t teach algebra, but have some limited experience with Hands on Equations. Always seemed to make sense to me. have you ever used this with your kids?


    • Thanks Joe! Its awesome to have so much encourgement from so many teachers I respect! I have had minimal “hands-on” experience with hands-on equations, but we use balance scales to represent the concept of equivalence. We did an interdisciplinary project earlier this year with science class. The students were learning about health and nutrition in science, so we used pan balances to show how to “balance their diets” with equal food consumption and exercise. Should I investigate hands-on equations further?


      • What I can say is that it made sense to me. As a self-confessed mathematically challenged learner… (documented here):
        …I use that as something of a benchmark. As in, if it makes sense to me, it might also make sense to someone else who struggles. We used it in the elem grades 3-5, kind of an end-of the-year, after the standardized test, “fun” activity, but we don’t anymore, and the district does not use it with any actual algebra classes. Wish there was a way I could get you some of the unused kits!

        Liked by 1 person

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