Monthly Archives: January 2018

Mobile Algebra

Following the introductory use of structure and emoji math to introduce systems, my teaching partner and I continued with mobiles as suggested by the authors of “An Emoji is Worth a Thousand Variables.” EDC has this great website, SolveMe Mobiles, that has 200 mobile puzzles like this:

mobile1

Each shape in this mobile has a value (or weight) and the total value (or weight) in this mobile is 60 (units). Go ahead and solve the mobile.

This mobile represents a system of four unknowns. Using traditional algebra symbols it might look like this:

mobile2

A couple of those equations have just one variable, so it may not be quite as intimidating to look at the traditional symbols. On the other hand, the mobile shapes are just so accessible to everyone!

We needed to move our students away from systems that had one variable defined for them, though, and the SolveMe site, as great as it is, always includes some kind of hint. So we started to make up our own mobiles.

 

As first, students used a lot of educated guessing to solve the mobiles. Then there was a breakthrough.

Take a closer look at the left-hand mobile.

mobile3

Students realized that they could “cross off” the same shapes on equal branches and the mobile would stay balanced. In the example above, you can “cross off” two triangles and one square. Whatever remains is equivalent, though it no longer totals 36. Therefore, two triangles equals one square. Using that relationship, some students then substituted two triangles for the one square in the left branch. Then they had a branch of 6 triangles with a total of 18. So, each triangle is worth 3. Other students used the same relationship to substitute one square for the two triangles in the right branch, resulting in a branch of 3 squares with a total of 18. So, each square is worth 3.

We were floored. We had never discussed the idea of substitution, but here it was, naturally arising from students reasoning about the structure in the mobile.

mobile4Looking closer at the center mobile, students used the same “cross out” method to find the relationship that 2 triangles equals 3 squares. If we’d been teaching the substitution method in a more traditional way, kids would have been pushed to figure out how much 1 triangle (or 1 square) was worth before making the substitution step. We knew substitution was happening here, but we didn’t invent this approach so we just followed closely to see where our students took us. Since 2 triangles equals 3 squares, some kids substituted 3 squares for the two triangles on the right branch of the mobile. Others made two substitutions of 6 squares for the 4 triangles on the left branch. Either way the result was a branch of 7 squares that totaled 14. It seemed quite natural to them.

What would you do with this one?

mobile5

Next up: Moving to traditional symbols. The final (?) post of this saga.

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Emoji math

The October 2017 issue of Mathematics Teacher included the article “An Emoji is Worth a Thousand Variables,” by Tony McCaffrey and Percival G. Matthews. (Note that you need to be an NCTM member to access the article without purchasing it.) The authors introduced their students to systems by using sets of equations comprised of emojis, similar to those puzzles that are found on Facebook. Lots of people, including those who say they hate math or they aren’t good at math or, “I’m not a math person” will do puzzles like these. They get lots of likes, answers posted in the comments, and shares –  probably because this doesn’t look much like math. Take a moment and solve the puzzle.

emoji

What if, instead of the emoji puzzle, I had posted this puzzle:

system

The two are actually equivalent, but the abstract nature of the second representation is enough to make our students who see themselves as not math people shut down.

My fall term teaching partner and I used the emoji approach with our students. We hadn’t discovered the website yet, so we gave our students this short worksheet. Many of our students had struggled with math prior to coming to Baxter Academy. They were in self-contained or pull-out special education settings or in pre-algebra classes in middle school. A few probably had something closer to algebra 1, but they certainly hadn’t solved systems of three or more equations. They had no problems understanding what the emoji puzzles were asking of them. They weren’t put off by the number of equations or the number of icons. They were able to explain their solution process clearly and with great detail.

In a follow-up exercise, asking students to make connections between the emoji representation and the more traditional representation, nearly 80% of my students saw and could articulate the direct connection between the icons and the variables.

If the emoji system is more engaging and more accessible, then why don’t we use more of them to introduce systems of equations? Is it because emoji systems seem to “dumb down” the mathematics? The authors of the article make the case against that view:

“The algebra represented in [the emoji system] is not dumbed down at all. Notice that the puzzle presents a linear system in three variables … First-year algebra students are generally not exposed to three-variable systems; indeed, when McCaffrey checked all the first-year algebra texts in his school’s faculty library, none included systems of three variables. Although McCaffrey’s students had never seen three-variable systems before this class, most found the puzzle intuitive enough to solve.”

It is important to transition from emojis to more formal algebra, but it’s not important to start with the most abstract representation – the one that leaves too many of our students behind.

 

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Using Structure to Solve Equations

Last November at the ATMNE 2017 fall conference, I attended a session where Gail Burrill highlighted some TI-Nspire documents that helped kids build concepts about expressions and equations. Sure, they’re targeted at middle level, but you use the tools that your students need, right, not where you wish they were?

Anyway, the one that really caught my eye was called “Using Structure to Solve Equations.” It was the perfect activity at the perfect time because my students were kind of struggling with solving (what I thought) were some fairly simple equations. Their struggle was about trying to remember the steps they knew someone had showed them rather than trying to reason their way through the equation. Just before learning about this activity, I had found myself using these strategies. For example, to help a student solve an equation like 3x + 8 = 44, I found myself covering up the 3x and asking the student, “What plus 8 equals 44?” When they came up with 36, I would follow up with, “So 3x must equal 36. What times 3 equals 36?” This approach helped several students – they stopped trying to remember and began to reason.

There were too many students who were still trying to remember, though. Their solutions to equations involving parentheses like 6(x – 4) – 7 = 5 or 8(x+ 9) + 2(x + 9) = 150 included classic distributive property mistakes. The “Using Structure …” activity was just like my covering up part of the equation and asking the question. But each student could work on their own equation and at their own pace. The equations were randomly generated so each kid had something different to work with. Working through the “Using Structure … ” activity helped kids to stop and think for a moment rather than employing a rote procedure.

In a follow-up to the activity, I asked kids to explain why 8(x+ 9) + 2(x + 9) = 150 could be thought of as 10(x + 9) = 150. My favorite response was because “8 bunches of (x + 9) and 2 bunches of (x + 9) makes 10 bunches of (x + 9).” From there, they saw it as trivial to say that x = 6. These were students who had previously struggled with solving equations like 3x + 8 = 44 because they couldn’t remember the procedure. After “Using Structure …” they didn’t have to remember, they just had to think!

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