In a previous blog about Emoji Algebra, I presented the idea of using pictures (i.e., emojis) for variables. I will now add to that idea by using the physical structure of a mobile to represent the balance (i.e., equality) that must take place between the two sides of an equation.
Imagine using mobiles that once hung above your childhood crib to learn how to solve algebraic equations.
The mobile on the left is balanced so both sides must weigh the same amount. What is the relationship between the orange trapezoids and the blue moon? Answer: One blue moon weighs the same as two trapezoids. In a different notation: moon = 2 * trapezoid. Or, with standard variables: y = 2 * x, where x is a trapezoid and y is a moon.
SolveMe Mobiles presents this visual, intuitive, and exciting way to present, reason about, and solve algebraic equations.
The mobile on the right of the first figure is also balanced so again both sides must weigh the same. The relationship between the blue heart and the red rectangle, however, is a little more difficult to figure out. Examining the mobile, however, we see that there is a red rectangle on the left side as well as the right side. Both red rectangles weigh the same so in a sense they “cancel out” each other. Ignoring them leaves us with 2 blue hearts = 4 red rectangles, which simplifies to 1 blue heart = 2 red rectangles. Students generally are able to “cancel out” the same number of like pictures on each side of a balanced mobile. Using the shapes as variables produces the sequence of steps on the left side of the next figure. The use of standard variables (i.e., x stands for blue hearts and y stands for red rectangles) produces the derivation on the right side of the next figure.
SolveMe Mobiles can get complicated and involve three or more shapes (i.e., variables), as is the case for the next figure. In this mobile, the number 60 indicates that the whole structure of the mobile weighs 60 ounces. (I chose ounces here as the unit of measure.) This problem uses four variables (i.e., four different shapes). Can you solve it by determining the values of the four different shapes? Note that since the entire mobile weighs 60 and the mobile is balanced, both sides equally weigh 30. Using the same reasoning, each strand of shapes weighs 15. (The solution is at the end of this blog.)
One corresponding system of algebraic equations using four standard variables (i.e., x, y, z, and w) looks like the following: x is the heart shape, y is the circle, z is the trapezoid, and w is the square.
I conducted an experiment with the 25 teen students in my summer math classes at Eagle Hill School. Every student had previously completed an Algebra I class. Each student was given four problems to solve: two mobile puzzles, including the one above, which possesses 60 ounces as its header; and two algebraic versions of the same mobile puzzles, including the above 4-variable system of algebraic equations. (Students were not told that the algebraic problems they were given corresponded to the mobile puzzles they were given.)
Students solved 91% of the mobile puzzles and only 18% of the algebraic versions of those same mobile puzzles. This result provides evidence that the mobile puzzles are easier to understand and solve than their algebraic counterparts.
In sum, the mobile puzzles presented by SolveMe Mobiles permit students to easily understand the relationships among four variables while a traditional algebraic presentation does not. Students who have never seen a 4-variable system were able to solve them using mobiles. The evidence of this small experiment points to the conclusion that students can more easily reach a deeper understanding of complex algebraic relationships by using mobiles than by using traditional algebraic equations. Students are capable of reasoning about 4-variable systems (and perhaps higher ones) if given the proper representation of that system. Further, it appears that the standard algebraic notation is actually obstructing the path to understanding and, if so, should be supplemented by a more intuitive method such as the mobiles. It is clear that the visual nature of the mobile with its hanging shapes is more intuitive than a set of interrelated algebraic equations using standard variables (i.e., x, y, z, and w). Finally, the mobiles visually present an image of balance between its two sides that seems to be a highly effective visual metaphor for expressing how two sides of an equation must be equal.
In brief, traditional algebra is good for computers, which mechanically manipulate the equations to find solutions. However, it is generally not good for humans, who are highly visual creatures capable of deep understanding when presented with the proper visual presentation. I applaud the cleverness of the mobile puzzles and encourage math teachers to seriously consider using them in their algebra classes.
Solution to 60-Header Mobile:
Purple trapezoid = 5
Red rectangle = 3
Blue heart = 2
Green circle = 9
Ethel McGinn told me about SolveMe Mobiles after seeing my blog on Emoji Algebra. Thanks, Ethel! She is a 2015 graduate of Assumption College and is currently a member of Teach for America—New Jersey.