August 11, 2006, 2:44 pm
By Chris Jackson
My daughters aren’t old enough to experience the joys of algebra yet, but when they are I plan to revisit Oliver Steele’s blog for some inspired thinking on how to explain algebra visually. Steele’s post Visualizing Basic Algebra begins with line drawings representing the associative property for addition:
and the commutative property for addition:
He then illustrates the commutative, distributive, and associative properties for multiplication using squares and cubes. These illustrations could be recreated on a kitchen table with building blocks; they’re tactile (unlike algebra). By making the conceptual visual, they provide that “Yes, I see it now!” moment.
Steele’s post continues by showing visualizations for some more complex algebra concepts: the Product of Alternates, Triangle Numbers, and this one for the Difference of Squares, which states that the difference between perfect squares always is odd:
Steele’s visualizations result from his asking the question, “What would a proof that stayed grounded in visuospatial concepts look like?” In a few years, when one of my daughters grapples with her first algebra problem, I plan to ask her, “What might the problem look like?”
I could’ve used this web site myself back in the day–especially to solve those dreaded algebra travel problems:
“Artie starts walking east at 4 km/hr on a long, flat recreation trail from the trailhead at 8:00 AM. At 8:45 AM Bill starts from the same trailhead running at 10 km/hr along the same path in the same direction as Artie. Once Bill catches up to Artie, they both continue running together at 10 km/hr. At the moment that Bill catches Artie, Carlos leaves the trainhead in his car travaling at a constant speed along a road that parallels the train to pick Artie and Bill up, and catches them after six minutes of driving. What time of day is it when the three of them meet, and how fast was Carlos driving?” (from experts.about.com/q/algebra)
I’d love to see a picture of that one.
Posted by Lisa Agustin on August 13, 2006 at 3:04 pm