At the February GeT Seminar, Justin Dimmel made a compelling case for the use of light from the sun and stars as a naturally occurring context to study the geometry of parallel lines. One outcome of his work has been the ongoing development of a mathematical tool called the SunRule. By combining the geometry of parallel lines which model the sun’s light, the mathematics of proportionality in similar figures, and the functionality of a slide rule, the SunRule allows the user to calculate the product of two quantities. Justin prompted us to consider the way this new tool could be used for mathematics teaching and learning, including teaching and learning for preservice mathematics teachers. I was interested in exploring the potential conceptions of multiplication that might arise from using the SunRule for computing products and making sense of multiplication, so I created a virtual version on Desmos for experimentation.

The SunRule leverages parallel light rays from the sun to create shadows along a ruled board. The shadows are cast by two gnomons, one (U) which is a single unit tall and the other (V) which can be set to any length (within the physical constraints of a physically instantiated SunRule). By tilting the board with gnomons attached, the angle (θ) at which the shadow “intersects” the board can be adjusted. The length of the shadow that the unit gnomon projects (U’) onto the board can be adjusted in this way, resulting in a proportional adjustment of the length of the shadow projected by the gnomon of variable height (V’).

The SunRule can be used to calculate the product of a • b by setting the height of V to a length of a and adjusting the angle of the board so that the length of U’ is b. The value of the product can be read as the length of V’, or the distance that the shadow of the variable length gnomon reaches along the ruled board. A series of products are illustrated in Figure 1.

Figure 1. Three SunRule states that illustrate the product of two values.

Once I understood how the SunRule could be used to compute products, I started to wonder what we can learn about multiplication with this model. My interpretation of the SunRule is that it scales the quantity U’ by a factor of V (as shown in Figure 2), although others might interpret it as the quantity V being scaled by a factor of U’. If both factors are integers, the product can be interpreted as V groups of the length U’ (as illustrated by the purple bars at the bottom of Figure 2), but the variable length gnomon can be set to any value between 0 and some maximum determined by the length of the rod being used. There is no reason for it to be set to an integer value. What happens if we set it to a non-integer value? What happens if we set V<1? What if we set U’<1? What if either factor is 0? What if either factor is negative? What if both factors are negative? What happens to the product as one factor slides continuously? What if both factors slide? If one factor is increasing at the same rate the other factor is decreasing, what happens to the product? What does the area of each triangle represent?! I would argue that many of these questions are best answered by playing with the SunRule eTool rather than considering static examples, though Justin might argue that playing with a real SunRule is even better than playing with an eTool. For now, sharing the eTool is easier than sharing the real thing, so try it for yourself: https://www.desmos.com/calculator/fj7zbbrkrd

What I have found most interesting is the relationship between the purple bar at the bottom of the diagram (let’s call that a U’ Number Line) and the horizontal axis which measures lengths of U. As we let U’ slide, the U’ Number Line is stretched or compressed, but the number of units displayed remains constant. As we let V slide, the length of a unit remains constant, but the number of units displayed changes. If we focus on the relationship between the U’ Number Line and the horizontal axis, we have a new (to me) representation of a double number line that can model multiplication of two continuous variables. It feels very similar to the classic number line representation of multiplication, which models a • b as a “jumps” of b length along a number line, but we no longer need to limit a to being an integer.

I invite you to play with and think about what the SunRule makes available for students to learn about multiplication. What does it obscure? What does this virtual approximation of the SunRule make available that the real SunRule does not? What does it obscure?