Daniel Scher, Ph.D., is a senior academic designer at McGraw-Hill Education. He has co-directed two NSF-funded projects: the Dynamic Number project and the Forging Connections project.

In his online article Simply Symmetric, Michael de Villiers observes that symmetry is a powerful but often overlooked tool for formulating proofs:

Most primary geometry curricula around the world introduce the concept of line symmetry fairly early, and sometimes also that of rotational, translational and glide reflective symmetry. … Continue Reading ››

In a prior blog post, I presented an uncommon method for solving the well-known Burning Tent problem. My solution, modeled on the approach in the Connected Geometry curriculum, used a dynamic ellipse to pinpoint the optimal solution. Now, I'd like to offer a related problem from Connected Geometry where … Continue Reading ››

In a previous post, I described two different approaches to solving the Burning Tent optimization problem. Now I'd like to offer a related problem that I assigned many years ago to my pre-service mathematics teachers at New York University.

How can you identify a lover of math? Casually mention a burning tent and notice if her first thought is how to minimize her path to a river and then to the tent to douse the flames. Here is a full statement of this classic geometry problem:

In a 2018 blog post, I presented George Gamow's pirate treasure problem, which can neatly be solved by capitalizing on the geometry of complex numbers. There's more treasure to be had, however, so get ready for another adventure!

An island contains a giant boulder, a lighthouse, a cave, and a jail. Among … Continue Reading ››

Of all the original games I've designed, Arranging Addends is among my favorites. On page 1 of the Web Sketchpad model below (and here), you're given five addends—1, 2, 4, 8, and 16—and asked to arrange them in the circles so that the sum of the numbers in each circle matches the values … Continue Reading ››

I'm a big fan of pan-balance puzzles in which you're given a two-pan balance and asked to use it to uncover a counterfeit coin or determine the weight of a coin. One classic example is the following puzzle:

You have 12 coins that all look exactly the same. One is counterfeit and is either heavier … Continue Reading ››

In last month's Construct a Building post, I presented any array model in which students construct the rooms and floors of a building as a way of representing multiplication. Now I'd like to follow up with a similar array model that allows students to take a problem they don’t know, like 8 × 7, and … Continue Reading ››

When Scott Steketee and I developed activities for the Dynamic Number project, we thought about ways that dynamic array models could help children to conceptualize multiplication.

Rather than presenting children with arrays that were fully formed, we thought it would be instructive for them to build these arrays themselves. That design goal led to the … Continue Reading ››