We were fortunate to have Dr. Michael Nakamaye from the University of New Mexico as a guest presenter at MaTCH. The focus of the session was on making sense of ratios as they related to the gears of a bicycle. Although many of us have experiences riding bikes, not many of us had thought about how the relationships between the gears make it easier or more difficult to pedal!

We were all pushed to reason about the mathematical relationships, providing a great connection to our pedagogical discussion, focused on proof and mathematical arguments, as described in the Common Core State Standards’ SMPs.

Single-Cut Geometry

This investigation into cutting out regular and irregular polygons using only a single cut of the scissors led to interesting discussions of symmetry, geometric centers and angle bisectors. The teachers in this session loved the use of representation and how the task might allow diverse students access into the mathematics. A discussion into the possible limitations and shortcomings of learning mathematics as memorized rules followed.

The Number Line Activity

In this session, using a human number line, we investigated transformational geometric motions related to operations on numbers. For example, what happens to one’s position on a number line when two is subtracted from your given number? What happens when your number is multiplied by negative one? What number would we get if we rotated 90 degrees on the plane when multiplying? This surprising investigation led us to relating transformational geometry with numbers on a number line and into discussions about quantitative reasoning and representations, Math Practice 2. Using a human number line and physical motion to represent numbers and operations on numbers encouraged us to think about those concepts in new and different ways.

Bicycle Tracks

Our first session of the year involved an investigation of the Bicycle Tracks problem, developed by mathematician James Tanton. As they worked through this task, teachers were encouraged to ask intriguing questions about paths created by the front and rear wheels of a bicycle as it moves down a length of paper, eventually leading to an investigation of tangent lines. Teachers were also introduced to open problem solving as an approach to learning mathematics, an important aspect of how they engage with mathematics in MaTCH sessions.