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.
Proofs of Impossibility
Very engaging for teachers, and applicable to a variety of classroom contexts, this session’s tasks involved deriving numbers using the four operations (+, -, x, ÷). This eventually led to the identification of numbers that were impossible to make and discussions on how we might prove that. Some of us found that it was more challenging to prove that an outcome is impossible than is possible! A variety of mathematical topics were also discussed, including order of operations and functions. Teachers were also introduced to a math task framework that might be used as a way to reflect on one’s practice.
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.
Flipping Pancakes Problem
Who knew there was mathematics behind flipping pancakes? Using a problem posed by mathematician Jacob Goodman in 1975, Dr. Manes and assistants Reckwerdt and Rader facilitated a session in which we investigated ordering, factorials, permutations and combinatorics within the context of this problem. Foam manipulatives provided a model for flipping pancakes and for finding the most difficult combination of different sized pancakes to place in an optimal order. Focusing on problem solving, a set of ten very different problem solving strategies were shared followed by an exploration of their accompanying resources from the MAA’s Curriculum Inspirations website.
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.
Stern-Brocot Trees – Finding Patterns with Rational Numbers
A special kind of binary tree, our investigation into the Stern-Brocot tree had us looking at patterns of rational numbers and generalizing those patterns to ascertain the mathematical structure behind the tree. The day also included an overview of the eight Common Core State Standards of Mathematical Practice and discussions around how these processes are integral to the work of mathematicians and the problems we collaborate to solve in MaTCH.
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.