In the problem What’s Your Angle?, students use geometric reasoning to solve problems involving two-dimensional objects and angle measurements. The mathematical topics that underlie this problem are attributes of polygons, circles, symmetry, spatial visualization, and angle measurement. In each level, students must make sense of the problem and persevere in solving it (MP.1). Each problem is divided into five levels of difficulty, Level A through Level E, to allow access and scaffolding for students into different aspects of the problem and to stretch students to go deeper into mathematical complexity.

PRE-K
In this task, students are presented with the task of examining the diagonals in different polygons. Their task involves determining the number of diagonals that can be drawn in a given polygon.

LEVEL A
In this level, students are presented with the task of examining the diagonals in different polygons. Their task involves determining the number of diagonals that can be drawn in a given polygon.

This level does not directly align to a standard, but it could be used to extend elementary-aged students’ thinking about the attributes of shapes (2.G.A.1). Making generalizations about geometric patterns builds a foundation for work students will do in high school as they describe and prove geometric relationships.

LEVEL B
In this level, students must investigate the number of diagonals in polygons. They need to find the number of diagonals that can be drawn in an octagon and search for a pattern to determine the number of diagonals in other polygons. They then write the pattern using a quadratic expression.

This level strengthens high school students’ thinking about generalizing patterns (F-BF.A.1a, SMP 7). This level builds on the numerical work in Level A as students describe the quadratic relationship between the number of sides of a polygon and the number of diagonals.

LEVEL C
In this level, students investigate spirographs. A spirograph is a geometric figure drawn from a finite sequence of terms. The students investigate the attributes and patterns found in spirographs.

This level does not directly align to a standard. It does provide students an opportunity to look at geometric patterns and might provide high school students an opportunity to apply some coding or programming skills to model the problem.

LEVEL D
In this level, students investigate a pool table problem. In the problem, pool tables come in different dimensions that are whole numbers in length and width. A pool ball is hit at a 45° angle and banks off a wall, then continues banking off walls until the ball finally lands in a pocket. The goal is to determine the relationship between the dimensions of the table, the number of banks, and which pocket the ball falls into.

In this level, students model the path of a ball on various rectangular pool tables and use their knowledge of reflections to predict the path (G-MG.A.3). Students look for patterns in the observations they make (SMP 7).

LEVEL E
In this level, students investigate making polygons and stars using a process similar to that used to make spirographs. Figures are generated using an iterative process that involves drawing a line segment, then rotating an angular distance. The process stops when one arrives back at the original starting position and in the original orientation.  Students are asked to predict the image given the turning angle. They are also asked to determine the turning angle given a figure.

Students use their knowledge of inscribed angles and their intercepted arcs (G-C.A.2) to solve problems involving the construction of polygons and stars. They also use their knowledge of the measures of interior and exterior angles of polygons and angles of rotation. The problems ask students to analyze and solve design problems (G-MG.A.3) while observing patterns and making generalizations (SMP.8).

PROBLEM OF THE MONTH