# Polly Gone

In the problem *Polly Gone*, students use polygons to solve problems involving area. The mathematical topics that underlie this problem are the attributes of linear measurement, square measurement, two-dimensional geometry, perimeter, area, and geometric justification. The problem asks students to explore polygons and the relationship of their areas in various problem situations.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 asked to use manipulatives and grid paper to create rectangular animal pens and find the pen with the most space for the animals.

LEVEL A

In this level, students are asked to create a rectangular animal pen with the greatest area using 40 cubes.

In this level, students solve a real-world problem involving rectangles with the same perimeters and different areas (**3.MD.D.8**). They must apply their understanding of area to find the dimensions of a rectangle with the greatest area for a given perimeter.

LEVEL B

In this level, students are presented with a triangular shape on grid paper composed of five smaller polygons. The students are asked to name and determine the area of each polygon. They are also asked to rearrange the shapes to construct as many different parallelograms as possible.

In this level, students determine the areas of triangles, squares, and parallelograms using grid paper. Students rearrange the shapes to create other shapes and determine the area by decomposing the shapes into smaller shapes (**6.G.A.1**).

LEVEL C

In this level, students are given a rectangle that is subdivided into nine smaller squares. The students are given the area of two of the squares and asked to determine the area of the remaining seven squares.

In this level, students deduce the area of a shape composed of many different squares. Students will have to reason about the side lengths of the squares to find the measures of the missing side lengths (**7.G.B.6**).

LEVEL D

In this level, the students explore concepts for maximizing area given a fixed perimeter. The students grapple with which polygon will produce the largest area as well as maintain a constant distance from the perimeter to design the playing surface in a sports arena.This problem will require students to connect their knowledge of polygons to circles.

In this level, students apply what they know about the circumference of a circle to design an arena in the shape of an *n*-sided regular polygon inscribed in a circle. They use their knowledge of circumference of a circle and area of triangles to determine the area of the regular polygon (**7.G.B.4, 7.G.B.6**).

LEVEL E

In this level, students are asked to construct a geometric figure from a square. An octagon is produced by drawing line segments from each vertex to its opposite midpoints. Students are asked to determine the area of the octagon in relationship to the area of the square. Students are asked to justify their solutions.

In this level, students use coordinate geometry (**G-GPE.B.4**) to solve a complex design problem (**G-MG.A.3**) in which they compare the areas of an octagon and a square. The problem involves students writing and solving systems of linear equations (**A-REI.C.6**), using the distance formula and the midpoint formula (**G-GPE.B.7**), recognizing symmetry to identify congruent triangles (**G-CO.B.7**), and calculating the area of an octagon by decomposing it into triangles.

PROBLEM OF THE MONTH

Download the complete packet of *Polly Gone* Levels A-E Here.

You can learn more about how to implement these problems in a school-wide Problem of the Month initiative in “Jumpstarting a Schoolwide Culture of Mathematical Thinking: Problems of the Month,” a practitioner’s guide. Download the guide as iBook with embedded videos or Download as PDF without embedded videos.

SOLUTIONS

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