Cut It Out

In the problem Cut It Out, students use properties of two-dimensional geometry to solve problems involving spatial visualization. The mathematical topics that underlie this problem are the attributes of polygons, symmetry, spatial visualization, transformations, patterns, functions, and fractal geometry. The problem asks students to use spatial reasoning to make sense of part of a visual image.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 will partition a square piece of paper into fourths. The teacher will lead them through making cuts in the folded corner of the paper, and the students will predict and discuss what the paper will look like when it is unfolded.

LEVEL A
In this level, students are asked to fold a square piece of paper into fourths such that the result is a smaller square. At the most folded corner, the students are to visualize cutting a tiny square out of the paper. Their task is to draw a picture of the unfolded square, noting where the hole(s) are located and the approximate size of the square.

In this level, students predict how different shapes will be reflected when a folded piece of paper with a cutout is unfolded. Students must use their knowledge of reflections to sketch what the resulting unfolded paper will look like (G-CO.B.6).

LEVEL B
In this level, students are presented with a similar situation that involves determining what the unfolded square looks like when different types of cuts are made. The students need to visualize the results of multiple cuts.

In this level, students predict how different shapes will be reflected when a folded piece of paper with a cutout is unfolded. Students must use their knowledge of reflections to sketch what the resulting unfolded paper will look like (G-CO.B.6).

LEVEL C
In this level, students work with the inverse challenge. Students are given visual images of the unfolded square piece of paper that has been folded a number of times and in which holes have been cut. The students must determine how many folds were required and how the cut was made.

In this level, students use their knowledge of reflections to determine how a paper should be folded and cutout to produce the given shape when unfolded (G-CO.B.6).

LEVEL D
In this level, the students explore a geometric fractal image. The fractal image is generated from folding a square paper and cutting out a small corner. The process is repeated, and the students draw the image and find the area of the paper and the perimeter of all edges at each stage. Using knowledge of patterns and functions, the students predict what the fractal looks like and its size in terms of area and perimeter at any stage of the fractal’s development.

In this level, students use a recursive process to create a fractal. At each stage, students calculate the area and perimeter of the fractal and then generalize the pattern for the nth stage (F-BF.A.1a). While there are no specific geometry standards addressed in this problem, students are using ideas of similarity and exploring fractals, which is an interesting geometric concept and can lead to further extensions and exploration.

LEVEL E
In this level, students design their own fractal, following some geometric pattern, and they compute formulas for generating the area and perimeter of a fractal.

While this level does not align to any specific standards, it provides an extension opportunity for students. Students design and create their own fractal and find the measurements for different aspects of the fractal at each iteration (area, volume, perimeter, etc.).

PROBLEM OF THE MONTH 
Download the complete packet of Cut It Out 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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