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© Noyce Foundation 2012
Problem
of
the
Month:
Cutting
a
Cube
The Problems of the Month (POM) are used in a variety of ways to promote problem‐solving and to foster the first standard of mathematical practice from the Common Core State Standards: “Make sense of problems and persevere in solving them.” The POM may be used by a teacher to promote problem‐solving and to address the differentiated needs of her students. A department or grade level may engage their students in a POM to showcase problem‐solving as a key aspect of doing mathematics. It can also be used schoolwide to promote a problem‐solving theme at a school. The goal is for all students to have the experience of attacking and solving non‐routine problems and developing their mathematical reasoning skills. Although obtaining and justifying solutions to the problems is the objective, the process of learning to problem‐solve is even more important. The Problem of the Month is structured to provide reasonable tasks for all students in a school. The structure of a POM is a shallow floor and a high ceiling, so that all students can productively engage, struggle, and persevere. The Primary Version Level A is designed to be accessible to all students and especially the key challenge for grades K – 1. Level A will be challenging for most second and third graders. Level B may be the limit of where fourth and fifth grade students have success and understanding. Level C may stretch sixth and seventh grade students. Level D may challenge most eighth and ninth grade students, and Level E should be challenging for most high school students. These grade‐ level expectations are just estimates and should not be used as an absolute minimum expectation or maximum limitation for students. Problem‐solving is a learned skill, and students may need many experiences to develop their reasoning skills, approaches, strategies, and the perseverance to be successful. The Problem of the Month builds on sequential levels of understanding. All students should experience Level A and then move through the tasks in order to go as deeply as they can into the problem. There will be those students who will not have access into even Level A. Educators should feel free to modify the task to allow access at some level.
Overview:
In the Problem of the Month
Cutting
a
Cube,
students use two‐ and three‐dimensional geometry to solve problems involving cubes and nets. The mathematical topics that underlie this POM are the attributes of polygons, faces, edges, vertices, spatial visualization, counting strategies, classification and geometric solids. The problem asks the students to examine a cube to analyze the attributes of a cube
© Noyce Foundation 2012
and how a cube can be cut into a flat pattern, as well as what flat patterns can be made into cubes. In the first level of the POM, students are presented with a model of a cube. Their task is to recognize and identify the attributes of a cube. In level B, students are presented with situations that involve determining the least number of cuts it takes to divide a cube into a single flat pattern or net. The students explain why it takes 7 cuts to make a cube into a net. In level C, students explain that any arbitrary 7 cuts do not determine a unique net and show multiple examples of nets that can be folded into a cube. In level D, students determine all the unique nets that fold into a cube and explain a valid process for determining all the unique nets that fold into a cube. In level E, students draw all the unique hexominoes and explain a valid process for determining all the unique hexominoes.
Mathematical
Concepts:
Spatial visualization plays an important part in real‐world experiences. From the most complex structures created by designers, architects, and construction workers to arranging the furniture in a room, spatial awareness, and visualization is essential. In this POM, students explore various aspects of spatial visualization, including designs in both two‐ and three‐dimensional space. This involves examining flat patterns as well as solid objects and understanding the relationship between the two objects. Students use polygons and develop understandings of their attributes both in the plane and on the surface of polyhedra. In addition to the geometric aspects of this POM, students seek to find patterns, count numbers of possibilities, and justify their answers. The mathematics involved in these aspects of the problem is often called discrete mathematics.
Problem of the Month Cutting a Cube Page 1
© Noyce Foundation 2004. To reproduce this document, permission must be granted by the Noyce Foundation: info@noycefdn.org
Problem of the Month
Cutting a Cube
A cube is a very interesting object. So we are going to examine it.
Level A:
Without holding a cube, try to picture it in your mind. How many sides (faces) does a cube have? How many corners (vertices) does a cube have? How many lines (edges) does a cube have? What can we say about the size of the sides (faces) and the lines (edges)? When you have made your guess (conjecture), then hold a cube and check (verify) your answers to the questions listed above. How might you be able to remember the parts (attributes) of a cube? Explain.
Level B:
A cube is like a box. You might think of it as a special type of cardboard box. We could cut up a cardboard box and make it into one large flat piece of cardboard. We often do that when we want to recycle the cardboard. The easiest way to cut a cardboard box is to cut along the lines (edges). How many cuts does it take to make the box into one flat piece? In other words, what is the least number of lines (edges) that need to be cut so that the cardboard is in one flat piece? Remember all the sides of the cardboard must remain attached in one single flat piece. What is the least number of cuts that need to be made? Explain how you determined your answer. Why do you think your answer is correct? Write a note to a friend to convince your friend that your solution will always work for every cube.
Problem of the Month Cutting a Cube Page 2
© Noyce Foundation 2004. To reproduce this document, permission must be granted by the Noyce Foundation: info@noycefdn.org
Level C:
When you cut a cube into one flat piece we call that piece a net. The reason we call it a net is because we can trace the pattern of the flat piece on a piece of paper or cloth material. If we cut out the pattern we can fold it back over the cube, surrounding it like a net. The nets that cover a cube can be cut into different patterns. One net looks like a cross. It has four faces in a column and two more faces on either side of that column. How would you cut the cube (which edges) to make the net into a cross pattern? Is there more than one way to cut the cube to make a cross? Find some different net patterns that would also cover a cube. Determine how you would have to cut the cubes to make them into new net patterns. Explain your methods. Are there ways to cut the cube so that it won’t make a net? Explain your thinking. Sometimes we might think two nets are different, but if you move one around it then looks exactly like the other net. How can you tell if two nets are different? Explain and define the difference.

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