Cutting a Cube
In the problem Cutting a Cube, students use two-and three-dimensional geometry to solve problems involving cubes and nets. The mathematical topics that underline this problem are the attributes of polygons —faces, edges, vertices —as well as spatial visualization, counting strategies, classification, and geometric solids. The problem asks the students to examine a cube to analyze the attributes of a cube and how a cube can be cut into a flat pattern, as well as what flat patterns can be made into cubes.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, the teacher asks the students to examine a cube and poses questions about the number of flat sides, corners, or edges the cube has.
LEVEL A
In this level, students are presented with a model of a cube. Their task is to recognize and identify the attributes of a cube.
This level supports Common Core standard K.G.B.4 as it has students analyze a three-dimensional shape and describe its parts and other attributes with informal language. This can begin the conversation for Standard 1.G.A.1 by having the students describe the defining attributes of a cube.
LEVEL B
In this level, 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.
This level supports Common Core standard 6.G.A.4 as students are introduced to the concept of creating a net for a cube.
LEVEL C
In this level, students are first asked to how to cut a cube to make a particular net. They are then asked to explain how to cut the cube to find other nets, explain why certain ways to cut the cube don’t form nets, and describe ways to tell whether two nets are different.
This level supports Common Core standard 6.G.A.4 as students find different nets for a cube.
LEVEL D
In this level, students are asked to find all possible nets for the cube.
This level supports Common Core standard 6.G.A.4 as students find all possible different nets for a cube. Students may also apply their understanding of rotations, reflections, and translations (8.G.A.1a) to explain why two nets are the same.They can look for and express regularity in repeated reasoning in order to help them determine all possible nets (MP.8) and construct arguments to convince others they have found all possible nets (MP.3).
LEVEL E
In this level, students find all unique hexominoes made up of six attached squares and determine the percentage that are cubes.
This level connects to Common Core standard 6.G.A.4 and 6.RP.A.3c as students find all possible hexominoes for a cube and determine the percentage of hexominoes that are nets. Students may also apply their understanding of rotations, reflections, and translations (8.G.A.1a) to explain why two hexominoes are the same. They can look for and express regularity in repeated reasoning in order to help them determine all possible nets (MP.8) and construct arguments to convince others they have found all possible hexominoes (MP.3).
PROBLEM OF THE MONTH
Download the complete packet of Cutting a Cube 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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