# Cubes Investigation: The Wonders of Cubic Growth – A Mathematical Exploration Activity

Hello fellow math enthusiasts! Today we’ll dive into a wonderfully immersive math activity that you can bring right into your classrooms. Aptly titled “Cubes Investigation: Cubic Growth,” this problem-solving task allows students to explore a tangible, thought-provoking problem related to cubic structures, geometry, and pattern recognition. If you’ve been searching for an engaging way to get your students thinking creatively about mathematical concepts, this activity is a must-try.

## Introduction to the Cubes Investigation Activity

Before we dive into the game play instructions, let’s set the stage for this activity. In the Cubes Investigation, students work in groups to discover how the surface area of different sized cubes are affected by painting and subsequent disassembly. The primary question posed to them is: “How many 1x1x1 cubes are painted on three faces, two faces, one face, or no faces when a larger cube is constructed, painted, and then taken apart?”

This problem involves a rich interplay of mathematical concepts such as counting principles, combinatorics, and geometry, making it a fantastic tool for fostering deep understanding in these areas. Not only does this activity engage students in critical thinking and problem-solving, it’s also a lot of fun!

## Gameplay Instructions and Scenarios

The first step is to divide your students into groups and provide each group with a plentiful assortment of unit cubes, graph paper, and colored pencils or markers.

Next, hold up a unit cube and tell your students that this is a cube on its first birthday. Ask students to describe the cube (eight corners, six faces, twelve edges). Encourage them to observe and note down the various characteristics of the cube, such as the number of corners, faces, and edges.

Once they’re comfortable with this, the students will then construct what the cube may look like on its second, third, fourth, and fifth birthdays. Each birthday signifies an increase in the cube’s size, which is achieved by adding more unit cubes. For example, a cube on its second birthday will be a 2x2x2 cube made from 8 unit cubes.

Now, pose the big question to your students: “If this cube was ten years old, dipped into paint, dried and then taken apart into the unit cubes, how many cubes have three faces, two faces, one face or no faces painted?” This question prompts students to explore and visualize how the painting process would affect the different unit cubes based on their position within the larger cube.

To aid their investigation, ask students to draw the cube on each of its birthdays up to ten. As they work, they should record the number of unit cubes painted on three, two, one, or no faces. Finally, students will display their findings in graph form to look for patterns.

## Observations and Patterns

Through the exploration, students should observe some exciting patterns:

1. Cubes with three painted faces are always at the corners. There are always 8 corners in a cube, regardless of its size.
2. Cubes with two painted faces occur on the edges. Excluding the corners, these increase by 12 with each “birthday”.
3. Cubes with one face painted appear as squares on the six faces of the original cube. These squares increase in size with each “birthday”, thereby increasing the number of unit cubes with one face painted.
4. The cubes with no faces painted are found within the larger cube. As the cube ages, the number of unpainted cubes (the cube within the cube) increases.

## Accommodations and Modifications

When implementing this activity, consider the diverse learning needs of your students. Here are some accommodations and modifications that could be made:

1. Visual aids: For visual learners, provide a tangible 3D model of the cube on each “birthday”. This can help students visualize how the number of painted faces changes as the cube gets larger.
2. Step-by-step guides: Some students might benefit from a step-by-step guide or walkthrough of the initial stages of the activity. This might include a detailed explanation and demonstration of how to construct the cube on its second and third birthdays.
3. Peer support: Pairing students who find the task challenging with those who grasp it quickly can promote cooperative learning. Students often can explain ideas to their peers in ways that are more readily understandable.
4. Scaffolded worksheets: Provide worksheets with guiding questions to help students organize their thoughts and findings. This can help them stay focused and make it easier for them to recognize patterns.
5. Extension activities: For advanced students, consider extension questions such as, “What happens when the cube reaches its 20th birthday?” or “Can you devise a formula to predict the number of cubes with a certain number of painted faces?”

## Making Connections to the Common Core State Standards (CCSS)

This Cubes Investigation activity is wonderfully versatile and aligns well with several Common Core State Standards (CCSS) for mathematics, notably:

• CCSS.MATH.CONTENT.5.G.A.1: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates.
• CCSS.MATH.CONTENT.5.G.A.2: Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
• CCSS.MATH.CONTENT.6.G.A.2: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism.

By exploring cubic growth through the Cubes Investigation activity, students gain a deep, hands-on understanding of these standards, providing a solid foundation for further learning in geometry, combinatorics, and algebra. So, why not bring this fun and engaging activity into your math class? It’s time to dive into the world of Cubic Growth and unearth the wonders of mathematical exploration!