Painted Cubes

Reflection:

The painted cubes task aligns with Standard of Mathematical Practice 8: look for and express regularity in repeated reasoning. To generalize the different number of painted cubes, students must look at other cubes and find patterns within their values. It begins by having students look at a 4x4 cubes and decide the different number of faces painted. When I attempted this problem I found it easier to look at smaller cubes, such as a 2x2 and a 3x3 to begin visualizing the different number of sides painted. I found it easier to find the total number of cubes to check my work that all painted cubes were accounted for. The only generalization I could find initially was that each cube had 8 cubes with three sides painted as all cubes (when side lengths are greater than or equivalent to 2) have eight corners. I then questioned how I could generalize the other painted sides in which I decided to create a table to look for patterns, aligning with SMP 8 to look for and express regularity in repeated reasoning.

This task was difficult to visualize in which I eventually had to pull out manipulative to create a 4x4 cubes in order for me to correctly find the different sides painted. Students who struggle thinking mathematically could be challenged by this task in which I would encourage students to build the cubes to better visualize the different painted sides. It was also difficult to see the patterns emerging until I created a table. Suggesting different models for the painted cubes, such as tables and graphs could enhance the students understanding of emerging patterns as well as aid in creating generalizations.

If I were to use this task in class, I would have manipulative available for students to aid in the visualization. Finding the different patterns aids in creating a high floor low ceiling as students should be able to find the different number of sides painted but can be engaged to dig deeper by creating generalizations for the different number of sides painted to find for any cube.

Source: https://www.youcubed.org/tasks/painted-cube/