The entire purpose of spending two days on this problem is to create an environment in which students can make sense of problems and persevere in solving them (MP1). After the first day of investigating this problem, some students will have made sense of it while other students have not. To create an environment in which students will persevere, tell them explicitly that they have all the time they need to make sense of this problem. When some students see other students already writing equations and finding patterns, they are often tempted to copy these or follow along without fully understanding. This is when you need to explicitly tell them that the goal is understanding, and work with them to find a way to represent the problem that makes sense to them, which means that they will need to keep working with the blocks, or keep drawing pictures.
The goal of the second day is to apply more abstract reasoning to the problem, which is MP2. In order to facilitate this process, ask students as they are working: “Do notice any patterns in these tables?” or, “Are these tables similar to the ones we have looked at on the warmups?” Additionally, as students start to find patterns or shortcuts to complete the data tables, ask them: “Does this shortcut match your data? Does it match the diagram? Does it make sense if you think about the cube?” There is a very strong connection between the rules and the cube: the rule for the cubes painted on two sides is linear, because those cubes literally form lines along the edges of the cubes. The rule for the cubes painted on one side is quadratic, because those cubes literally form squares on the sides of the cube. The rule for the cubes painted on 0 sides is cubic because those cubes form a cub inside of the outer layer of cubes. This is a perfect opportunity for students to go back and forth between the abstract and the quantitative.
Lastly, the task of making a generalization about the end behavior of the basic polynomial functions involves attending to precision (MP6) through communication.
The big goal of the middle of today’s lesson is for students to identify patterns in the data table and to use the physical representation of the problem to write algebraic rules to represent each column of data. Students are eager to find patterns and sometimes make wild guesses about what kind of data is being shown. To address this, ask them to explain how their data matches the situation, using a cube or a diagram. The actual shape of the diagram helps develop the rules: the cubes that are painted on two faces fall on the edges of the large cube; the cubes that are painted on one face form squares on each face of the large cube and the cubes that are painted on 0 faces form a small cube inside of the large cube.
To structure this work time, ask students to start by comparing data tables with each other. It is important to circulate quickly and immediately upon starting work time to make sure that each student has a starting point. There will be some students who do not know how to get started, or who got frustrated with the problem to the point that they are unwilling to work on it. It is important to check in with these students right away to make sure that they don’t get off task. If there are several students feeling this way, you can quickly pull them to one table and work as a group with them to build a colorcoded cube. If this is not logistically feasible, you can even just show them the image of the colorcoded cube and ask them to attempt to build a 4by4by4 cube that is also colorcoded. This direction should be enough to get them back in the game.
In order to create the equations, notice the following: there are 12 edges of the large cube and each of those edges includes cubes that are painted on 2 faces (the 2 comes from the fact that the two corner cubes are painted on 3 sides). There are 6 faces of the large cube, and each of those 6 faces includes an square of sidelength of cubes that are painted on one face. There is one inner cube and that cube has sidelength . It is worth noting that each of these equations is a horizontal shift of an equation of the form , so it has the form . In order to see this, it is important that students complete the first row of the data table.
Also, students want to use patterns to complete the data table, even before finding the rules. The second image shows one students use of the 2^{nd} constant differences to fill in the rest of the data table for cubes with 1 face painted.
The extension for students who figure out all these equations with time to spare is to prove algebraically that , which is true about each row of the data table. The last two images below show student work to tackle this extension.
Because students make different amounts of progress towards solving the Painted Cube problem in class, the lesson ending depends on how much time you have and how much progress students made. One option is to simply ask students to share their work with a partner and discuss the rules they found. If students have made progress past this point, they can start work on the End Behavior Generalization, which should seem familiar after working on the WarmUps from yesterday and today.
If you decided to transition from the Painted Cube Problem, this can be done with small groups of students or with the whole class, depending upon every student’s progress. For students who are ready to transition, the question about end behavior is: How can we use the coefficient and the exponent of the function to determine its end behavior? Students can use the warmup examples from both yesterday and today to attempt to formulate a generalization. Most students “see” the idea relatively quickly, but have trouble articulating it, because they do not have practice creating “If___, then ____” statements. Even if they create some such statements, they do not realize that 4 such statements are needed to fully describe this generalization. You can provide students with a simple table like the one below to help them organize their thoughts, but don’t do this until they have struggled to think it through for a few minutes.
The Coefficient 
The Exponent 


Even 
Odd 

negative 



Positive 

