Designing Packages

How are the Process standards illustrated throughout the lesson?
Problem Solving:
The students have been presented with a problem to solve. It is their mission to arrange 24 wooden blocks into a rectangular prism that would require the least amount of packing material for shipping. To solve the problem, the students must examine the concept of surface area as it applies to a three-dimensional object. In addition, the students have the opportunity to reflect back on discoveries and use them to make a recommendation as to the best packing arrangement.  Not only were the students successful in finding the best solutions, but they also came up with solutions that were unanticipated by the teacher (the staircase-style irregular prism).
Reasoning and Proof:
The students were not given a specific method to calculate the surface area of the rectangular prisms. Some students used addition, some multiplication, and some counted them by hand. In the end, most students came up with similar calculations of surface area. The teacher had the students explain their processes to each other. Their common results lent proof to the fact that multiple methods could solve the problem successfully. One student even came up with an unanticipated explanation for why the surface area decreased as the prism became more cuboid, explaining that more faces of each cube were hidden within the interior of the prism.
Communication:
The teach was careful to ensure that students used appropriate terminology when communicating their methods and results, paraphrasing when necessary or having the students repeat their description using terms such as length, width, height, etc.
Connections:
Connections within mathematics were evident as students referred to multiplication tables to identify all the numbers that, when multiplied, equal 24. Another student noted that all the prism dimensions were factors of 24. The connection to the real world was evident in the use of packaging as a practical application of the need to calculate surface area.
Representation:
Students were encouraged to organize their data but not told specifically how to do it. The groups all developed tables to communicate their findings. As the tables were constructed unsystematically, it made the students' observations a bit challenging to compare. In future investigations, I am curious if the teachers would challenge the group as a whole to come up with a more coherent way to communicate their findings through table representation.

How do you see the Standards of Mathematical Practice being portrayed?

1. Make Sense of problems and persevere in solving them. This investigation required the students to address a problem and see it through to a solution. The problem offered enough structure to ensure that all necessary concepts were addressed, but enough flexibility to allow students to come to their own unique understanding. 
2. Reason abstractly and quantitatively. Some students appeared immediately capable of leaping into the realm of equations, multiplying length by width to find the area of a face and then multiplying the resulting product by the number of same-dimension faces. Other students counted the faces cube by cube and only later noted the efficiency of multiplication. This activity supported learners at various phases of development. 
3. Construct viable arguments and critique the reasoning of others.  Students were instructed to organize their work without being told specifically how to go about it. The students knew they would be asked to justify their recommended solution, so it was in their best interest to keep their results reasonably well organized. After the groups had come to their conclusions, their recommendations were discussed as a group. Students presented their arguments; their classmates challenged them on their conclusions. Through the process, students were able to identify misconceptions.
4. Model with mathematics. Students used dimensions to describe their prisms, in addition to sketches. 
5. Use appropriate tools strategically. All the groups constructed tables to communicate their findings without explicit instruction to do so. 
6. Attend to precision. As mentioned previously, the teacher ensured that appropriate terminology was being used, such as length, width, height, surface area, and volume. 
7. Look for and make use of structure. The teacher explicitly encouraged the students to look for patterns in their results and the results of their classmates. Some students noted patterns in the numbers such as factors of 24.  He also set up a large scale display of the cube arrangements in order of largest to least surface area. By doing so, he modelled a method for identifying structure. 
8. Looking for and expressing regularity in repeated reasoning. More advanced students identified the efficiency of multiplication to determine surface area. Less advanced students came to that conclusion more gradually. While the teacher did not expressly teach this, he did pointedly draw attention to the different methods used by the groups.