Given a piece of plywood 150 cm x 300 cm, design a doghouse that can be made from the piece. Make your doghouse as large as possible. Show what your finished product looks like and give good enough information so that someone else could use your plans to build a house for their dog.
This task is taken from the NCTM Curriculum and Evaluation Standards. It can be found under Standard 13: Measurement on page 118. The class is a multi-age sixth-seventh group of 56 heterogeneous students, working in cooperative groups. We had been doing a lot of measurement activities and I was looking for a task that would extend their thinking to a new application where they would have to resort to models since the actual dimensions were so large. Asking for the largest house possible, necessitates trying a variety of models and calculating the volume of each. This encourages the use of a table or chart for organization. It also calls on students to consider the practicality of some of their designs. This task allowed much mathematical communication as I had the students working in groups of three or four. They used grid paper to represent the sheet of plywood and many converted from centimeters to decimeters in order to have a one-to-one correspondence. Many of the models were odd shaped and required some research into calculating volume especially by the sixth graders. The groups brainstormed ways to cut the wood and piece it together to make the largest looking structures. Many used grid paper and cut out rectangles 15 x 30 squares and experimented with various possibilities. Many then taped the pieces together to make models. By comparing the volume of each model, they determined the largest house. Some groups then made some judgments that the largest possible shelter would not be appropriate for protecting a dog for one reason or another. As most students own dogs, there was considerable expertise! It became apparent to most groups that they needed a chart of some sort to organize their data once they were done. This task took the better part of a week of 40-minute classes to complete. Some groups worked outside of class time to complete the task on time. Students had done mechanical drawings and blueprint designs in Technology Education a few weeks beforehand and recalled that training in their work on this task. It could be adapted to create something other than a doghouse based on an interdisciplinary unit of study you might be doing at the time. Have a lot of grid paper, scrap oak tag (old manila folders) and tape on hand.
Having the students work in groups enables them to come up with a wider selection of designs and provides many hands for making the models in less time. Students would have been content to work on this task for weeks if time allowed. They kept coming up with new ideas for further designs to push for the largest possible design. It became quite a contest as groups posted their current record volume numbers on the flip chart. There was much discussion about what constitutes a humane dog shelter. Many individuals thought that digging an enormous hole and covering it with the entire piece of wood was the best idea. Their peers disallowed this option.
- Calculators
- Rulers
- Grid paper
- Oak tag (old manila folders)
- Scotch tape
Using a 10 cm strip to cut four 5 cm x 75 cm legs to support the rest of the sheet (like a table) will give a volume of 3,262,500 cm3. This is about the largest reasonable design found by this group. Cutting the sheet in half to form two 150 cm x 150 cm halves and leaning them together (tent style) will give a volume of 1,687,555 cm3.
This group tried only one design. They drew a diagram of the completed house, but did not label the pieces on the sketch of the full sheet of plywood. Their diagrams lack any meaningful labels - there are no units and only sketchy numbers. There is no evidence of calculating the volume of this design. There is no discussion as to how they selected this to be the largest possible house. When prompted, their second attempt accomplished little.
This group did a comparison between two designs, but stopped there. The two designs were similar (both were tent-like and had only one cut of the wood). The calculations are accurate. They speak of "area" when they mean volume in paragraph three. They do offer diagrams of their models, but omit any measurements. Their offering of a graph does little to enhance one's understanding of their work. They attempt no observations, connections or generalizations.
This group constructed three models from sketches they had drawn. While they do not document it well, they had done some rough calculations to discover that none of the traditional designs would approach their numbers for volume and they were going for largest possible volume regardless of doggie comfort! Their consciences did catch up with them as time went on, as witnessed by their decision to only build number one. Their calculations are correct and their reasoning was good. They have a good variety of math language used accurately throughout. This is a group that really benefited from working together, as they would challenge each other to try for even greater volume. They were completely involved for days. Unfortunately, their 3-D models did not copy well. They were made from oak tag and colored beautifully.
This group designed four houses of various shapes and sizes all based on criteria established at the outset regarding dog comfort. Doghouse B and C show their understanding of difference of volume based on subtle measurement changes. More mature students might have made an observation based on these models and their choice of design D. As they approach cubic design, the volume continues to increase. This concept is demonstrated nicely in their organizational chart. They move with ease through the metric conversions to suit their needs. They use a variety of good math language including symbols throughout the solution. They attempt a connection to their "real world", but it falls short of being mathematical. Their strength is in their explanation of how and why they do as they do and the practicality of their decisions throughout the task.
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