Introduction
Math has always been a subject that has raised the interest of very few students. It mainly consists of writing down problems and formulas and talking about how it applies to the real world. In this article that is all changed. Math can be fun for many, have hands on work, students can actually see how it applies to real life, it can bring out the artist in some students, and get the attention of students who love history and seeing how things are made.
The lesson provided is appropriate for grades nine through twelve and can be modified to fit any classroom. This lesson will get students’ attentions and help students who have trouble in math get involved, have fun and realize that math can be applied to real world examples. In this lesson students will be provided blueprints with the English system of a given building and asked to convert the building to a nonstandard unit of measure of their own design. There is a discussion about common standard units of math, why they are important and why they are used. Students will create a ruler using appropriate body parts as units of measurement and marking those units on a dowel rod. Afterward, students will create a conversion rate between their system and the English system. They will then create a scale model of the building according to the blueprints provided. To wrap up the entire lesson students will present their structure, giving a brief history of the building they recreated, explaining any sources of error they might have encountered, and reflecting on ways that their building could be improved in a presentation before the class.
Overall, this lesson will take about a week to do, which seems kind of lengthy but it has a great trade off. Doing a lesson like this allows for more students to become involved in math, and draws in students who are artistic and hands-on learners. Also by putting students into groups it allows those who better understand math to help those who don’t quite understand.
Warm up
You can use any introduction to this lesson as you wish, but for this lesson, a snippet of history was used to bring this topic into focus. Take the following as an example: King Egbert was the first king of England (called Wessex) in the 9th century A.D. Let’s assume that the king wishes to construct a castle for himself and he conscripts workers from throughout his kingdom. Each worker goes out and cuts stone blocks the measure of his foot and brings the finished blocks back to the building site. The students can simulate this process by cutting out squares from a 11 x 17 sheet of paper the length and width of their feet. Then bring each cutout to the front of the class to have them “assembled” on a wall or board. Clearly, there will be a problem putting the “blocks” together because not all blocks were cut uniformly—each person’s foot has a different measure to it. How can this problem be solved? Students might propose that one person’s foot is used as the ‘standard’ of measure. Now we can assume that the King uses his own foot as the basis for all lengths. This is how the actual measure for the English foot came about. Many of our measurements are based on measures taken from the human body.
The Procedure
The first step to the process is to have students create their “cubit” ruler. To do this, each group of students should have a dowel rod that they can break down into smaller units of measure. It is recommended that dowel rods be at least 24” in length as the goal is to base the cubit off of the length of their forearm (elbow to fingertip) or from the length of their arm (shoulder to wrist). The overall finished project will be 2’ x 2’ or smaller so a 24” dowel rod works well, as does a 36” rod. The teacher can select whether the “cubit” will be measured from elbow to fingertip or shoulder to wrist. For the trial run of this lesson, shoulder to wrist was used and it measured to be approximately 24 inches. The students should mark off their cubit and smaller measurements in pencil before making final marks in fine-tipped black marker. The smallest unit of measure should be one-half inch or smaller, since many of the building measurements will have parts measuring one inch or less. After the cubit has been measured out, smaller units of measure will be based on other parts of their body (such as feet, hands, fingers, finger nails, etc.). After their ruler has been created, they need to determine the appropriate conversion from the English system to their cubit. This can be done as easily as lining a ruler up against their dowel rod and finding points where the two units share whole measurements. The trial run had the cubit measuring 24”. From there, the ruler was broken down into a “half-cubit” (measure of a foot), a “court” (length of a finger which was roughly 4 inches) and further using fingertips, and finger joints. The students are encouraged to name each of their smaller units for a more personal feel. The students will fill out a conversion chart for the length of an inch, a foot, and a yard. If their rulers go far enough, they can convert a half-inch as well. Later they will be asked to take a blueprint of an ancient structure and convert it into their system of measure which could yield measurements as small as a half-inch when they construct the actual building.
After creating their rulers and finding conversions from the English system to the cubit system, they should be given a copy of the blueprint that their group will be working on. As homework or an in-class assignment on a following day, they need to convert the measurements given on the blueprints in the English system into measurements in their cubit system. This portion needs to be checked by the teacher prior to beginning construction of the actual work. If a conversion is off, it can cause a big problem during the building process. Note that some error will occur regardless, and students will reflect on these errors at the end of the unit. It will take approximately 10-20 minutes to convert the measurements. After the conversion has been completed, students need to determine the ratio of their construction to the actual building itself (as per the measurements on the English blueprints).
After the blueprints have been converted, they can begin constructing the actual building itself. Three blueprints have been provided, but teachers can design their own if they so choose. Each group can use whatever materials they wish to construct their buildings. Typical materials might include paper-towel cores, popsicle sticks, cardboard, construction paper and Play-Doh. The sky is the limit here, so long as the students can shape or cut the materials to fit their project. This is likely going to be one source of error that students can reflect on later. Construction of the actual building will take several days, and teachers can either require students to divide the parts up among themselves and work at home, set aside class time to work on them, or a combination of both. The sample project took approximately 20 hours to complete and so groups of 3 or 4 work best for this project. If students are dividing up the work to do at home, it is recommended that students divide the work in such a way that if one student’s measurements are off, it won’t affect the overall look of the structure. This might mean that one student takes one set of parallel walls, all of the pillars, an entire floor or even the building foundation and grounds. Teachers might pose this solution to students, but each group should have the freedom to do as they see best. Remember that the final project should fit within a 2-foot by 2-foot area.
After the project is completed, students will need to prepare a few minor details before they present their building. The first thing is that they need to research their structure. What was its original purpose? What people-group built it? When was it built? List one additional interesting piece of information about the structure. Second, students need to try to identify sources of error. They should measure their building in the English system, multiply this by their ratios (the reciprocal) and compare that to the dimensions on the blueprints. If their measurements are accurate, these numbers should be the same. If not, then they need to answer the question ‘why?’. Last, students need to prepare a 3-5 minute presentation about the structure, what they learned and difficulties they encountered. This scenario can even be put into a clever theme as if the students were taking the roles of building contractors who were contracted by a museum that desires a diorama be made from a structure from history for the opening of a new centerpiece for the next year. Students would be presenting their final piece as a bid for the use of their structure and the best group’s structure would be on display. This could add meaning to what they are doing and set the project into a more real-world setting.
The Sample
The sample project was similar to the above instructions but the building selected was Bodiam Castle, constructed in East Sussex England in 1385 by Sir Edward Dalyngrigge. Its purpose was to be the home of the knight and a fortification against the French during the Hundred Years War. Exact measurements for the structure were not available and so mock measurements were used with the castle being approximately 100 feet long by 92 feet wide. The first step was to create the cubit ruler which measured two feet in length and was broken down into subunits called courts, robs and tips. Each tip measured to be 1 inch in length. After the ruler was constructed a ratio was determined for construction in order to create a structure that measured 2 ft. x 2 ft. or less. The castle was estimated to be 1:48 the original building (see pictures). The scale for our construction was such that each foot in length according to the blueprint would mean 1 “tip” by our cubit (approximately 1 inch). These measurements were then divided by 4 meaning that 1 tip was to represent 4 feet. From there, the rulers were used to measure out cardboard pieces for the walls and towers. Popsicle sticks were used to form the ceiling of the castle while construction paper was used to form the castle grounds, moat and bridges. The towers for the castle were constructed out of paper towel cores and the structure was overlaid in paper with stone images to give it more of a realistic feeling. All of this was laid onto a poster board that measured 24” by 28”. There are areas of error in our model. The primary errors stem from the fact that the cubit ruler doesn’t convert exactly into the English system, which gives small measures of error in the dimensions. Second, the materials used were not extremely exact which left various gaps when piecing the walls and towers together. A third location of error is that the paper towel cores did not measure out to be what was expected and so 2 cores were used for each tower to create a rounded effect. The last source of error is found in cutting down the cardboard into appropriate sizes and creating the walls themselves. Thus walls were not perfect rectangles. The overall result is that the model is not quite 1:48th the predicted size as estimated. Where the castle should have measured 25 inches by 23 inches, the overall dimensions of the model were 24 inches long and 22 inches wide meaning that or model was within 1 inch either direction to the 1:48 scale.
Conclusion
This lesson plan can fit easily into any unit dealing with a number of different topics, although it is designed to cover ratios and unit conversion. As mentioned earlier, the goal is that the project should fit within one week’s worth of class and the teacher would do well to have the final day for presentations and a question and answer segment in case there are problems the students encountered that weren’t ironed out earlier.