The Oresmian Coordinate System - Bridge

Step One: Making the Coordinate System

TEACHER NOTE: The easiest way to use the coordinate system is to give the students two parameters in the design task (e.g. strength and price) and create the axes. These constraints should give the students a direction but not an ultimate goal, by giving them a direction students will always have the possibility to improve their design. This type of design task and way of using a coordinate system increase the students’ abilities to problem solve. The coordinate system can also be used to construct knowledge.

TEACHER NOTE: (Optional Anticipatory Set) Before class, review and print copy of Activity 1 for each student: Coordinate Place Four and Independent Worksheet 1: Coordinate Dot-to-Dots. Found in the Math Learning Center, Bridges in Mathematics, https://www.mathlearningcenter.org/sites/default/files/pdfs/SecB4SUP-C5_GeomCoordSys-201304.pdf

Essential Question: What are Cartesian Coordinates? What is the origin of the name?

Fun Fact: This lesson plan is named after the French natural philosopher and mathematician Nicole Oresme, who used cartesian coordinates before Descartes, from whom they get their name. Oresme used the coordinate system to plot physical attributes, observed in nature, whereas Descartes used it to describe and understand principles of algebra.

1. In whole class, review Khan Academy Introduction to Coordinate Plane: https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview-hist-alg/v/descartes-and-c artesian-coordinates
2. Explain the rules of the Coordinate Place Four game. Give students a few moments to examine the game.
3. Explain to students that the game will help them learn to locate positions on a grid by naming coordinates. These are ordered pairs of numbers, or in this case, ordered pairs of letters and numbers.
4. Students will pair-share ideas about where the class should place the first red marker. Ask students to identify the location of the markers as precisely as possible.
5. In small groups, students will complete Worksheet 1. On each grid, students will take turns drawing and numbering the dots at each of the ordered pairs on the list.
6. Teacher will walk around the room to each group, checking for understanding and correcting any misconceptions students may have.

Step Two: Introduce the Iterative Design Process

Essential Question: How can you use the iterative process to construct knowledge to improve designs?

TEACHER NOTE: Keep the introduction as short as possible, the students need to create their own experiences and will learn through this. However, sharing some information at the beginning is essential. Review “The Iterative Designer: An approach for structuring a learning activity with a 3D printer, (Reference Iterative Design PDF )

Explain the Goal– Inform students that the goal of the lesson is to go through an iterative process. Students will learn how designs can be developed and improved over time. The teacher will guide the pupils through six phases and help them construct knowledge to improve their design. First introduce the students to the design task. Hereafter, the pupils will begin the iterative process and design a solution. (see Chart on p. 1 of Iterative Design handout).

Introduce the Process -Before students start, they need to know that they will be working through several

iterations, and that the learning activity extends beyond the initial print. Explain to them how they are going to design, print, test, and evaluate their bridges several times, Students also need to know what to do in each phase and how much time they have, in order to regulate their process themselves.

Introduce the Task – Finally students need to know the design task and which problem they are solving. Finding a good design task can be hard, and several considerations are necessary. First of all, it is an advantage if the object is small and fast to 3D print, being able to print each iteration within an hour is an advantage. Furthermore, the task should be to solve a problem, but with multiple possible approaches. This can be easier if the task contains some constraints or goals which gives the pupils a direction for their design. Introduce the actual coordinate system you will use for analysis in step five so the students know what the goals are: price and breaking weight are the 2 axes in the coordinate plane.

1. Introduce the Design Challenge to the students.

   “In this task students will design a highway bridge where two cars can drive over the bridge and four trucks, two in each direction, can drive under. The bridge must be as strong as possible to avoid accidents and collapses, however, the price must also be taken into account.”

2. Students will create a model of a bridge using the six phases of the iterative process:
   • Introduction
   • Design/Re-Design
   • 3D Printing
   • Testing
   • Evaluation
   • Presenting
3. Explain to students that designing in a 1:500 scale produces bridges of an adequate size allowing comparison, both with their own previous designs but also with other students’ designs.TEACHER NOTE: Another reason for designing a 1:500 scale model of their bridge design is to keep the printing time down. If you do not have any experience with the printer, you can practice 3D printing the truck and car designs.It will be an advantage to give students models of trucks and cars in the right scale. Both teacher and students can draw them to practice skills in the design tool. Or, students can use the predefined models. These predefined STL models are imported into the workspace of the modeling software, the pupils can use them to “model around” easing the process of getting scales right. However, using these models will limit their use of math, so you have to decide on the learning goals for the activity.
4. Have students develop a design for future highway bridges through an iterative process in groups of 2-3. Review stages of iterative process for student understanding and correcting misperceptions.
5. Allow each group of students to start their own prints, so they can be in control of the process. Often the bridges need to be rotated in order to 3D print This is also a good opportunity to introduce this concept to the students.

Step Three: Calculate the price

1. While the model is being printed, students will apply ratios, measurements and cost-benefit analysis to help calculate the budget of their bridge in real life scale.

TEACHER NOTE: There are several different approaches to calculating the price of a bridge. In this lesson, we start by finding the volume of the print, by seeing the 3D printer as a machine which arranges a long cylinder (the filament) into a new shape (the design). When starting the 3D printer the program will most likely inform you about the length of filament which will be used on this design, and the height (h) of the cylinder. On the spool of filament you can find the diameter (most likely 1,75mm or 2,85mm), which can be converted into the radius (r).

V model  = π · r2  · h

• Volume of Cylinder (model of bridge)= π ⋅ r^2 ⋅ h.

Since the bridge is designed in 1:500 scale (s) on all three axes:

V bridge  = π · r2  · h · s3

• Volume of Bridge=(π ⋅ r ^2⋅h)⋅ 500 ^3.

The next step is to multiply the volume of the real life bridge with the price (p) of the material in which the bridge will be built. This lesson used the price of Steel Fiber Concrete on 1.831 Danish Kroners pr. m3

P bridge  = π · r2 · h · s3 · p

• Price of Bridge=(π ⋅ r^2 ⋅ h) ⋅ 500^3 ⋅ (Price of material pr.m^3)

Example: With a filament radius (r) of 0,00175 meters (Converted from mm to keep units the same), and 2,5 meters of filament used (h), a scale factor of 1:500 (s), and a price per m3 alculation will be as follows:

And the price of materials for building this bridge in real life is 191.406$.

TEACHER NOTE: When adapting lesson, teacher should decide to price materials to construct a bridge in real life.

Step Four: Test the strength

1. After the bridge is finished it is time to test the strength of the Have students place weight on the 3D printed model to get an understanding of the durability of their design. Designate an area of the classroom as the testing zone. (Make sure all 3D printed support material is cleared away for accurate testing).
2. Use stacks of paper to measure the durability, but other heavy elements would do just as well. Just be prepared that some of the bridges hold over 100kg.
3. Have students share the weights and take turns in testing the durability of their design.
4. Tell students not to be afraid of failing, because the purpose is to investigate failures– and learn from them. Tell students the bridges MUST FAIL, and weight should be added until it breaks.
5. Once a student’s bridge collapses, the student should record the failing weight for the evaluation process

Step Five: Evaluating and Presenting

Essential Question: How does failure improve the design process?

1. Review the coordinate plane, checking for student understanding and misconceptions.
2. In whole class, tell students “when you know the price of the bridge and the strength of the model write it on a post-it and place it on the coordinate system.”
3. Remind students that the X-axis is the price and the Y-axis is the strength, which means that a strong design with a low price places the design in the top-left part of coordinate system.
4. In small groups, help students place their post-it and failed model on the coordinate system. Then, ask students where their next iteration should be placed and discuss what they can do to achieve that goal.
5. Have students study weak spots on their design, in order to find where their design needs improvement.

TEACHER NOTE: This is usually the time where the teacher can have very interesting and deep conversations with the student, where knowledge construction can be achieved. Remember not to give your own suggestions on how exactly to improve the design, but have the pupils discover this for themselves.

Step Six: Presentation

1. After several iterations, the students will end the lesson with presentations of their work.
2. Students debate which of their designs are the best and give reasons.
3. Extend this activity by testing the models to see which one can support the most weight.
4. Provide students with a third piece of paper and have them list other structural elements that would increase their bridge support. At what cost?
5. Ask students what structural elements they might add to make their bridge support even more weight.
6. Host a class competition to find out which bridge is the strongest.

Formative Assessment: Students will be able to explain the process they used to determine the type of bridge that would work best; how they constructed it; and the results of their weight load test.

More documentation on Facilitation / Procedures: The iterative designer.compressed.pdf