This is the first part of a series of projects intended for first year calculus students. In this series, students will learn how to use different mathematical equations to assess the properties of solids and containers. My apologies if the "X" and "x" thing is confusing, but in some situations formatting makes using an asterisk impossible, so big X is a variable, and little x is multiply.
In this first project we will be finding the generic equation for a lidless box we will 3D print by using a standard geometric equation. For instructions and equations read further.
I optimized this thing to be printed in ABS with maximum settings. I would recommend a skirt with an offset of 0mm (otherwise known as a brim) with 2-5 loops for ABS. As can be seen from the photo, an increased infill may be helpful as well.
For this project I used a free program from Autodesk called TinkerCAD. I have long been a user of AutoCAD which is a desktop drafting program from the same company. I used TinkerCAD to make it easier for educators and learners to be able to more easily replicate what I did. TinkerCAD is awesome because it's not only free, but it's easy to use. It also runs in your browser and has built in instructional lessons.
The first part of this project will start with a standard rectangle. I wanted dimensions that wold be easy to work with, take a minimal amount of plastic, and print easily. Therefore I used the box tool to create an object that was 100x60x2mm.
Next I used the box function again along with the hide function in order to make my "cuts" into my object. For this example I used X=20mm.
At this point it is easy to see the dimensions X = 20mm will give use. We have a base of 60x20mm, two flaps with dimensions of 60x20mm, and two flaps with dimensions of 20x20mm.
For the purpose of this experiment, imagine that we have a material such as paper or cardboard with dimensions of 100x60mm. Our lidless box is made by cutting out squares from the corners as seen in the "How I Designed This" section.
The corners are then folded up to create a lidless box. If you will notice the image on the left looks a lot like the object we created, which will allow students to visualize this part of the project and tie it in to the next parts as well.
Help students visualize the math that they will be using and make it easy to see the real world applications.
Even though the overall project is designed for Calculus students, this part of the project can be used for algebra students as well.
- Teachers and/or students who are creating a lidless box using their own values of X will need to have access to a computer with internet access and be logged on to a free TinkerCAD account.
- Tinkercad is easier to use with a mouse than with a tablet or trackpad, so computer mice are recommended.
- No previous 3D design experience is required for the students, although familiarity with Tinkercad would be helpful.
- The instructor should be comfortable answering modeling questions about Tinkercad and be able to advise students to avoid design features that might cause printing difficulties (overhangs, delicate features, etc).
- It is helpful to have students form into working groups of two or three so that they can collaborate on the design process. Having students work in groups is also helpful if you have a limited number of computers and/or limited 3D printer access.
Step 1: Standard Formula
To find our generic equation, we will be starting with the standard formula for volume:
V = area x height
Step 2: Establish X
For the purpose of this experiment, X will be an unknown equal to the length of one of the sides of the squares that we "cut" out of the original 100x60mm object.
Step 3: Use X To Find Generic Equation
Taking the standard formulas V = area*height and area = widthxheight, then subtracting X from the width and height to represent the squares we cut out, we are given:
V = (100 - 2X)(60 - 2X) * X
Step 4: Substitute for X
V = (100 - 2(20))(60 - 2(20)) x 20
= (100 - 40)(60 - 40) x 20
V = 24,000mm^3
We have now used our conceptualization along with algebra to come up with an equation that will give us the volume of a lidless box created by starting with a 10060mm rectangle with squares cut out of the corners with dimensions of XX. No matter how big or small our squares are, we will be able to easily find the volume of the lidless box by using our equation:
V = (100 - 2X)(60 - 2X) * X