Summary

This is the complete 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.

Print Settings

Printer:

CraftBot

Rafts:

No

Supports:

No

Resolution:

optimum

Infill:

30% square

Notes:

I optimized these things 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. I would also suggest an increased infill ratio. Notice on one of the prints the top layer is missing in areas. That is because the gap of the infill was too great. So I went from 19% triangle to 30% square.

How I Designed This

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.
www.tinkercad.com

Concept

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.

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. If you would like to create your own that has easier equations later, try 160x100x2mm.

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 part 2 of this project I started out with our product from the part 1. If you remember, we used X=20mm which gave us a base of 60x20mm, two flaps with dimensions of 60x20mm, and two flaps with dimensions of 20x20mm. I used what we learned from part one to break down our different parts.

(Remeber that the height of the base is 2.0mm)
To make this into a mold, I created a channel around the base that consisted of:

a border sunk down to 1.0mm that was 2.1mm thick (extra .1mm makes all the pieces fit together real nice on max resolution, but you may need to add more if you will be using low resolution)
a border around the first border that was 2.0mm thick and 2,0mm tall

Remember that you can adjust your snap to .1mm in the lower right hand corner of TinkerCAD.

Next I extended all the flaps from being 20mm tall to being 21mm tall to make up for the channel. I also had to extend the width of the small sides to 22mm (from 20mm) to fill in the rest of the space.

Project: Using derivatives

Objective

In this project students will be able to go all the way from a solid understanding in algebra, through good foundation to what calculus is all about. They will get the tools they need to be able to visualize what they are doing. They will be able to touch ,interact with, and change the things they are solving equations for. This project is intended to take learning out of the books and put it into the hands of students.

Audience

This project as a whole is intended for calculus teachers and their students. Parts of it, however, would be useful for students of other subjects such as algebra, statistics, and even art. In fact, it would be beneficial for teachers to work together on this project. Algebra teachers could teach the first part of this lesson one year, so when those students move on to calculus they can make the connection between the two. Art students could also have fun with making the molds, silicone, and castes.

Preparation

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.
Teachers and students will need access to a computer with a spreadsheet program, preferably excel.
Teachers should be comfortable answering questions about the spreadsheet program.
Students don't need any prior experience with excel.

Teachers will need to get all supplies ahead of time:

100% Silicone caulking
Cost: $3-$5
Caulking is available in all types and brands, but what you need is simply 100% Silicone caulking. Any brand will do, so buy whatever is on sale. The white kind is better than clear, as it's easier to tell when it's thoroughly mixed. This forms the bulk of your mold. Get a caulking gun too, if you don't have one already.
Plastic Cups, Silverware, and Straws
Cost: $3-$5
The silicone won't stick to the cup after it's cured, and you can throw it away when you're done. The straws are helpful if you don't have any pipettes or eyedroppers.
Corn Starch
Cost: $2-$4
The cornstarch is essential to this. It will absorb the moisture in the mix and allow it to dry throughout.
Teachers will need to make sure they have access to graduated cylinders with mL/MM3 markings.

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. 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 3: Substitute for X to find the volume

V = (100 - 2(20))(60 - 2(20)) x 20
= (100 - 40)(60 - 40) x 20
= (60)(20)(20)

V = 24,000mm^3
= 24ml

Step 4: Rinse and repeat

Now that we have our formula to find V, we can start to plug in X values and start to compile data:

V = (100 - 2X)(60 - 2X) * X

Now establish your X values. For the following parts of this project, you may want to make models with X = to 2mm, 4mm, 6mm, etc... Or you can make them for X = 5mm, 10mm, 15mm, etc... depending on how many groups of students there are or how precise you would like your data to be.

Step 5: Print

Print out your thing(s). For tips on settings, see the top of this project. After you things are done, pop them off your print bed and remove the loops if you used them. Sometimes some slight sanding of the edges with a high grit sand paper may be useful in getting the "flaps" to sit in place better. Also, I designed this thing so the addition of rubber-bands would add to the stability as well as hold the flaps together better.

Step 6: Fill 'em up

Fill your mold(s) with your choice of substrate. Some fun things to try:

Sand
Kinetic Sand
Play-Doh
Clay
Silicone
You can add silicone to the bottoms and sides of your flaps to make your thing water tight if you would like to add a liquid.

Step 7: Make permanent castes

This is where the cups, mixing utensils, corn starch, and silicone caulk come in handy. Following is a cheap and easy way to make silicone castes.

Your ratio determines how long your cure time is. A ration of 1:1 corn starch to caulk should give you about 5-10 minutes cure time. This should be about right for this project. You will want your mixture to be about the consistency of cake frosting when it is all mixed together. It is important that you mix it thoroughly. If you don't then the caste will not dry evenly and in extreme cases cold cause it to crack.

Use what ever item (silverware, pipettes, etc...) you have to spoon/pour your material out with. Make sure it is even with the top of the mold. You can use a flat object (such as the back of a knife) to run along the top of the mold and wick away the excess. Parts this small should be dry in 1-2 hours.

Step 8: Verify equation

Using the castes we just made, we will be verifying the equation we came up with by measuring the volume of our castes using a submersion method. This is where you get out your graduated cylinders!

Place the graduated cylinders on a table visible to all students. Add just enough water so the castes from part 3 can be submersed. Make sure you record where the water level is BEFORE you submerse any castes, and in-between submersing each one. Submerse the castes that you just created, one at a time.

To find V, you must take the water level before submersion, and subtract it from the water level after submersion. Compare the volumes that you recorded from this part to the numbers that you calculated from your equation. If all goes well, your numbers should match up pretty well.

Step 9: Chart your data

If you have not done so already, assign each group to find the rest of the X values you will be using. For my own, I used all even numbers 2-30. My data was as follows:

X Volume
2 10752
4 19136
6 25344
8 29568
10 32000
12 32832
16 30464
18 27648
20 24000
22 19712
24 14976
26 9984
28 4928
30 0

You should record your values in a similar manner.

This part will also verify the math we have done so far. In Excel, type "X Values" into column A1, and "Volume" into column B1. Next type all X values into the "X value" column. In box B2 (to the right of the X value "2") use the equation from part one of the project to have Excel automatically find the volume. You should type in something that looks like this:

=(100-(2xB2))*(60-(2xB2))xB2

Use the graph (scatter) feature to make a graph of your data. Based on the spreadsheet and the graph, have students make predictions as to what value(s) of X will give the greatest volume.

Step 10: Using derivatives to find max Volume

In order to get to the point where we can use derivatives, we need to first have a function to derive. Luckily, we already have an equation for volume as a function of X, from part 1 of this project. Another way to say this is f(x). Therefore, we have:

f(X) = (100 - 2X)(60 - 2X) * X

Now we just need to simplify:

f(X) = (100 - 2X)(60 - 2X) X
= (6000 - 200X -120X +4X^2) X
= 4X^3 - 320X^2 + 6000x

For simplicity sake and as it relates to the problem at hand, a derivative is equal to the slope of f(X) at any given point of X. I.e. if the you were to pick the point X = 5 on the graph of f(X), and the slope of the graph at that point were 2, then on the graph of DY/DX where X was equal to 5, Y would be equal to 2.

There is a long version of finding derivatives and creating proofs that most teachers get a thrill of putting their students through. I am not one of those, so I will jump straight to the short version. To find a derivative, use this formula:

f(X) = n^r
DV/DX = rn^(r-1)

Therefore we have:

f(X) = 4X^3 - 320X^2 + 6000x

So....

DV/DX = 12X^2 - 640X + 6000

So let's think about what we need to find out, and what we have learned. We need to find out where V, or f(X), is greatest. We know that DV/DX is equal to the slope of f(X). If you think back to the graph we made in part 5, it appeared as though the slope at the greatest value of V was 0. Therefore, in order to find 'critical points' on a graph of f(X), we should be able to set DV/DX equal to 0. Since I was easy with the derivative, I made this part messy. Let's try it, the long way:

DV/DX = 12X^2 - 640X + 6000
12X^2 - 640X + 6000 = 0
6000 + -640X + 12X^2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(1500 + -160X + 3X2) = 0

Ignore the factor 4.

1500 + -160X + 3X2 = 0

Begin completing the square. Divide all terms by
3 the coefficient of the squared term:

500 + -53.33333333X + X2 = 0

Move the constant term to the right:

500 + -53.33333333X + -500 + X2 = 0 + -500

Reorder the terms:
500 + -500 + -53.33333333X + X2 = 0 + -500

Combine like terms: 500 + -500 = 0
0 + -53.33333333X + X2 = 0 + -500

The X term is -53.33333333X. Take half its coefficient (-26.66666667).
Square it (711.1111113) and add it to both sides.

Add '711.1111113' to each side of the equation.
-53.33333333X + 711.1111113 + X2 = -500 + 711.1111113

Reorder the terms:
711.1111113 + -53.33333333X + X2 = -500 + 711.1111113

Combine like terms: -500 + 711.1111113 = 211.1111113
711.1111113 + -53.33333333X + X2 = 211.1111113

Factor a perfect square on the left side:
(X + -26.66666667)(X + -26.66666667) = 211.1111113

Calculate the square root of the right side: 14.529663152

Break this problem into two subproblems by setting
(X + -26.66666667) equal to 14.529663152 and -14.529663152.

Subproblem 1
X + -26.66666667 = 14.529663152
X = 14.529663152 + 26.66666667
X = 41.196329822

Subproblem 2
X + -26.66666667 = -14.529663152
X = -14.529663152 + 26.66666667
X = 12.137003518

X = {41.196329822, 12.137003518}

Normally we would plug both of these numbers in to see if either or both is what we were looking for. However, since we know that anything above 30 will result in a negative number, we will just use 12.137003518.

(100 - 2X)(60 - 2X) * X
(100-2(12.137003518))(60-2(12.137003518))12.137003518
32835.28

Just as your students should have predicted in previously, the greatest value of V was around X = 12. The greatest value of V was close to 32832 as well.

Results

Students now have gone from creating a lidless box, to calculating it's formula, and even to proofing that their formula works. After they did this and were able to actually touch what they were solving for (our castes), they were able to use their new derivative ability to solve the problem at hand. Students will now be able to see just how handy, not to mention cool, derivatives can be, as well as understand their importance and usability.

Calculus project complete: Lidless box