Demonstrating Circle Theorems

An inscribed angle that intercepts a diameter is always 90º.

An inscribed angle that intercepts a diameter is always 90º.

You can move any of these vertices, but the two triangles are always similar.

You can move any of these vertices, but the two triangles are always similar.

Using this tool and a few rubber bands, you can illustrate the properties of chords and the angles they form. 

 

PRINT DETAILS

Material: PLA

Resolution: .2mm

Raft: None

Supports: None

Infill: 10%

Shells: 3 (for added strength, necessary when using a rubber band)

Boat & Lighthouse

boatAndLighthouse

This model was based off of a question from the August 2015 Common Core Geometry Regents. When I used this question to review with my class, many students had trouble understanding what the question was describing. This model allows me to show the student and let them experiment with the angle relationship in a very hands on way: by attaching a rubber band between the hook on the ship and the hook on the lighthouse, the elastic rubber band will tighten as the ship moves, forming the angle of elevation.

Print Details

Material: PLA

Resolution: .2mm

Raft: None

Supports: None

Infill: 10%

Shells: 3 (for added strength, necessary when using a rubber band)

Volume of a Pyramid

The volume of a pyramid with a given height and base area is exactly 1/3 the volume of a prism with the same height and base area. This can be illustrated with three congruent, right pyramids if the pyramids have a square base and a height equal to the width of the base. (You can not make a prism out of three congruent pyramids if the pyramids do not have these dimensions, but you can use Cavalieri's principle to generalize this special case.)

 

Printing Details

Raft: off (can be on)

Supports: off

Material: PLT

Sliding Ladder

slidingLadder

A sliding ladder is the subject of many trigonometry and calculus problems. (While the question of who would ever let a ladder slide so dramatically is not covered in either class.) Often, such a problem requires students to compare two states of the moving ladder so a teacher or textbook accompanies the problem with a couple of right triangles to illustrate the change. Sometimes this is sufficient convey the idea, but the actual relationship between the changes legs and angles of a right triangle with a constant hypotenuse is, to me, much more interesting than two static sketches can convey. (omg, does this problem embody the rate of change of the tangent function? Eeeek!)

I designed this model so that I could actually show students what is happening and let them play with it and make their own observations before we drew the sketch. 

Printing Details

Raft: off (can be on)

Supports: off

Material: PLT

This print can be done using two colors or, if you have a single extruder, you can print the ground with one color and switch to red after the base is completed. The ladder needs to be printed separately (in Tinkercad, select each model (the wall and the ladder) and when you download for 3D printing, check the "Download the selected shapes" box. Note that the ladder, as is, is a bit flimsy. It is designed to slide through the slot in the ground, but it could stand to be a teenie bit wider and thicker.

Lines and Planes

Intersections of parallel planes and two planes perpendicular to the same line. The model on the left was filed down. The one on the right has not been filed yet.

Intersections of parallel planes and two planes perpendicular to the same line. The model on the left was filed down. The one on the right has not been filed yet.

Gaining the ability to print models like these was one of many reasons I even considered buying a 3D printer. I have been teaching geometry for nearly a decade and in that time I've resorted to crumby 2D sketches and sheets of printer paper to illustrate the relationships between lines and planes. It was time I really showed my students what I was talki about.

Designing a printable model of a plane that intersects two parallel planes does present some difficulties. Clealry, while a geometric plane has no thickness, such a model cannot exist. It took some experimenting to find the right thickness for sturdy lines and planes. Secondly, to avoid copious scaffolding, I've had to design models that exhibit that classic 45° slant. Finally, for clarity (and fun) I made two-color models. These prints will look best if printed on a dual extruder. If you have a dual extruder but have never tried a two-color print, you can find my directions here. The models can also be printed in a single color.

 

Printing Details

Raft: off (can be on)

Supports: off

Material: PLT

Two-color print