I have a section of 8th grade engineering/mathematics. The class is meant to be a support/enrichment environment rather than instructional, they have a separate math teacher for pre-algebra/algebra. We have been doing a lot of fun engineering projects, but i would like to work more mathematics in. Rather than going through engineering by discipline (chemical, electrical, mechanical, etc) the class is project-based where we learn about a specific topic and then complete a project with the engineering design cycle in mind. For example, we did a unit on water and the water crisis where students built water desalinators and DIY water filters. Then we learned about hydropower and dams and did some calculations of energy potential given the size of a waterfall and plant and compared it to how many houses it could power. I am open to any type of math that can be worked into an engineering curriculum because I have been given freedom to write my own curriculum, which is still in progress. Also, any one-off ideas where a little bit of engineering knowledge and a little bit of math can come together in an hour or so can be fit in between units.
Here are two activities I have used. The second does not fit in an hour, however.
(1) Give the students maybe two different sizes and thickness of paper. Ideally, the two are quite different. Their goal is to design a paper airplane that flies the longest distance when launched from a standard height. Depending how ambitious you are, you could even brush up against the Reynolds number. At a simpler level, if the paper sizes/weights are quite different, the winning designs will be different. Scaling up the best lightweight design does not yield the best heavyweight design.
(Image from here.)
(2) This is hardly original, but I gave highschool students standard cardboard rectangles. Their task was to cut and crease one rectangle to fold into a single structure—no glue or tape, all one piece, gripped with slots etc.—that folded over and protected a single egg. Then we dropped them from a height of about 5m to see which protected the egg from cracking. The requirement that it be all one piece, without attachments, requires considerable geometric ingenuity (and lots of trial and error!).
I don't have an appropriate illustrative image. This will have to suffice:
(Image from here.)
This activity took all morning, approximately 3 hours.
Make a horizontal cantilevered beam. Place it against a horizontal vs. vertical grid. Put progressively heavier weights on the end of it. Trace the shape of the cantilever for each weight. Fit a polynomial to the shape of the curve. Notice whether the deflection is proportional to the weight.
Use plastic Slinkies to illustrate S-waves and P-waves.
Confirm that the period of a pendulum really is nearly constant.
Demonstrate leverage using a balance-scale.
Demonstrate leverage using a hydraulic jack.
Drop something just dense enough that you can measure its terminal velocity.
Use a Scotch Yoke, a piston, and a chart recorder to show how circular motion can be converted into a sine wave.