This page is for collecting ideas and drafts of experiments for a Girl's Science Day at Columbia. This day has groups of no more than 12 girls, 6th-8th grade, doing 35-minute experiments.

## Possible experiments (final selection process)

- Diffraction
- We'd get lasers of two colors and things with different shapes of holes cut out in them. Girls would predict the shapes of diffraction from different shapes of holes. They could compare the size of the diffraction pattern for different colors of lasers. Open questions: What is it they will predict/test? Can we make this rigorous enough (not just "playing") without using math?

- Cake-cutting algorithms
- In this experiment, we would first divide girls into groups of two. We would give them simulated cakes (print-outs) and explain how to divide the cakes evenly using cut and choose. We then have them try to break the system, and thus show that people only harm themselves. We would then expand them to groups of three, and have them try to fairly divide the simulated cakes themselves. We have them try to break their algorithms (show what happens when you try to cheat the system). We then have them try to devise an algorithm for four people
- We conclude by explaining various ways this could be used (the cake cutting algorithms book has various real-world examples), talk about other extensions (reducing the number of cuts, envy-free), talk about algorithms in general and how they are the basis of things like computer programming.

### Graph theory (Seven Bridges of Königsberg)

Make a course of bridges and land following the historical Seven Bridges of Königsberg problem. This could be done either outdoors using chalk on the ground or else indoors using tape. Have the girls try to walk the course, starting from wherever they choose, crossing each bridge once and only once. Once everyone has had a chance to try, discuss why it can't be done. Introduce the concept of edges and vertices and reformulate the problem in an abstract way.

Next, divide the girls up into two groups, corresponding to the blue and red castles in the Wikipedia article. Have the blue team try to figure out where to add an eighth bridge so that they can walk the bridges without allowing the red team to walk the bridges, too. Meanwhile, take the red team to a second course -- perhaps be a similar course where it's possible to walk the bridges starting from one location. Then the red team could try to figure out how to walk the new course successfully. After the blue team has finished adding their bridge, help the red team try to figure out where to add a ninth bridge to reverse the situation. Meanwhile, take the blue team to the second course.

At the end, do some kind of wrap-up. Thoughts?

- We will explain what a graph is: a connection of edges and vertices
- We will pull out rope, and use rope to overlay the map on the ground
- We will explain how a vertex with an odd number of edges must be either the first or last stop on your path, and thus there can’t be more than two vertices with an odd number of edges to have a Eulerian path. We will also explain the requirements for a closed path
- We will provide handouts with assorted landforms and bridges on them, and ask students to figure out which have eulearian paths.
- We will go back to the bridges of Königsberg and ask “Where do you have to add a bridge so that you can walk across all of them?”
- We will provide some of the challenges on the Wikipedia page.
- If we have left over time, we’ll have students draw out their own landforms on the ground and encourage each other to find Eulerian paths.

There are many possibilities listed above: we’ll decide the exact plan later, and can always improvise as we go along.

Here is our actual lesson plan:

- Draw two courses: Königsberg and Königsberg with one added bridge (so that the course was solvable). "Long ago, in Europe, there was..."
- Divide students into two groups. Each does one of the courses, then they switch.
- "The one day Euler came..." Draw vertices and edges ("points" and "connections").
- Discuss even/odd connections. Have the mini island-pictures which demonstrate this concept.
- Go back to the maps and discuss even/odd bridge counts.
- Explain: The graph makes it simpler. That makes it easier to solve.
- If there is time, try adding a bridge to one of the courses to be able to walk starting at a certain place and ending at another certain place.
- This lesson plan took 35-40 minutes with 11-20 students. They really liked it.

## Possible experiments (preliminary ideas)

- Optics: Reflection/Refraction
Graph theory: http://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg

- Something related to möbius strips
- Non-Newtonian fluids
- Knot theory
Monty Hall problem: three doors, pick one, another opened, do you switch?

- Trigonometry: make a primitive sextant, calculate the height of buildings and width of things.
- Symmetry: something demonstrating the power of symmetry arguments in deduction.
- Something game-related.

## Other sources of experiments

Wisconsin MRSEC site: http://mrsec.wisc.edu/Edetc/nanolab/index.html

- Look at various puzzle books and convert them to experiments
- Look at 6th-8th grade science books