This week we are going to take a look at applied math. Learners will be given a ramp (this can be a ruler), a ball that fits on the ramp, measurement device (ruler) and a timer. With these instruments learners can investigate the relationship between distance and time as the ball rolls down the ramp. I encourage teachers and facilitators not to give too much procedure here and let students come up with the way they will execute an experiment and record their data. I usually go with a 10-15 minute “sweat-it-out” period before I give them any hints.
This exercise is a great intro into calculus concepts like “instantaneous velocity” and derivatives of functions that model falling bodies. I think this is fun to do with middle schoolers and up. You can plot your functions and then draw the tangents to the curve and find slopes (this is fun in groups).
If students are stuck, hints may look like this:
Have you set up an experiment that you can repeat to get multiple samples?
What time intervals are you going to try to record? how far has the ball rolled with zero seconds? 1 second? 2 seconds?
Once you have your data, can you plot it to see if there is a relationship? Is it a direct variation?
If you plot the relationship, what type of curve is it? Can you create a function for that curve?
Here is a sheet I had for a class that students received after completing their experiments:
I love Martin Gardner’s work and books that brought math to so many people in a fun and engaging way. One of the topics he covered was Soma Cubes. This week learners can create and play with this wonderful seven piece puzzle that was invented by Mr. Piet Hein during a lecture on physics. I love this puzzle because there are so many questions to ask and ways to solve it. There are a few options for creating your own:
Option 1: Wooden cubes
I ordered wooden cubes and found they aren’t perfect, but do the job with students. You can get them at craft stores or amazon (affiliated link).
Option 2: Sonobe Origami
You can make a Soma cube with a lot of folding. I would recommend doing this with teams and older students (or as an adult). The folds need to be exact. That being said I have seen 9 and 10 year olds do beautiful origami Soma cubes. The best tutorial that I was able to find is on the Luck Paper Scissors Blog here.
Here are some questions/exercises:
How many unique ways can you solve it? Is there a systematic way to track your solutions?
Are there combinations that will never have solutions (ex: starting with one or two pieces in a particular way)
What other symmetric constructions can you create?
Tag is a great way to get moving, and it isn’t just for kids. I have had my highschool groups play with just as much zeal as the 5 year olds. For this week I encourage learners to create outdoor tag games that incorporate a mathematical concept.
Here is an example using the power of 2 in tag:
Start of tag – 1 person is “it”
That person has to tag 2 people to become “not it”
Now two people are “it” and they must tag 2 people to become “not it”
The game ends when everyone is “it”
With chalk learners can draw an “it” tree to show the exponential growth of the game (1,2,4,8,16,32… or 2^0, 2^1, 2^2, 2^3, 2^4,…)
The Dragon Curve is a fractal that is well explained in this numberphile video. This week learners can create a dragon curve with a strip of paper, Lux Blox, Python programing, Legos or toothpicks.
For paper, I would suggest using a strip of thin paper. Thicker paper doesn’t produce as many folds as thinner paper. You fold the paper in half, and just make sure you fold from left to right. You can tape, glue or pin your dragon curve down when you are down. One question I like to ask students is: “does the length of paper change how many folds you can get? If so, how?” It is a fun experiment to run.
If you have never used Python, then I recommend going here. For kids there is a great DK book here.
With python, I would encourage learners to think about how the algorithm would look to create the Dragon Curve. There needs to be a loop for each iteration, but what does that look like? Here is my code (copy and paste it into a py file), but I encourage learners to try first. If you notice that I have an input for angle, it’s because I liked playing with the angles of the dragon curve to create different patterns and variations of the curve. You can hardcode it to 90 degrees if you wish. Play and you never know what you will find.
This week we are going to look at density in a two-dimensional sense. The idea is to create two dimensional images using various densities of points. The medium and approach can vary for the classroom. Some ideas are:
Sand art on a stick surface using different densities of sand (try light colored sand and a dark surface or vice-versa)
Pointillism with pens, pencils or markers to create a peice
play with shading and contrast
practice drawing shapes and objects first
Pea gravel on asphalt to create images
Moving densities with people to create a moving scene or mandala (this takes some choreography)
Paint with round pencil erasers as the point/dot maker
Round stickers on a contrasting surface
For larger grains or objects learners can measure the density of different areas by calculating how many grains/objects are in a given area (ex: grains of sand per square inch)
Students can calculate the density for various areas of their projects and note observations. Classrooms can discuss and play with density functions, look at density maps (ex: population density), look at pointillism art, and/or use apps that change photos into pointillism sketches (pointillist is the one I use).
This week learners can get hands on with plotting. I encourage learners to investigate the history behind the Cartesian Coordinates (it’s interesting – I was just reading about it in Infinite Powers by Steven Strogatz).
The idea is to plot with D&D figures, chalk, legos, or watercolors. Make art out of plots! This is a great activity for pre-algebra and algebra students. Younger students can learn as well but can focus more on finding ordered pairs (x,y). Below are four activities for plotting:
Activity 1: Hit the monster (game it up!)
Use a gridded mat (like what is used in D&D), large graph paper or overhead projector
Draw Axes on the grid and define the quadrants and scale
Place or Draw monsters throughout the plane
Have students devise functions that can hit/intersect monsters
This can be timed or not timed
Students can work in teams
This can be a D&D math mission if you are gamifying your lessons
If there is only one or two learners then smaller graph paper can be used
Activity 2: Cartesian Lego
Decapitate as many minifigures as possible for this activity (other round 1×1 pieces will work as well.
My students used a large gray sheet and black flats for the cartesian coordinates
Make plots of various functions and then see if others can “name that function”
Activity 3: Watercolors (or other art media)
Create plot families using watercolor flash cards
add characters, color, and comics
label the backs with the family the plot belongs to.
Activity 4: Plotting in a large room
With masking tape in a large room you can make a grid
Have students plot functions with beanbags or rope
Students can toss a beanbag and then try to figure out the coordinates
This can work at an outdoor park if you can grid off an area without creating a tripping hazard