# The Reflective Educator

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This is another post in my series on math in the real world.

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When plants lean over due to being pulled by gravity, they often form a similar shape. With some exploration, we can determine what shape this is (at least approximately). First, I opened up one of these pictures and embedded it in Geogebra. Next, I added some points to my diagram, following along the shapes of one of the plants.

Next, I exported these points over to MS Excel, so I could find a regression on the points. A quick glance at the shape the curve seemed to be representing suggested I should try fitting the points to a parabola.

The shape does appear to be a parabola, however, I know from experience that not all parabolic shapes are what they appear. For example, a hanging line is actually a catenary.

What would you have to do to confirm that this shape is a parabola? Is it possible that it is only approximately a parabola?

Thanks to the @OpenCulture blog, I got to listen to a very interesting interview with Keith Devlin. Keith argues that kids need algebraic reasoning, and arithmetic, to a point. He doesn’t say kids need to be able to do pencil and paper algebra, in fact, he has a very interesting argument for using spreadsheets more often in schools. Listen here:

This is another post in a series I’m doing on math in the real world.

Image credit: mvplante

There is a lot of different types of mathematics in family relationships.

For example, each generation you go back, the number of ancestors you have increases exponentially. This works, of course, since we all have a lot of overlap on our ancestors, and eventually everyone is related to just one person, a woman named Eve who lived in Africa many years ago.

You can also look at the probability of relationships forming, based either on interest, or on type of friendship building activity in which you participate. When we want to form relationships, we tend to participate in high probability activities, like drinking with friends at a club, or discussing books during a book club. My friend noted that the probability of a couple forming strong relationships with other couples where they have similar interests, everyone gets along with each other, and each member of the couple has a compatible schedule is actually rather low.

If you look at the relationships of the families themselves, you can draw graphs of the relationships where the circles in the picture above represent people, and the lines between the circles represent the relationships between the people. Would you say that this is a functional family, or not?

I’m happy to report that the recording from my Reform Symposium presentation on Interactivity and Multimedia in Math is available to be downloaded here. Almost all of the recordings for the other presentations are up as well, which you can access here.

I’ve also uploaded my presentation slides here, so that you can download it and look at it yourself. Finally, if you are interested in further reading from my blog related to my topic, see these two links:

Update: Taking advice from @shamblesguru advice, I’ve converted my presentation, using a screen-casting program, into YouTube format.

I’m very interesting in finding ways mathematics is present in the world outside of the classroom, which I’ll call the "real world." Obviously what students do in the classroom is part of the real world, but too often in math instruction school math is completely separate from the contexts kids experience in their day to day lives. I’d like to build a collection of resources for math teachers so that we aren’t all scrambling in the dark looking for ways to incorporate more contextual learning in our teaching. Note that I’m not at all opposed to teaching mathematics which is highly interesting but has no real world context. I’m just opposed to teaching math which neither has context or any pizazz.

Here are some of the ways you can share your thoughts and suggestions on real world math.

• Join the Flickr group here. Make sure that each of the photos you include has a description of why it represents a mathematical idea. Your students could also join this group and share their own ideas!

• Blog about real world mathematics. Use the tag ‘realmath’ in your blog post (all one word) so that other people can find it by searching. Post a link to your blog post on Twitter (or any other social network) with the hashtag #realmath.

• Post ideas under #realmath directly on Twitter (no blog post required).

• Contact me through the form above and (assuming it’s appropriate!) I’ll post your idea about real world math here.

Please suggest other ways we can share ideas for context based mathematics and I’d be happy to include them here.

This is another post in a series I’m doing on math in the real world.

Nature has lots of things which look random, but which are influenced by environmental factors. For example, when pine cones fall from a pine tree, they will tend to fall in a ring around the tree, but will bounce when they hit the ground, so the terrain impacts the distribution of the pine cones on the ground. Barnacles are essentially randomly placed on rock, but on shore-lines they are almost always found on the side of the rock facing the ocean.

Generally we tend not to understand randomness very well. When we create something that we want to be random, we tend to over-emphasize filling of empty space and spreading out the information, and under emphasize the number of pseudopatterns and clumps in the data.

Here’s a classroom idea. Have students go around and take pictures (or record in a journal, perhaps with a pencil drawing) of things which look random. See if they can find examples of things which are actually random, and things which are evidence of human activity. In the slideshow above, you should spot one example of a rock collection which most decidedly not random.

This is another post in a series I’m doing on math in the real world.

The amount of mathematics required to design, plan, and construct one of these houses is amazing.

The plans have to be done in scale, and the building is often shown in both perspective and orthogonal views. The budget for a housing project is often estimated using square footage formulas, but can be quite complicated when more carefully calculated. Ensuring that the house is up to building code, or is LEED certified can involve more sophisticated calculations and analysis. For custom designs, the customer needs to be involved in the process, which means that good communication skills are crucial.

When creating the plan for actually creating the real version of the house, each aspect of the project has to be carefully scheduled. This is so that one isn’t trying to wire the house for electricity after one has laid the drywall. Creating this schedule often involves collaborating (with multiple contractors), problem solving, and logical reasoning skills. The design itself has to be checked for structural flaws. For example, it is also critical to check at this stage that load bearing walls which actually be able to bear the weight of the floors above them.

The construction itself involves measurement, and understanding tolerances in measurement. Some measurements also involve using trigonometry, or the Pythagorean theorem. Workers have to plan carefully, and when they run into issues, creative solve problems.

In your school there are a few ways to use construction problems. Students could use a 3d design program like Google Sketchup to create a model of their house. From this model students could calculate the measurements of the house, check for structural integrity, and even create a careful budget of the cost of the house. Students could even create a design, and then implement their design (either in miniature, or possibly at full scale if you have a useful project that needs to be completed).

This is another post in a series I’m doing on math in the real world.

The growth of trees is actually a fairly mathematical process that at least involves fractal theory, graph theory, and topology. You can actually generate very realistic looking trees using a computer. See the video below for an example of simulated tree growth.

Here’s an idea. Take your kids outside and find some trees (even bushes or ferns will do in a pinch). Explore (and catalog) what rules different trees seem to follow as they branch. See if you can follow those same rules with pencil and paper to produce tree-like drawings. For bonus points, take some pictures of some younger trees, and use your rules to predict where the next branches will start, then follow up in a year to see if you were right.

This is the second in a series of posts on math in the real world.

Which piece of cake should you choose?

This is a problem that often happens at birthday parties. Should you pick a corner piece, a side piece, or one of the middle pieces?

The answer depends on whether you want to optimize for volume of cake, volume of icing, or a balance between enough icing, and enough cake. You can either view this as a volume and surface area problem (or use calculus to determine the optimal piece of cake).

It’s not hard to see that a corner piece has the most available area for icing, but it is less clear whether a middle piece or a side piece has more icing. How much of a difference does the angle in the cake make?

One obvious way to bring this problem (literally) into your classroom is to make a cake and share it with the students, which is more useful if you talk about the shapes of the pieces in advance, and get kids thinking about which piece of cake they should take, depending on their preferences. At the very least it is likely that students will never look at a birthday cake quite the same way again.

This is the first in a series of posts on mathematics in the "real world."

Should you pay to play these games?

Aside from the obvious answer, that these games are fun, and so whether or not you win, the games have an intrinsic "fun" value, one wonders how fair these games are to play. It’s easy to deduce that the games are unfair (why would an organization trying to make money use them otherwise?) but how unfair are they?

This is essentially an expectation problem in the wild, and attempts to analyze games of chance like these by mathematicians led to the formation of probability theory itself.

One way to answer this question is to go to a carnival where there are a bunch of these games, and record some experimental data. You can either use your own money to play all of the games, or stand around surreptitously with a notebook and let other people do the experiment for you, while you record results.

Another way would be to set up your own versions of these games, then host a carnival at your school (or in your classroom), and have participants play the games (perhaps using monopoly money?). If they don’t record results, the activity is fun, but not very mathematical, so I strongly recommend some record keeping takes place during a school sponsored carnival event. You will also need some time at the end for people to analyze, then discuss their results. I recommend, if you do this as a school activity, checking in with your students about common misconceptions about probability that they have, and make sure their analysis of the results spends at least a bit of time debunking some of those misconceptions.