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Math in the real world: Marshmallow constructions

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

Building materials

My wife, son, and I  went to a kids science event at SFU today, and at one table they had some marshmallow diagrams set up to demonstrate molecules. They let the kids play with the marshmallows and toothpicks, so my son made a giraffe. When we got home, he helped himself to some marshmallows and toothpicks and continued to make things with them.

Simple diagram

 

My son noticed that the most stable form included triangles (with some help from mommy), so he started to construct everything with triangles. When he moved into three dimensions, he noticed that the tetrahedron was the most stable of the forms he could build and so his construction soon began to look very mathematical in shape.

More complicated diagram

 

Now in his most complex form, he has started to build a three dimension tesselation. If he hadn’t been called away to dinner, or if we hadn’t been running low on toothpicks, I’m sure he would have continued the pattern.

Very complicated diagram

 

This activity involves both 2d and 3d geometry, tesselations, sequences and other patterns. Can you think of other mathematics that can be found in this activity?

Workplace mathematics

Here are some videos on mathematics in the workplace I’d like to watch later, when I get a chance. Thanks to Gary Davies and Lorri Carroll for sharing them with me.

Learning Origami

Origami swan

I started learning origami again this past weekend. So far I’ve built a swan, and a couple of paper airplanes that are more advanced than what I usually make but none of it has been particularly complicated to make. I’ve often thought that origami would be a fun hobby, but that I wouldn’t find much use for it in my teaching.

Today, I watched a TED talk (thanks to @BobbycSmith for sharing it with me today) that definitely changed my mind. Origami is way up there now on my list of things I need to learn.

Math in the real world: Balloons

This is part of a series of posts I’m doing on math in the real world.

Balloons in an office

 

The first question I thought of when I saw these balloons in my colleagues office was, how many of those would I need to be able to float? Clearly, this is a math problem, and one students can actually test themselves (I would recommend using inert ballast to test student guesses, rather than actual students). Students would first have find out the amount of weight one balloon can lift, and then use division to determine how ballons would be required to lift their weight.

If you want to make this problem much more complicated (and more of a calculus problem), you would point ouf that the density of air decreases as the balloon lifts, lowering its buoyancy, and putting a limit on how far the balloons will actually lift the student.

The shape of the balloons in this picture is also mathematically interesting, as is the shape of other balloons. Why do balloons form the shape that they do? How do the manufacturers of balloons know in advance what shape the balloons will have before they fill them up with helium?

Numeracy for preschoolers

Count 10 Read 10Bon Crowder has started an initiative to embed numeracy in the early lives of children via their parents, which she calls Count 10, Read 10. The basic idea is to split up the 20 minute of reading for parents into 10 minutes of numeracy and 10 minutes of literacy every day. 

Most parents aren’t reading to their kids daily (only about 48% in the US do) which is hurting their abiliy to learn how to read when they get to school. Unfortunately, an even smaller percentage of parents engage in daily numeracy building activies. If you think not being read to impacts your ability to be successful in school, imagine what happens if you can’t count.

Only 45% of adult Canadians are numerate, "demonstrat[ing] skills and knowledge associated with the ability to function well in Canadian society." By comparison, 52% of adult Canadians demonstrate the minimum levels of literacy required for a person to function well in today’s society. Neither of these numbers is very impressive, but clearly our society is doing a slightly better job preparing people to be literate.

The importance of an early start in numeracy has been well established. While the relationship between the ability to do math and being numerate is not completely clear, the relationship between early numeracy and later numeracy should be. Parents can have a strong impact on the numeracy of their children, and should engage in early numeracy building activities.

One issue, besides of course having time to do these activities with their children, is that many parents don’t know many strategies for building numeracy. Just as educators provide strategies for parents to use to develop early literacy skills, we should do the same to help parents with early numeracy strategies for children.

As a parent with a strong sense of numeracy, and an educator, I have some activities I’ve done with my now 4 year son which you are welcome to share with parents.

My son, wife, and I count everything. We count stairs as we climb them, we count plates as put them out on the table, we count down from 10 when we pretend to blast off in our rockets, and up to 10 when we play hide and seek. We count by twos, we count by fives, and we count by tens. We talk about the relative size of numbers, and use language like less than, more than, and other mathematical comparison language.

Playing chess

We play dice games, like Backgammon or Parcheesi and recently even more advanced dice rolling games like Titan. My son counts up the two dice by himself to see how far he gets to move, and then counts to move his pieces. We play Chess together, and my son’s favourite part of this game is making up rules for how the pieces can move. We play card games together, like Go Fish and War which not only let my son see both the numerals, and a representation of the number on the cards themselves, but also look for comparisons between numbers.

My son bakes and cooks in the kitchen with both of us and is learning about ratios in food, and fractions in baking. We split cookies into halves when sharing, and cut sandwiches into quarters. We talk about food and how old my son is in terms of fractions. He knows he was once four and a half, then four and three quarters, and now he is four and eleven twelfths. While he doesn’t know what eleven twelfths means yet (although he does understand halves and quarters), the fact he has heard about fractions being used in context allows him to start developing some meaning for them.

We build patterns together. We’ll stack blocks into stair cases. We’ll talk about the shapes of blocks using their names (like pentagon) and together we will explore the similarities and differences between his shapes. We make circles out of his train tracks. One of his favourite toys is his magnetic blocks, which he builds into many different types of shapes.

We also play number games like "How can I get to __?" How this game works is that given a number, you try and figure out different ways to get that number by adding smaller numbers together. For example, 1 + 1 + 1 + 4 = 7. 1 + 2 + 1 + 1 + 2 = 7, and so on. I even recently taught my son how to play Nim, which is a great game for teaching about looking ahead.

The point is, my son is immersed in a world of numbers and his ability to see the world through numbers later in life is greatly increased.

Math in the real world: Sound

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

Vi Hart explains much of the mathematics behind noise in great detail, so watch her awesome video below. Thanks to @delta_dc for sharing it with me.

 

Notice her use of Audacity? I think we could quite easily turn this into a lesson plan… perhaps related to fractions, or to sine and cosine waves.

Math in the real world: Roller coasters

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

 

When my son and I were on the roller coaster, I was again in awe about how quickly even a small roller coaster like this travels, and how it doesn’t drive right off the tracks.

Roller coasters have to be constructed fairly carefully, and follow some mathematical rules in their construction. They need to first be concerned about how to make the roller coaster safe. They need to calculate exactly how fast it will travel through the loops and turns, and how much of an angle they will need to prevent the roller coaster from taking a dive during those turns. They need to watch out that they don’t cause the participants of the roller coaster to pass out during a turn as they experience additional forces on their bodies!

The various costs associated with a roller coaster need to be calculated as well. There’s the cost to build, maintain, and operate the roller coaster. There’s an additional cost to pay for insurance for the roller coaster, which means an actuary needs to examine the probability of a problem occurring for any given roller coaster. The operator of the roller coaster needs to determine, given the cost to operate the roller coaster, etc… what they should charge to make a return on their investment, and attempt to maximize their profits.

While you could use a roller coaster simulator to explore some of this math, it’s a lot more fun to experience it in person…

Math in the real world: Leaning Plants

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

 

View all pictures

 

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.

Graph of points - parabola regression model

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?

 

Math in the real world: Relationships

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

5 generations of women

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.

Disfunctional family

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?

RSCON3 presentation online

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.