Ordering Games

Ordering games involve estimation and comparison, two essential skills for mathematics.

Here are some ordering games I gave to groups of students at Greenwich High School.  Teams raced to arrange the slips of paper from smallest to largest.  (No calculator)

Some groups were very enthusiastic about this game; others not so much.  In the future it might help to do one “practice” round with numbers like 1,2,3,4,5, so that each group could get a feel for what they are trying to do and what a successful ordering looks like.

Each ordering game below is geared to a different skill level.  The cards are not necessarily in order as pictured.

Focus on numbers and exponents:

Focus on shapes and area:

A little bit of trig:

Finding areas enclosed by curves.  I gave this to students in a calculus class, but it could also be great for eager students who haven’t learned calculus yet.  It can be completed without any calculus.

I feel I have only scratched the surface of what can be done with ordering games.  It would be cool to create decks of cards (several different decks) like this.  You’d need an ordering game of 13 items.  The concept of suits (clubs, diamonds, hearts, spades) would be preserved.  Just as in a normal deck, there would be 4 copies of every number.  However, math skills would be required to sort the cards.  For instance, instead of relationships like “a jack is higher than a 10” you’d have relationships like “a card with log(11) is higher than a card with sin(89 degrees).”  Students could then play classic card games with each other, using the mathematical card decks.  They’d have to swap out decks periodically to avoid getting too familiar with one deck and memorizing the orderings.

Eventually students could design decks for each other.  This would involve thinking of 13 numbers that are obscure enough to make people think, but “easy” enough that no laborious calculations would be needed to compare them to one another.

#MathProblemOfTheDay: Skateparks and Concavity

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Skateparks are a great place to observe concavity.

A curve is concave up if its slope is increasing. This could mean the curve has a positive slope and it’s getting steeper, or it has a negative slope and it’s getting flatter. In both cases, we say the slope is increasing because it’s getting more positive. (Think about how -10 –> -7 is an increase just the way 7 –> 10 is.)

A curve is concave down if the opposite is true: the slope is decreasing. Either it has a positive slope and it’s getting flatter…. or it has a negative slope and it’s getting steeper.

Visually, you can think of a positive slope as an uphill, and a negative slope as a downhill.

A point of inflection is where the concavity changes from down to up, or from up to down.

Where is the point of inflection on this skateboard ramp?

P.S.  What would be the concavity of a straight line?

#MathProblemOfTheDay: keeping warm in the snow

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Which position would keep you warmer?

The answer is probably intuitive to you. That’s why I love this image – it connects simplicity and intuition to a more complex topic: optimization.

Optimization is an example of applied calculus. It sounds all fancy but it can be expressed simply. You want to optimize one thing (in other words, maximize something you want, or minimize something you don’t want), while keeping another thing constant (this is called a constraint).

In this photo, I want to minimize surface area to keep warm. (Why will minimizing surface area keep me warm??) My constraint is my volume – I can’t change the size of my body. It stays constant no matter how I position myself.

Ok, now imagine we’re no longer dealing with the human body. Take a piece of clay. What would you mold it into, in order to minimize surface area? How do you know you’ve succeeded?