When are we gonna use this?

I fear that far too often we have math instruction turned around backwards. We teach concepts and skills as an end unto themselves, when really these are tools to solve problems. They were human inventions created to understand the world around us and to be able to articulate that understanding to others. When math concepts are presented within the context of a problem, students are much more eager to engage in deep thinking. Furthermore, they will invent the math necessary to solve the problem if given the time and resources to do so.

A few weeks ago, as the Space Shuttle Endeavour was being prepared to move from the Vehicle Assembly Building (VAB) to Launch Pad 39A, I couldn't help but present an intriguing problem to my 5th grade math students: How fast does the crawler move as it transports the shuttle "stack" to the launch pad?

This particular math group is in a rational numbers unit. In this unit, rational numbers, in all their forms, are explored as different ways to notate the same ideas, whether using any non-zero denominator (fractions), denominators that are powers of ten (decimals), or denominators of one hundred (percents). They are comfortable with these concepts and understand that, operationally, if they can work with the number in one form, they have to be able to work with it in another form. When this particular problem was presented, they had not yet encountered divisors that were not whole numbers, though they had explored decimals in the quotient and have moved from expressing remainders as R# to R/div to a decimal expansion. They have also worked with non-terminating decimal expansions and know how to use a vinculum to express these ratios.

The problem was not presented as, ok, let's divide with decimals. Rather, it came in the midst of our following the final three flights of  the NASA shuttle program. As Endeavour was moved from the VAB to launch pad, the question came from them.

Students: "How far is the launch pad?"
MW: "A few miles."
Students: "How fast does the crawler move?"
MW: "Not very fast."
Students: "Why?"
MW: "It's extremely heavy, and vibrations could tear things apart."
Students: "So HOW fast?"
MW: "Let's see if we can figure it out?"

We then set out to collect the data. From a couple of NASA Tweets, we found that Endeavour started its rollout at 7:56 p.m. and arrived at the launch pad at 3:49 a.m. We also found that it was 3.4 miles between the VAB and pad 39A. We talked about the fact that speed was in miles per hour, meters per second, kilometers per hour, feet per second, etc. In all cases a unit of distance over a unit of time. Aha! A ratio! They immediately recognized they needed to find how much time had elapsed. They also recognized that the time was not in a decimal form. Hmmm. The first subtraction was start time minus end time. Well that doesn't make sense... Ah, there's that pesky midnight in the middle of it all. They decided to use what we term "subtraction by adding up" to determine the elapsed time. 7:56 to 8:00, 8:00 to 12:00, 12:00 to 3:00, 3:00 to 3:49. 0:04 + 4:00 + 3:00 + 0:49. Now the task of turning a hr:min into a decimal. Equivalent fractions! Eureka! n/60 = x/1000.

Now we have the time as a decimal and the distance as a decimal. Set up the ratio 3.4/7.883. Ew! Decimal numerators and denominators. Equivalent fractions to rescue again! The students already know that the decimal moves when multiplied or divided by powers of ten. So by multiplying both numerator and denominator by 1000, they have two whole numbers in their ratio. Brilliant! They are also familiar with n/d = n ÷ d. With a couple of reminders of these basics, they quickly set up the problem and find that the crawler moves at a rate of approximately...

I'll leave it for you to finish up.

This was an exciting math period where my students identified the math that they needed to solve the problem. Some of which they had at their ready disposal, some of which needed to be discovered. At the end of it, they now had a model for how to work with these numbers that could be translated to a wide variety of situations. Because they now OWN the methods, they are able to hold onto the concepts and reconstruct the math as needed.

Fractions & Probability

This comes by way of @NCTM daily problem tweets. It is a great investigation for probability. By constructing outcome grids to determine the all the possible outcomes vs all the desirable outcomes for each of the situations, students can then compare fractions using benchmarks and concrete materials or drawings to determine which is more probable.


When three 6-sided fair dice are rolled, which type of sum is more likely: a multiple of 3; one more than a multiple of 3; or two more than a multiple of 3?

Fractions

Fractions are among the most abstract concepts presented in elementary school math. We need to ensure that a sound conceptual base is established. This means taking time and providing numerous, concrete experiences for children to construct an understanding of these concepts before every formalizing them.

Here's an example of work adding fractions in a 5th grade classroom.

Students were asked to find the sum of the fractions 1/4 + 1/2 + 2/8. Using materials, they were able to find that these three fractions had a sum of 1 whole.

Students also noticed that they could trade in the fourth in for two eighths and the half in for four eighths, leaving them with 8/8, another form of 1 whole. Likewise, they could trade the eighths and half in for fourths, or trade the fourth and eighths in to make a half. Thus proving that it was equivalent to 2/2, 4/4, or 8/8... all equivalents of 1.

Without ever thinking about "common denominators" the students were dealing with the concept. 

In a more challenging problem, they were asked to think about the sum of 4/5, 7/8, and 2/3. Without worrying about the exact sum, they reasoned that 2 < sum < 3. From earlier explorations, they already know that when a fraction was missing a unit fraction, it was as close to 1 as possible. Thus, three fractions close to one added together, must get close to three. It would have to be close to two because each of the fractions was at least 1/2.

Once assembled, children were easily able to see that these three fractions had a sum of about 2 1/3.

Powerful reasoning, with no need for "common denominators."

Yes, they will be exposed to the algorithm for adding fractions, but they will first construct it through many more investigations such as these. In the last example above, the student noted, "I could find the answer by thinking about 3 - 1/3 - 1/5 - 1/8. That's like taking away about 2/3 from three."

Female Math Stereotypes

This article is more than a year old, nothing new, but here is some research to back up what many of us who teach math have long known. Teaching math is different than teaching other subjects. One has to really love it and have a deep understanding and appreciation for the subject in order to be effective in the classroom. 

Believing Stereotype Undermines Girls' Math Performance: Elementary School Women Teachers Transfer Their Fear of Doing Math to Girls, Study Finds

ScienceDaily (Jan. 26, 2010) — Female elementary school teachers who are anxious about math pass on to female students the stereotype that boys, not girls, are good at math. Girls who endorse this belief then do worse at math, research at the University of Chicago shows.
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Happy New Year

2011 is a prime number year.
What's more, it's the sum of eleven consecutive primes!
2011=157+163+167+173+179+181+191+193+197+199+211

Football Math

So I'm sitting here at Giants Stadium, waiting for my Redskins to take the field. Thinking about the numbers in football. Like all sports it's full of statistics.
Football scores are quite complex. Six points for touch down. One point for the extra point after a touch down. Or, you could run the ball in after a touchdown for two points. Then of course there is the three points for a field goal. Finally, the saftey for two points. 6, 3, 2, 2, 1.
So I'm thinking, what kinds of numbers can the final score be? Could it be prime? What about square or cubic? What would the score look like written in Binary?
Well, I suspected the Redskins would lose. They have a lot of players out with injuries in key places and the quarterback has been struggling. That, with the fact that the Giants have been doing very well lately.
So the final score was 7-31. Both prime numbers. How many different ways could these scores have been achieved using the posibilities given above?

Binary

So boil it all down to one and zero and you have the simplest possible system. Two digits. Yes or no.

The Indian scholar Pingala is credited with the first known use of a binary system. He used a system of dashes, a short dash and a long dash to represent 1 and 0. The Chinese used hexagrams to represent numbers. These are squares made of a stack of 6 lines that are either solid or broken in the middle to represent 1 and 0. Then finally in 1703, Leibnitz uses the 1 and 0 digits as we know them and formalized the binary system as a practical device for computation.