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."
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.