I like to start this lesson by comparing the Japanese and US national debt, only I leave out the units and just provide the numbers and question:
Which number is larger?
Currently the US debt is about 16.738 trillion (see resource link) and the Japanese National debt is at 1 quadrillion. I would write the numbers in standard form next to each other and leave space between them for the inequality sign.
The goal here is to allow students to show their understanding of very large numbers. They can compare them using scientific notation and use reasoning around the powers of 10 to show that one number is larger than the other, but they can use any method at their disposal. For example, they might simply divide the larger number by the smaller and show and reason that since the quotient is larger than 1, the first number is larger than the second (quite a mouthful).
The question is open ended and meant to encourage a variety of responses. I circulate and look for errors in reasoning. I share these (anonymous) mistakes with the class and ask for feedback. The goal is to make sure that each student is able to make mistakes and then reflect upon them.
Perhaps the best part of this discussion is the US debt > Japanese National debt (because of the currency rate). I will not reveal this right away, but ask the class to think of scenarios in which the smaller number could in fact be a larger amount. I will encourage them to provide simple digestible examples, but stress that the context is key. For example, 2 dollars > 100 pennies and 2 miles >8,000 feet. I am simply asking them to imagine how the context and units could change their answer.
After we discuss the article blanks, I would return to the start up.
"So which number is larger?"
Again I would return to their previous arguments (usually quoted on the board) and then deepen the process by naming each number. Once we agree that a quadrillion is certainly larger in the abstract sense, I add context by writing the units next to each number:
1,000,000,000,000,000 JPY 16,738,000,000,000 USD
"What do you think these units mean?"
After a few guesses, "what if I told you that these units have to do with money from two different dollars?"
After we discuss that JPY = Japanese Yen and USD = United States Dollar, I would ask the questions again, "which number is bigger?" This time I would ask them "why might these units change your answer?" After we discuss the idea that not all currencies are equal, I would ask again them "what do you need to solve this question?" The wonderful thing is that I know I am running the conversation right if they ask for the right into. They should ask for "conversion rates" or some version of this statement. I follow by giving the rates and then make sure they use those rates to critique the last line of the article: "Compared with Japan, the United States national debt is a mere $16 trillion or so. But if you convert that number into yen, it comes to about 1.5 quadrillion. So it’s good to have a currency that conserves its zeros. Of course, that also means the total American debt is even larger than Japan’s (though not, it should be noted, as a percentage of gross domestic product)."
To start the investigation, I pass out the article by Johnathan Schwartz in the New York Times and blank out many key numbers and comparisons. I usually number the blanks for review at the end. For example, in one line he writes, "A paltry million is the numeral one followed by six zeros. A billion? Nine zeros. A trillion is getting up there: 12 zeros. But the mighty quadrillion has 15 of them."
I would blank out a few key points here and label them by putting numbers in the blank: "A paltry million is the numeral one followed by 1 zeros. A 2 ? Nine zeros. A trillion is getting up there: 3 zeros. But the mighty quadrillion has 15 of them." This allows me to ask them, "what did you get for blank 1?"
The students have 5 minutes to review the article and make annotations. They write out their thoughts and questions and prepare for discussion. I meet with our humanities' teachers and make sure the instructions I give match their instructions surrouning non-fiction reading. Our students may not be familiar with reading in math, but they are very familiar with reading. I don't need to reinvent their teaching in reading, but I need to use the same language to tap into the work from humanities.
Their job is to eventually fill in all the blanks and justify their reasoning. I also ask them to extend certain blanks with their own comparisons. For example, in the article Schwartz writes, "Neil deGrasse Tyson, the astrophysicist and director of the Hayden Planetarium at the American Museum of Natural History, helpfully offered a few other ways to think about a quadrillion. It would take you 31 million years to count to a quadrillion — one number per second, never sleeping,” he said in an e-mail, adding that “a quadrillion yen, stacked in 1,000-yen notes, would ascend 70,000 miles high.”
I would ask them to make their own comparison, like how high would 1 quadrillion pennies stack? We can plug these amazing measurements into a search engine and see how this measurement corresponds to other things in the universe. We can ask questions like, "could 1 quadrillion pencils reach the sun?"
As a real challenge, I might ask tem to respond to this amazing estimation: "He also wrote, though it is not clear how he would know such a thing, that “the total number of all sounds and words ever uttered by all humans who have ever lived is about 100 quadrillion.” I would provide them with speaking rates and estimations of human existence to verify this estimation.
The article is full of teaching opportunities that surround the math practice standards. You will find student reasoning and modeling with large numbers in order to argue and critique the reasoning of others. It can be a really exciting lesson.
In this part of the lesson, I place the article on the project and discuss responses to each of the blanks in the article. I like to test their reasoning here and deepen their thinking around their responses.
For example, the line about time is classic: "“It would take you 31 million years to count to a quadrillion — one number per second, never sleeping." I would leave the "31 million blank" and ask them to verify how we know that makes sense, but I would follow up by asking how long is a million seconds, a billion seconds, a trillion seconds, etc? The amount of time taken by each new number group reflects the amazing distance between values when one number is 1000 times larger than another. I usually list this out in a table and always manage to astonish students as we jump from the amount of days in a milion seconds and then the amount of years in a billion seconds.
I try and cover all the aspects of the article but make sure I take time to discuss their findings on comparing the JPY and USD. To supplement the discussion and talk a bit about currency, I would show the interactive graphs on Bloomberg Market Data. I like to use the trend of currency over the year, scroll through the rises and falls in value and ask questions around this data.
"What does each axis represent?"
"When the line goes up what does that mean?"
"Why would it fluctuate so much?"
"What time of year would you have used USD currency to by JPY?"
"What time of year would you have used JPY currency to by USD?"
In this discussion I hope to get at more than just the algorithm for converting, but also the reason that estimation is a key element in converting currency.