In my last blog on the Migrant Crisis I asked students to complete the following analogy:
A penny is to $10,000 as _________ is to $1 trillion. The value you just put in the blank is like a penny to the U.S. government’s budget.
This example illustrates the importance of proportional reasoning, starting with ratios and highlighting the importance of units as they relate to conversions and scales. It is a perfect segue to Chapter 3 in my course on Units, Conversions, Scales, and Rates; and I used this example to engage my students in connecting the ideas from Chapter 2 on Ratios and Proportions to the new material. We start with a simple ratio,
1¢ : $10,000
comparing one cent to $10,000, which for most of us is a sum that resonates in terms of our salaries and budgets. We can scale this ratio and represent it in many different ways:
This scaling of the ratio is not to be confused with proportionality, we are simply representing the ratio in different ways using the idea of equivalent fractions. At this point our students are in their comfort zone, set up a proportion and cross multiply, but they have to grapple with the number word “trillion”. This derails many of them, and is something we need to reinforce with examples like this one. Solving for the unknown in the proportion gives the number 100,000,000. And many of our students are happy to put this down as the answer, ignoring the units entirely.
We discuss that the units of this number will be cents, just like in the different ratios shown above. Now we have to convert the 100,000,000 cents into dollars. To do this we need the conversion factor, 100¢ : $1. There are now two ways to proceed.
- Set up a proportion.
Solving for the unknown will give 1,000,000 and once again we have to use the proper units from the proportion, $1 million.
- Use dimensional analysis and cancel units.
Where the units, ¢, cancel on top and bottom.
So now we have solved the problem, one penny is to us as $1 million is to the federal government (their budget is in trillions). Thus we have created a scale:
1¢ : $1 million.
A scale allows us to build a model (mental in this case) of a situation that is too large for us to grapple with, in this case the federal budget. None of us can truly comprehend the scale of the federal budget which is measured in trillions of dollars. But our scale allows us to wrap our minds around these gargantuan numbers. If someone asked you to borrow a quarter, would you expect to be paid back? A quarter to us is $25 million to the federal government!
25¢ : $25 million.
Many of us would buy a friend a cup of coffee for $1.50 without thinking twice about it, that’s $150 million to a congresswoman. If you see a penny do you pick it up? Many of my students would not (except those who have heard the refrain: “find a penny pick it up and all day long have good luck”). If a senator sees a $1 million lying around in a bill she may not “pick it up” either and just let it go to some pork barrel project if it means the bill will pass. Or she may buy a friend a “cup of coffee” for $150 million :O)