This is Maura, teaching for Ethan today. The objective was to cover absolute and relative change. This is one of the few times when we give a nice formula in a box. The basic point:
absolute change = new value – old value
relative change = absolute change/old value
When we want to describe how something has changed (usually over time), we have some choices: we can use simple subtraction to make a comparison, or we can put it in a context by finding the relative change, which we then usually write as a percentage. The second approach – finding relative change – often gives us more information.
In class today we looked first at the population of the United States. Our first question was “what is the current population?” We experimented with a Google search, which gave a not-too-helpful graph, and we also went to the U.S. Census page on population. Our answer was that the population is about 310 million. Writing “about” allows us to round – all we’re interested in is a ballpark figure – and using “million” means that we don’t have to write all those zeros.
Then we compared the U.S. population in 1990 with the U.S. population in 2000. In 2000, the U.S. population was 281.4 million; in 1990 it was 248.7 million. The easy way to describe this growth is using absolute change: there was an increase of 32.7 million people in that time period. By looking at relative change, we had a context for this growth: it was about 13%. Then we looked at part of publication from the Census Bureau (http://www.census.gov/prod/2001pubs/c2kbr01-2.pdf) which included a graph of absolute and relative change in decennial censuses since 1950. The text on the first page stated that “The 1990 to 2000 population increase was the largest in American history.” That’s true if we use absolute population increase. If we use relative change, it turns out that the jump in population from 1950 to 1960 was the largest. So this was a nice example of how relative (or percentage) change can give really different information from absolute change. Here’s the page we looked at:
I felt that the class was following until one student asked, “what question are we trying to answer?” Great question, and it helped bring me back to what we were trying to do. The last sentence in this Census page is complicated. It talks about the decreasing population growth from 1950 to 1960. I was trying to get the class to think about how a growth can be decreasing. Doesn’t that seem like a contradiction? That student’s question pushed me to focus a bit more, and also to slow down and make sure the class knew what I was looking at. Not sure it quite worked, but I plunged on. I asked: “has the population of the U.S. increased or decreased since 1950?” We only had this graph, which shows population change. Some students said yes, it has increased; others said “no – the graph shows that the bars are getting smaller.” But it’s important here is that you pay attention to the columns of the graph and what they represent. The columns represent growth, and as the decades march on the growth decreases – until we get to 1990.
This took a good two-thirds of the class time. In the remaining time, we looked at an article from yesterday’s Boston Globe about arson. The link is: http://www.boston.com/news/local/massachusetts/articles/2010/09/08/scientists_challenge_massachusetts_arson_convictions/
We skimmed the article – there is not much that is quantitative in it but there is a good graph. The students identified an important quantitative piece: that the number of structural fires ruled arson fell by about 70% from 1984 to 2001. We verified that and realized that in this case we get a negative absolute change, and a negative relative change, which makes sense if the number decreased. The article discusses possible reasons for this decrease. Perhaps, as the fire marshals claim, it’s due to better education and awareness. But the scientists interviewed claimed that the science of determining when a fire is caused by arson has changed, and that the techniques are now more refined. This is what has led, they claim, to the change in the number of fires declared arson. We had a brief discussion of this point – that when quantities change, we tend to look for a reason. In this case, different groups (with different interests) are promoting different reasons.
We calculated the change in the number of structural fires (quite small – around -3%) and then as a last exercise verified the calculation that in 1984, arsons were 20% of structural fires. It’s always good to try out new ideas in contexts where you know what the answer should be. This gives the freedom to try different approaches, and it was good to see a couple of different approaches to verifying the 20% figure.
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