MATH THAT MAKES ENTS

Transcription

1 Since 1984, PNC Bank has been tracking the annual Christmas Price Index: how much it would cost to buy all of the gifts in the song, The 12 Days of Christmas. If you d like to learn more about how they calculate the price of, say, 10 Lords a Leaping, visit com. YEAR PRICE 199 $13, $14, $14, $15, $15, $14, $16, $1, $18, $18, $19, $21, $21, $23, $24, $25, Question: In which of these years was there the largest percentage increase in over the previous year s price? 1998? 2006? 2011? 2012? Extra: Consider all of the changes since 1998 instead of just the four listed above. Which was the largest percentage change? MATH STANDARDS ALIGNMENT Grade : Ratios & Proportional Relationships CCSS.Math.Content..RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 1

2 Personal Finance Big Ideas: Inflation, What is Money, Time Value of Money METHOD 1: LOGICAL REASONING First I took the important information out of the chart, since we don t need the whole chart. We only have to look at five years. I made a list of them and the increases on those dates. 1998: $13, > $14, Difference: $ : $18,348.8 > $18, Difference: $ : $23, > $24, Difference: $ : $24, > $25, Difference: $ After I listed the old price and the new price for the year, I also subtracted to find the difference in the prices. I used a calculator to help with the subtracting because I didn t want to make a little mistake and get the final answer wrong. Percentage increase is calculated by dividing the increase by the original price. For example, if something cost $1.00 and the price went up to $1.25, the percentage increase would be 25%, since 25 is 25% of 100. I thought about how I would calculate the percentage increase for each of the 4 increases. 1998: out of 13, : 51.2 out of 18, : out of 23, : 1168 out of 24, I m trying to figure out which of those would be the biggest percent. I can already rule out the middle two. We re dividing smaller numbers by bigger numbers, so neither 2006 nor 2011 can be a bigger percent change than The contest is between 1998, when we divide a smaller number by a smaller number, and 2012 when we divide a bigger number by a bigger number. 1998: , = = 6.5% 2012: 1168 out of 24, = = 4.8% The largest percent increase of the four target years was the first one, Extra: We could calculate all percentage increases, but that s tedious. (I could, though, type the info into a spreadsheet and let the computer do the work, which would be less tedious.) Instead, I think I can apply the thinking I used to the first part of the problem looking for big changes with smaller original prices. First I wrote down the differences for all of the price increases (actually, I had the computer do it). See the table on the next page. Then I looked for which differences were likely to yield the biggest percent increase (because they were big differences paired with smaller original prices). I noticed that the biggest change was from 2002 to 2003, when the price increased by over $2000. The only time it was even close to that was from 2009 to 2010, when the price increased by $193. But when the price increased by over $2000, it was also associated with one of the smallest original prices in the table, since 2002 s price was the third smallest. So since 2002 to 2003 had the largest increase compared with one of the smallest original prices, I m convinced 2002 to 2003 was the largest percent increase ever. 2

3 YEAR PRICE DIFFERENCE 199 $13, $14, $ $14,940.1 $ $15, $ $15,48.81 $ $14, $1, $16, $2, $1, $ $18,348.8 $1, $18, $ $19,50.25 $ $21, $1, $21, $ $23, $1, $24, $ $25, $1, METHOD 2: USE A FORMULA Some students might find a formula for percentage increase that looks something like this: new-original original 100 = percentage increase We re not going to include an example of such a solution here. It s fine to use such a formula, but it s worth looking closely at why it works and how it s related to the work done in the solution shown above. It might be a good opportunity to remind students that we don t really need formulas to solve most math problems, but that knowing them offers us shortcuts to reasoning things out every time. Formulas can sometimes obscure the logic behind a particular concept. This is a good example of that. Using the formula is simply a mechanical process. Reasoning out the answer helps us see why the answer actually makes sense, and in a problem like this, can actually save time. 3

4 4

5 5

6 6