Monday, July 30, 2007

Percent Word Problems Revisited -- Making It Easier?

Since there has been some discussion about the complications in the previous post on percents, how do we as educators deal with adversity and transform it into a teachable moment. We're explaining a difficult problem that students are struggling with, so we try to explain it again, but to no avail. What do experienced educators do?
(A) Abandon the problem - it was simply too hard or they're not ready for it yet. Perhaps assign it for extra credit?
(B) Make the question simpler by removing some of the complexity. Consider a scaffolding approach?
(C) Re-think what prerequisite skills were needed?

In this case, let's re-work the problem as follows:

In the senior class, there are 20% more girls than boys. If there are 180 girls, how many more girls than boys are there among the seniors?

What do you think the results will be now that we've concretized the problem?
Do you think some students will make the classic error of taking 20% of 180 and subtract to obtain 144 boys? You betcha!! There's no getting around the issue of recognizing the correct BASE for the 20% in my opinion. Whether you consider the boys to be 100% and the girls to be 120% or you let x = number of boys and 1.2x = number of girls, the central issue is recognizing that you cannot take 20% of the girls! Sometimes students need to just use algebra as a tool - it's a great one! By the way, one student used algebra with ratios to solve this:
Boys = x
Girls = 1.2x
Difference = 0.2x
Therefore, the difference/girls = 0.2x/1.2x = 1/6 and 180 divided by 6 equals 30!

How would you have reworded the question to make it more accessible?


Jackie Ballarini said...

Hmm, to make the new question more accessible for a student I might ask the following:

Will the number of boys be more or less than the number of girls?

If there are 180 girls, does 10 boys meet the constraints? 170? 20? 160...? And for each asking... how do you know?

For the original question (before the number of girls was added), I might first ask:

If there are 100 boys, how many girls are there?

...50 boys, 20 boys, 150 boys...

Then after they were comfortable with this, move to giving only the number of girls, then to only the total number of students. Hopefully we could then return to the original problem with no numbers given.

Then I'd give them an entirely "new" problem: If the number of adults is 25% more than the number of students, what percent of the people are adults?

My hope of course would be that I wouldn't need to provide this level of scaffolding.

Anonymous said...

"There's no getting around the issue of recognizing the correct BASE for the 20%..."

Our pre-algebra class just finished a chapter on percents, so I thought my son might have a chance at your first percent question. Nope! He jumped without thought to the conclusion that 60% of the class must be girls. After I explained the significance of the word “than”, he solved this follow-up problem just fine.

I used to be surprised at how often the trouble with word problems is in understanding the English, not the math. Awhile back, I took a detailed look at the trouble of identifying the correct base in percent problems:
The Search for 100%
By the time I had finished pulling together the list in that post, I was no longer surprised to see students flub percent problems. What a mess our language can be!

Unknown said...

Thanks for these posts! Very enlightening. As you might guess from my comment on the previous post, I'm going with choice (C).

Dave Marain said...

Thanks, jackie, Denise, Mr. Person--
Your comments are compelling and this was the kind of reaction I was hoping for. Somehow we need to help our students come to grips with the varieties of semantics used in percent problems. Some teachers still use 'IS OVER OF" - I always poll my classes to check this. I learned that way but this method is limited when one encounters the more sophisticated examples or even discounts where the "OF" is missing (then of course we teach how to do % discounts as a separate model!).

Mr. Person's comments struck home in particular. % increase, % decrease, % more, % less, % change always involve a difference divided by something. The 'something' is the "ORIGINAL AMT" in % increase-decrease problems" but it's usually the quantity that comes after "THAN" in % more-less" problems. The key is again understanding 'what' is being compared to 'what' since every percent problem is nothing more than a ratio problem. The bottom line is that students need to do MANY MANY different types to gain facility. The readers of this blog should get together and develop a unit on percent word problems with abundant examples, instructional strategies, the whole 9 yards! Unless of course you have found the 'perfect' resource! Game?

Unknown said...

What else is interesting is that students have trouble determining the difference between the actual percent increase and the result of that increasing. I got a lot of answers to a percent increase problem where the answer was 60%, but students wrote that the *increase* was 160%. There's a subtle English language difference there that I just can't seem to make stick.

Dave Marain said...

I've seen that error many times! I try to explain it with money. If your original salary was $100, after a 60% increase, your raise would be $60 but your new salary would be $160. The problem is that students seem to forget that the original salary is 100%! Thus the new salary is 160% OF THE ORIGINAL but THE INCREASE IS
160% - 100% OR 60%.
If we can somehow get the message across that your starting value is always 100%, this might get through, but the wording of the question (and their comprehension) is also critical.
If the question asks:
Your new salary is what percent OF the original, then it's 160% (not commonly asked for).
If the question asks:
Your new salary is what % MORE than your original, then it's 60%.
% MORE and % INCREASE always involve subtraction, if that helps.
Hey, no matter what we try there's always a learning curve that varies from student to student. Good luck!

Jackie Ballarini said...

IS over OF? I do not use this for the same reasons you stated above Dave.

I haven't found a perfect resource. So, unless anyone else has, you can count me in on the unit development.

Anonymous said...

"The readers of this blog should get together and develop a unit on percent word problems with abundant examples, instructional strategies, the whole 9 yards!"

And a list of prerequisite skills, too. I would love to see a unit like this! Singapore math has a lot of good material, but it is scattered over several books. For example, when we come back together this fall, I'm going to give my Math Counts students Singapore NEM2's true-false quiz, with questions like:

"If A is 5% more than B, then B is 5% less than A.
If A increases by 8% and the decreases by 8%, the result is equal to A.
A increases in the ratio 5:4 means A increases by 25%.
If the percentage profit is a%, then the cost price is (100-a)% of the selling price."

These kids are just starting algebra, so all those A's and B's will confuse them a bit. But I am hoping this will lead to some interesting class discussion.

Doctor Pion said...

The simplest version, which would test the prerequisite knowledge, would be to ask the same question with the number of BOYS being given. No algebra, just "percent more." Then you can go on to the algebra problem.