Friday, April 3, 2009

Two SAT-Type Percent Problems Appropriate for Middle School as well...

Version I (Level of difficulty 3 - medium)

With a special promotion, Al received a 60% discount on a new stereo system and paid $x. Sylvia bought the same system (same original price) but only received a 20% discount. In terms of x, how much (in dollars), did Sylvia pay? Assume x > 0 and disregard sales tax.

(A) 4x (B) 3x (C) 2x (D) 4x/3 (E) x/3

Version II (Level of Difficulty 4 - medium/hard)
Grid-In Type

Maury purchased a new electronic game system with a 25% off coupon. His friend bought the same system (same original price) with a 40% off coupon. If his friend paid $45 less for the system, how much did Maury pay (disregard sales tax)?

For the answers, suggested solutions, strategies and discussion, click Read more...

Level I problem
Answer: (C) 2x

Possible Solutions:
Method I ("Plug-in" SAT Strategy - Student-preferred?)
Let original price = $100.
Then Al's discount was $60, so he paid $40. Thus x = 40.
Sylvia's discount was $20, so she paid $80, which means she paid twice Al's price or 2x.

Method II (conceptual)

Al paid 40% of the original price, Sylvia paid 80%, therefore Sylvia paid twice as much as Al.

Method III (traditional - Algebraic)
Reasoning as in Method I, Al paid 40% and Sylvia paid 80% of the original price.
Let y = original price (before discounts).
Then Al paid 0.4y = x. Solving, y = 2.5x.
Sylvia paid (0.8)(2.5x) = 2x.

Level II Problem
Answer: $225
Note: There is a mental math method which will be discussed later.


  • What are some of the factors which might make the second problem more difficult?
  • Should these types of percent questions be included in prealgebra texts for middle schoolers?
  • The first question appears to be fairly straightforward. Which part of the problem would cause the most difficulty for students in your opinion?
  • Note that in the ever-popular 'plug in a number method', the ever-popular $100 was chosen for the original price. Can you think of an occasion where you might not want to substitute $100 in a discount problem?
  • Students also need to recognize that I plugged in $100 for the original price rather than replace x by $100. They often feel they must substitute a number for the variable given, which, in this case would lead to more work.
  • There are other approaches here. Please share different methods you have used!
  • How would we assess their learning of the concepts here? Include one of these on the next quiz or test? Use these as warmups occasionally? Develop a worksheet of similar problems for homework?


Anonymous said...

For version II, a 15% difference was worth $45, 1% = $3, original price was $300, 3/4 of that is $225.

what would a kid do?

.75x - .6x = 45?

it's not so bad, but I don't know how easy it is to write.

Knowing that 40% off means 60% of the price, I bet that that's a huge sticking point.


Dave Marain said...

You nailed the common difficulty students have with the Version I problem! For Version II, there is a part:part ratio approach that is underemphasized IMO:

Starting as you did--
15% of the total equals $45.
Instead of determining the total,
compare (i.e., make a ratio) the "difference" of 15% to what Maury paid which is 75%.
15:75 = 1:5 so Maury paid $45 x 5 = $225.

This is what I meant by a part:part approach instead of part:whole. How many secondary students, never mind middle schoolers, would approach it this way in your opinion?

Actually the simpler version I is based on exactly this principle! One person paid 40% of the original price, the other paid 80% of the original price so the part:part ratio is 1:2 or x:2x.

I've posted other questions like this in the past (see "percent word problems" and "ratios" in the index) but my instinct tells me that our students need lots of reinforcement in this area.

Curmudgeon said...

One of the considerations for the difficulty rating is the availability of the answers - and this isn't always helpful.

Had $300 been one of the answers given, this question would have been just as "hard" as the grid-in version, since many kids would have gravitated to that answer.

I tell my kids all the time, "The difficulty of an SAT question is in small part objective difficulty, part obscurity of the !AHA! thought, and part the ready availability of the first number you'd logically get in solving it. Of course, you want the second number, which is the answer to the question. Always re-read the question! Remember, level and placement in the test is based on the percent who got it right, not on the difficulty."

D.L. in this case would be a 3 if $300 were not in the list because you'd get $300 as the original cost but not find it in the A-E and thus be "prompted" to go looking for the question and then the real answer. D.L. would be a 4 if $300 were there.

D.L. for this as a grid-in is 4 because there are so many places to make an error that most kids will get it wrong under time pressure, not because the problem is all that difficult.

For the record, I did the same mental 15%=$45, so total=$300 and M=$225 thing that Jonathan did. In the first, I used $100 and compared the two numbers.

Dave Marain said...

Excellent points, Curmudgeon. Your students are fortunate indeed to have a teacher who is so knowledgeable about the subtleties of this kind of standardized test.

There is no question that the distractors among the choices are a significant factor in the difficulty level of the problem and your comments captured the essence of that.

Another type of SAT problem doesn't require that Aha! insight necessarily nor is the difficulty coming so much from the choices. The difficulty instead comes from complexity of the number of steps required to solve it. A typical example of this is:

Find the perimeter of a square whose area is the same as that of a circle whose circumference is 6.

This type of question absolutely has appeared on the test with this much complexity. I should post that one as a separate article!

A couple of more points...

(1) I'm still hoping that someone will comment on the underlying mathematics here, particularly my thoughts about the part:part approach which I rarely see stressed in texts or classrooms.

(2) I'm sure you realize that my intent in publishing posts about SAT-type problems goes far beyond that test. I spend far more time developing the question than the choices which is why it's easier for me to give a "grid-in" example. Most commenters in the past have focused more on my selection of choices (distractors) than the problem itself!

(3) You are aware of something that most are not. The difficulty ratings of questions published by the College Board are determined by actual student performance on that question when it was given in an experimental section (i.e., "field-tested"). I do not have the luxury of such field testing so my difficulty rating are highly subjective and open to discussion and criticism!

Again, thank you for your insightful comments. Have you considering registering for my upcoming contest?

Curmudgeon said...

Good Afternoon

You said "I do not have the luxury of such field testing so my difficulty rating are highly subjective and open to discussion and criticism!"

Au contraire. You have more than one section of math, or at least you can use them again next year with the next crop of students. It's just a data-recording problem.

You give them to one class or yearly cohort and start building your difficulty level from that group of students. A hard (level 5) MC question is one that fewer than 15% get right; 4 ~ 15% - 35%; 3~ 35%-55%; 2~ 55%-75%; 1~75%-95%; if more than 95% get it right, throw it out.

As with the real test questions, actual difficulty might have nothing to do with it. It might be a poorly worded question, or too many steps, or just that few bothered to try it and of those, only a part got it right.

If you pose it as a grid-in question and record the most logical wrong answers that your students give, you'll have your multiple choice distractors for the next batch of students, too.

Finally, with respect to the part:part solution ...
I can't really see any kids using that approach on the first go-round on this type of problem. After doing a bunch of them, I would mention it and expect maybe a third to grasp it as being a shortcut or a new idea and the other two-thirds to see it and understand it, but not "trust" it. They would stick to the part:whole method. It's more concrete for them. This in lower level high school classes. Stronger groups would have probably gone right to the part:part solution with a quick prompt.

Curmudgeon said...

So, instead of a SAT Difficulty Level, it would be a "Dave Marain Difficulty Level 5," or DMDL5, pronounced "Dim-Dull-5"

Or not.

Dave Marain said...

Gee, thanks, for coining a new term for my unofficial difficulty rating. I will take that in the spirit in which it was given!

While I still work with some SAT groups, I do not have the luxury of collecting as much anecdotal evidence about student performance on my questions as in the past. I'll leave it to my readers to test these questions with their students.

However, I do have the benefit of nearly 40 years of anecdotal evidence from working with thousands of students. While my "difficulty ratings" are not based on careful statistical analysis, I find that I can predict with reasonable accuracy which of my questions will challenge different ability levels of students. Writing questions that discriminate the 750-800 SAT student from the rest comes from years of writing math contests for my state. One learns from experience what constitutes a difficult problem.

So, while I was very clear about the subjectivity of my ratings, I also have a base of experience to back these up. You may not con-"CUR" but perhaps that's how you got your name!

Regarding the part:part issue...
I agree that currently it is more likely that only the strongest students would consider that line of reasoning. However, you may have missed my point. I am arguing for increased emphasis on both part:part AND part:whole thinking in middle schools. When we provide models like "the number of boys is half the number of girls", we are beginning to address this. This kind of thinking is standard fare in Singapore Math early on. Students work with models to represent relationships among parts as well as to the whole.

The fact that only our strongest students would consider this approach is probably more related to the fact that it is rarely taught. N'est-ce pas?

Curmudgeon said...

Rarely taught? Probably, for three reasons.

The first is that teachers are under pressure to "finish" things, to check off the standard and move on, to "get through" the material. As soon as the kids "know" the material, it's time to move on. There is rarely time for the extensive exploration that I seem to recall from my own days.

Standards-based educational theory says that you need to pick and choose your topics until you get each kid to the understanding point but says nothing about total mastery. "Drill and kill" is an epithet. "Drill and Practice" is unknown.

Many people have also fallen for the "spiraling" fad and never quite complete a thought before they're on to the next one. "We'll spiral around to this again in that module in next year's course" is probably the dumbest thing ever to come out of edumacation colleges.

I can't tell you how many times (because I've lost count) I and the other teachers have been told that we needed to get our "bubble" kids over the line into the passing zone.

"To hell with knowing math, just know enough for the test" seems to be the rallying cry.

The upshot is that you can mention these things to the better students who will get it easily and gain an even better understanding. The weaker students just trundle along.

The second reason is that weaker students are resistant to trying a second or third idea. They want to understand THIS one. A second approach is confusing. It takes a while before they get comfortable with multiple approaches and some never get much beyond "I'm not a good math student. I'll learn this but only to a point."

It takes a determined teacher to ease them into this, but she can't have anyone breathing down her neck to do test-prep.

The worst reason is that a fairly large percentage of our teachers don't really understand math to the level required. They've either bought into failed and worthless education theory or they simply are stupid.

I've related the story of the fourth grade teacher in the in-service this year ...
Me: "You say you want to be a guide on the side not a sage on the stage, yet you're teaching 4th graders. You still need to teach them things. They have to memorize 4*3 for example."
Her: "No. We should be teaching them how to look that up on Google. Did you know that Google will give you the answer to that? It's true." She leaned back, satisfied that she had put one over on me.

Is it any wonder that her kids arrive with no understanding of fractions, decimals, percents, operations? "Where's the fraction key?" "Sorry, that calculator doesn't have one, the problem you're doing doesn't need one and you wasted more time than if you just looked at it and solved it."

And some elementary and middle-school teachers "just don't DO math, tee-hee-hee." They SEEM to do math - they have lots of test-prep bubbling exercises from the publisher of the math book, but they fundamentally don't understand the nuances of the material. "Let's see what the calculator says."

In a different in-service this year, the presenter was showing how pre-schoolers and elementary kids learn math. She had lots of visuals and was "teaching" the teachers as if they were students. The aides were getting questions wrong and the elementary teachers weren't doing too much better.

This explains a lot.

Dave Marain said...

May I call you C.M.??
I'm not going to say something inane like "I feel your pain." Most of the angst you expressed is a reflection of what I was feeling when I retired. You're describing a system that is in dire need of systemic change, not tweaking.

I will not apologize however for my advocacy of national standards in math. It is unconscionable that students across the country are not learning the same content - concepts, skills, procedures, terms, definitions,... This is truly inequitable.

However, I have also been preaching "LESS IS MORE" for the same period of time. Finally, NCTM has taken this position with their Curriculum Focal Points document. William Schmidt (of TIMSS renown) stated this obvious fact 15 years ago when he described our math curriculum as "one inch deep, one mile wide."

We cannot expect students to learn math well if we fill our textbooks each year with every topic under the sun. If, for example, our teachers could focus on the essential ideas of ratios and ALL students were required to solve a range of problems from the basic to the more challenging, then most students would eventually learn ratios and be able to handle fractions. BUT facility with ratios and fractions requires facility with division which requires mastery of multiplication, etc.

"Jack, you will learn the times tables. The facts you got wrong in class today, you will have to write five times each for homework tonight. Tomorrow, I will ask you to answer just those." I know there are some teachers out there who are doing this. Is everyone?

While base ten blocks, unifix cubes and learning software have their place, we all know that nothing replaces repetition. Some students need far less than others but all students need some.

Yes, C.M., there are serious teacher preparation issues out there. Read my comments at the bottom about Finland. Yes, C.M., I share your feelings about spiraling, although there are aspects of spiraling which make sense.

I am also concerned as you are that standards-based learning has devolved into learning only for the state assessment. While we are moving toward national standards, we have to rethink how we will evaluate student learning and the bottom line is: "WHAT IS THE REAL PURPOSE OF TESTING?" What you and I and millions of other teachers see is that testing has little to do with helping children improve. It has everything to do with POLITICS:


Remember that quote:
"In other countries, education is seen as an investment; in the US, it is seen as an expense."

Perhaps this administration will "see" it differently. I truly hope so. In Finland, teachers, have to take additional years of training beyond college before they officially are certified. This additional 2-3 years culminates in a year of working in a laboratory school with real students. A true internship. And, by the way, THE GOVERNMENT PAYS EVERY PENNY FOR ALL THIS ADDITIONAL TRAINING. This is how Finland has turned around its system in the past 20 years.

INVEST IN EDUCATION, INVEST IN OUR CHILDREN, INVEST IN OUR FUTURE. Don't look for short-cuts, folks. There are none. Expedient solutions lead to students who only care about the bottom line, the grade, not about learning. This "get results without really earning it" mentality is pervasive in our society. In the worst case, this mentality produces the AIGS, the Enrons and the Madoffs of the world.

Ok, now you got me to rant too. I guess I needed that catharsis. otherwise it sounds like I'm just pontificating about challenging our best and brightest with all these problems I'm writing. But there's so much more to it, C.M...

P.S. I have a feeling that your comments and mine are not being read by most of my readers. I'm thinking I should copy them into a separate post and really incite a riot! I think I will do that unless you state an objection! THANK YOU!

Anonymous said...

"P.S. I have a feeling that your comments and mine are not being read by most of my readers."

or that swamped readers are waiting for the weekend...

interesting stuff


Dave Marain said...

Actually, Jonathan, many do not get to see comments in general. Only a few of my blog readers also subscribe to my comments feed. I may republish curm's comments and my replies. He made some excellent points.