kitchen table math, the sequel: severely fragmented knowledge

Thursday, June 24, 2010

severely fragmented knowledge

Looking through the SAT prep posts, I found this problem that, just 12 days ago, I couldn't begin to do.
If 20 percent of x equals 80 percent of y, which of the following expresses y in terms of x?

(A) y = 16% of x
(B) y = 25% of x
(C) y = 60% of x
(D) y = 100% of x
(E) y = 400% of x

The Official SAT Study Guide 2006
ISBN-13: 978-0874477184
p. 550

Today it seems simple and obvious.

This is a classic case of inflexible knowledge. I've practiced solving many, many literal equations over the past few years, a skill I learned and practiced in high school, too. I had no trouble doing it then, and I've had no trouble doing it now.

So, in theory, I 'know' how to do this problem.

And yet when I encountered a literal equation involving percent, I was mystified.

What fresh hell is this?

from cranberry:
Funny, I didn't need to set up a formal equation for this. If 20% of x = 80% of y, y must be four times as small as x. Thus, 25% of x.
That's the kind of thing years of doing bar models does for you.

Not that cranberry has spent years drawing bar models.


Catherine Johnson said...

question: What fresh hell is this?

answer: It's the same old hell as always; I just don't recognize it.

Catherine Johnson said...

Just to be obsessive here...looking back at the earlier thread, I see that I could do this problem but the answer in the book was wrong.

(And...uh...guess what? The answer in the book is not wrong. sigh.)

NONETHELESS, the first time I encountered a literal equation expressed in terms of percent, I was flummoxed.

Anonymous said...


Are you saying that the answer is not (B) y = 25% of x?

Because if that is not the answer, I'd like a clear explanation of the math that arrived at a different answer.

cranberry said...

Funny, I didn't need to set up a formal equation for this. If 20% of x = 80% of y, y must be four times as small as x. Thus, 25% of x.

Dee Hodson said...

Hi Catherine
I ma not sure how to post a new thread? Sorry- for the life of me I can't figure it out.
I was wondering if anyone out there has heard of, or has opinion sof, Renzulli Learning. A dear friend is moving her children to Queens and she would love some feedback t this program-
Dee Hodson

Catherine Johnson said...

cranberry - good for you!

If I'd done bar models sometime in the past year, I'd see that off the bat, too...

Catherine Johnson said...

Answer is 25%

Catherine Johnson said...

Dee - you need to sign in up at the top right, then hit "New Post," also at the top right.

It's pretty easy after that.

If you have trouble, shoot me an email:

cijohn @

LexAequitas said...

Princeton Review test prep

"of" in these problems means multiply.


.2 * X = .8 * Y

Then it's straight algebra:

(.2/.8) * X = Y

.25 * X = Y

25% OF X IS Y.

cranberry's way is great, of course, as long as you have easy numbers (which, admittedly -- the SAT almost always has).

Allison said...

but how do you know if means multiply?

How about this:

What percentage is 18 of 25? or
what percentage is the same ratio as 18 of 25?

Catherine Johnson said...


I had a moment of revelation working a problem of the first type - not because I mistakenly set it up as 18 x 25, but because I was struck by the fact that in this problem "of" does not mean "multiply."

lgm said...

Hmm. The first step in problem solving is to understand the problem. Polya suggests drawing a figure. When you first approached it, what went through your mind? Were you able to draw a figure?

One thing that helps with problems like this is to know factoring and double digit mental math operations so well that you don't have to think for that part.

lgm said...

Another thing that can help in figuring your approach is to use simpler numbers. (simpler meaning easier for the solver to use and work with mentally). Perhaps this one would be easier to visualize if it was:
If 50 percent of X equals 100 percent of Y, how would you express Y in terms of X..

Catherine Johnson said...

I had trouble drawing this one....

I think I have the same problem with drawing a model that I do with simply solving a literal equation: I mainly drew fraction problems when I was working my way through Challenging Word Problems.

I'm having trouble transferring 'training' amongst bar models.

Catherine Johnson said...

OK, I did it.

But it's obvious I have inflexible bar-model knowledge, too.

Catherine Johnson said...

I probably need to work my way through Singapore Math.

In fact, I do need to work my way through Singapore Math.

Crimson Wife said...

Catherine- if there are certain topics that would like to learn how to do the Singapore way without having to get all the Singapore books, check out the Math Mammoth "blue" series of e-books. They are written by Maria Miller, the author of the "Home School Math" blog. I'm using them to supplement Singapore in certain places where my DD is having difficulty making the conceptual leaps Singapore requires.

lgm said...

I'd encourage you to work through SM and then switch over to Dolciani pre-algebra. They'll stretch your visual and problem solving skills.

In the meantime, you might enjoy playing with thinking blocks which is free over at

bky said...

Catherine -- in the problem, "What percent is 18 of 25?" the "of" actually does mean multiply, essentially. The problem is in the grammar. Reword the question as a statement:

"18 is 'what percent" of 25'."

Let W stand for the fraction (rather than percent) and you can translate this as:

18 = W x 25

Solve for W = 18/25 and get the fraction; then convert to decimal. If the question had asked for "fraction" rather than "percent" then in fact "of" would directly correspond to multiplication. So there is a multiplication, but, clearly (now) it is not 18 times 25. It's "what percent" times 25.

bky said...

Opps. LexAequitas had already made my point. Sorry for the redundancy.

Allison said...


The reason you can solve this problem is because you already know how to solve this problem.

For students who don't know how, who don't really have it mastered, the "of is multiply" trick doesn't work.

That's because they won't understand how to reword the question, because they won't understand what's being asked. If they do understand what's being asked, then they know what to do, and what it means, without the rule.

That's the problem with the tricks that we who are experts developed. The tricks are shortcuts because we already know what to do. It's not helpful to provide those rules to those who aren't experts. The rules mislead and confuse.

To a 7th grader who is afraid of fractions, thinking clearly enough to rewrite the "of" as division in one case and "of" as multiplication in the other is a bridge too far.

bky said...

Allison -- I don't see this as a trick at all, just the opposite. I have taught my kids at home to translate English into math and then follow their nose. That means a sentence becomes an equation, with letters for quantities you don't know. Then do whatever arithmetic you need to do to calculate the thing you want. To me this is the anti-trick.

I am not sure what you mean when you write, "If they do understand what's being asked, then they know what to do, and what it means, without the rule." What rule do you mean? I am not proposing any rule. If the kids understand what the question asks -- well, from my perspective that is the same as saying the kids could write the equation 18 = Wx25 (and know the difference between the fraction W and a percentage that represents it). If they don't understand what is being asked, then it is still time for teaching, not testing.

By the way, my kids are used to the idea of rephrasing questions as statements from diagramming sentences and otherwise identifying parts of speech and so on. It is good to study grammar.

Catherine Johnson said...

clearly (now) it is not 18 times 25. It's "what percent" times 25

right - I was just making the point that I think a lot of kids, taught to translate 'of' into 'times' would multiply 18 by 25.

Catherine Johnson said...

Crimson Wife -- thanks for the book reference.

lgm- "I'd encourage you to work through SM and then switch over to Dolciani pre-algebra. They'll stretch your visual and problem solving skills."

That is an excellent idea.


Does anyone have advice about which book to start with in Singapore Math?

I own the whole series, so I could start from book 1 --

LynnG said...

I recommend 3B or 4A -- just to see how they introduce bar models. Scan through the problems related to bar models in 3B. If you've got the books, start there, but you probably don't have to work through all the problems of the entire book -- just focus on the bar models and then move on.

lgm said...

Start at the beginning. A big key is practicing number bonds until you know them so well that the possibilities just pop up automatically when you see a number. The mental math practice is worthwhile too.

My 7th grader's approach to the above problem is 20X to 80Y scales to 1X to 4Y so (visualizing those 5 peices) 25% X = Y.

Crimson Wife said...

FWIW, bar models are actually introduced in Singapore 3A, chapter 2.

There's a lot of repetition of topics from year to year in the Singapore books (a "soft spiral" approach). Also, there's a fair number of units on measurement that I'm not sure are all that relevant to SAT prep.

This is why I think the single-topic Math Mammoth books would be better. You can select only the ones for the areas in which you know you need work (fractions, percents & ratios, etc.) rather than having to go through a bunch of different Singapore books.

Catherine Johnson said...

I've never even heard of Math Mammoth.

Catherine Johnson said...

If Amazon weren't down I'd go look it up.

Catherine Johnson said...

I worked my way through all of Challenging Word Problems Book 3 a few years back.