kitchen table math, the sequel: help desk - absolute value

## Sunday, June 5, 2011

### help desk - absolute value

Problem 85 in Perfect 800: SAT Math:
In an amusement park, regulations require that a child be between 30" and 50" tall to ride a specific attraction. Which of the following inequalities can be used to determine whether or not a child's height h satisfies the regulation for this ride.

(A) | h -10 | < 50
(B) | h - 20 | < 40
(C) | h - 30 | < 20
(D) | h - 40 | < 10
(E) | h - 45 | < 5

update: The book's answer is C, which is wrong.

Anonymous said...

start from 30<h<50

try to shift the endpoints to be +- something. Since they are 20 apart, the endpoints need to be +-10, which would require subtracting 40

-10 < h-40 < 10

|h-40| < 10

Catherine Johnson said...

Thanks!

That was my answer, but the book says C.

SteveH said...

Believe in yourself.

Catherine Johnson said...

loll!!

If I have time, I'll type up the author's explanation, which is entirely algebraic & beyond my energy level at the moment...

I'm relieved. I only recently discovered there was such an animal as an absolute value word problem, & it was just a few weeks ago that I grasped the logic of setting one up -- so to have suddenly think I had completely blown it was a little horrifying.

Catherine Johnson said...

Do high school algebra courses teach absolute value word problems these days?

I had NEVER heard of such a thing!

Somehow I thought math book authors wrote up absolute value inequalities & math students solved them and that was the end of it.

I'm 99% positive C. has never seen such a thing, either.

He insists he knows how to set up and solve an absolute value problem, but he absolutely does **not**!

Niels Henrik Abel said...

Another way to think about it:

40 inches is obviously in the exact middle between 30 and 50 inches. Therefore the kid's height has to be within a 10-inch margin on either side of 40 inches.

==> |h - 40| < 10

Niels Henrik Abel said...

Personally, my first inclination would be to approach it algebraically, as did GSWP. Set things up as a (double) inequality, and convert it an inequality that is equivalent to absolute value.

Stacey Howe-Lott said...

Hi Catherine -
This is lifted from the Blue Book, Test 5, Section 8, question number 9 on page 669. The correct answer is D. The best explanation I've found is from the Khan Academy. Here's the link to his video explanation: http://www.khanacademy.org/video/sat-prep--test-3-section-8-part-2?playlist=SAT%20Preparation. Go to the 5:35 mark.

Catherine Johnson said...

0 inches is obviously in the exact middle between 30 and 50 inches.

That's how I thought of it & how I set it up.

pckeller said...

"It has to be within 10 units of the middle of the range" is definitely the clearest, most insightful approach. But for test-takers who don't think of that...

You can make up different heights, as yourself whether that kid should be allowed to ride, and then go through the answers to see if the rule gives the proper answer. Depending on what numbers you try and in what order, it takes a few tries. For example, try 51, 49, 31 and 29...

Catherine Johnson said...

Hi Philip!

That's how I would have solved the problem a couple of months ago - !

Anonymous said...

Actually, I thought of the 40+-10 first, but then worked out the slower approach as being more explanatory. There is a difference between solving problems and explaining the solutions to problems. It is often better to provide a slower approach that seems less "magical" to students.

Michael Weiss said...

Not to be nitpicky, but the wording of the problem is a teeny bit ambiguous: Is a 30" tall child allowed to go on the ride? Or a 50" child? Presumably the answer is yes, in which case the < symbol should be "less than or equal to".

PWN the SAT said...

I'm a little late to the party here, but I think it's helpful to remember a simple conversion: Say |x| < 2. That means x could be 1, 0, -1, and a bunch of non-integer values, of course. In other words, if you nix the absolute value brackets, you could rewrite this -2 < x < 2.

The same conversion can be applied to all of the answer choices. For brevity's sake, I'll just do the correct answer:

| h - 40 | < 10
-10 < h-40 < 10
(-10 + 40) < h < (10 + 40)
30 < h < 50

As was pointed out above, on SAT questions like this the middle value of the range will always be inside the absolute value brackets, and the amplitude of the range will be on the other side of the < sign.

I wrote some more examples like this a while back.

pckeller said...

But it's easy to underestimate how challenging that is for many students. Also, to go from

-10 < h-40 < 10

to the next step, it is clear that you add 40 to every term. If a student can't see why, you can explain that you are trying to get the variable by itself. But that is NOT the algebraic problem this problem presents. We have to START from

30 < h < 50

and figure out what to do next. It is a much more subtle point to see that subtracting 40 from each term will give you "boundaries" that are additive inverses, ready to be expressed in an aboslute value equation. I'm betting that of those who get this problem right, most will either make up numbers or have already seen this type of problem so they know the find-the-distance-from-the-middle approach.

In general, this relates to a constant challenge in helping kids to prep. A student asks "How would you solve this" but often the question they mean to ask is "how could I solve this?" -- me and not you with your total algebraic fluency and your vast recall of previous tests!

PWN the SAT said...

@pckeller I agree with you completely. When I teach this kind of problem to a kid, we plug-in heights (or weights of the boxer, etc.). That method has its own difficult mental leaps when the question is written well (plugging in values that should work won't eliminate every choice, so now you've gotta pick values that SHOULDN'T work and eliminate choices that DO work with them...), but I agree it's the most intuitive way.

Once we've slogged through that, I'll decide whether to present the solution I mentioned above, based on the amount of consternation I'm sensing from the student.

Bottom line for me: when there exist multiple ways to skin the cat, I'll present as many as I feel the student wants, and let him or her choose which fits best.

Anonymous said...

Here is how I remember to do this problem from school (similar to above):

+/-(h-40)<10 so

+(h-40) < 10 ; h<50

and

-(h-40)< 10 ; -h+40< 10; -h<-30 and

h>30

therefore 30<h<50

Thanks for posting this problem. I love to randomly through these in for my tutoring students who are headed for the SAT.

Catherine Johnson said...

It is often better to provide a slower approach that seems less "magical" to students.

Definitely.

Catherine Johnson said...

I've got to post Elizabeth King's explanation. It was fantastically helpful for me.

Catherine Johnson said...

We have to START from
30 < h < 50
and figure out what to do next. It is a much more subtle point to see that subtracting 40 from each term will give you "boundaries" that are additive inverses, ready to be expressed in an absolute value equation.

right - and I still have to work with this ---

Jen said...

**Not to be nitpicky, but the wording of the problem is a teeny bit ambiguous: Is a 30" tall child allowed to go on the ride? Or a 50" child? Presumably the answer is yes, in which case the < symbol should be "less than or equal to".**

It's not nitpicky, it's part of the problem, however your conclusion was incorrect. Between means just that -- between them, so it doesn't include 30" or 50".

Jen said...

Regarding the question of whether problems like this are taught in school or not --

Isn't that the stated purpose of the SAT? (Note: not saying I believe all their statements!) Their idea being that this isn't something you should study for, but rather something that a solid base of knowledge will let you figure out. Thus their fondness for problems that aren't just like ones done in textbooks (at least until they're all teaching to the test(s)).