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Chris Padilla
10-01-2013, 11:52 AM
A friend of mine sent this to me. It was a math question on his daughter's math test:

****
Insert parentheses and addition signs to make this equation true.
Remember that the parentheses can indicate multiplication.

5 4 2 3 = 32
****
There were a few spaces between each number but it isn't showing here.

Stephen Tashiro
10-01-2013, 12:08 PM
There were a few spaces between each number but it isn't showing here.

Do the spaces prohibit using 5 + 4 + (23) ?

Chris Padilla
10-01-2013, 12:23 PM
That was the solution, Stephen!

Now do you think that this is an "appropriate" "math" question? A buddy of mine posted this on his Facebook page and no one could solve it. When we found out the solution, a lot of people went crazy!

Personally, I found it interesting and "out of the box" thinking. I liked it.

Eric DeSilva
10-01-2013, 12:48 PM
It is an interesting puzzle if posed as a word game, but it isn't mathematics. Putting two things together without an intervening operator in mathematical notation indicates multiplication, not what was done here. In other words, in mathematical notation, "ab" means "a x b," not "a x 10 + b."

Chris Rosenberger
10-01-2013, 12:50 PM
That is very useful in everyday life.
I am always running into strings of numbers & wounder how I can make equal other numbers.
Schools are great these days.

David Weaver
10-01-2013, 1:11 PM
Personally, I can't find anything wrong with it. It's a question, and as far as we know, the teachers may have gone through questions of that type as part of class.

Otherwise, the outside of the box type questions like that are more IQ questions than the standard school "i tell you something and then you take a test and tell me the same thing".

I'm reminded of a saying one of my professors used to say...."Learning is Painful". Sometimes there are questions only a few students get, and maybe only a few more of them understand why they didn't get them after that. I know a lot of people don't agree with me, but I don't think every question asked needs to be one that 80% of students get correct.

Larry Browning
10-01-2013, 1:13 PM
A friend of mine sent this to me. It was a math question on his daughter's math test:

****
Insert parentheses and addition signs to make this equation true.
Remember that the parentheses can indicate multiplication.

5 4 2 3 = 32
****
There were a few spaces between each number but it isn't showing here.

Maybe I am not up on all math punctuation, but I have never heard of a parentheses indicating multiplication. Star(*) indicates multiplication. AFAIK multiplication is never implied. At least in my world (computer programming)
So far I haven't come up with the solution.

Larry Browning
10-01-2013, 1:27 PM
Do the spaces prohibit using 5 + 4 + (23) ?

Wait a minute!!!! I call fowl. No where did it say you could remove spaces. In only allowed inserting parentheses and addition signs. Plus, if parentheses indicates multiplication of the numbers inside then 5 + 4 + (2 3) would equal 15 correct?

Chris Padilla
10-01-2013, 1:30 PM
I agree that this isn't rigorous, rudimentary math but a careful reading of the problem statement doesn't say one can't take two of the single digits and "create" a two digit number from them. It further doesn't state that you can do this. As I biked into work this morning (a 50 minute commute), I pondered this "math" problem over and over and couldn't find a solution. When I saw the answer (that the teacher was looking for) posted, I was floored that I didn't think to do this. Like most of us, I suspect, it would not have occurred to us to make a two-digit number. Fundamentally, rigorously, it is not correct but it certainly isn't prohibited. Interesting stuff.

Dan Hintz
10-01-2013, 1:30 PM
Maybe I am not up on all math punctuation, but I have never heard of a parentheses indicating multiplication. Star(*) indicates multiplication. AFAIK multiplication is never implied. At least in my world (computer programming)
So far I haven't come up with the solution.

Computer programmer here (but also an engineer)... multiplication was always implied/assumed throughout my engineering career. Blocks could be separated by (), [], or even {}. It was only when those symbols were used as operators that the multiplication symbol '*' was used to keep things straight.

Chris Padilla
10-01-2013, 1:34 PM
Maybe I am not up on all math punctuation, but I have never heard of a parentheses indicating multiplication. Star(*) indicates multiplication. AFAIK multiplication is never implied. At least in my world (computer programming)
So far I haven't come up with the solution.

It is pretty fundamental, Larrry. Have you ever heard of PEMDAS? Parentheses Exponent Multiplication Division Addition Subtraction (Please Excuse My Dear Aunt Sally) These are orders of operation so if you see something like 5 * 4 + 6 / 3 = ??? Certainly, I could make it clearer: (5*4)+(6/3) = ??? or even ((5)(4))+ (6/3) = ???

Chris Padilla
10-01-2013, 1:36 PM
Wait a minute!!!! I call fowl. No where did it say you could remove spaces. In only allowed inserting parentheses and addition signs. Plus, if parentheses indicates multiplication of the numbers inside then 5 + 4 + (2 3) would equal 15 correct?

Well, it didn't say you could remove them. It also didn't say you could NOT remove them. hahaha :) In this case, the parentheses were used to "group" the 2 and 3 together to "create" 23. Like I said above, fundamentally, this is not correct but it solves the problem.

Chris Padilla
10-01-2013, 1:38 PM
That is very useful in everyday life.
I am always running into strings of numbers & wounder how I can make equal other numbers.
Schools are great these days.

Government math perhaps? :D

Larry Browning
10-01-2013, 1:48 PM
I agree that this isn't rigorous, rudimentary math but a careful reading of the problem statement doesn't say one can't take two of the single digits and "create" a two digit number from them. It further doesn't state that you can do this. As I biked into work this morning (a 50 minute commute), I pondered this "math" problem over and over and couldn't find a solution. When I saw the answer (that the teacher was looking for) posted, I was floored that I didn't think to do this. Like most of us, I suspect, it would not have occurred to us to make a two-digit number. Fundamentally, rigorously, it is not correct but it certainly isn't prohibited. Interesting stuff.
It is clearly stated that that you must only insert parentheses and addition signs. It did not say you could remove a character. If I used the logic that since it didn't say I couldn't do something like removing a space between numbers, then I could do other things that it didn't say I couldn't do, like changing a number to a different number, or re-arranging the order. A space is a valid character just like a number is a valid character. I say the solution is bogus and violates the restrictions put on the problem. BO O O O GUS!

Larry Browning
10-01-2013, 2:05 PM
Well, it didn't say you could remove them. It also didn't say you could NOT remove them. hahaha :) In this case, the parentheses were used to "group" the 2 and 3 together to "create" 23. Like I said above, fundamentally, this is not correct but it solves the problem.

If the logic is, it didn't say I couldn't remove a character and somehow that is ok to do, then it also didn't say I couldn't change the numbers all around and even remove some of them. So using this logic the correct answer could be 32 = 32. All I did was remove some of the numbers and spaces and then rearranged the 2 and the 3. The instructions didn't say I could not do that.
The poor space character just gets no respect these days:eek: We can just throw it away and ignore it any old time we want!



this is not correct but it solves the problem.
Yes, this is true!

Man! I sure hope that was a bonus question that didn't count as part of that poor girl's grade on the test.

Eric DeSilva
10-01-2013, 5:03 PM
I'm going to go further than saying "this isn't a math problem" to saying "the solution presented is incorrect." The problem could be phrased as a symbol logic problem--can you arrange these symbols in a way that creates a mathematically valid equation. But, the problem is currently stated as a mathematics problem--it was on a "math" test and relies exclusively on the language of mathematics. What people seem to forget here is that mathematics is a different language--if you are going to state a problem in that language, you are implying a solution in that language.

I previously noted, in the language of math, we imply multiplication all the time when two symbols are placed adjacent with no operator--"ab" is "a times b." Therefore, in the language of mathematics, two discrete numbers adjacent to one another should be multiplied together. The "solution" presented relies on concatenation. The operator used to denote concatenation in mathematics is a double bar, since concatenation is useful in some work with prime numbers and other number theory--in other words, "5 || 4 = 54" is a correct mathematical expression. However, the problem implicitly restricts you to combining certain numbers using addition--denoted by "+"--and multiplication--denoted through parentheses. The problem does not allow you to use concatenation through parentheses, and therefore a solution relying on parentheses is wrong.

This math teacher should be de-certified.

Rick Potter
10-01-2013, 5:23 PM
Well,

Did she get it correct?

Phil Thien
10-01-2013, 5:36 PM
I applaud the effort of the teacher(s), but I think this actual "puzzle" falls a little short. I think the instructions need to be more clear, maybe some examples demonstrating that space elimination is acceptable.

I'd encourage the teachers to do more of it, they will improve.

Bruce Boone
10-01-2013, 7:18 PM
(5+4)2/3 =(3)(2) That's as close as I come if other operators are allowed. I agree. The spaces argument is pretty weak. It belongs on a logic test, not a math test.

Jason Roehl
10-01-2013, 7:39 PM
Wait a minute!!!! I call fowl. No where did it say you could remove spaces. In only allowed inserting parentheses and addition signs. Plus, if parentheses indicates multiplication of the numbers inside then 5 + 4 + (2 3) would equal 15 correct?


(2 3) is not used to represent 6 when speaking of parentheses and multiplication. When numbers are used some sort of operator is needed between them, such as the 'x' (better is the Greek letter chi--I think that's the one that looks sort of like a cursive x), asterisk, a raised dot, or even two sets of parentheses--(2)(3) or even just 2(3) = 6. However "2a" implies "2 times a".

Okay, now I'm going to throw all this back in the mental attic where it can resume its dust-collecting experiments.

Rick Christopherson
10-02-2013, 1:14 AM
A buddy of mine posted this on his Facebook page and no one could solve it. And there is your answer right there. I am betting that your buddy didn't post anything, but instead "Shared" a posting or picture seeking the answer. It is just a scam to get other people to like or share the posting in order to increase ranking.

Charlie Velasquez
10-02-2013, 2:33 AM
(2 3) is not used to represent 6 when speaking of parentheses and multiplication. When numbers are used some sort of operator is needed between them, such as the 'x' (better is the Greek letter chi--I think that's the one that looks sort of like a cursive x), asterisk, a raised dot, or even two sets of parentheses--(2)(3) or even just 2(3) = 6. However "2a" implies "2 times a".

Okay, now I'm going to throw all this back in the mental attic where it can resume its dust-collecting experiments.

Have to go with this. In the 80's the National Council of Teachers of Mathematics recommended that numbers be written without commas, and a space was to be used between the periods. So 34,265,117 would be written 34 265 117. I think the rationale was that when programming computers commas delineated separate values, while spaces added nothing to the place value. So maybe the NCTM felt this would help negate confusion for those that would later turn to computers. Not many teachers followed suit- many teachers were notoriously stubborn and taught the way and stuff that they were taught (see, we are still using and teaching the imperial units of measurement...). Or, the rationale may have been just to better align ourselves with the rest of the world as most non-English speaking countries did not use commas to identify periods...

Frank Trinkle
10-02-2013, 8:22 AM
The Europeans confuse issues even more. They don't use a decimal point for monetary tenths (cents). They use a comma.
Therefore 120Euros and 25 cents is expressed as €120,25 rather than €120.25. Takes a while to get used to!:confused:

John Coloccia
10-02-2013, 9:23 AM
I could just write:

32 = 32

Because it didn't say I can change the numbers to be whatever I want, but it also didn't say I CAN'T.

That's the problem with this sort of smarty pants puzzle. Where do you draw the line at how far out of the box is too far? In this case, it's entirely unclear that putting parentheses around two numbers magically turns them them into one number. All I did was reverse two numbers and get rid of two. If I can magically get rid of spaces, why can't I get rid of numbers too? I don't even have to move any around. I can simply say:

3=3

having gotten rid of all the superfluous characters (numbers and spaces).

Chris Kennedy
10-02-2013, 9:32 AM
Easy:

5 4 2 3 becomes (5)(4)(2)(3) and I multiply them together, mod 88, and I get 32.

As someone who does math research, these kind of "cute" puzzles annoy me. They aren't math, and I wouldn't even say they are logical.

Chris

Prashun Patel
10-02-2013, 9:45 AM
Jeez, why are you guys so worked up about it? The problem made us all think. It made my 9 year old son shout "hey, that's not fair" when he heard the solution, but he was smiling and thought it clever.

What's wrong with a little cheek? Why are these problems 'annoying'? The thought exercise is way more valuable that way than if it were written so that we could all solve it easily:

"5_4_23 = 32. Fill in the blanks - and only the blanks - with plus signs to make this equation true."

Phil Thien
10-02-2013, 9:58 AM
And there is your answer right there. I am betting that your buddy didn't post anything, but instead "Shared" a posting or picture seeking the answer. It is just a scam to get other people to like or share the posting in order to increase ranking.

I guess I'm pretty naïve about facebook, do people actually increase rankings there by getting others to share postings? How does that work?

Dan Hintz
10-02-2013, 9:58 AM
Jeez, why are you guys so worked up about it? The problem made us all think. It made my 9 year old son shout "hey, that's not fair" when he heard the solution, but he was smiling and thought it clever.

What's wrong with a little cheek? Why are these problems 'annoying'? The thought exercise is way more valuable that way than if it were written so that we could all solve it easily:

I think that's the issue right there... it's a "clever" solution. Occasionally my daughter will bring home one of these and I just roll my eyes. The intent is to teach the kids the subject of mathematics, not to show how cutesy the problem creator can be. Offer those problems as "extra credit", and make the rules of solving the problem crystal clear (or specify they've been left intentionally vague), but don't parade them as real problems and make the child suffer looking for a solution. It sends the wrong message and has the child questioning if they understand the core subject.

John Coloccia
10-02-2013, 10:34 AM
I think that's the issue right there... it's a "clever" solution. Occasionally my daughter will bring home one of these and I just roll my eyes. The intent is to teach the kids the subject of mathematics, not to show how cutesy the problem creator can be. Offer those problems as "extra credit", and make the rules of solving the problem crystal clear (or specify they've been left intentionally vague), but don't parade them as real problems and make the child suffer looking for a solution. It sends the wrong message and has the child questioning if they understand the core subject.

Exactly. If you lay out rules that say "Add + signs and Parentheses" then you expect to be able to solve it with + signs and (). Make the rules clear:



5423 = 32
Make the above statement true. You may insert + signs and parentheses, but you may not change the order of the numerals.

And then you can simply say

5+4+23=32

That actually makes a lot of sense and you don't even need the parentheses.

(2 3) doesn't equal 23, so the whole thing is just stupid, incorrect and doesn't belong in a classroom. What a stupid thing to teach a child. "When you have two numbers that have spaces between them, when you put parentheses around them they become one number". There's no other reason for the parentheses, and in this case their use is so completely wrong that this doesn't belong anywhere near any classroom. There are so many ways to "think out of the box" that actually results in correct answers without all of this nonsense, why would anyone use this particular example?

Eric DeSilva
10-02-2013, 10:35 AM
One key about mathematics and the language of mathematics is that any value should be able to be replaced by a variable without ambiguity. When you have symbols separated by spaces--"3 2"--you should be able to replace that with "x y, where x = 3 and y =2," because abstraction is one of the fundamentals of mathematics. Honestly, we never used actual numerical values when I studied higher math because everything was a symbol--actual values generally were reduced to constant generically labeled "K" or something. In my math studies, I don't think I did anything with an actual Arabic numbers from 8th grade on. So when you use spaces in formulation of a problem--NCTM be damned(*)--the indication is that each symbol is a unique value. Unique values placed next to each other without a space denotes multiplication--"xy" where x=3 and y=2 is "x times y" or 6.

(*) if the NCTM had succeeded, what they would have done is to create ambiguity as to whether certain strings of symbols were one value or not. But, if that was commonly adopted, people writing in the language of math would therefore be unable to use numerical values and spaces without resolving the ambiguity. For example, a 2 dimensional matrix is typically denoted with curly braces and values separated by rows of numbers separated by spaces (the only place I can recall where standard math notation involves Arabic numerals separated by spaces). I'll separate rows with a comma, so something like {1 0, 0 1}. If NCTM got their way, it might be "valid" to write a matrix like {100 101 102, 0 1}, but it is ambiguous, because that it could mean either {100,101 102, 0 1} or {100 101,102, 0 1}. So, I'll call the proposal silly on that basis. Like Texas trying to say pi should be 3.

Prashun Patel
10-02-2013, 11:31 AM
Kids learn how addition and multiplication earlier than 6th grade. If the problem were stated as clearly as you guys wish, I think I'd be rolling my eyes at how basic the question is, and how kids are not being pushed.

As an engineer, Dan, I'm sure you'll attest to the fact that if we only solved problems using the rules, we'd never make real progress. This was a clever - but not unreasonable - stretching of the rules. I totally applaud anyone who asks that kind of question, those who can answer it, and those who fail at answering but are internally encouraged to think that way next time.

If ALL the questions are like this, I see your point. But it's one, cheeky question.

Chris Padilla
10-02-2013, 11:32 AM
And there is your answer right there. I am betting that your buddy didn't post anything, but instead "Shared" a posting or picture seeking the answer. It is just a scam to get other people to like or share the posting in order to increase ranking.

Sorry, Rick, that is incorrect. It was a question on his daughter's math test and he simply, verbatim, copied down the text and asked anybody if they could solve it. I'd say about 15-20 of us posted that we couldn't solve it. When the teacher got back to him with the solution, most folks went ballistic...as some on here have done! So interesting the problem and the reactions!!

Chris Padilla
10-02-2013, 11:33 AM
Well,

Did she get it correct?

My friend has twin girls in 6th grade and one of them figured out the answer the teacher was looking for.

Dan Hintz
10-02-2013, 11:49 AM
As an engineer, Dan, I'm sure you'll attest to the fact that if we only solved problems using the rules, we'd never make real progress. This was a clever - but not unreasonable - stretching of the rules. I totally applaud anyone who asks that kind of question, those who can answer it, and those who fail at answering but are internally encouraged to think that way next time.

I think you're stretching definitions here, Prashun. You break "rules" in engineering when the "rule" no longer makes sense for the problem at hand... which means the "rule" wasn't appropriate in the first place. In math class (at least at the 6th grade level), we're trying to teach kids how to handle mathematical operations, we're trying to teach them the "rules". These are "rules" that shouldn't be broken until you start getting into theoretical constructs. Confusing the kids with how the rules work serves no purpose for the task at hand. If you want them thinking outside of the box (which is a good thing), tell them so, but don't mix it in with the regular stuff and then wonder why the kids are confused.

Chris Kennedy
10-02-2013, 12:49 PM
As an engineer, Dan, I'm sure you'll attest to the fact that if we only solved problems using the rules, we'd never make real progress. This was a clever - but not unreasonable - stretching of the rules. I totally applaud anyone who asks that kind of question, those who can answer it, and those who fail at answering but are internally encouraged to think that way next time.


You don't break rules arbitrarily in mathematics. If you do, you end with with contradictory and paradoxical constructions.

That doesn't mean that mathematics is hidebound. Some of the most amazing mathematics has resulted from mathematicians looking at a standing assumption and asking themselves -- does that have to be true or can we construct a logically consistent system without that assumption? Case in point -- hyperbolic geometry. But that takes a fair understanding of how things work before you can do that.

I don't see how this exercise encourages good or critical thinking. All it does is encourage making arbitrary changes to get what the desired ends. In some sense, it is mathematical "the ends justify the means."

Chris

Prashun Patel
10-02-2013, 12:53 PM
Then we agree to disagree.

Larry Browning
10-02-2013, 1:21 PM
My main problem with this "puzzle" is that it is presented as a mathematical problem that can be solved with mathematics. When in reality it cannot be solved with mathematics.
The solution presented as the correct one is actually incorrect in a mathematical context. It only solves the problem with some sort of illogical logic. To me this is NOT something I would want my children's MATH teacher to be presenting them as actual math. They are confused enough by actual math without being taught this little piece of crap! I see this as some silly puzzle the teacher saw in a magazine somewhere and thought it was cleaver, so he/she put it on a test. How is this helping with the education of our children?
Sorry for the rant, but I take the education of our children VERY seriously, and to me this kind of thing just sets us back.

Duane Meadows
10-02-2013, 1:23 PM
I say 2 + 3 = 6, you say it's 5. Unfortunately, too many people think we are both right, that there are no absolutes! And we wonder why only 50% of HS grads, give or take, can even pass college entrance exams!

Larry, +1:)

Pat Barry
10-02-2013, 1:34 PM
I suspect that this question was a review item for something that had been discussed and demonstrated in class by the teacher. In other words, I don't suspect that this was a final exam question, completely out of the blue. It was there for the kids to think about potential solutions to demonstrate they learned from the classroom examples. Assuming this is all true, we are all way too worked up about the purity of the mathematics involved.

Larry Browning
10-02-2013, 1:52 PM
I suspect that the origin of this was much more like I described. Plus, if they had been discussing and demonstrating the logic needed to solve this puzzle in a math class, then this is even worse than I had first thought!

My point is that this has no place in a MATH class, where MATH is supposed to be taught. This is not math. It could have a place in some other subject (it escapes me as to which one), but not math, especially when it is presented as a math question.

Chris Padilla
10-02-2013, 3:14 PM
I suspect that this question was a review item for something that had been discussed and demonstrated in class by the teacher. In other words, I don't suspect that this was a final exam question, completely out of the blue. It was there for the kids to think about potential solutions to demonstrate they learned from the classroom examples. Assuming this is all true, we are all way too worked up about the purity of the mathematics involved.

I suppose I could inquire a bit more as to the environment in which such a question was asked but I think this horse is dead. Suffice it to say, the devil is likely in the details. I don't think some have given the kids enough credit here to understand the problem. I don't think it will be detrimental to their mathematical education ASSUMING (and I think that is a reasonable assumption) that the teacher explained the premise behind the question as Pat alluded to above.

I gave this problem to my daughter (also in 6th grade) and she was unable to solve it like the teacher wanted. I explained the answer to her and she immediately declared the unfairness of the question. I agreed with her and proceeded to explain to her that she was correct but that sometimes creative thinking is warranted in some situations.

Larry Browning
10-02-2013, 3:38 PM
I think I am ready to declare this horse dead as well, as I have contributed many of the same blows several times! Even after he was already dead. My blood pressure has come down to normal again and I would prefer it to stay there.
I think I figured out the subject where this question could be asked. Political Science. The question could read: "How would a typical politician make this equation true?"

Mark Blatter
10-09-2013, 3:11 PM
I came up with;

(5-4) + 2 + 3 = 3 * 2

Larry Browning
10-09-2013, 3:23 PM
that's about as good of an answer any. Since it didn't say you couldn't add a minus sign.

BTW: I am SO over this!