6th grade math question help

[NCJessieRN]

I know this is easy but my brain this morning is not high functioning! Here is the question:

 

A train leaves the station every 12 min. Another train leaves every 9 min. If both trains just left on parallel tracks, when will both leave the station together again? 

 

The answer is 72 but I can't help my son figure out why! Thanks! 

[MamaTX5]

The first train would leave at minutes 12, 24, 36, 48, 60, and 72. The second train would leave at 9, 18, 27, 36, 45, 54, 63, and 72. I don't know why the answer is 72, because they would also leave together at 36 minutes, unless it is asking for the second time they would leave together. (?)

[Chrysalis Academy]

Yeah, it sounds like a Least Common Multiple problem - but why would the answer be 72?? 

 

What curriculum is this from?

[Elisabet1]

You are looking for the lowest common multiple. And 36 is the answer.

[kiana]

As written, I believe the answer key is wrong and the answer should be 36.

 

The only hypothetical possibility I can see for '72' being in the answer key is that they are talking about two single trains -- and thinking of it as train 1 going from station A to station B in 12 minutes, then back to station A in 12 more minutes, so it'd really be leaving every 24 minutes, and similarly for train B. But if this is what they meant, then the question is extremely badly worded.

[MerryAtHope]

I know this is easy but my brain this morning is not high functioning! Here is the question:

 

A train leaves the station every 12 min. Another train leaves every 9 min. If both trains just left on parallel tracks, when will both leave the station together again? 

 

The answer is 72 but I can't help my son figure out why! Thanks! 

 

I also think it's 36. I would solve it this way:

 

Factor each number into prime numbers:

 

9 = 3 x 3

12 = 3 x 2 x 2

 

Count how many times each factor is used per number:

9 has two 3's

12 has one 3, two 2's

 

Use the largest count of factors for each number, and multiply (so use two 3's, not one)

 

3 x 3 x 2 x 2 = 36.

[Steven]

No, the answer is 72, assuming that no trains left the station at 0 minutes. On one track the train leaves at 12, 24, 36, 48, 60, and 72 minutes. On the other track, the trains leave at 9, 18, 27, 36, 45, 54, 63 and 72. But the problems states that "both trains just left on parallel tracks." So the current time is 36 minutes, which is the first time that both trains leave at the same time. The question is "when will both leave the station together again?" The next time they both leave at the same time is at the 72 minute interval. 

[maize]

Agreeing with others that the answer should be 36. Math books do have errors at times. What book is this? Sometimes errata are available on the publisher's website.

[wendyroo]

No, the answer is 72, assuming that no trains left the station at 0 minutes. On one track the train leaves at 12, 24, 36, 48, 60, and 72 minutes. On the other track, the trains leave at 9, 18, 27, 36, 45, 54, 63 and 72. But the problems states that "both trains just left on parallel tracks." So the current time is 36 minutes, which is the first time that both trains leave at the same time. The question is "when will both leave the station together again?" The next time they both leave at the same time is at the 72 minute interval. 

 

Why would we assume that no trains left the station at 0 minutes?

 

We are the ones who are deciding to define minute 0 as just now when the two trains both left the station on parallel tracks, so of course the trains left at minute 0.

 

Since the problem says the trains just left (aka it happened ~now) and then asks "when will both leave the station together again?", it is actually just asking how much time elapses between the events.  They could leave now at "minute 0" and leave together again at "minute 36" or they could leave now at "minute 150" and leave together again at "minute 186".  How we label the times doesn't matter.

 

Wendy

 

[Steven]

Why would we assume that no trains left the station at 0 minutes?

 

We are the ones who are deciding to define minute 0 as just now when the two trains both left the station on parallel tracks, so of course the trains left at minute 0.

 

Since the problem says the trains just left (aka it happened ~now) and then asks "when will both leave the station together again?", it is actually just asking how much time elapses between the events.  They could leave now at "minute 0" and leave together again at "minute 36" or they could leave now at "minute 150" and leave together again at "minute 186".  How we label the times doesn't matter.

 

Wendy

 

Well, I agree the question is poorly written. But the question does not explicitly state that both trains first leave the station at a given time (e.g., 6:00 AM) and then every 12 and 9 minutes thereafter.

 

I am guessing that the problem is trying to illustrate common multiples, and because zero is not a multiple of 12 or 9 and because multiples are not cyclical, but continue to every higher numbers, we don't assume both trains leaving at 0 minutes nor do we keep resetting the elapsed time to zero every time the two trains leave together. 

 

The question would be better formulated if it stated something like: A train station opens its doors at 6:00 AM and Train A leaves the station at 6:12 and thereafter every 12 minutes. Train B leaves the station at 6:09 and then every 9 minutes thereafter. If both trains have just left the station together for the first time, how many minutes after the station opened will both trains again leave at that same time? (72 minutes)

 

Or they could have asked: Passengers Mark and Debby are madly in love with each other and will not leave the other alone on the train platform. Mark rides Train A, which leaves every 12 minutes. Debby rides Train B, which leaves every 9 minutes. They arrive at the station just at both of their trains are pulling out. How long do they have to wait until they both catch a train at the same time? (36 minutes)

[wendyroo]

Well, I agree the question is poorly written. But the question does not explicitly state that both trains first leave the station at a given time (e.g., 6:00 AM) and then every 12 and 9 minutes thereafter.

 

 

Sure it does.

 

"A train leaves the station every 12 min. Another train leaves every 9 min. If both trains just left on parallel tracks, when will both leave the station together again? "

 

​In blue we have an explicit statement that both trains first leave the station at a given time...just now.

 

In red we have a statement that if the first train left just now then in will leave again in 12 minutes and that if the second train left just now that it will leave again in 9 minutes.

 

The question in green is very straight forward.  When will both leave the station together again?  36 minutes from now.  "Just" is the only explicit time the problem gives us to work with, so the only answer that means anything is one given in relationship to "just".

 

Wendy

[Steven]

Sure it does.

 

"A train leaves the station every 12 min. Another train leaves every 9 min. If both trains just left on parallel tracks, when will both leave the station together again? "

 

​In blue we have an explicit statement that both trains first leave the station at a given time...just now.

 

In red we have a statement that if the first train left just now then in will leave again in 12 minutes and that if the second train left just now that it will leave again in 9 minutes.

 

The question in green is very straight forward.  When will both leave the station together again?  36 minutes from now.  "Just" is the only explicit time the problem gives us to work with, so the only answer that means anything is one given in relationship to "just".

 

Wendy

 

Actually, any multiple of 36 would correctly answer the question as you have formulated it, since the problem does not ask when the trains will next leave together, but simply when will both leave the station together. So it would be true to say they will both leave together in 72 minutes (or 108 minutes). 

 

The bottom line is that the question is poorly worded and open to many interpretations. I don't know, but I am guessing that the problem is intended to teach about common multiples (not just least common multiples). Based on that assumption, I offered an explanation of how the authors could come up with 72 as the answer. 

 

If I were explaining this to my daughter, I would do as MamaD4 did by creating a number line: Train A will leave at the 12, 24, 36, 48, 60 and 72 minute marks; Train B will leave at the 9, 18, 27, 36, 45, 54, 63, and 72 minute marks. I would not include trains at zero minutes because zero is not a multiple of 12 or 9.  If the student wants to have the trains leaving the station together at zero on the number line, just say: "Oh, the station just opened, so you need to give the passengers time to get on the trains, so no train leaves at zero minutes. Thus, if we have just seen the two trains leaving together for the first time of the day, we must be at the 36 minute mark, so the next time they will both leave together will be at the 72 minute mark.

 

This will probably satisfy most students, but If it does not, just admit that the wording of the problem is not very clear and that there are many ways to answer this question correctly. You could note that the trains will always leave together at any number that is a common multiple of 12 and 9.* Then you could say, "Here is a better version of the problem with only one correct answer: 

 

The train station opens its doors at 6:00 AM and Train A leaves the station at 6:12 and thereafter every 12 minutes. Train B leaves the station at 6:09 and then every 9 minutes thereafter. If both trains have just left the station together for the first time, how many minutes after the station opened will both trains next leave at that same time? (72 minutes)"

 

*Actually this is not necessarily true. Let's assume that the train station opens at 6:00. Train A first leaves at 6:12 and then every 12 minutes. But Train B first leaves at 6:04 and then every 9 minutes after that. It is still a true statement to say that Train A leaves every 12 minutes and Train B every 9 minutes (thus meeting the condition of the original problem), but their first common departure will not happen for hours. But once they do leave together, then they will continue to leave together every 36 minutes. 

 

[wendyroo]

Actually, any multiple of 36 would correctly answer the question as you have formulated it, since the problem does not ask when the trains will next leave together, but simply when will both leave the station together. So it would be true to say they will both leave together in 72 minutes (or 108 minutes). 

 

The bottom line is that the question is poorly worded and open to many interpretations. I don't know, but I am guessing that the problem is intended to teach about common multiples (not just least common multiples). Based on that assumption, I offered an explanation of how the authors could come up with 72 as the answer. 

 

If I were explaining this to my daughter, I would do as MamaD4 did by creating a number line: Train A will leave at the 12, 24, 36, 48, 60 and 72 minute marks; Train B will leave at the 9, 18, 27, 36, 45, 54, 63, and 72 minute marks. I would not include trains at zero minutes because zero is not a multiple of 12 or 9.  If the student wants to have the trains leaving the station together at zero on the number line, just say: "Oh, the station just opened, so you need to give the passengers time to get on the trains, so no train leaves at zero minutes. Thus, if we have just seen the two trains leaving together for the first time of the day, we must be at the 36 minute mark, so the next time they will both leave together will be at the 72 minute mark.

 

This will probably satisfy most students, but If it does not, just admit that the wording of the problem is not very clear and that there are many ways to answer this question correctly. You could note that the trains will always leave together at any number that is a common multiple of 12 and 9.* Then you could say, "Here is a better version of the problem with only one correct answer: 

 

The train station opens its doors at 6:00 AM and Train A leaves the station at 6:12 and thereafter every 12 minutes. Train B leaves the station at 6:09 and then every 9 minutes thereafter. If both trains have just left the station together for the first time, how many minutes after the station opened will both trains next leave at that same time? (72 minutes)"

 

*Actually this is not necessarily true. Let's assume that the train station opens at 6:00. Train A first leaves at 6:12 and then every 12 minutes. But Train B first leaves at 6:04 and then every 9 minutes after that. It is still a true statement to say that Train A leaves every 12 minutes and Train B every 9 minutes (thus meeting the condition of the original problem), but their first common departure will not happen for hours. But once they do leave together, then they will continue to leave together every 36 minutes. 

 

I would not teach it that way.  Sure, it leads to the answer in the book, but only by spinning an intricate story using made up parameters (we are counting minutes from when the "station opens", trains do not leave immediately upon opening).  As you pointed out in the last paragraph, a different story with different made up parameters would lead to a different answer.  If we count from the "station opening", which isn't an event even alluded to in the actual problem, then we can come up with all sorts of answers depending on what story we tell about when the trains leave.

 

If the next day's problem is:

Bob has 2 pies and he goes out and buys 2 more pies.  How many pies does he have altogether?

 

If the books says the answer is 5, do I want the kids to resort to creative writing (well, maybe his neighbor also gave him one pie and the problem just doesn't mention it) in order to shoehorn the problem into the prescribed answer?  No.

 

I do agree that since the original problem did not ask for the next time the trains leave together, that 72 minutes is a valid answer.  As is 108 and 36 and other multiples.  I think most people would answer 36 minutes, because that is how most people interpret the word again in that context.  If I ask my mom when she is going to see my brother again, I expect she will answer next week or next month, not Christmas 5 years from now even though that answer is equally true.

 

Wendy

[kiana]

I'd much rather teach my kid that the answer book may not always be right than try to spin a story about why it is.

 

Sometimes the answer book is wrong.

[NCJessieRN]

WOW! Thanks so much for all of the replies! My son chose 36 on his quiz and got it wrong and it says the correct answer is 72. We are using Course 1 Mathematics Common Core by Pearson through Connections Academy this year. I was having him make quiz corrections and could not figure out why he got this answer wrong. All of the replies really help and I will ask his teacher to explain further. I was just going crazy trying to figure it out! Thanks again! 

[kiana]

Asking the teacher sounds like a great idea, because if the answer is supposed to be 72, the question is really badly worded -- more of a "guess what I'm thinking" than a math question. 

[Steven]

Why am I not surprised that this is a problem written by the Pearson for the Common Core?!! On her blog, Out in Left Field (http://oilf.blogspot.com), Katherine Beals has been noting numerous badly worded problems in Common Core math materials. For those interested in Common Core math issues, I highly recommend her blog. 

[MrSmith]

I have a question:  Why is 0 not a multiple of 12 or 9?

 

From http://mathforum.org/library/drmath/view/60913.html:

Let x and y be integers.  Then x is a multiple of y if there exists another integer z such that x = y*z.

That is the definition.  Now let x = zero and let y be an arbitrary integer.  Can we find an integer z such that 0 = y*z?  I think you'll see that choosing z = zero will do the trick every time.  So zero is a multiple of every integer.

Did I misinterpret this explanation?

 

 

[kiana]

I have a question:  Why is 0 not a multiple of 12 or 9?

 

From http://mathforum.org/library/drmath/view/60913.html:

Let x and y be integers.  Then x is a multiple of y if there exists another integer z such that x = y*z.

That is the definition.  Now let x = zero and let y be an arbitrary integer.  Can we find an integer z such that 0 = y*z?  I think you'll see that choosing z = zero will do the trick every time.  So zero is a multiple of every integer.

Did I misinterpret this explanation?

 

Nope. Your interpretation is correct. Good catch -- I'd skimmed that post and didn't notice that.

[Steven]

Yes, it is true that zero is a multiple of every number, but zero is not a positive number, so it is not a positive multiple. At the 6th grade level, we are really only dealing with the positive multiples, not the zero-multiple or the negative multiples--otherwise the least common multiple of any two numbers would be zero (or some infinitely large negative number). Can anyone cite a grade school level book that specifically teaches zero as a multiple in its coverage of multiples and least common multiples? Or even mentions this possibility?  Personally I would be surprised it there is. For example, the glossary in Singapore Math defines lowest common multiple as "the smallest number that is a common multiples of two numbers. It then list the multiples of 2 starting with 2 (not zero) and of 4 starting with 4. Finally it identifies 4, 8, and 12 as common multiples and 4 as the lowest common multiple. 

 

Do not mistake me--I am by no means defending this poorly worded problem. I think it is an example of the poorly constructed and test materials being pushed on students in the name of Common Core. But it think it is reasonable to point out that with the restricted level of mathematical information typically given to 6th graders, 72 is a feasible answer for this problem and could realistically be the answer the author intended. When we covered multiples in Singapore Math, for instance, the book did not include zero as a multiple. Nor did the book go into a discussion of why zero is not the least common multiple of any two numbers. Rather the book assumes that the student is dealing with positive multiples. I think the author of this problem is making the same assumption. 

[kiana]

First, I would still not say "0 is not a multiple of 9 and 12" but rather "0 is not a positive multiple of 9 and 12" in my explanation of the problem. I'm surprised that Singapore Math doesn't just say "the smallest positive number" as this definition would be mathematically correct regardless of whether you assume negative numbers have been introduced or not. However, I don't have a copy available to check for myself. 

 

Second, I still do not think that excluding 0 is defensible. It states that "both trains have just left". It does not state that the station has just opened. If, in your hypothetical explanation of the problem, we are at 36 minutes, then 72 minutes later will be 36 minutes from now. 

[Cosmos]

I don't think it's a badly worded question at all. A crucial skill is learning how to translate word problems into mathematical language and vice versa. The problem says nothing about multiples or zero or clocks. We who are solving the problem have to impose our own mathematical structure and then solve the problem. Then we have to go back to the problem itself to be able to answer the question within the problem's own context. The only event mentioned in the problem is the time when both trains left at the same time. We could call that time t = 0. Or t = 36, or t = -57. Any of those are fine, though some are more conventional choices than others. Say we choose t = 2 for when the trains left simultaneously. I think everyone agrees that the next time the trains leave simultaneously will be 36 minutes later. So in my frame of reference, that would be at t = 38. But to answer the question, I MUST translate that answer into the language of the problem. There is no t = 0 in the statement of the problem. I MUST state my answer in reference to the only known event of the problem, i.e. "both trains just left". So my answer must be: "36 minutes after they first left together", or "36 minutes from now". Even an answer of 36 is somewhat incomplete.

[kiana]

I don't think it's a badly worded question at all. 

 

Oh, I think it's badly worded if they're trying to form a question which results in an answer of 72. 

 

If the answer key is just wrong, then I don't see any issues. 

[wendyroo]

Rather the book assumes that the student is dealing with positive multiples. I think the author of this problem is making the same assumption. 

 

I don't particularly care if they accept 72 as a valid answer, but 36 is equally correct and arguably shows a higher level of understanding, so there is no justification for marking that incorrect.

 

​Whether or not they want to acknowledge that zero is a common multiple does not change the fact that it is.  Not that I understand your reasoning that you would only reach the answer 36 is you knew that zero was a common multiple.

 

Say we ignore zero...

 

9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108

12: 12, 24, 36, 48, 56, 60, 72, 84, 96, 108

 

So since we know that both trains just left together, let's start counting from a common positive multiple - let's choose 72 (because there is nothing preventing us from choosing any common multiple we want).

So 72 minutes is right now when they leave together.  When will they leave together again?  36 minutes from now.

 

Wendy

[Steven]

Oh, I think it's badly worded if they're trying to form a question which results in an answer of 72. 

 

If the answer key is just wrong, then I don't see any issues. 

I completely agree. Unfortunately, I believe that the author was indeed trying to form a question which results in the answer 72 and would mark an answer of 36 as wrong. And as I have already pointed out, any (positive) multiple of 36 correctly answers the question as written.

 

I think the very first response to  this question showed the thinking of the author: the author is (in my opinion) thinking of elapsed time, the trains leave at 9, 18, 27, etc. and 12, 24, 36, etc. He observes the train leaving together at 36 minutes of elapsed time and foresees that they will again leave together at an elapsed time of 72 minutes. 

 

I am not saying that this is the only way to read the problem, so an answer of 36--or any multiple of 36-- is (in my opinion) not wrong. But I do not think it is the answer the author is looking for and I do not think that the author would acknowledge an error in the answer book. But, I tend to be cynical.