The thinking skills necessary for this at first seem nominal- but as one delves deeper into the scenario, it ends up really making a student think and apply vocabulary, order of operations thinking and logic. The answer we got through algebrafication is 5*5*7.
I got the same answer as Lori - the twins are 5 years old and the other sister is 7. I also agree that at first glance this problem seemed like it would be really easy to solve. But upon closer inspection it took a little bit of thinking. I was thinking at first that I needed to double the ages of the twins and then multiply it by the other sister's age. So I was going to say there couldn't be a solution since anything doubled would be even and then an even number times any other number would also be even so the product could NOT be 175 (odd). But then I realized that 175 is the product of all of their ages and since there are twins that meant a SQUARE NUMBER not double a number! So then I listed factor pairs of 175 until I got to one with a perfect square (7 x 25). ?
The third sister is between 2 weeks old and 40 years old with not enough information to determine exactly what their age is right now.
Longer answer:
In thinking about this problem and reading responses I wonder how often we adults fall under the assumption trap. When we read, "....the product of their ages ..." most of us immediately think of integer values and yet, other than that exact moment of time every year we were born, our age is always a rational number. With that in mind, I offer the following response to the question.
Assuming the twins were the first number given (and notice I fall back to integers for simplicity) I offer the second number as the age of the third sister.
Twins = 1 3rd sister = 175 (obviously this has not happened in any recorded modern time so I have to subjectively throw this combination out)
Twins = 2 3rd sister = 43 years 9 months old (possible, but mom pushing the biological clock with those twins! Still not a very likely possibility)
Twins = 3 3rd sister = 19 years 5months 10 days old (finally a range that seems 'normal')
Twins = 4 3rd sister = 10 years 11 months 8 days old
Twins = 5 3rd sister = 7 (as has already been shown to be the answer most likely given to this problem)
Twins = 6 3rd sister = 4 years 10 months and 8 days old (for some reason we always assume the 3rd sister is older, don't we?)
Twins = 7 3rd sister = 3 years 6 months and 24 days old
Twins = 8 3rd sister = 2 years 8 months and 23 days old
Twins = 9 3rd sister = 2 years 1 month and 27 days old
Twins = 10 3rd sister = 1 year 9 months old
Twins = 11 3rd sister = 1 year 5 months 10 days old
Twins = 12 3rd sister = 1 year 2 months 17 days old
Twins = 13 3rd sister = 1 year and almost a half month old
Twins = 14 3rd sister = 10 months 20 days old (how much longer can this go on ? )
Twins = 15 3rd sister = 9 months 10 days (seriously... does this keep going on...)
Twins = 16 3rd sister = 8 months 6 days old
ok lets just jump to something outrageous to see if this might end at some point
Twins = 30 3rd sister = 2 months 10 days old (at this point the biological clock is ticking on this 3rd kid, isn't it?)
Twins = 40 3rd sister = 1 month 8 days old
Twins = 50 3rd sister = 23 and half days old
Twins = 60 3rd sister = 16 days old (at this point mom would be in her 70s for this conception so we probably need to stop)
So I would say that the twins would range from 1 to 60 years old with 50 being a more likely maximum (of course this assumes much on the lower end as far as when does age start ....) The question is, of course, asking for the age of the 3rd one in which case I might be a bit befuddled about how low to start. Maybe 2 weeks old to around 40?
Do you see how some of us can't take standardized tests? I would certainly have answered this question as I have done so above and with most assessment algorithms, I would have likely been deemed incorrect. In our efforts to contrive "meaningful" math problems we end up neglecting the contextual meaning of our questions, specifically in thinking about how people may process the scenario first and then how they might look at the math within. For example, in the problem given the concept of age and "How old is someone" really is an assumption that I feel is unfair to students. From early on in grade school they start playing around with, "I am 8 and A HALF!" or, " I am practically 11 because I am 10 years 11 months old..."
Does this problem get fixed by putting in something like, "Assuming we measure their ages in integer values only..."?
In a bigger picture question, after observing thousands of math questions that immediately get me questioning the context before ever looking at the math, I wonder how we mitigate students' current need to differentiate between real math applications where they benefit from using what they learn from "...this is an assessment environment in which you need to throw common sense or what you know of a situation out the window to better focus on what skill you think the test maker is trying to get you to show?" As a student and as a teacher, this aspect of mathematical assessment has always caused frustration for me.
I am the same way. Recently, I was taking the GRE, and during the math portion, I ran out of time because I was thinking - what is the point of this question what are they trying to ask me? Do they want one answer because it could have multiple answers? Because I was focused on what I noticed and what do I wonder, I ran out of time. So I understand your dilemma - but my try it Tuesdays are for fun and not to stress you out. I hope you had some fun! Thank you for sharing your thoughts.
This problem is a good example of how reading is as much a part of problem solving as math. Once, the words product and sum are recognized and the impact of the word twins - the formula is set. (x*x) * y = 175. Then the order of operations takes over. Thanks for sharing your wisdom everyone!
Maureen, I agree! These problems, along with probability problems that state things like of no greater than 54, which means less than or equal to 53 - seem to confuse me. My stubborn brain has to convince itself that the wording and the math states the same thing. It is hard for this native speaker.
Comments
The thinking skills necessary for this at first seem nominal- but as one delves deeper into the scenario, it ends up really making a student think and apply vocabulary, order of operations thinking and logic. The answer we got through algebrafication is 5*5*7.
I got the same answer as Lori - the twins are 5 years old and the other sister is 7. I also agree that at first glance this problem seemed like it would be really easy to solve. But upon closer inspection it took a little bit of thinking. I was thinking at first that I needed to double the ages of the twins and then multiply it by the other sister's age. So I was going to say there couldn't be a solution since anything doubled would be even and then an even number times any other number would also be even so the product could NOT be 175 (odd). But then I realized that 175 is the product of all of their ages and since there are twins that meant a SQUARE NUMBER not double a number! So then I listed factor pairs of 175 until I got to one with a perfect square (7 x 25). ?
Short answer:
The third sister is between 2 weeks old and 40 years old with not enough information to determine exactly what their age is right now.
Longer answer:
In thinking about this problem and reading responses I wonder how often we adults fall under the assumption trap. When we read, "....the product of their ages ..." most of us immediately think of integer values and yet, other than that exact moment of time every year we were born, our age is always a rational number. With that in mind, I offer the following response to the question.
Assuming the twins were the first number given (and notice I fall back to integers for simplicity) I offer the second number as the age of the third sister.
Twins = 1 3rd sister = 175 (obviously this has not happened in any recorded modern time so I have to subjectively throw this combination out)
Twins = 2 3rd sister = 43 years 9 months old (possible, but mom pushing the biological clock with those twins! Still not a very likely possibility)
Twins = 3 3rd sister = 19 years 5months 10 days old (finally a range that seems 'normal')
Twins = 4 3rd sister = 10 years 11 months 8 days old
Twins = 5 3rd sister = 7 (as has already been shown to be the answer most likely given to this problem)
Twins = 6 3rd sister = 4 years 10 months and 8 days old (for some reason we always assume the 3rd sister is older, don't we?)
Twins = 7 3rd sister = 3 years 6 months and 24 days old
Twins = 8 3rd sister = 2 years 8 months and 23 days old
Twins = 9 3rd sister = 2 years 1 month and 27 days old
Twins = 10 3rd sister = 1 year 9 months old
Twins = 11 3rd sister = 1 year 5 months 10 days old
Twins = 12 3rd sister = 1 year 2 months 17 days old
Twins = 13 3rd sister = 1 year and almost a half month old
Twins = 14 3rd sister = 10 months 20 days old (how much longer can this go on ? )
Twins = 15 3rd sister = 9 months 10 days (seriously... does this keep going on...)
Twins = 16 3rd sister = 8 months 6 days old
ok lets just jump to something outrageous to see if this might end at some point
Twins = 30 3rd sister = 2 months 10 days old (at this point the biological clock is ticking on this 3rd kid, isn't it?)
Twins = 40 3rd sister = 1 month 8 days old
Twins = 50 3rd sister = 23 and half days old
Twins = 60 3rd sister = 16 days old (at this point mom would be in her 70s for this conception so we probably need to stop)
So I would say that the twins would range from 1 to 60 years old with 50 being a more likely maximum (of course this assumes much on the lower end as far as when does age start ....) The question is, of course, asking for the age of the 3rd one in which case I might be a bit befuddled about how low to start. Maybe 2 weeks old to around 40?
Do you see how some of us can't take standardized tests? I would certainly have answered this question as I have done so above and with most assessment algorithms, I would have likely been deemed incorrect. In our efforts to contrive "meaningful" math problems we end up neglecting the contextual meaning of our questions, specifically in thinking about how people may process the scenario first and then how they might look at the math within. For example, in the problem given the concept of age and "How old is someone" really is an assumption that I feel is unfair to students. From early on in grade school they start playing around with, "I am 8 and A HALF!" or, " I am practically 11 because I am 10 years 11 months old..."
Does this problem get fixed by putting in something like, "Assuming we measure their ages in integer values only..."?
In a bigger picture question, after observing thousands of math questions that immediately get me questioning the context before ever looking at the math, I wonder how we mitigate students' current need to differentiate between real math applications where they benefit from using what they learn from "...this is an assessment environment in which you need to throw common sense or what you know of a situation out the window to better focus on what skill you think the test maker is trying to get you to show?" As a student and as a teacher, this aspect of mathematical assessment has always caused frustration for me.
I am the same way. Recently, I was taking the GRE, and during the math portion, I ran out of time because I was thinking - what is the point of this question what are they trying to ask me? Do they want one answer because it could have multiple answers? Because I was focused on what I noticed and what do I wonder, I ran out of time. So I understand your dilemma - but my try it Tuesdays are for fun and not to stress you out. I hope you had some fun! Thank you for sharing your thoughts.
This problem is a good example of how reading is as much a part of problem solving as math. Once, the words product and sum are recognized and the impact of the word twins - the formula is set. (x*x) * y = 175. Then the order of operations takes over. Thanks for sharing your wisdom everyone!
Maureen, I agree! These problems, along with probability problems that state things like of no greater than 54, which means less than or equal to 53 - seem to confuse me. My stubborn brain has to convince itself that the wording and the math states the same thing. It is hard for this native speaker.
So, I *started* with assuming the ages are integers. So if the product is 175, welp, 25 is a perfectly good square and that times 7 is 175, so boom.
However, is that the only answer? What if they aren't integers? Infinite possibilities 'cause I just square one age and divide 175 by the answer.