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DC is on rates and unit rates, something I don't really fully understand. I mean, I can often "figure out the problems" but I have a feeling I'm missing something with this problem. DC will get some help at school tomorrow but wanted me to try to explain it to her tonight. Added bonus is that DH and I are at odds over how to solve this problem. Give it a shot! Extra points for those able to explain the method  especially if the method relates to the subject of rates/unit rates.Â
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"Kevin can walk 2/3 of a mile in 2/5 of an hour. He lives 1 mile from school. Us this information to complete the statements. At this rate, Kevin can walk X miles in 1 hour. At this rate, it will take Kevin X hours to walk to school."Â
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Here's how I'd solve it:
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If Kevin can walk 2/3 of a mile in 2/5 of an hour, how many miles can he walk in 1/5 of an hour? 1/5 is half of 2/5, so I will divide the amount he can walk in 2/5 of an hour (2/3 mile) by 2. 2/3 divided by 2 is 1/3.
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If Kevin can walk 1/3 of a mile in 1/5 of an hour, how many miles can he walk in 1 hour? How many fifths are in hour? 5. So I would multiply the amount he can in 1/5 of an hour (1/3 of mile) by 5. 1/3 times 5 is 5/3 miles or 1 and 2/3 miles.
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So the answer to the first statement is Kevin can walk 1 and 2/3 miles in 1 hour.
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Since it's 1 mile to school, how many 1/3 miles is that? 3. So I would take the amount of time that it takes to walk 1/3 mile (1/5 of an hour) and multiply that by 3. 1/5 times 3 is 3/5 hours.
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So the answer to the second statement is At this rate, it will take Kevin 3/5 of an hour to walk to school.
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I love these kinds of problems.
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It's YOU!! Love it! Â Ok, off to read how you did it...Â
Ok, I've read it  it's official  unit rates are totally over my head. And, HHM, you have just sided with my DH, thankyouverymuch.Â
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We may be keeping this between the two of us (and all of MDC).Â Â I will, however, be sharing your explanation to DC.Â
ETA I just say the first question. 1/3 of a mile per 1/5 hour so 5/3 miles or 1 2/3 miles
Ding, ding, ding!! We have a winner. THIS I understand.Â
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This question was just so funky for me. I wanted to go to converting the fractions to decimals but then I got really confused... Â Ok, keeping in mind that this is NOT my homework. Â But, it wasn't really DC's either. It was school work that the teacher didn't finish going over and he asked the kids to take the work home. I don't think he realized who much DC didn't understand the lesson and I'm SURE it would have been fine for DC to go in with it unfinished because she didn't understand.Â
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But, you know what happened? DH and I just got totally obsessed with trying to figure it out. It was pretty hysterical. CLEARLY he and I need a break from family life. Good thing we're going way tomorrow  woot!Â
But wait  is it just a coincidence that the answer is 3/5 and 5/3? Â Yes, exposing my utter lack of mathematical intuition. My poor children. ;) Â Ok, and one more funny thought  I am the one who posted that whole thread about what constitutes doing your child's homework. I got a lot of good responses but the best one would have been, "As if!"Â
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Wait  one more question. Why did you both know to stick in the fractions realm? Â I asked my brother and he spit out an answer in like 30 seconds that I think was "right" but it was in numbers/decimals. It was something like 1.6/.4 or something like that. Would that have been "correct" or is math at this stage very much about sticking to the method? Â (I do realize that this is a better question for the teacher but I'm looking for a more general answer about math education/principals). Â
My method is to divide distance (2/3mi) with time (2/5hr) to get miles/hr.
When you're dividing fractions, you just flip the divisor and multiply it.
So that is:
2/3 x 5/2 = 5/3mi/hr
so 5mi/3h = 1mi/x hr
x= 3/5hr or 36min
My guess is that this may be what DC is taught today in school. Â I'll tell her about your way of approaching the problem right now  thanks! Â
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Give me a few minutes while I caffeinate.
(Your brother is wrong.Â Well, your brother was close, but the teacher probably wants the exact answer.)
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I can't resist.Â I love explaining these things. Â It is NOT a coincidence that the 3/5 and the 5/3 popped up.Â The problem was carefully designed so that things cancel to make the solution faster.Â This problem is actually easier to solve if you DON'T simplifiy or convert into decimals.Â The way I solve it is to keep track of the units. That way, I don't really even have to think.
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Here is my attempt at explaining:
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Rate needs to end up in miles per hour.Â So, we have 2/3 mile, and we have 2/5 hour.Â Because we want to end up with miles per hour, we probably want to divide the thing with the mile, by the thing with the hour. So that is why you need to take the 2/3 and divide by the 2/5.
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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/3) mileÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/3)Â Â Â Â Â Â Â mileÂ Â Â Â Â Â Â Â Â 2 * 5Â Â Â Â Â Â Â Â Â mileÂ Â Â Â Â Â Â Â Â Â Â 5Â Â Â Â mile
Rate =Â  =Â Â Â Â Â Â Â = Â Â Â  =Â Â Â Â Â Â 
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/5) hrÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (2/5)Â Â Â Â Â Â Â Â Â hrÂ Â Â Â Â Â Â Â Â Â Â 3 * 2Â Â Â Â Â Â Â Â Â Â hrÂ Â Â Â Â Â Â Â Â Â Â Â 3Â Â Â Â Â hr
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Great!Â The rate ended up in mph!
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Next, to figure out the time.
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Well, time needs to end up in hours.Â We've got something in miles per hour (rate), and we've got something in miles (distance).Â So if we take the something in miles, and divide it by the something in miles per hour, we will get hours.Â So that is why it makes sense to use
time = distance / rate:
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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â
Â Â Â Â Â Â Â Â Â Â Â Â Â distanceÂ Â Â Â Â Â Â Â 1 Â Â Â mile Â Â Â Â Â Â Â Â Â Â Â Â Â (1 mile) (3 hr)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3
time = Â =Â Â Â Â Â Â = Â Â Â =Â Â Â Â Â hr
Â Â Â Â Â Â Â Â Â Â Â Â Â Â rateÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 5Â Â Â Â Â Â Â mileÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â (5 mile)Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 5
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3Â Â Â Â Â Â Â Â Â hr
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But because we think in minutes, instead of in hours, you now have the question:Â How many minutes isÂ time = 3/5 hr ?Â
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Well, we know that 60 minute = 1 hour, so you can just take your time in hours (the 3/5 hr), and multiply the numerator and denominator by the same time interval:
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Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 3Â Â Â Â hrÂ Â Â Â Â Â Â Â Â Â Â Â 60 minÂ Â Â Â Â Â Â Â Â Â Â Â 3 x 12 Â hr x min
time =Â Â Â Â Â Â  x Â Â =Â Â Â Â   Â = 36 minutes
Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 5Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1 hrÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 1 x 1Â Â Â Â Â Â Â Â Â hr
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BTW, to figure out the rate (rate = 5/3 mph) in decimals, you just take the 5 and divide it by 3, and you get rate = 1.67 mph.Â
Â Â Â Â Â Â Â Â Â But I assume the teacher would prefer to be given the answer in the form of rate =Â 5/3 mph.
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I apologize if this is as clear as mud, but I promise you that it is 100% correct.Â
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By now, this answer probably comes to you too late, but just think how smart you'll sound when your dd gets home from school this afternoon!Â
Watch out, ya'll...my DC has like 8 more years of math and MDC members are my new favorite resource. Screw Kahn Academy (j/k, I love him too!). Â
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Wait  one more question. Why did you both know to stick in the fractions realm? Â I asked my brother and he spit out an answer in like 30 seconds that I think was "right" but it was in numbers/decimals. It was something like 1.6/.4 or something like that. Would that have been "correct" or is math at this stage very much about sticking to the method? Â (I do realize that this is a better question for the teacher but I'm looking for a more general answer about math education/principals). Â
My guess is that your brother probably spit out 0.6 / 1.67.Â Your brother probably saw the 5/3, and converted it to decimals, which is 1.67, and he probably saw the 3/5, and converted it to decimals, which is 0.6.Â Then it looks like for some reason he mistakenly thought that you have to divide these two numbers, so that is where the 0.6 / 1.67 came from.Â (That's actually wrong.)
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But how did we know to keep it all in fractions and not to mess with the decimals?Â
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Two reasons.
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REASON 1.
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Well, very few people can do 0.6 / 1.67 in their head.Â They have to resort to fractions or you have to resort to a calculator. (Many people have a different opinion, which is fine, but I happen to believe that reliance on a calculator is an unhealthy thing.Â I prefer to resort to fractions.)Â If you have fractions, you can cancel things out until you cancel it down to something simple that you can do in your head.
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For example, if I see 0.6 / 1.67, this is what I do.Â I think to myself that 1.67 is the same as 5/3, and that 0.6 is the same as 6/10.Â So then when I want to calculate 0.6 / 1.67, this is what I do:
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0.6 Â Â Â Â Â Â (6/10) Â Â Â 6x3 Â Â Â Â Â Â (2x3) x 3Â Â Â Â Â Â Â Â 9
 =Â Â Â Â  = Â =Â Â  =Â Â Â .Â
1.67 Â Â Â Â Â (5/3) Â Â Â Â 5 x 10 Â Â Â Â Â Â 5x(2x5) Â Â Â Â Â 25 Â Â Â Â Â Â
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Now, what is 9/25Â in decimals?Â Well, if you have 1/25 of a dollar, you have four cents.Â So if you have 9/25 of a dollar, you have 36 cents, which is the same as $0.36 .Â So 9/25 = 0.36
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So 0.6/1.67 = 0.36
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REASON 2.
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Remember I said that your brother saw the 5/3 and said it is the same as 1.67?Â Well that's not entirely true.Â it's close to 1.67, but of course, 5/3 is actually 1.66666666666
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So when your brother converted to decimals, that guarantees that he is not going to get the exact answer, just close to the exact answer.Â And for most real life situations, that will be just fine.Â (But it won't be exactly correct for those people who care about that kind of thing. This may include the teacher.)
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Now you will really sound smart when your daughter gets home from school :D
Yepâ€”what they said. Just rest assured that the math only gets harder. I did this one in my head. I did have to stop and think about it, but I've had practice. Dd1 is in 7th grade now and this was pretty easy compared to her homework, most of which I can no longer do in my head at all.
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Fractions are both easier and more accurate in this case than decimals since you're dealing with 2/3rds which can't be expressed as a simple decimal. You have to put one of those lines over the Â repeating digit in the decimal form to be accurate with 1.6666666666666666666...
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Dd1 has a problem of the week each week which is often a logic problem. This problem reminds me of that, but a bit more straightforward. Here's one of her problems of the week that I remember. It's a classic logic puzzle, so if you've seen it before you know the answer.
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Ok, you have 3 drawers of socks labeled "white", "black", and "mixed" (meaning, both white and black), but someone has played around with the labels andÂ all the drawers are now labeled wrong. Your job is to close your eyes and pick one sock out of one drawer and then open your eyes, look at the sock you pulled out, and relabel the drawers correctly. You can't look in any of the drawers. How do you do this?
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In class these past few weeks they've been doing volume, such as volume of a square based pyramid vs a triangle based pyramid as well as rates and conversions. I am not an exceptionally mathy person, so DH usually helps her with this stuff.
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"All you fascists are bound to lose" â€” Woody Guthrie
I took algebra in college not TOO long ago (like maybe 4 years ago) so when DC gets to that maybe I'll be back on track. I swear, I don't remember anything like what she's learning now. I spoke with another parent who has a child who switched from the same school as my DC and she said that this child is just not starting to click with this school's curriculum but only by working with her parents much more than my DC does now. I tried to do some work again with DC tonight and it was over my head again. ;) Â At least she says she understood it. Fun, fun, fun!Â
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