When the wrong conversion factor is used.

[daijobu]

 

There are pts in gallon.  How many gallons are in pts?

 

The way I teach this is to start with:

 

 

And then you can create different conversion factors:

 

Divide both sides by  to get

 

 

 

and so

 

Similarly, another conversion factor can be generated:

 

 

Then I tell students that newbies will sometimes select the wrong conversion factor, generating something that looks like this:

 

  which does not answer the question, but it is technically correct.  

 

My question is how do we refer to these quantities with strange units, that aren't of use to anybody?  Do we have a name for them?  

[Miss Tick]

We had a similar discussion last night starting with something like 1 hot potato per ( Hand width x (blink of an eye) squared) and converting to Pascals and the myriad ways it can go wrong. If there isn't an extent name for these quantities, perhaps we can come up with our own!

[Jean in Newcastle]

In logic, it would be a non sequitor because the answer "does not follow" (ie. answer) the actual question.  The question asked was how How many gallons are in 48 pints.  The answer given is How many pints are in one gallon. 

[Dianthus]

The answer is pt ^2 so should indicate it's incorrect. I stress ending with the correct units.

I would teach this way, to make equivalent ratios and solve for the unknown.

8 pt / 1 gal = 48 pt / x gal

Then cross multiply

(8 pt * x gal) = (1 gal *48 pt)

X = 48/8 = 6

 

Or we do unit multipliers and cancel units until we have desired

Cm to mile is 

1 cm * (1 in/ 2.54 cm) * (1 ft/ 12 in)*(1 mi/5280 ft) . 

So for yours:

48 pt * (1 gal/ 8 pt). The pts cancel and left with 6 gal.

[EKS]

I think there is something wrong with the example you gave above.

We want to know how many gallons are in 48 pints.  If I was to do this using conversion factors, I would do the following:

We have 48 pints, and there is 1 gallon per 8 pints.

48 pints x 1 gallon/8 pints = 6 (pints)(gallons)/(pints)

You cancel pints and are left with 48/8 gallons = 6 gallons.

(pints)(gallons)/(pints) is what I would call an "intermediate."  My students would probably never see it because they would do the canceling before they did the multiplication.  

Before explaining how to manipulate units in this way, I would be sure that students had an intuitive understanding of how to do these sorts of problems as well as facility with doing them using cross multiplication.

[Not_a_Number]

On 10/13/2022 at 10:18 AM, EKS said:

Before explaining how to manipulate units in this way, I would be sure that students had an intuitive understanding of how to do these sorts of problems as well as facility with doing them using cross multiplication.

Yes, I'd also emphasize this. 

[FreyaO]

On 10/13/2022 at 10:18 AM, EKS said:

Before explaining how to manipulate units in this way, I would be sure that students had an intuitive understanding of how to do these sorts of problems as well as facility with doing them using cross multiplication.

I have never taught this, so I don't really have any advice for the OP. But I have met multiple graduate students with STEM degrees who lack exactly this intuitive understanding... it's really disheartening.

[EKS]

3 hours ago, FreyaO said:

I have met multiple graduate students with STEM degrees who lack exactly this intuitive understanding... it's really disheartening.

For anyone who is wondering what it means to have intuition about this, here is an example of what I'm talking about:  We want to know how many gallons are in 48 pints.  Gallons are bigger than pints; therefore, the number of gallons will be less than 48.  Ah ha! the student with this thought should think.  One gallon is 8 pints.  How many groups of 8 pints are in 48 pints?  I shall divide 48 by 8 to find out!

I used to manage a biochemistry lab that employed lots of recent biology, biochemistry, and chemistry grads.  It was extremely unusual to find one who could figure out unit conversions or dilution factors at all.  Of the ones who could, the vast majority relied on an inflexible algorithmic approach that had to be written down.  These folks were usually unable to detect when they had made a mistake.  For example, if they were asked to dilute a stock solution of something at 10 mg/mL to 1 mg/mL (a 1/10 dilution, or one part stock solution in ten parts total), they would not see anything amiss if they were to come up with needing to perform a 1/10,000 dilution.

I can think of only two people who, fresh out of college, were able to do the sort of thing I described above with the gallons and pints problem. 

Edited October 15, 2022 by EKS

[Not_a_Number]

2 hours ago, EKS said:

I used to manage a biochemistry lab, and the vast majority of recent grads with biology degrees (and a significant number of folks with chemistry and biochemistry degrees) had no clue.  

For anyone who is wondering what it means to have intuition about this, here is an example of what I'm talking about:  We want to know how many gallons are in 48 pints.  Gallons are bigger than pints; therefore, the number of gallons will be less than 48.  Ah ha! the student with this thought should think.  One gallon is 8 pints.  How many groups of 8 pints are in 48 pints?  I shall divide 48 by 8 to find out!

I used to manage a biochemistry lab that employed lots of recent biology, biochemistry, and chemistry grads.  It was extremely unusual to find one who could figure out unit conversions or dilution factors at all.  Of the ones who could, the vast majority relied on an inflexible algorithmic approach that had to be written down.  These folks were usually unable to detect when they had made a mistake.  For example, if they were asked to dilute a stock solution of something at 10 mg/mL to 1 mg/mL (a 1/10 dilution, or one part stock solution in ten parts total), they not see anything amiss if they were to come up with needing to perform a 1/10,000 dilution.

I can think of only two people who, fresh out of college, were able to do the sort of thing I described above with the gallons and pints problem. 

Yep. I had exactly the same kinds of issues when teaching college calculus. It was really disheartening. 

It had nothing to do with units, either. Kids would NOT be able to think about a number like 1/0.0001 without actually calculating it out. (Is it big? Small? Positive? Negative?) This meant that there was no intuition at all when it came to things like asymptotes for functions, and people had to rely on very inflexible algorithms. 

I've been teaching elementary-aged kids for a while now, including kids who aren't mine and aren't at all mathematically gifted, and I am absolutely sure that with good teaching and without actual disabilities, kids WILL have intuitions about things like that. We don't teach it like that, and it's a crying shame. 

[EKS]

1 minute ago, Not_a_Number said:

Kids would NOT be able to think about a number like 1/0.0001 without actually calculating it out.

With a calculator!  No thinking something like how many 0.0001s are in 1?  A whole lot!  Oh and 0.0001 is 1/10,000?  Hmmm...maybe there are 10,000 of them!

[Not_a_Number]

Just now, EKS said:

With a calculator!  No thinking something like how many 0.0001s are in 1?  A whole lot!  Oh and 0.0001 is 1/10,000?  Hmmm...maybe there are 10,000 of them!

That would require realizing that the quantity 1/0.0001 actually answers a QUESTION! 

Some of them could do it without a calculator! They could use a memorized algorithm. But they couldn't ballpark the answer at all, which of course made the point I was making fall really flat (we were talking about something like the function 1/x near 0....) 

[Not_a_Number]

Wow, this discussion is reminding me why I'm homeschooling my girls 😂. DD10 has been required to do this kind of thinking her whole life, and she's now pretty much into high school math and is continuing to be mindful and intuitive about it. 

[mom31257]

@Not_a_Number, dd said no one in her chemistry for nursing class could do math in their heads. She thanked me so much for making sure she could think! 

I teach conversions by cancelling units using conversion factors as @EKSshowed above. I show kids how you want the desired unit to be in the numerator of the last fraction and all other units need to be cancelled. 

 

 

[EKS]

16 minutes ago, mom31257 said:

I teach conversions by cancelling units using conversion factors as @EKSshowed above. I show kids how you want the desired unit to be in the numerator of the last fraction and all other units need to be cancelled.

That is definitely something that kids should learn, but it is also really good for them to have the sense of whether the wanted number is going to be more or less than the original number--at least for easy problems like the one presented here.  

[wendyroo]

1 hour ago, EKS said:

That is definitely something that kids should learn, but it is also really good for them to have the sense of whether the wanted number is going to be more or less than the original number--at least for easy problems like the one presented here.  

The more or less game is something I start playing with my young kids as soon as they understand measurement and have been introduced to various units.

Is it more miles to the library or feet to the library? Once they realize that it would be more feet, we look it up to practice with numbers: it is about one and a half miles, so that is about 8000 feet. And then the vital question, would it take longer to walk one and a half miles or 8000 feet? It is hard for young brains to accept the idea that one and half miles could be the same as 8000 feet...but that is the crucial concept.

We play the same game with all sorts of units. How many cups vs. Tbs to fill a glass. How many grams vs. ounces to balance with our apple. Eventually, how many square cm vs. square inches to cover the tablet screen.

I think older students who struggle with conversions might still, on some deep conceptual level, not fully accept or intuit that the converted units represent the same thing, the same reality, just expressed in two different ways.

Edited October 16, 2022 by wendyroo

[FreyaO]

19 hours ago, EKS said:

For anyone who is wondering what it means to have intuition about this, here is an example of what I'm talking about:  We want to know how many gallons are in 48 pints.  Gallons are bigger than pints; therefore, the number of gallons will be less than 48.  Ah ha! the student with this thought should think.  One gallon is 8 pints.  How many groups of 8 pints are in 48 pints?  I shall divide 48 by 8 to find out!

I used to manage a biochemistry lab that employed lots of recent biology, biochemistry, and chemistry grads.  It was extremely unusual to find one who could figure out unit conversions or dilution factors at all.  Of the ones who could, the vast majority relied on an inflexible algorithmic approach that had to be written down.  These folks were usually unable to detect when they had made a mistake.  For example, if they were asked to dilute a stock solution of something at 10 mg/mL to 1 mg/mL (a 1/10 dilution, or one part stock solution in ten parts total), they would not see anything amiss if they were to come up with needing to perform a 1/10,000 dilution.

I can think of only two people who, fresh out of college, were able to do the sort of thing I described above with the gallons and pints problem. 

Maybe we are the same person, because that's exactly my experience. And when you have protocols with instructions in nM, but a solution stock in mg/dL..... What this young adult does may mean weeks of work down the drain, thousands of dollars wasted. After the first couple of students, I realized it's a systemic problem, so I sent them home and gave them a weekend to find the right answer, without much luck. That was even more disheartening. Maybe they felt under pressure in my presence, but they were also not motivated or able to think with plenty of time provided.

[EKS]

1 hour ago, FreyaO said:

Maybe we are the same person, because that's exactly my experience. And when you have protocols with instructions in nM, but a solution stock in mg/dL..... What this young adult does may mean weeks of work down the drain, thousands of dollars wasted. After the first couple of students, I realized it's a systemic problem, so I sent them home and gave them a weekend to find the right answer, without much luck. That was even more disheartening. Maybe they felt under pressure in my presence, but they were also not motivated or able to think with plenty of time provided.

I finally started asking job candidates during their interview with me two questions:

1. What is a difference between DNA and protein? It can be any difference.
2. How would you prepare one liter of a 1 M sodium chloride solution?  The formula weight of sodium chloride is 58.

You would not believe how many biology majors could not answer the first question.  I mean, seriously?  Biochemistry majors had a better time with it.  But I would say that overall maybe half of the candidates would get this one. 

Only a handful of people were able to answer the second question.  This was out of dozens of candidates over the years.  For anyone wondering, the answer is to put 58 grams of sodium chloride into 1 liter of water (well, it's a little more complicated than that--the final volume needs to be 1 L, but I would have accepted anything along these lines).

This was in the 1990s.  I wonder how it is now.