Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

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Visually Explaining Shared Work Problems

Posted by mark schwartz on June 3, 2016


In class we had worked through how to solve shared time problems. Some of the students seemed puzzled by the procedure (partly because it has fractions in it) and they expressed it as “I can’t see that”. It struck me that seeing it actually might make a big difference for some students. After all, many of them claim to be visual learners and if I could provide a non-Algebraic approach, it might give students a better way to “see” the relationships. It’s not completely free of Algebra but it did make a difference for many students. One student actually said “Hey, this is Geometry not Algebra” and talking to him later it turns out that in high school he really did well in Geometry but not Algebra and he has always tried to set up Algebra problems geometrically. As it turns out, in Smyth’ 1859 Elementary Algebra (page 39), he cited the importance of seeing the work in one day (or one month) units. Again, the word “seeing”, so I tried to help students see it.

The Story.

Let’s step through this using the following problem:

Danny takes 4 hours to do a job alone. His brother Mike can do the job alone in 6 hours. If they work together, assuming no gain or loss of efficiency, how long will it take them to do the job?

(Before going on, I’d like to note that this will be discussed in slow-motion math which results in a lot of words to describe a somewhat simple procedure. The point is that this is a discussion with the class and most of these words were spoken throughout the demonstration.)

The following represents the amount of time it takes to do the whole job and notice the notation.

|←     time for job done     →|

|←       The whole job          →|

|←     the whole job done  →|

Let’s start with Danny, who can do the whole job by himself in 4 hours. The question is how much of the job can he do in 1 hour? Collectively, the class (well, almost everyone) responded   . Visually, this is

|← 1 hr.→|

|←       The whole job          →|

|← 1/4  →|

Now, Mike can do the whole job by himself in 6 hours. The question is how much of the job can he do in 1 hour? Again, collectively the class responded 1/6. Visually, this is

|←   1 hr.   →|

|←       The whole job          →|

|←   5/12   →|

The 5/12 comes from 1/4 + 1/6 but it’s still a total of 1 hour of time, since they’re working at the same time; we don’t add the time just the amount of job done.

Now what? Consider this: if 5/12 of the work can be done in 1 hour, how long will it take to do the rest of the job? Visually, this is

|←   1 hr.    →|←             x hrs.  →|

|←           The whole job            →|

|←   5/12    →|←          7/12        →|

Amt. of job to do

Notice the additional notation of “Amt. of job to do”. And further, notice (with a little prompting!) that we have a proportion (!), which taken from the figure is

1/(5/12) = x/(7/12)

Yes, I know fractions again but students know how to solve this proportion. Solving for x, the result is 1 2/5 hrs. So, the total time for the job is 1 + 1 2/5 hrs, a total of  2 2/5 hrs. This is the initial hour plus the remaining time to complete the job.

And now, a very interesting visual display of a problem not likely to ever show up in anyone’s text. The problem is

If Adam can do a job in 30 minutes and Bob can do the same job in 20 minutes, how long would it take them to do the job together?

Typically, these kinds of problems are used with reference to hours not minutes, although minutes or hours will work. I’ll use the hours because this introduces fractions and it makes the problem more interesting.  Here’s the questions to consider:

  1. In 1 hour, how much of the job can Adam do?
  2. In 1 hour, how much of the job can Bob do?

Mathematically, Since Adam can do the whole job in 1/2 hour, he can do the job twice in an hour and since Bob can do the job in 1/3 hour, he can do the job three times in an hour. So, a total of 5 jobs can be done in 1 hour and here’s how it looks

|←                       1 hr.                 →|

|←1 job→|←←|←x→|→→|→→|

|←                 5 jobs                     →|

Look at this carefully before you go on. It looks a little weird but it’s correct. It the previous example, there was some part of the whole job left to do, and it was found by setting up a proportion and adding the result to the 1 hour already spent on the job. In this case, we have an excess of 4 “jobs”, which will have to be removed, which means removing the amount of time from the 1 hour it took to do those 4 jobs (breathe…). So, now we have to set up a proportion, get the answer and “add” it to the 1 hour. I stated “add” because the excess has to be removed and it will be identified as a negative amount of time – the amount to be taken away.

Using the same proportion procedure, the proportion looks like this:

1/5 = x/ ̶  4

Solving, x =  ̶  4/5. The amount of time for Adam and Bob to do the job together is 1  ̶  4/5, or  , 12 minutes.

I told you that slow-motion math takes a lot of words, but the point is that the visualization of the problem helps students actually “see” how all the Algebra works, and why. I was asked if Algebra problems can always be set up geometrically, and I said that we’ll see when we get to other kinds of problems.


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