Egyptian Fractions Target Game

Recently I had the opportunity to play around with a novel middle school level calculator that “does fractions”. [For those who are curious, it is the Sharp EL-E300 model.] In fact, not only does it “do” them in the normal vertical format, but also it shows more than one fraction at a time in the display window. “Neat-o torpedo”, I thought. “Now what could I do with this clever instrument that would capture the interest and attention of my students while at the same time teach them some good stuff about this time-honored bug-a-boo of math at this school level?”

After spending some time reflecting on the matter, I decided I wanted to create a game that would involve various concepts: addition of 2 (or 3) common fractions, perhaps their subtraction as well, comparing relative sizes of 2 or more fractions, and decimal equivalents of fractions would be nice, too. Since estimation and number sense are hot topics these days, I wanted to include them also. And there was one more thing: how about a little historical connection thrown in for good measure!

The latter goal was easily met: Egyptian Fractions. (See my page on Egyptian Math for more on this topic.) From there, things just took a natural course, and I developed the little game that you will see described below. By the way, that’s an Egyptian fraction up there on the left side of the title bar; its modern form is given on the right.


Rather than presenting a lot of complicated rule descriptions of how to play, I think I’ll just describe how two students might go about a sample game. Our players are the ubiquitous John and Mary.

Play begins when their teacher announces that the “target fraction” will be “7/19“. The timer is set for 2 minutes. At the word “GO!“, John and Mary must each try to find two Egyptian fractions whose sum is as close to 7/19 as possible, without exceeding it, before time runs out.

If you recall your math history, you will know that John and Mary are looking for two fractions of the form

1
---.
x

Another name for such fractions is unit fractions, perhaps because the numerator is the unit “1”. However you choose to call them, the addition
procedure for such a pair is really quite easy. Observe:

                  1     1      b      a     b + a
                 --- + --- = ---- + ---- = -------
                  a     b      ab     ab     ab

In other words, the sum of two unit fractions is equal to the sum of the two denominators over the the product of the denominators. For more about all this, go to this WTM article: Fraction Addition.

[Recall that in Egyptian fraction work, all denominators must be distinct values, no repeats. So a is NOT equal to b.]


The two minutes are up. Let’s see how John and Mary are doing now. John chose 3 and 30 as his denominators, whereas Mary chose 4 and 11. So their respective sums are these:

          1      1     33     11             1      1     15
         --- + ---- = ---- = ----    and    --- + ---- = ----
          3     30     90     30             4     11     44

The next step is to determine which sum is closer to the target fraction (7/19) without going over. This can be done in two ways: fractionally or decimally. If speed of play is more important to you, then the decimal approach is probably better; otherwise, doing it by fractions could provide a different view of the situation. First, let’s look at the decimal way.

We convert the 3 fractions (target and 2 sums) to their decimal forms by simple division in our calculators, obtaining

target: 7/19 = 0.368421053

John: 11/30 = 0.366666667

Mary: 15/44 = 0.340909091

It’s clear that both players have obtained fractions whose values are less than the target’s value. It is considerably more helpful if we round our decimal expressions to some convenient level. Here I would recommend “to the nearest 1000th“, giving us these numbers. target: 7/19 = 0.368

John: 11/30 = 0.367

Mary: 15/44 = 0.341

Mere inspection, or simple subtraction, allows us to declare a winner here: John. His error was approximately “0.001” (“Very good, John.”) while hers was approximately “0.027” (“Not bad at all, Mary.”) “How about another game, kids? Try 13/47 as your target fraction.”


While they are busy on that, let’s return to the first game and discuss it some more, particularly the other, fractional, way to compare the results. This time we need to subtract the fraction sums from the target fraction. That is not so difficult as you might believe if we use the formula that was presented in the page, Fraction Addition, or rather a minor alteration of what was given there, namely:

                    a	    c	    ad - bc
                   ---  -  ---  =  ---------
                    b	    d	     bd

Using this formula on our target and sums fractions, these results are produced:

 7     11     210 - 209      1
  ---- - ---- = ----------- = -----
  19     30        570        570
  7     15     308 - 285      23
  ---- - ---- = ----------- = -----
  19     44        836        836

Again the results appear rather lopsided, even in fraction form, that 1/570 is smaller than 23/836. Checking it in general can be done in 2 ways: fractional and decimally. And again, decimally wins for ease of computation. The 2 differences have these decimal values:

1/570 = 0.001754385964912 or about 0.002

23/836 = 0.02751196172249 or about 0.028

[These figures agree, allowing for rounding of course, with what we found earlier.]

But to do it by fractions requires a bit more time; here’s how I would proceed. I like the “cross products” test.

                          836          13110
                    1         23
                  -----  <  -----
                   570       836

And since 836 is definitely less than 13,110, the left fraction is smaller than the right fraction. [I leave it up to you to ferret out the why’s and wherefore’s. :>) ]


Well, it’s about time to check in on our game players and see who is the winner this time. How’d you do, kids?

John says he used 4 and 50 as his denominators, while Mary chose 5 and 14. So the sums this time are

1      1      54      27             1      1     19
--- + ---- = ----- = -----    and    --- + ---- = ----
4     50     200     100             5     14     70

Getting right down to the nitty-gritty (i.e. decimal comparison strategy), we have these figures:

target: 13/47 = 0.276595745

John: 27/100 = 0.27

Mary: 19/70 = 0.271428571

While John’s sum produced a nice fraction with a terminating decimal form, and pretty close to the target fraction we might add, it’s easy to see that Mary’s choice is just a little bit better. (Girls get the point here!)


Extensions & other comments

Throughout the whole discussion of this game use of the calculator has been assumed at all moments. If one has access to a model which handles fractions, things are a bit more interesting. But such models are not required; the game can be played with even simple four-function types. The players just have to “know” a bit more mathematics. (And that’s not a bad thing either, is it?)

The time element of the game is enhanced if one’s calculator has a “replay” feature (as does the Sharp EL-E300 refferred to at the start of this page). Changing the numbers to investigate a “better choice” is more effecient with such calculators. And after one has gained a lot of experience there is a “trick” that is useful to recognize to make one’s search more efficient. Namely, after all is said and done, the error value (and not the sum of the fractions) is the most important information to be obtained. So one could combine and streamline the computation by using parentheses, or even not using them. For John’s work in the first game it would look like this:

7/19 – (1/3 + 1/30)
or 7/19 – 1/3 – 1/30

Now, by using the cursor keys and the replay feature, the denominators can be adjusted or changed rapidly, then the difference is more quickly
computed, even in decimal form.

Finally, one way to make this game a bit more challenging would be to require three fractions to be added. Now, things really start to get interesting.

Postscript

While playing around with the target fraction in the first game, I noted this:

1/5 + 1/6 + 1/570 = 7/19, exactly

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