"ALMOST" MATCH GAME

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John has \$120 in his bank account and saves \$8 each week.
Mary has \$230 in her account and withdraws \$6 each week.  After
how many weeks will they have the same amount?  What will that
amount be?

Are you having a sensation of "deja vĂș" just about now?  That is, do you feel
you've seen this before?  Well, the answer should be: "yes" and "no".

Yes: because the action of saving and withdrawing money is present once
again.  For the "no" part: at least the money numbers for the persons are different.
(Then there is a slight change in the title of this page, if you noticed it.  The word
"Almost" must have some purpose or it wouldn't be there, right?)

So, perhaps, just perhaps, the goal of equal balances can't be reached this
time.  If you have considered this as a possibility, you are well on your way to being
an intelligent and observant problem solver.  This problem is therefore actually "more
realistic" than the previous problem: "UP-down Match Game".  If you think about it,
it is not too likely that if two people behave in such a way as described in that problem,
that their balances would someday "come out the same!"

In this new situation, as you will soon discover as you work on it, the balances
do not become equal.  However, they do come close to each other at some future
point, "passing like ships in the night".

Our chart strategy is again the order of the day, but this time it might be useful
to employ an extra column, to see some important information as it develops.

Now we can alter our style of answer conclusion by stating something similar
to this:

John and Mary will not have equal balances,
but they will be very close in Week No. 8,
where the difference is the smallest, \$2.

Teacher Commentary

It is recommended that for this problem the students not be warned in advance
that the money amounts do not eventually coincide.  This is intentional with the hope
that they will "discover" it in the natural course of doing the solving process.  This
little unexpected outcome is a good experience in watching the flow of data carefully
and not assuming that just because the two problems appear to be the same, that they
are indeed the same.  This prepares them to "expect the unexpected" in math problems
and to always be alert for subtle or great differences that might arise.

One of the main objectives of this lesson is to recognize the concept of
"inequality", and more specifically, when the two balances are unequal by the
least amount.  In Week 7, John's amount was less than Mary's.  But in Week 8, the
situation was reversed!  In fact, the absolute value of the differences was even
smaller.  This was just the way it happened this time; problems can be easily
constructed  in which the minimum difference occurs on the earlier week (see
Appendix: #2 below).

[It should perhaps be noted that negative numbers were given in the chart.
That was a result of always subtracting Mary - John during its construction.  Whether
a given class should utilize this concept should depend on the level of the students
involved.  It is not essential to this lesson's topic as a whole.  Certainly it does
provide a natural situation for giving negative numbers an important use!]

After reflecting on the "tricky wording" of the problem as given at the top,
it might be suggested for students to come up with a new wording that does not
deceive" the reader.  An example might be this:

John has \$120 and saves \$8 each week.  Mary has \$230
and withdraws \$6 each week.  After how many weeks
will the amounts of money they have be the closest?
And what is the difference between them?

Now that equal-amount problems and different-amount problems have been
covered, students can now be encouraged to write their own.  Writing one's own
creations had to wait until this point; it might have been a little difficult for young
learners to directly write equal-amount problems any sooner.  Now the element of
surprise and wonder can enter the process.  Whatever numbers are used, there
is a meaningful outcome.  If the balances match, good; if not, well that's okay too.

This then brings up the next natural question: can one know in advance when
type of outcome will result?  The answer is Yes, but finding the "trick" is a problem
to solve in itelf.

Appendix

Here are a couple of extra problems to get you started.  Answer data in red.

1.  John: \$234         saves: \$6      w =  9   d = \$10   \$288 & \$298
Mary: \$370  withdraws: \$8      w = 10  d = -\$4    \$294 & \$290

2.  John: \$180         saves: \$7      w = 11  d =  \$6    \$257 & \$263
Mary: \$351  withdraws: \$8      w = 12  d = -\$9    \$264 & \$255