University of Missouri-Rolla

When you have completed the problem, don't let your answer get lost
in a lot of scratch-outs, scribblings, and scrawls. The grader (or your
boss) will look for the answer first, then will check on how you
arrived
at it. You've invested time and effort in coming up with that answer,
and
right or wrong, it represents YOU. Put it out there, bold and
confident,
for all the world to see - they'll either find it anyway, or mark the
problem
wrong. If your answer is right, you should be proud of it. If it's
wrong,
there is nothing to be ashamed of if you've given it your best effort -
finding out why it's wrong is part of the learning process. If you
don't
find out why it's wrong, or you haven't given your best effort to solve
the problem, you have wasted the time and effort you have invested.

Whenever you perform numerical calculations in a problem, it is a good idea to also do a rough or approximate calculation, preferably "in your head". This provides a check on the calculation, especially on decimal points and powers of ten where errors often occur, and also helps develop skills in estimation.

**R**ead and **R**e-state

**E**valuate and **E**xplore

**A**rrange and **A**ttack

**D**o it and **D**efend
your answer

**I**dentify the Principles **I**llustrated

**T**ry a different **T**actic

**Read** and **Re-state**:
Carefully **read** the problem so that you understand it well
enough
to **re-state** it in a different way, even if it's no more than
"They're
telling me this and this; and they want to know that." Making a
quick sketch based on the information given is often a good way of
re-stating the problem.

**Evaluate** and **Explore**:
**Evaluate**
what is given to see if it contains "hidden" information.
**Evaluate**
the question in terms of information that might be helpful in coming up
with an answer. The information given is a starting point, the answer
is
the destination - there should be a logical path between them. **Explore**
your resources - text, notes, study materials - to find the path, which
is usually an equation or a set of equations. If you run out of ideas
on
working from the starting point to the destination, try working back
from
the destination to the starting point.

**Arrange** and**
Attack**: **Arrange** the given information in the symbols
and units required by the equation or path. Identify and define any
intermediate
quantities you may need to calculate. If the path is complex, sketch
out
your plan of **attack**.

**Do it** and **Defend**
your answer: **Do** the necessary calculations. State your answer
clearly,
boldly, and legibly, including units and a reasonable indication of
precision.
Assume that someone is challenging your answer, and prepare to **defend**
it in terms of the sign, magnitude, and units of the answer. Go back
over
the logic of your solution, the mathematical operations involved, and
the
numerical calculations.

**Identify** the Principles **Illustrated**:
Instructors assign problems to help you learn concepts, principles,
and/or
skills. Go back over the statement of the problem and the path you
followed
to the solution to **identify** the concept or principle or skill
it
was designed to
**illustrate**.

**Try** a Different **Tactic**:
See if you can find a different way of attacking and solving the
problem.
During the Evaluation and Exploration stage, you often think of several
different routes or **tactics** you may take to reach a solution.
Go
back and **try** one or more of those. If it leads to a different
answer,
go back over both solutions to locate the flaw. If the two paths lead
to
the same answer, decide which is the cleaner, more direct path in terms
of the principles and concepts involved.

**Level 1:example
1practice
exercise**

**Level 2:example
2practice
exercise**

**Level 3:example
3practice
exercise**

**If today is Friday, what
day
of the week will it be four days after yesterday?**

**Solve the problem to your own satisfaction, then
think
about how you arrived at that answer before clicking on continue.**

**To many people, this problem appears so easy that
they
think they should be able to answer it without even thinking!!
Some
may make a (slightly educated) guess. Others may have counted on
their fingers or poked their finger at the days on an imaginary
calendar.
Some may even have created a makeshift calendar by writing the days of
the week on a scrap of paper and marking them off to arrive at an
answer.
A few will have mentally performed some or all of these operations,
arriving
at the answer without any physical movement whatsoever.**

**While the problem may appear easy, the answer is
not
immediately obvious. You have to go through some thought
processes
to come up with the answer. This may be a light workout for your
mental muscles, but it is exercise and practice so that you can go on
to
greater challenges. It also lets you look at the problem -
solving
tactics you are presently using and to compare them to a formal
strategy.**

**If today is Friday, what
day
of the week will it be four days after yesterday?**

**Read
and Re-state:**

** This
problem clearly requires careful reading. You must pay close
attention
to the words today, yesterday,
and after. You must think about
what
it says - If today is Friday - so we'll
just
pretend that today is Friday. Then
it
mentions yesterday, so that would have
been
Thursday.
Now we've got to figure out what day comes four
days after Thursday.**

**Evaluate
and Explore:**

** This problem is stated simply in
a straightforward manner, so there isn't a great deal to explore.
It is clear that the answer must be based on the sequence of days in
the
week:**

**SundayMondayTuesdayWednesdayThursdayFridaySaturday**

**If you can hold this sequence in your head while
working
on the problem, that's great. However, if you find anything
awkward
or confusing about it, make a note on a piece of scrap paper.
Unless
you're specifically told NOT to do this, it's perfectly OK.**

**SMTWThFSat**

**Making notes of information and/or making a
sketch
of the situation that the problem is addressing is an excellent tactic
in starting to solve a problem. It helps you organize your
concept
of the problem, which will form the basis for your solution. It
often
makes you go back and read the problem more carefully.**

**Arrange
and Attack:**

** You may be able to work directly
from the note above, wrapping around from Saturday to Sunday or (in a
different
problem) from Sunday back to Saturday, but there's nothing wrong with
writing
the days of the week in a more convenient manner. Since we want
to
count forward from yesterday (Thursday), let's start the sequence on
Thursday:**

**ThFSatSunMTW**

**Do
it and Defend
your answer:**

** Now the problem is simply a
matter
of counting days:**

**ThFSatSunMTW**

**yesterdaytoday**

**1234**

**So the answer is**

********* Monday
*******.**

**Identify
the Principles Illustrated:**

** The obvious principle of this
problem
is the sequence of days in the week. Since this discussion is
about
problem - solving, there are other principles involved: careful
reading,
translating the words into a thought process, defining a route to the
solution,
etc. If you have correctly identified the principles involved,
you
should be able to construct a similar problem based on these
principles.
Maybe,**

**If today is Sunday, what
day
of the week was it two days before tomorrow?**

**or perhaps something even more complicated.**

**Try
a Different Tactic:**

** If you made a note and worked the
problem on a piece of scrap paper, try starting over and working it in
your head, just to build up some muscles there. If you worked it
in your head, write the notes and check your work. The big
advantage
of the notes is that they give you a basis to check your work, and if
you've
made a mistake there is a possibility for someone to show you where
you've
gone wrong. A very good device here is to think of how you would
explain to someone else how the problem might be worked.**

**Here's a problem to consider:**

**If the time is now 8:45
Tuesday
evening, what will be the time in 6 hours and 30 minutes?**

**Solve the problem to your own satisfaction, then
think
about how you arrived at that answer before clicking on continue.**

**If the time is now 8:45
Tuesday
evening, what will be the time in 6 hours and 30 minutes?**

**Read
and Re-state:**

** All problems must be read
carefully.
This one states that the time is now 8:45 PM.
It asks what the time will be in
6 hours and
30
minutes. This then seems to be a problem of adding time.**

** Before leaving this step, you
should
pay special attention to how you are expected to state your answer:**

**Evaluate
and Explore:**

** What is it that makes this
problem
difficult? If it asked us to add 1
hour
and 10 minutes, the answer would be
obvious.
The problem is that we can't add time like ordinary numbers - minutes
add
up to :59, then the next minute is :00
in the next hour. Hours add up to 12:,
then the next hour is 1:, and we change
from
AM
to PM or from PM to AM.**

** Troublesome
thought: Is midnight 12:00
AM or 12:00
PM? Does
midnight Tuesday come before or
after
noon
Tuesday?**

** Proper
Response: That's interesting, but not relevant to solving THIS
problem.
I'll try to remember to think about it later.**

**Arrange
and Attack:**

** Make some notes:**

time now: 8 : 45 PM

time to add:6 30

**Do
it and Defend
your answer:**

** Carry out the calculation in
parts:**

time now: 8 : 45 PM Tuesday

time to add:30

new time: 9 : 15 PM Tuesday

time to add:6

new time: 15 : 15 ? ? This is confusing.

time now: 8 : 45 PM Tuesday

time to add:30

new time: 9 : 15 PM Tuesday

time to add:3This will get us to midnight, and there are 3 hours left over.

new time: 12 : 15 AM Wednesday

now add:3

new time: 3 : 15 AM Wednesday

**Try
a Different Tactic:**

** Go back to the step above where
it became confusing. The new time was 15:15
Tuesday(?) evening(?). When the hour went past 12,
it became Wednesday morning and the hours
started over so the hour is now (15 - 12 = 3) 3,
and the time is 3:15 AM Wednesday.**

**Here's a problem to consider:**

**Using only an unmarked cup
and
a large unmarked bowl, how would you mix water and antifreeze to make a
solution which is protected from freezing to exactly 0 ^{o}F?**

**Evaluate
and Explore:**

** Obviously, the desired solution
can be prepared by putting 3 quarts of antifreeze in a 9-quart
container.
However, the problem specifies that it must be solved using only an
unmarked
cup and an unmarked bowl, so we have no way to measure quarts.
Looking
a bit more closely at the table, there are other mixtures which will
give
protection to 0 ^{o}F: 4
quarts in a 12-quart radiator, 5
quarts in a 15-quart radiator, 6
quarts in a 18-quart radiator, etc.**

**Arrange
and Attack:**

** Now the problem is to physically
mix the right amount in order to get a volume fraction of 1/3 of
antifreeze
in water, using an unmarked cup and bowl.**

**Do
it and Defend
your answer:**

** Using the cup, we simply measure
out a cupful of antifreeze into the bowl then add two cups of water.**

** Nagging
thoughts:**

** - do we
rinse the antifreeze out of the cup before measuring the water?**

** - do you
really get three cups of mixture when you mix a cup of antifreeze with
two cups of water?**

**Some problems can't be done perfectly - we just
have
to do the best we can with what we've got!**

**Identify
the Principles Illustrated:**

** The principle was properly
identified
in the Evaluate and Explore step - the
freezing
point of a mixture of antifreeze and water depends on the concentration
of antifreeze. In this case, the concentration is most easily
expressed
as the volume fraction.**

**Try
a Different Tactic:**

** There isn't an apparent alternate
way to solve this problem within the given restraints. However,
you
can think about how the problem would be solved without these
restraints
- perhaps if you had a graduated cylinder. If you haven't already
done so, try to visualize the actual process of measuring and mixing
the
components. Sample exercise 3 below is an
attempt
to help with this step.**