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Chemistry is the study of matter, its composition, its structure and the reactions that change matter from one form to another. There are two broad areas in chemistry: first, descriptive chemistry. This is the collection of data and discovery of information. Secondly, there is the using of this information to form laws and theories which explain our observations, and the using of these laws and theories to predict the future behavior of matter.
Mathematics, or the abstract treatment of quantities, is of paramount importance to chemistry. We must constantly answer such questions as how much will we need? How long will the reaction take? We must measure matter, express our measurements in numbers, and be able to manipulate these numbers to form conclusions, or solve problems. However, this should not frighten the student because high school chemistry usually involves only simple arithmetic, simple algebra and proportions. Most of the difficulty students encounter is not In the math, but in setting up the problem.
Since our activities in science involve the solving of problems, the sooner we devise a problem solving method that works, the sooner the students will be comfortable as chemists.
G. Polya has written several books on this subject, and I found his book, How to Solve It especially helpful in developing our chemistry problem solving plan. This plan can be applied to any problem in any area and should meet with a high degree of success. There are four steps involved.
Step 1

Understand the problem. We must read it carefully to discover what it asks us to find. This is the unknown. We must discover what facts are given to us. These are the data.

Step 2

Formulate a plan. We develop a suitable plan for soIving our problem, using the data we have been given. We ask ourselves such questions as Is this a familiar problem? Do I have sufficient information to answer it?

Step 3

Carry out our plan. (1) Carefully list our data. (2) Write down any formula we are going to use. (3) Substitute our data into the formula. (4) Make careful calculations, being sure to include units. By carefully writing down each step of our solution, we avoid careless errors and we will be able to check our work.

Step 4

Check our work
. Recheck each step, being sure we have used correct formula, substituted correct numbers in using the formula, that our computations are correct and that we have included correct units.

The importance of including units and handling them correctly should be stressed.. It means nothing to say we have 5. We must say we have 5 dogs, or 5 apples, or 5 grams of water. The following things should be remembered when dealing with units.

1. Unlike units must be changed to common units before calculations can be made.

2. Units are manipulated in the same manner as numbers. That is, they must be squared or canceled during calculations.

3. The sign “/” is read as “per” and means “for every” as in 60 miles/gallon.
Examples: addition and subtraction
____
4 liters + 7 liters = 11 liters
____
10 grams 5 grams = 5 grams
Multiplication and division
____
3 cm
^{
2
}
x 2 cm = 6 cm
^{
3
}
3 hours x 50 miles/hour = 150 miles
(figure available in print form)
Now, let us illustrate our method with a very simple problem.
Problem: Find the area of a room 400 meters long and 100 meters wide.
Step 1

Understand the Problem What is our unknown? Look for such clue words as “find”, “how much” and “how many”. In this case we are asked to find the area, and the data given includes length and width.

Step 2

Plan
We know that area of a rectangle can be computed by multiplying the length times the width, using the formula A = L x W.

Step 3

Carry out plan
or
make calculations

Data length= 400 meterswidth = 100 meters
Formula A =L x W
substitution of our data into formula
A 400 m x 100 m = 40000 m
^{
2
}
Step 4

Check our work
Did we use correct formula? Yes,


Did we substitute correct data? Yes.


Are our calculations correct? Yes.


Did we handle units correctly? Yes.


Our Problem is complete.

Since all scientific measurements and calculations are made in the metric system, some time should be taken at this point to review the metric system. Students should not be asked to change measurements from the English to the metric system. They should, for all intents and purposes, forget about the English system in science, except for using it for a rough approximation of the relative size of the metric units. It has been my experience that the following facts should be included in the review of the metric system.
meter the unit for measuring length in the metric system. One meter is just slightly longer than one yard. (39.37 inches)
liter
the metric unit for volume. One liter and 1 quart are about the same size. ( 1 quart = .95 liters)
gram
the metric unit used to measure mass. One kilogram is equivalent to about 2 pounds. ( 2 pounds .91 kilograms)
The following list of prefixes can be used with any of the above three units to indicate how many of the units we are dealing with.
milli

1/100

centi

1/100

deci

1/10

deka

10

hecto

100

kilo

1000

Students find the system easier to work with if they understand what the units are based upon, and how they are related to each other. The metric system is a decimal system. Each unit is a multiple or a dividend of 10. There are 10 millimeters in each centimeter, and 10 centimeters in each decimeter, and 10 decimeters in a meter etc. It follows that there would be 10 x 10 millimeters in a decimeter, or 100.
The meter is the length of a certain number of wavelengths of the orange red line in the spectrum of krypton 86. Water is then used as the medium for transferring length to mass. One cubic centimeter of water is assigned a mass of one gram. (This is at 4°C, the densest point of water.) This gram of water is then said to have a volume of one milliliter. Because water is used in the transition from one dimension to the other, we can say that one cubic centimeter of water, one gram of water, and one milliliter of water all refer to the same quantity.
Before assigning problems in metrics to the students, it is necessary to review with them the operations of multiplying and dividing by 10’s and multiples of 10. To multiply by 10 or multiples of 10, you move the decimal one place to the right for each zero in the number you are multiplying by. To divide by units of 10, you move the decimal one place to the left for each zero in the divisor.
To add or subtract decimals, the decimals must be lined up directly underneath each other, because this aligns the units correctly. To multiply with decimals, you multiply as usual, then count the total number of digits that appear after the decimal point in the numbers to be multiplied, and place the decimal that many places from the left in the answer.
Students should be given several objects to measure in the metric system. They should familiarize themselves with using this system by finding areas and volumes of such things as their desk top, milk cartons, books, etc. Some sample metric problems and their solutions are below. Students should probably be reminded of the formulas used for computing areas and volumes of common objects.
area of a rectangle length x width area
area of a rectangle length x width area of a circle ¹ x radius
^{
2
}
volume of a rectangular box = length x width x height
volume of a cylinder = ¹ x radius x height.
Example 1 How many centimeters are contained in 14 meters?
Step 1

Unknown the number of centimeters in 14 meters Data we have 14 meters.

Step 2

We know that the prefix centi means 1/100, and that there are therefore 100 centimeters in each meter. Since we are converting from a relatively large unit meters to a relatively small unit centimeters we know that we will have more of the smaller units and therefore we must multiply.
In going from a larger unit to a smaller unit. we multiply the number of large units given by the number of small units in each large unit.

Step 3

unknown centimeters in 14 meters


data 1 meter= 100 centimeters


14 meters = 14 m x 100 cm/m = 1400 cm

Step 4

Check work.

Example 2 How many grams are in 76.3 milligrams.
Step 1

Unknown grams in 76.3 milligrams

Step 2

The prefix milli means 1/1000. Since we are changing from a smaller unit to a larger unit, we will have fewer of the larger units, and must therefore divide. a larger unit we
must divide the numbe
r Of small units in each large unit.

Step 3

Unknown  grams in 76.3 milligrams


data 1 gram contains 1000 milligrams

(figure available in print form)
Step 4

Check work. Be sure you have the number of smaller units per large unit correct, that your multiplication and division are correct, and that units have been handled correctly.

Enough problems of this type should be completed by the students so that will be familiar enough with the metric system to use it in handling density problems.