In this module we discuss simple building blocks. But even though the blocks are simple they can be used to describe incredibly complex and often beautiful things. Our basic building blocks are

• Numbers
• Addition
• Subtraction
• Multiplication
• Division
• Functions

When you are teaching calculus you will also use two more sophisticated building blocks.

• Differentiation
• Integration

Numbers

Because we are using Mathematics as a language to talk about phenomena in the real world, we rarely use numbers just for their own sake. Usually numbers represent things -- people, height, weight, force, speed, and so forth. Thus, numbers usually have associated units -- for example, if we are talking about height in the United States we are likely to use feet or inches and in the rest of the world, meters or centimeters. Here are some typical examples of units.

• For length -- inches, feet, miles, centimeters, meters, and kilometers.

• For temperature -- degrees Fahrenheit, degrees Celsius, and degrees Kelvin.

• For prices -- dollars, cents, francs, and centimes.

• For area -- square inches, square feet, square miles, square centimeters, square meters, and square kilometers.

• For time -- seconds, minutes, hours, days, and years.

• For velocity -- miles per hour and kilometers per hour.

We often need to convert from one unit of measurement to another unit used to measure the same kind of thing -- for example we can convert from feet to inches by multiplying by 12 inches per foot since there are 12 inches in each foot -- for example,

Although this is often simple mathematics, in the real world it can be very complicated and even controversial. For example, the exchange rate used to convert prices from one country's currency into another country's currency is often a matter of national pride and international trouble. It is not unusual for a country to maintain an official rather high exchange rate for its currency while a thriving black market has another lower exchange rate for the same conversion.

Click here for a table showing the conversion factors for various currences. Suppose you read that a new calculator is being sold in France for 1,500 French francs. What is the price in U.S. dollars?

As this is being written, the conversion factor for converting United States dollars paid in one year to United States dollars paid in another year is a matter of intense controversy. Because of inflation a dollar bought less in 1997 than it did in 1996. The table below was obtained from the Bureau of Labor Statistics.

Consumer Price Index-All Urban Consumers

Series Catalog:

Data:

This table helps us compare the purchasing power of a dollar in various different months and years to its purchasing power during the base period of 1982-84. For example, notice that the entry for March 1997 is 160.0. This means that $1.60 in March 1997 had the same purchasing power as $1.00 did during the base period.

The debate about how to convert dollars from one year to dollars in a different year is taking place on several different stages -- in the debate about the consumer price index and in the debate about capital gains taxes.

When an investor sells stock she pays a tax on her profit -- the difference between the price at which the stock is sold and the price at which it was purchased. Suppose an invester bought 1,000 shares of stock at $50.00 per share in 1994 and sold them in 1996 for $60.00. She would pay a tax based on a profit of $10.00 per share -- that is, a total profit of $10,000. One argument that is made for a lower tax rate for capital gains (this kind of profit) is that this computation mixes apples and oranges -- because of inflation, a 1996 dollar only had the buying power of 0.945 1994 dollars. According to this argument her profit per share is really

or 7.09 1996 dollars. Thus, her real total profit would be $7,090 rather than $10,000 and her tax should be based on $7,090.

With a little cleverness you can use your computer algebra system to keep track of units. We illustrate how this can be done using the TI-92. If you have a TI-92 you should follow along using it. You can also follow along using Maple or Mathematica by clicking on the appropriate CAS button in the navigation frame at the top of this window. Because the TI-92 material is included in this browser window, the TI-92 button above is inactive. You can still use the TI-92 help button, however.

Begin by choosing one basic unit for each kind of measurement -- for example, we will choose feet as our basic unit for length and seconds as our basic unit for time. We will use the corresponding variables -- feet and seconds -- in our computer algebra system. If you have used either of these variables then you will have to "clear" or "undefine" it. To clear a variable in the TI-92 go to the VAR-LINK menu, select the variable to be cleared as shown in the screen at the right.
and then press F1 to see the screen at the right. Notice that Delete is highlighted in the pull-down menu. Press ENTER.
A dialog box will appear to confirm that you want to delete the variable. See the screen at the right. Press ENTER to delete the variable that you selected earlier.
Notice that this variable disappears from the list of variables as shown in the screen at the right.
Now you can do simple calculations involving these basic units as shown in the screen at the right.
You can define additional units in terms of these basic units as shown in the screen at the right.
and now your computer algebra system will do conversions automatically as shown in the screen at the right.

Set up your computer algebra system to work with the following units -- fluid ounces, cups, pints, quarts, and gallons. Use fluid ounces as your basic unit of measure. Make up several problems involving these units and use your computer algebra system to solve them.

Addition

Addition is used to model or represent many different ideas. Here are some examples.

• Counting the result of combining two disjoint sets of things. For example, if someone buys 5 apples on Monday and 10 apples on Tuesday then for the two days they have purchased a total of 15 apples.

  If 5 textbooks are unusable because they are out-of-date and 10 books are unusable because their bindings are broken then we cannot conclude that a total of 15 books are unusable unless we know there is no overlap -- that is, none of the out-of-date books also have broken bindings.

• We often use numbers to represent location and movement. The figure below shows how numbers are used to represent location. We begin by picking one reference point, called the origin. Then we mark points at equally spaced intervals to the right and to the left of the reference point. Points to the right (east) are traditionally labeled with positive integers and points to the left (west) with negative integers. Points in between these points are labeled with real numbers. The red dot in the figure below is located at the point marked by the number 1. The blue dot is located at the point marked by the number -4.

  

  The figure below shows how numbers are used to represent movement. The red arrows all indicate a movement 2 units to the right (east) and are represented by the number 2. The blue arrows all represent a movement 1 unit to the left (west) and are represented by the number -1. Traditionally, positive numbers represent movement to the right and negative numbers represent movement to the left.

  

  Numbers representing location and movement have associated units of length -- for example, feet or meters. We can use addition to represent the result of moving an object. For example, if an object starts at the location 3 inches to the right of the origin (represented by the number 3) and is moved 2 inches left (represented by the number -2) then it will end up at the location one unit to the right of the origin (represented by the number 1). In the figure below the red dot represents the initial location of the object; the purple arrow represents the motion; and the blue dot represents the final location of the object.

  

Give three examples of things that can be modeled by addition. Explain clearly what the numbers involved represent and indicate the kinds of units used. At least one of your examples should involve numbers representing different things that are measured using the same units.

Subtraction

Subtraction is used to represent or model many different ideas. Here are some examples.

• Withdrawing money from a bank account. For example, if you withdraw $25.00 from a bank account that has $200.00 then after the withdrawal the balance in the account would be $175.00.

• Finding the movement necessary to move from one location to another. For example, if you are located 5 miles east of your home and want to travel to a point 7 miles west of your home then you must travel 12 miles westward. This problem can be represented algberaically by
  -5 + x = 7

  or

  -5 + _____ = 7.

  Whichever notation you use the answer is 12.

Give three examples of things that can be modeled by subtraction. Explain clearly what the numbers involved represent and indicate the kinds of units used. At least one of your examples should involve numbers representing different things that are measured using the same units.

Multiplication

Multiplication is used to represent or model many different ideas. Here are some examples.

• Determining the area of a rectangle. For example, the area of a rectangle whose length is four inches and whose height is twelve inches is 48 square inches.

• Determining the volume of a right parallelepiped. For example, the volume of a right parallelepiped whose length is two inches; whose height is three inches; and whose thickness is four inches is 24 cubic inches.

• Determining the distance traveled by an object moving at a constant speed during a time interval. For example, a car traveling at 60 miles per hour for three hours will travel 180 miles.

Give three examples of things that can be modeled by multiplication. Explain clearly what the numbers involved represent and indicate the kinds of units used.

Division

Division is used to represent or model many different ideas. Here are some examples.

• The average velocity of a moving object over a given period of time. For example, if a car travels 200 miles over a period of four hours then its average velocity is 50 miles per hour.

• The actual velocity of an object moving at a constant velocity over a period of time. For example, if an object travels 200 miles at a constant velocity over a period of four hours then its velocity is 50 miles per hour.

• The average rate of growth of the population of a city over a period of time. For example, if the population of a city increases by 50,000 people over a period of ten years then its average rate of population growth is 5,000 people per year.

Give three examples of things that can be modeled by division. Explain clearly what the numbers involved represent and indicate the kinds of units used.

Functions

We use functions to represent certain relationships between two different quantities -- for example, the radius of a circle and the area of a circle. A function is used when one of the quantities determines the other. For example, the radius, R, of a circle determines its area, A, according to the function

The TI-92 screen at the right shows how these two functions can be defined and used on the TI-92. Notice the use of units. It is almost exactly as expected. The only difference is that when we determine the radius of a circle from its area the units are |feet|. This oddity is a result of the way in which units calculations are implemented in some computer algebra systems and the fact that these systems work very carefully with square roots.

The (positive) square root of a number x^2 is not x but rather |x|. On some computer algebra systems unit calculations are implemented using variable names for the unit names. Because variable names can also represent numbers the square root of feet^2 is |feet|.

We used approximate calculations in this screen.

Functions represent underlying relationships between quantities. In order to understand functions it is useful to look at some basic relationships, some of which can be represented by one or more functions and some of which cannot be represented by functions.

• The relationship between the area of a circle and its radius can be represented by two different functions because the area can be determined from the radius and the radius can be determined from the area.

• Suppose that in your state children are enrolled in kindergarten if they were born between September 1 and August 31 inclusive five years before the current academic year. Thus, if a student's age on September 1, 1997 was x years his or her grade in the 1997-98 school year would be [x - 5], or the greatest integer in x - 5. This function determines grade level from age but there is no function that determines age from grade level since the age of a student in grade level 1 might be, for example, 6 years, 6.1 years, 6.5382 years, or any age between 6 and 7 except 7.

• Can the relationship between the length of the sides of an equilateral triangle and its area be expressed using one or more functions? If so, find the function(s). Explain the units in which each of the quantities involved is measured.

• Can the relationship between the length of the hypotenuse of a right triangle and its area be expressed using one or more functions? If so, find the function(s). Explain the units in which each of the quantities involved is measured.

• If you are familiar with the TI-CBL, then record the temperature at a particular spot in your classroom during the class period using a TI-CBL. The links below lead to programs and instructions for the TI graphing calculators explaining how to record temperature measurments.
  

  If you are not yet familar with the TI-CBL then just draw a rough graph indicating how the temperature in your classroom might vary over the course of a class period. Can this relationship be expressed by a function? Explain the units in which each of the quantities involved is measured.

  answer

• If you are familar with the TI-CBL, then record the light intensity at a particular spot in your school playground during one day using a TI-CBL. If you are not yet familar with the TI-CBL then just draw a rough graph indicating how the light intensity at a particular spot in the school playground might vary over the course of a day. Can this relationship be expressed using one or more functions? Note these function(s) probably cannot be expressed using simple algebraic formulas. Explain the units in which each of the quantities involved is measured.
  

Differentiation

Differentiation and division are closely related. They can both be used to determine the rate at which one quantity is changing compared to another -- for example, we often use them to determine the velocity of a moving object, or the rate at which its location is changing compared to time. When this rate is constant, simple division will do the job. For example, if an object is traveling at a constant velocity along an interstate highway and at 2:00 it is located at mile marker 123 and at 4:00 it is located at mile marker 247 then its velocity would be 62 miles per hour. Notice that location and the change in location are measured in units of length and that time and the change in time are measured in units of time so that the quotient is measured in units of length per time.

When the rate of change is not constant then simple division must be replaced by differentiation -- for example, the height of an object that is dropped from a height of ten feet at time t = 0 can be described by the function

and its velocity by the derivative

The TI-92 screen below shows how you can work with these functions using a computer algebra system. You may also want to look at the TI-92 help module on differentiation.

• The height of an object dropped from an initial height of 30 meters at time t = 0 is given in the metric system by the function
  h(t) = 30 meters - 4.9 (meters / sec²) t²

  Find the height and velocity of the object 1 second after it is dropped. Find the height and velocity of the object 2 seconds after it is dropped.

• On the moon the height of an object dropped from an initial height of 30 meters at time t = 0 is given in the metric system by the function
  h(t) = 30 meters - 0.82 (meters / sec²) t²

  Find the height and velocity of the object 1 second after it is dropped. Find the height and velocity of the object 2 seconds after it is dropped.

Integration

Multplication and integration are closely related. In fact, integration can be thought of as generalized multiplication. Consider, for example, the problem of determining area. For a simple rectangle the area is height multiplied by width or H(b - a) in the left side of the figure below.

When the height of a figure varies from point-to-point and is given by a function h(x) as shown in the right side of the figure above the area is given by the integral.

Each symbol used in the integration notation above has a meaning and should be thought of with associated units.

• The integrand h(x) dx should be thought of as a product with each factor -- h(x) and dx -- having associated units. When we work with area usually both units are the same -- feet or inches -- but when we use integration to determine displacement, for example, the units are different -- for example, h(x) might be measured in meters per second and dx in seconds. The units for the integrand are the product of the units for the two factors.

• The limits of integration a and b are measured in the same units as x and dx.

The screen below illustrates how integration and units are used on the TI-92. You may also want to look at the TI-92 help module on integration.

• Find the area of the region trapped between the curve y = sin x, the two axes, and the line x = 1. Assume that units along the x-axis and the y-axis are measured in inches.

• Find the distance travelled by a baseball dropped on the moon from a height of 30 meters during the first second after it is dropped.