which in the judgment of the teacher are of particular value to them. In all cases pupils should be shown the explanation of the short method because the explanation is an aid to mastery of the process, because it leads to an appreciation of our number system, and because it frequently points the way to applications of algebraic processes. Problem Solving It is highly important that pupils find the number relations in a problem situation and express them mathematically before they proceed to the computation. The indicated number relationship between the question and the data is in reality the answer which needs to be expressed in simpler form through computation. The final computation has relatively little value as compared with the thinking which is necessary to formulate the equation, graph, or table from the problem situation. Pupils can be stimulated to greater activity in problem work if they are encouraged to select for a given problem the method of solution which in their judgment is most economical and which at the same time shows the required relation relatively accurately. In many problems the reasoning should be expressed in the form of an equation because then both the teacher and the class can determine quickly the merits of the proposed solution. The Use of Formulas The place of the formula in general mathematics is stated in simple terms in Chapter VII. The pupils have been prepared for this chapter incidentally in their work with general problems (solving problems without figuring); in the derivation of mensuration rules by converting the given figure into a given rectangle and then stating the derived rule as a formula; and in the use of the three formulas for the determination of any side of the right triangle when the other sides are given. It is intended that pupils should learn to use and evaluate formulas in this year's work without the introduction of algebraic SUGGESTIONS TO TEACHERS vii technique. Formula interpretation and manipulation of the character developed in the above chapter should give pupils power to grow naturally into the more general mathematics of the ninth year and should prepare pupils to read with understanding the easy formulas of physics which they may encounter in their general science. The Function Idea The function idea is stressed in this book through the straight line graph and through the formula by frequent recurrence to the what-happens-if type of question, as exemplified in the formula for the area of the circle, A=TR2, "what happens to A if R is doubled?" Teachers of second year junior high pupils may be certain that the function idea is taking hold if they find these pupils getting the what-happens-if attitude toward formulas and problems. Square Root In the extraction of square root the method of approximation by long division is used in this book because the process is readily rationalized and effectively remembered. As a consequence this method may confidently be expected to function in later life. Pupils should also be given the opportunity to read square roots from the table on page 234. The Metric System The chapter on the metric system is designed to accomplish two things: first, to show how the system is constructed from one basic unit (the meter) and how the standard units are easily translated into the equivalent English units; second, to give a reading knowledge of the more common units of measure through problem solving. Specific Objectives 1. To reach and maintain through practice tests and short methods such proficiency in arithmetical computation as will satisfy the standards set by experts in the teaching of arithmetic. 2. To give some practical knowledge of the arithmetic of banking, of thrift, of investments, of transportation, and of travel. 3. To develop more power in problem solving through the use of the graph and equation solution. 4. To apply the knowledge of the fundamental processes in the development of algebraic computation chiefly through the formulas derived from the mensuration studies of Book One and through the equation used in problem solving. 5. To give some skill in the extraction of square root. 6. To develop the principle of the Pythagorean theorem through intuitive geometry and to apply this principle to right triangle problems. 7. To develop the principle of similar triangles and to apply it in the determination of unknown heights and distances by indirect measurement. 8. To develop the function idea through the construction and use of line graphs and through the application of the what-willhappen type of question to formulas and problems. 9. To provide opportunity for the pupil to explore mathematics beyond arithmetic far enough to enable him to ascertain whether , it is profitable to continue his mathematical studies. |