« PreviousContinue »
I gladly give credit to Professor C. H. Currier of Brown University for many valuable suggestions.
IMPORTANT FEATURES OF THIS BOOK
1. Syllabus plan; proofs of theorems to be found in
the class. 2. Every definition suggested as needed; these also
have to be found, but need not be written. 3. Propositions arranged in groups, each group with a
single purpose. 4. Emphasis upon fundamental methods in each group. 5. Summaries of methods in each group. 6. Exercises applying these summaries. 7. Comprehensive sets of review questions on definitions,
fundamental principles, and methods. 8. Notes giving a general survey of Geometry, a gen
eral survey of each chapter, and methods of proof for each theorem. In some cases special proofs
are given. 9. Dictionary of geometrical terms, answering all ques
tions on definitions. 10. It follows closely the recommendations of the National
Committee, and is suitable for both elementary and
advanced classes. 11. It is especially adapted for reviews and can be used
with any text-book. Only seven proofs may not fit the logical order of the class text-book, and these are explained in the Notes, page 86. These notes should be consulted regularly. Pages 83, 84, and 85 are recommended for drill immediately before college examinations.
12. It is ideal for beginning classes if the teacher believes
that the pupil should be given "the notion that he is discovering for himself that which he is being taught.” Here no other book is needed, but the proof of each theorem marked with an asterisk should be written in a notebook. This work is not harder for the pupil and does not take more time.
SUGGESTED CLASSROOM PROCEDURE Drill with books open: 1. Stating definitions, summaries, etc., with illustrations. 2. Stating a method of proof for each theorem. 3. Stating a method and outline of proof for each theorem. 4. Stating a method and complete proof for each theorem.
The work should be distinctly heuristic. Let the
students find out everything if possible. A plan that gives good results is: 1. The theorem is read very carefully. The figure with
out aid lines is on the blackboard. 2. What is given? First pupil answers. 3. What is to be proved? Second pupil answers. 4. What are methods of proving this thing? Third
pupil answers. 5. What does the first method require? 6. Can we get this requirement from the given conditions?
Aid lines should be drawn when needed. 7. If not, try another method similarly. 8. After a method has been adopted, the outline or proof
is built up, each student taking his turn. Many theorems not marked with an asterisk may be treated as exercises.
All exercises need not be taken by any one class.
Model theorem, Article 41. 1. Pupils read the theorem to themselves. 2. What is given? 111' and interior angles u and u on
the same side of t. 3. What is to be proved? Lut 20 = 2rt. 4. 4. What are the methods of proving this kind of thing?
The basic principle, Art. 26. (State it.) 5. What does this method require? One straight line
meeting another straight line, making two adjacent
angles. 6. Can we get this requirement from the given condi
tions? Yes, more than once. Put x for the interior
angle adjacent to v. Proof: First statement, 2 x + 20 = 2rt. 6. Reason, Art. 26 in words. Second statement, Zu LX. Reason, Art. 37 in words. Third statement, Lut 20 = 2rt. 6. Reason, Article 7b (p. 103).
See what the notes have to say.
In a beginning class each pupil should write down each step of the proof as soon as it is stated orally. This proof should be put in a notebook during a study period. Prepared loose leaf geometry sheets are very convenient for this purpose. See the arrangement of Art. 264, p. 97.
RESULTS CLAIMED BY ITS USE
1. Better view of geometry as a whole; better view of each topic by itself.
2. Increased power of concentration. 3. Increased power of analysis, classification, and method. 4. Increased power of building up a proof. 5. More interest. 6. After a few weeks the pupil can go ahead more rapidly than by the old method.
DRILL BOOK IN PLANE GEOMETRY
1. Explain Solids, Surfaces, Lines, Points, Geometrical Figures.
2. Explain Position, Form, and Size. What does Geometry investigate?
3. Explain Straight line.
5. Explain Angle. What kind of motion does it represent?
6. How can lines or angles be proved equal at first? 7. State six general axioms of equality. 8. Define Straight Angle. 9. Define Adjacent Angles. IO. Define Right Angle and Perpendicular. II. Define Acute Angle and Obtuse Angle.
Define Supplementary angles. 13. Define Complementary angles. 14. Define Vertical angles.
15. Define Equal figures. Define Similar figures. Define Congruent figures.
16. Explain Superposition.
17. Define Parallel Lines.
20. Only one straight line can be drawn through two given points. (Axiom.)
21. Through a given point, only one straight line can be drawn, making a given angle with a given straight line, and having this angle on a given side of each line. (Two cases.)
Two straight lines can intersect in only one point. 23. All straight angles are equal.
24. Through a given point, only one straight line can be drawn perpendicular to a given line. (Two cases.)
25. Through a given point only one straight line can be drawn parallel to a given straight line.
26. If one straight line meets another straight line, the two adjacent angles are supplementary.
27. If two adjacent angles are supplementary, their exterior sides form a straight line.
28. Only one straight line can be drawn under certain conditions. State four methods.
State one principle of angle-sums.
State a method of proving two straight lines lie in one straight line.
1. Illustrate by a figure:
a. A straight line extended its own length.