Answer Key for our 11th and 12th Grade Workbook

Description

Our Making Math Meaningful^™ workbooks are specifically designed to lead the students along a path of discovery.  The teacher needs to carefully guide the students along this path.  The problems contained in these workbooks are NOT randomly ordered or selected. They are chosen to promote the development of mathematical thinking in a very conscious way. As teachers of math, our most important task is to develop mathematical thinking in our students.  This thinking needs to be flexible and creative.

Who can use this workbook?  Most of the author’s teaching experience is within the Waldorf school system.  Yet this workbook can be effectively used by any teacher or homeschooling parent who wishes to teach meaningful, age-appropriate math.

Problem-Solving Exercises!  The eleventh grade year in math should be exciting and mind-expanding.  Puzzles, games, and problem-solving can be a welcome change from the normal routine; they are also an important component of a math curriculum in order to develop students’ thinking capacities.

This is why we have included an entire unit on problem-solving exercises at the end of this workbook.

Additionally, you can many puzzles and games in our book, Puzzles, Games and More!

Blind procedures vs. discovery and mathematical experiences.  As we mention in our High School Source Book, we feel that all too often the standard approach to teaching math is based on “blind procedures”; students simply follow procedures without any understanding of what they are doing.  In contrast, we have tried to put together our high school workbooks so that, whenever possible, the students are led down a path of discovery.

We also feel that mainstream mathematics emphasizes procedural skills too much.  Often, 90% of classroom time (or more) is spent having the students slog through countless “problems”, all in the name of checking off topics on the LIST, and ensuring ourselves that our students have good “skills” (at solving meaningless problems).  In contrast, we believe it is important to have a healthy balance between procedural skills and real “mathematical experiences”.

Main lessons also provide opportunities for the “big picture”, where you can integrate history and philosophy with mathematical topics.  This develops wonder and enthusiasm, and shows how math is a human endeavor.

We encourage you to read more about all of this in the introduction of our High School Source Book.

Regarding Skills.  While our 9^th Grade Workbook focuses largely on skills, the purpose of this 11^th/12^th Grade Workbook is not just skills.  The purpose here is also to experience the beauty of mathematics and to develop mathematical thinking capacities.

Why an eleventh & twelfth grade combined workbook?  This workbook is designed to be used by either an advanced or regular math track class.  The two calculus units are intended for an advanced class, while the two statistics units are more designed for a regular math class.  With the exception of the statistics units, an advanced math class ought to be able to work through most of this workbook over the course of the eleventh grade year and the first half of twelfth grade.  A regular math track class in eleventh or twelfth grade can simply pick and choose from the many units in this workbook and from units in the second half of our 10th Grade Workbook (e.g., “Exponential Growth”, “Sequences & Series”, “Math & Music”).

The units covered in this workbook are:

1. Cartesian Geometry, Part I
2. Trigonometry, Part II
3. Complex Numbers, Part I
4. Cartesian Geometry, Part II
5. Possibility & Probability, Part II
6. Trigonometry, Part III
7. Cartesian Geometry, Part III
8. Logarithms, Part III
9. Complex Numbers, Part II
10. Trigonometry, Part IV
11. Cartesian Geometry, Part IV
12. Calculus, Part I
13. Calculus, Part II
14. Statistics, Part I
15. Statistics, Part II
16. Problem-Solving Exercises