The cumulative and coherent study of mathematics is, in fact, a microcosm of the entire curriculum and reflects in its expanding field the workings of the scholarly mind in a manner analogous to that which we examined in the field of arts and letters.
—David Hicks, Norms and Nobility
SUBJECT: Higher mathematics
TIME REQUIRED: 5 hours per week for each year of study
Aclassical education considers competency in higher-math skills—algebra, plane geometry, and geometrical proofs—to be part of basic literacy. The classically educated student will complete courses in geometry, first-year algebra, and second-year algebra. This coincides with the bare minimum demanded by most colleges for admission.
Most classical educators suggest that students who have no particular bent for mathematics and no plans for a career in science be allowed to “rest on their oars” after completing basic upper-level mathematics requirements. Ideally, the student would take at least one additional mathematics course after Algebra II; this will give him an edge both in college admissions and in general mathematical and scientific literacy. If this is impossible or if the student has spent extra years struggling to reach the Algebra II level, the additional mathematics course can be eliminated. Generally, students who don’t plan to use advanced mathematics in later life and who aren’t trying for selective college admissions can aim for math-free junior and senior years (grades 11 and 12).
Of the math programs described in Chapters 6 and 15, four—A Beka Book, Saxon, Math-U-See, and Singapore—extend through the high-school years. Developmental Math is not advanced enough to stand alone as a high-school math program, but the books can be very useful for providing extra practice and checking comprehension. In addition, the Teaching Textbooks and Chalk Dust Math programs described in Chapter 15 offer high-school courses.
The A Beka program includes Algebra I (ninth grade), Plane Geometry (tenth grade), Algebra II (eleventh grade), and a two-semester advanced elective course that covers trigonometry in the first semester and analytical geometry in the second. As in the early grades, the presentation is clear and drill-intensive.
Saxon, which began its algebra sequence with pre-algebra (Algebra 1/2) in the middle-grade years, continues with Algebra I, Algebra II, and Advanced Mathematics. Rather than separating geometry out into a different course (a custom peculiar to American mathematics), Saxon covers geometry over the three years of this sequence. By the end of Advanced Mathematics, the student has mastered a full geometry course. Saxon then offers three upper-level electives: calculus, physics, or trigonometry.
Math-U-See is a good option for parents who feel the need for step-by-step teaching but don’t plan on using a tutor. This manipulative-based curriculum presents concepts on video and then asks the student to complete workbook exercises. Algebra I is designed for ninth grade; Geometry, for tenth grade; Algebra II for eleventh grade. A trigonometry elective is available for the twelfth grade.
The Singapore high-school math sequence, designed to run from grades 7 through 10, consists of a four-book sequence: New Elementary Mathematics 1–4. These four books cover introductory, intermediate, and advanced algebra and geometry. Unless you’ve been using Singapore all along, this would not be a good program to start with; it doesn’t follow a standard U.S. math sequence, and it depends on problem-solving methods taught in the primary and middle-school books. You would probably want to list these four years as Algebra I, Algebra II, Geometry, and Trigonometry/Pre-Calculus on the high school transcript, to show that the student has met U.S. college admissions requirements. (NEM covers some, but not all, trigonometry concepts.)
Teaching Textbooks, described in Chapter 15, offers interactive math instruction from Pre-algebra on through Algebra 1, Algebra 2, Geometry, and Pre-Calculus. Chalk Dust Math, another program that makes use of lectures on CD/DVD, also offers these courses, as well as Trigonometry and Calculus 1.
As with junior-high courses, the program you choose will depend on the student’s learning style and your own preferences. If you intend to use a tutor or are comfortable with high-school math, Saxon, A Beka, or Singapore may suit you; if you need a course that provides the student with more tutorial support, Math-U-See, Teaching Textbooks, or Chalk Dust may be a better fit.
Whichever program you use, continue to give the student problems from a second program (as suggested in earlier chapters) to check comprehension. If she can transfer skills learned in one program to problems presented in a different way, she’s thinking mathematically, rather than learning by rote.
The entertaining narrative math series Life of Fred (Chapter 6) can still provide a valuable supplement to the high-school math student. The volumes Beginning Algebra and Geometry are followed by Advanced Algebra, Trigonometry, Calculus, and Statistics; even students who do not intend to go on through a full math course in these advanced topics can benefit by working through these books.
POSSIBLE SEQUENCES
The Ideal Saxon Sequence
In Chapter 15, we outlined several possible tracks for the home-educated student. If you’re using the Saxon home-education courses, you’ve probably followed this sequence:
|
Seventh grade |
|
Algebra 1/2 |
|
Eighth grade |
|
Algebra I |
|
Ninth grade |
|
Algebra II |
The student who’s been on this schedule should now continue with the Saxon Advanced Mathematics course in tenth grade. This program weaves together topics in algebra, geometry, trigonometry, discrete mathematics, and Euclidean geometry. Any student who has managed to finish Algebra II by ninth grade should take this course. Because Saxon integrates geometry into the algebra programs, the student will not have completed a full geometry course until he finishes Advanced Mathematics. A full geometry course is important to fulfill college admission requirements. Students who want to do well on the PSATs (which also serve as the National Merit Scholarship qualifying test), taken in the fall of the eleventh-grade year, should have completed Advanced Mathematics before taking the test; both the PSATs and SATs are heavy on geometry.
After Advanced Mathematics, the student can drop math or continue with Advanced Placement options. Saxon also offers an eleventh-grade calculus course designed to prepare home-educated students for the Advanced Placement examination.
The mathematically gifted twelfth grader can elect to take the Saxon physics course, a mathematically focused program for Advanced Placement students. Or he could take a trigonometry course. Saxon doesn’t offer trigonometry, but the student can take a course at a local university or community college. Alternately, he can enroll in one of the correspondence or online courses that we discuss later in this chapter.
The sequence for high school, then, becomes
|
(Eighth grade) |
|
(Algebra I) |
|
Ninth grade |
|
Algebra II |
|
Tenth grade |
|
Advanced Mathematics |
|
Eleventh grade |
|
Calculus (elective) |
|
Twelfth grade |
|
Physics (elective) or trigonometry (elective) |
The Saxon advanced-math course covers the trigonometry necessary to do well on the SATs. If the student chooses to take an SAT II subject test in math, an advanced, full course in trigonometry should be taken rather than (or before) physics (use the A Beka program or a community-college class).
The Slightly Slower Saxon Sequence
Some students using Saxon math take an extra year to begin algebra, as we outlined in Chapter 15. These students will follow this pattern:
|
(Seventh grade) |
|
(Math 8/7, the extra drill book for those who aren’t quite ready to move on to pre-algebra) |
|
(Eighth grade) |
|
(Algebra 1/2) |
|
Ninth grade |
|
Algebra I |
|
Tenth grade |
|
Algebra II |
|
Eleventh grade |
|
Advanced Mathematics |
|
Twelfth grade |
|
Calculus (elective) |
You might also find yourself on this schedule if you’ve pulled your high-school student out of a public or private school. Although this is a good college prep program, it has one drawback: the student doesn’t finish Advanced Mathematics until the end of eleventh grade, which means that he will not have full knowledge of geometry when he takes the PSATs.
Should you worry about this? The PSATs have two functions: they serve as practice for the SATs taken in the senior year, and they also qualify students for the National Merit Scholarship program. If your high-school student consistently tests above the 90th percentile, a National Merit Scholarship is within the realm of possibility. In this case, you should plan on doing a catch-up course in geometry the summer before the eleventh-grade year so that he’ll be fully prepared for the test.
You can use the A Beka geometry text and work through it with a tutor, or use an online tutorial or correspondence course (see Chapter 27 for details).
A Beka Book Sequence
If you find Saxon frustrating or if you’ve been using A Beka all along and are happy with it, use the A Beka upper-level math program. If you follow the suggested sequence, you will be a year behind the full-speed Saxon program. The Basic Mathematics I text provides a review of arithmetic topics and introduces a few advanced concepts. A strong mathematics student could skip this level and go into pre-algebra in seventh grade, which would accelerate the Beka program to match the Saxon program. You may also wish to use a tutor, since A Beka was designed for classroom use and includes little in the way of teacher guidance.
|
Grade 7 |
|
Basic Mathematics I |
|
Grade 8 |
|
|
|
Grade 9 |
|
Algebra I |
|
Grade 10 |
|
Plane Geometry |
|
Grade 11 |
|
Algebra II |
|
Grade 12 |
|
Trigonometry with Tables, one semester/Analytic Geometry, one semester |
Remember, you should plan on completing geometry by the end of the tenth-grade year in order to score well on the PSATs.
Math-U-See
The Math-U-See program is excellent for visual learners and parents who need extra teaching support. The student should have finished the two-year Advanced Mathematics course, which covers seventh and eighth grades (see Chapter 15 for a full explanation of this level).
|
Grade 9 |
|
Algebra I |
|
Grade 10 |
|
Geometry |
|
Grade 11 |
|
Algebra II |
|
Grade 12 |
|
Trigonometry |
This suggested sequence isn’t for students who wish to specialize in math, since it leaves no room for physics, calculus, or other advanced math electives. However, it provides a good, solid knowledge of high school mathematics for the non-specialist.
Singapore Sequence
Students who have already used Singapore should be on the following sequence:
|
Grade 7 |
|
New Elementary Mathematics 1 |
|
Grade 8 |
|
New Elementary Mathematics 2 |
|
Grade 9 |
|
New Elementary Mathematics 3 |
|
Grade 10 |
|
New Elementary Mathematics 4 |
This leads the student through all algebra, geometry, beginning trigonometry, and pre-calculus skills. A calculus or full trigonometry course from another program could be added for students who wish to do upper-level electives.
If the student is on a slower sequence (beginning NEM 1 later than seventh grade), she’ll run into the same difficulty as the student on the slower Saxon sequence: standardized testing will begin before she has covered all of the required mathematics. Consider supplementing with a focused geometry course from another program before the test.
Chalk Dust Sequence
Students who began the Chalk Dust pre-algebra in seventh grade will follow this sequence:
|
Grade 7 |
|
Pre-algebra |
|
Grade 8 |
|
Algebra 1 |
|
Grade 9 |
|
Geometry |
|
Grade 10 |
|
Algebra II |
with the option of moving on to:
|
Grade 11 |
|
Trigonometry |
|
Grade 12 |
|
Precalculus with Limits |
Students who move more quickly also have the option of finishing the sequence with the Calculus of a Single Variable course.
Teaching Textbooks Sequence
Because Teaching Textbooks allows students to complete a seventh-grade year of arithmetic before moving on to pre-algebra, students can follow this sequence:
|
Grade 8 |
|
Pre-algebra |
|
Grade 9 |
|
Algebra 1 |
|
Grade 10 |
|
Geometry |
|
Grade 11 |
|
Algebra 2 |
|
Grade 12 |
|
Pre-calculus |
OUTSIDE HELP
Online Resources
No matter what program you use, you may find yourself needing help. Cornell University sponsors a website with links to all the best online math resources: Ask Dr. Math (a free question-answering service staffed by Swarthmore College students and faculty), Calculus Help, Math Homework Help, Interactive Online Geometry, software tutoring packages, and more. Go to www.tc.cornell.edu/Edu/MathSciGateway to explore the options. You can also check out the University of Pennsylvania math help site at www.math.upenn.edu/MathSources.html. The Living Math website offers articles, reviews of programs, and links to reference and information sites. You can access the site at www.livingmath.net.
Correspondence Options
Seton Home Study School offers both Saxon algebra and a standard Houghton Mifflin geometry course by correspondence. You pay the tuition fee; they provide explanations, lesson plans, tests, and the final grade. Seton is Catholic-oriented and academically demanding. See Resources for contact information.
The most extensive high-school correspondence program is offered by the University of Nebraska at Lincoln. Their Independent Study High School is used both by home schoolers and by small rural schools that want to broaden their range of courses. Over 130 courses are available, including a full range of mathematics classes. See Resources for contact information.
The University of Oklahoma offers by correspondence Algebra 1 and Algebra 2, using the Saxon texts. Other high-school math courses include modern geometry, trigonometry, pre-calculus, and analytic geometry. These will provide the student with more feedback than the Saxon home-study kits.
Keystone National High School offers a full range of high-school courses, including AP courses, by traditional correspondence or through online instruction; you can view the course options at www.keystonehighschool.com.
A number of other colleges and universities offer correspondence courses that are open to high-school students. For a full list, see Chapter 43.
For advanced mathematics, you can also enroll your high-school student in a local college or university introductory math course. This is known as “concurrent enrollment” and has the advantage of proving that the student is capable of college-level work. Consider this option for eleventh-and twelfth-grade math electives.
ADVANCED PLACEMENT
Students completing calculus and trigonometry as advanced electives may qualify for Advanced Placement credit. The standard Advanced Placement test is given every year in March. The College Board offers Advanced Placement credit in both calculus and physics. To qualify for credit, all you have to do is score well on the exam. For detailed descriptions of the exams, you can call the College Board AP information line at 212-713-8066 or visit their website at www.collegeboard.com (search the site for “AP”). According to the College Board, home schoolers are permitted to take Advanced Placement exams at their local school; call your nearest high school, and ask to speak to the person in charge of the exams.
For further discussion of standardized tests, see Chapter 40.
SCHEDULES
All college-bound students should complete two years of algebra plus a geometry course. This sequence should begin in eighth or ninth grade.
Saxon Sequence
Algebra I
Algebra II
Advanced Mathematics
Elective: Calculus
Standard Sequence
Algebra I
Geometry
Algebra II
Electives: Pre-calculus, Calculus, Trigonometry, Statistics
Singapore Sequence
NEM 1 (algebra, geometry)
NEM 2 (algebra, geometry)
NEM 3 (algebra, geometry, basic trigonometry, pre-calculus)
NEM 4 (algebra, geometry, basic trigonometry, pre-calculus)
RESOURCES
For publisher and catalog addresses, telephone numbers, and other information, see Sources (Appendix 4). We suggest that you contact these publishers of math materials and examine their catalogs closely before deciding on a curriculum. Most publishers will help you place your child at the most appropriate level. We have listed basic curricula first and other resources (online tutorials and correspondence schools) second.
Math Curricula
A Beka Book Traditional Arithmetic series. Pensacola, Fla.: A Beka Book.
Order from A Beka Book. A Beka Book also offers a wide range of mathematics teaching aids. Ask for a copy of their home-school catalog. Parents don’t need the curriculum/lesson-plan books for each level; these give tips for classroom teaching.
Basic Mathematics I (seventh grade/optional). $20.25.
Basic Mathematics I Teacher Key. $25.00.
Student Test and Quiz Book. $6.75.
Teacher Test/Quiz Key. $9.50.
Pre-Algebra (seventh/eighth grades). $20.25.
Pre-Algebra Teacher Key. $25.00.
Student Test and Quiz Book. $6.75.
Teacher Test/Quiz Key. $9.50.
Algebra I (eighth/ninth grades). $23.75.
Algebra I Solution Key. $42.00.
Algebra I Student Test and Quiz Book. $10.50.
Algebra I Teacher Edition. $42.00.
Algebra I Teacher Test and Quiz Key. $11.50.
Plane Geometry (ninth/tenth grades). $23.75.
Plane Geometry Solution Key. $58.75.
Plane Geometry Student Test and Quiz Book. $10.50.
Plane Geometry Teacher Test and Quiz Key. $11.50.
Algebra II (tenth/eleventh grades). $23.50.
Algebra II Solution Key. $43.00.
Algebra II Student Test and Quiz Book. $10.50.
Algebra II Teacher Test and Quiz Key. $11.50.
Trigonometry with Tables (eleventh/twelfth grades, one semester).
$20.00.
Trigonometry Answer Key. $11.75.
Trigonometry Solution Key, Selected Problems. $37.00.
Trigonometry Student Test and Quiz Book. $6.50.
Trigonometry Teacher Test and Quiz Key. $9.50.
Analytic Geometry (eleventh/twelfth grades, one semester). $20.00.
Analytic Geometry Solution Key, Selected Problems. $37.00.
Analytic Geometry Student Test and Quiz Book. $6.50.
Analytic Geometry Teacher Test and Quiz Key. $9.50.
Chalk Dust Math. Sugar Land, Tex.: Chalk Dust Company.
Order from Chalk Dust Company.
Algebra 1. 6 DVDs, textbook, and solutions guide. $354.00.
Geometry. 9 DVDs, textbook, and solutions guide. $299.00.
Algebra 2. 8 DVDs, textbook, and solutions guide. $429.00.
Precalculus. 14 DVDs, textbook, and solutions guide. $534.00.
Calculus. 11 DVDs, textbook, and solutions guide. $429.00.
Trigonometry. 9 DVDs, textbook, and solutions guide. $419.00.
Developmental Math: A Self-Teaching Program. Halesite, N.Y.: Mathematics Programs Associates.
Each level includes a workbook and teacher’s edition. Buy from Mathematics Programs Associates. Good for reinforcement and comprehension testing. A full description of the twenty levels available can be obtained by calling 888-MPA-MATH or visiting www.mathplace.com.
Level 13. Decimals, Fractions, and the Metric System: Concepts and Basic Skills. $20.00.
Level 14. Fractions: Concepts and Skills. $20.00.
Level 15. Fractions: Advanced Skills. $20.00.
Level 16. Special Topics: Ratio, Percent, Graphs and More. $20.00.
Level 17. Algebra 1: Signed Numbers. $20.00.
Level 18. Algebra 2: Equations. $20.00.
Level 19. Geometry 1: Foundations of Geometry. $30.00.
Level 20. Geometry 2: Two-Dimensional Figures. $30.00.
Life of Fred. Reno, Nev.: Polka Dot Publishing.
Order from Polka Dot Publishing.
Beginning Algebra. $29.00.
Fred’s Home Companion: Beginning Algebra. $14.00.
Advanced Algebra. $29.00.
Fred’s Home Companion: Advanced Algebra. $14.00.
Geometry. $39.00.
Geometry Answer Key. $6.00.
Trigonometry. $29.00.
Fred’s Home Companion: Trigonometry. $14.00.
Calculus. $39.00.
Calculus Answer Key. $6.00.
Statistics $39.00.
Statistics Answer Key. $6.00.
Math-U-See.
This program has a number of different levels and workbook/video/DVD/manipulative combinations. For prices, explanatory material, and brochures, call the Math-U-See national number (888-854-6284), which will transfer you to a local representative; or visit the website at www.mathusee.com. Each teacher kit includes an instructional video or DVD and a manual; each student kit contains a text and test booklet. The manipulative blocks are the same set required for earlier Math-U-See curricula. New “Honors Enhancement” books are offered for each level, $10.00 per book.
Pre-Algebra.
Teacher Kit. $50.00.
Student Kit. $20.00.
Algebra/Decimal Inserts. $20.00.
Manipulative Blocks. $30.00.
Algebra 1.
Teacher Kit. $50.00.
Student Kit. $20.00.
Geometry.
Teacher Kit. $50.00.
Student Kit. $20.00.
Algebra 2.
Teacher Kit. $70.00.
Student Kit. $20.00.
Trigonometry.
Teacher Kit. $70.00.
Student Kit. $20.00.
Saxon Secondary Mathematics. Orlando, Fla.: Harcourt Achieve.
Order from Saxon. Prices are for the Saxon Home Study Kits. Visit the Saxon website (www.saxonhomeschool.harcourtachieve.com) for other combinations and options.
Math 8/7. $84.50.
This is the transitional book for seventh-grade students who aren’t ready to begin pre-algebra. If your sixth grader went through Math 7/6 without unusual difficulty, you can skip Math 8/7 and go straight into Algebra 1/2.
Algebra 1/2. $99.00.
Pre-algebra for seventh grade; also used by eighth graders who did the Math 8/7 book in seventh grade.
Algebra 1. $102.00.
For eighth grade; also used by ninth graders who did Algebra 1/2 in eighth grade.
Algebra II. $99.00.
For ninth grade. Used by tenth graders who did Algebra I in ninth grade.
Advanced Mathematics. $108.00.
For tenth grade. Used by eleventh graders who did Algebra II in tenth grade.
Calculus. $74.50.
For eleventh or twelfth grade. Treats the topics covered in an Advanced Placement AB-level program as well as some of the topics required for a BC-level program.
Physics. $69.50.
For any student who has completed Algebra 2.
Singapore Math, U.S. edition.
Singapore Math workbooks and textbooks can be ordered from Singapore Math (an independent dealer, not the program publisher). The U.S. edition uses American weights and money (the previous edition for sale in the U.S. does not). Visit singaporemath.com for additional options, combinations, and courses.
New Elementary Mathematics 1 (seventh grade)
Textbook. $21.50.
Workbook. $8.30.
Teacher’s Manual. $9.00.
Solution Manual. $27.00.
Handbook for Secondary Math Teachers (7–10). $29.00.
New Elementary Mathematics 2 (eighth grade).
Textbook. $21.50.
Workbook. $8.30.
Teacher’s Manual. $9.00.
Solution Manual. $27.00.
New Elementary Mathematics 3 (ninth grade).
Math Textbook 3A. $12.50.
Teacher’s Manual 3A. $6.50.
Math Textbook 3B. $12.50.
Teacher’s Manual 3B. $6.50.
New Elementary Mathematics 4 (tenth grade).
Math Textbook 4A. $12.50.
Math Textbook 4B. $12.50.
Teacher’s Manual 4. $15.00.
Teaching Textbooks. Oklahoma City, Okla.: Teaching Textbooks.
$184.90 for each course. Order from Teaching Textbooks.
The Pre-Algebra Teaching Textbook.
The Algebra 1 Teaching Textbook.
The Geometry Teaching Textbook.
The Algebra 2 Teaching Textbook.
The Pre-Calculus Teaching Textbook.
Online Tutorials and Correspondence Schools
Cornell University website with links to online math resources: www.tc.cornell.edu/Edu/MathSciGateway.
Keystone National High School, 920 Central Road, Bloomsburg, Pa. 17815; 866-382-1228; www.keystonehighschool.com.
Living Math: www.livingmath.net.
Saxon Math online help: www.saxonpub.com.
Seton Home Study School, 1350 Progress Drive, P.O. Box 396, Front Royal, VA 22630. Call: 540-636-9990. Web: www.setonhome.org.
$175.00 plus book fee for a single course. Offers Saxon Algebra 1 through AP Calculus.
University of Nebraska–Lincoln Independent Study High School. P.O. Box 888400, Lincoln, NE 68588-8400. Call them at 866-700-4747. Or visit the website at www.nebraskahs.unl.edu.
The University of Oklahoma Center for Independent and Distance Learning, University of Oklahoma High School, 1600 Jenkins, Room 101, Norman, OK 73072-6507. Call them at 800-942-5702, or visit the website at www.ouhigh.ou.edu.
Also see the full list of correspondence-course resources in Chapter 43.