The Well-Trained Mind

15 THE LANGUAGE OF REASON: MATH

Mathematics possesses not only truth, but supreme beauty—a beauty cold and austere, like that of a sculpture.

—Bertrand Russell

SUBJECT: Mathematics and algebra, grades 5–8

TIME REQUIRED: 45 to 60 minutes per day

During the logic stage, the study of mathematics goes from arithmetic (mathematical operations such as adding, subtracting, dividing, multiplying, and so on) to mathematics (understanding how numbers relate and why).

The child’s mind also makes the transition from the mental image mode (picturing objects to go along with numerical symbols) to the symbolic mode (using numerals alone). Until this transition is complete, the abstract operations demanded by pre-algebra and algebra are impossible. A problem such as 9 × 2 simply requires you to picture two sets of nine objects. But a problem such as-5x =-15 requires you to deal with symbols that have no easily pictured reality behind them. If I don’t know what x is, how can I picture it? And what mental image can I make of a negative number?

In math, the fifth-and sixth-grade years complete the transition to symbolic thinking. During these years, the student solidifies her grasp of mathematical operations (addition, subtraction, multiplication, and division). She will also be introduced to more abstract concepts: negative numbers, percentages, probabilities, and decimals. She’ll begin to do more complex word problems, ones that will require both logic and abstract mathematical reasoning.

The fifth-and sixth-grade math curricula should involve plenty of practice and no use of hand-held calculators. Until the transition to the symbolic mode of thought is complete, the student must continue to practice math operations.

We suggest that you also do some practical, hands-on math work during these years. The middle-grade student grows easily impatient with material that doesn’t seem to have any logical connection to real life, which is why the National Council of Teachers of Mathematics suggests that middle-grade math curricula place “math in the context of students’ everyday lives…giving students hands-on activities”1 and real-life problems to solve.

Most math curricula can be finished in a year if you do four lessons per week and set aside one extra day to do testing, consumer math, a real-life math problem, or math games. We’ve suggested a few consumer math and math game books at the end of this chapter. Real-life problems might include

figuring out the family’s grocery budget for a week (or a month), or finding the best buys at the grocery store
figuring out expenses and profits for a kid-run home business—grass cutting, pet tending, baby-sitting, baking
balancing a checkbook (better now than in college)
figuring out the monthly and yearly interest on a credit-card debt (ditto)
calculating the area of a room, a wall, or the entire house for wallpapering, carpeting, or another home project
figuring out the cost of driving the car to and from a special event

figuring out how much a restaurant meal would cost if it were cooked at home

calculating the cost in work hours of movie tickets, concert passes, or other types of entertainment

altering a recipe so that it serves a different number of people—for example, reducing a six-person dish so that it will now serve two or (more complicated) rewriting a four-person recipe so that it will now serve nine or eleven

working out the itinerary for a family trip, complete with routes, timetables, and scheduled stops

Your own family life will yield plenty of additional problems. Try to stay alert for those times you use numbers, measurements, or calculations, and then ask yourself whether this problem is within the reach of your young math student.

HOW TO DO IT: FIFTH AND SIXTH GRADES

Somewhere in the logic stage, you may find yourself stymied as you teach your child mathematics. Math, unlike history or reading, requires the mastery of the language of symbols. If you feel uncomfortable speaking this language, you won’t be able to guide your child in its use.

Unless you decide to use the Math-U-See program, with its teaching videos, or the Calvert correspondence course, you may eventually need a mathematics tutor. In Chapter 43, we discuss the numerous tutoring options available to home-school parents: home-school co-ops, local college students, online tutorials, and more.

Whether you use a tutor or teach the material yourself, you’ll need to choose a curriculum and supervise overall progress. For fifth and sixth grades, we recommend one of the following:

Saxon Math

In the middle grades, the Saxon texts continue to teach mathematics incrementally, introducing new concepts one at a time, with plenty of drill and application. The middle-grade home study kits include the texts, an answer key for you, and examinations (with answers) that you can give the child throughout the year. The examinations start with Math 5/4, which is typically used in fourth grade; in grades 1 through 3, students complete oral and written assessments that are not formally graded. Home-schooled students need to practice taking examinations without “peeking,” begging you for extra help, or getting up for chocolate milk. Without this practice, the transition into college work will be unnecessarily difficult.

Saxon is systematic, simple to teach, and mathematically excellent. The Saxon publishing company cultivates the home-school market, providing help for home schoolers via e-mail, a website, and an 800 number. If you used Saxon for grades 1 through 4, you probably finished Math 5/4 at the end of the fourth-grade year. Continue with Math 6/5, the fifth-grade book, and Math 7/6, the sixth-grade book. If you haven’t used Saxon before, request the home-study catalog. A diagnostic test is bound into the center of the catalog; the score card will tell you which Saxon level is right for your child.

Math-U-See

The Math-U-See program is based on a series of teaching videos in which concepts are demonstrated using manipulatives; the student also works with these manipulatives when completing workbook exercises. If you’ve been using Math-U-See, you’re probably over halfway through the Elementary Curriculum; the last three levels of this curriculum, Delta (division), Epsilon (fractions), and Zeta (decimals and percentages), will take you through the fifth-or sixth-grade year, depending on how quickly your student moves through the levels. After finishing Zeta, Math-U-See moves to a standard progression: pre-algebra, first-year algebra, geometry, second-year algebra, trigonometry. You’ll want to begin pre-algebra in seventh or eighth grade and continue on through the progression.

A Beka Math

A Beka Book is used by many home schoolers. The program is comprehensive and drill-intensive. As we’ve noted before, don’t try to do all the problems. You must pick and choose among the A Beka lessons as well if you want to spend one day per week doing real-life math, as we suggest. Note: A Beka is a Christian publishing house, so you’ll find Psalms interspersed with some lessons.

Developmental Math

The Developmental Math program, published by Mathematics Programs, provides a simple math option, in which the child works at her own pace instead of sticking to a daily lesson plan. However, there’s just not enough practice in this program (used alone) to produce mastery, especially in the important early middle-grade years. Developmental Math is best used along with another program; it would be an excellent choice for a child who attends regular school but is struggling and in need of supplemental work. Because the books are divided by skill and are affordable, you could also use them for extra work in a particular area where a student simply needs much more practice.

Singapore Math

Singapore Math, described in detail in Chapter 6, is a program that focuses on teaching mathematical thinking. The fifth-and sixth-grade years of the Singapore Primary Math program consist of one course book and two workbooks for each half of the year; the program focuses on math concepts and on teaching problem-solving and application. Students just beginning Singapore should usually begin with the B book of the semester preceding their grade year.

The coursebook and workbooks alone will not provide enough drill for most students to master thoroughly the facts needed to lay a foundation for upper-level math; invest in the Extra Practice and Challenging Word Problems books as well, or supplement with drill from another program.

Calvert Math

The Calvert math program, offered as an independent correspondence course (Calvert’s other programs are part of their entire grade package), is a solid traditional mathematics course for grades 5 through 8. Taking it by correspondence provides you with daily lesson plans and good teacher support from Calvert tutors (the “Advisory Teaching Service”). Calvert also supplies all manipulatives and supplies, along with tests and quizzes; Calvert awards the grade and keeps an official transcript.

If you feel that you need additional help to teach math, as well as grad ing support, this program is a very good choice that will build excellent math understanding.

Right Start Math

The elementary levels of Right Start Math end with Level E, which should finish out the fourth-grade year. The next level of the course, Intermediate Mathematics, acts as the transition between arithmetic and mathematics; students are intended to progress from the end of Intermediate Mathematics on into a pre-algebra course.

Many students should be able to transition from Intermediate Math into one of the courses listed in the Upper-Level Math section that follows. However, we would offer one caution: the Intermediate Mathematics course takes less than two years to finish, so a student who begins it in fifth grade will be finished in mid-sixth grade. This is young to begin a pre-algebra course. Although some children will be ready to move on to more abstract work, many more need an extra year (or more) of maturity before starting an upper-level program.

You can use Intermediate Mathematics and then plan on going into the sixth-grade year of one of the programs that then progresses on into pre-algebra: Saxon, A Beka, Singapore, or Teaching Textbooks (perhaps moving a little more quickly than usual so as to finish up the sixth-grade year on time). Alternatively, you can go directly from Level E of Right Start Math into the fifth-grade level of any of these programs.

Teaching Textbooks

Teaching Textbooks is a relatively new program that combines texts and software to provide a tutorial experience for the student: the courses are designed for independent study in grades 5 and up. Students read the text lesson, watch a DVD lecture, work practice problems, watch a DVD explanation of the solutions to the practice problems, complete a problem set, and then watch a final solutions lecture: this back-and-forth method allows the student multiple opportunities to understand the material. Teaching Textbooks is not yet a tried-and-true method, but response from the home-schooling community has been very positive and test scores seem to be high. Teaching Textbooks is not dependent on parent exper tise, and offers a seamless transition from arithmetic on into high-school level mathematics.

Other Useful (Fun) Stuff

As a supplement, consider the entertaining narrative math series Life of Fred (discussed in Chapter 6, Chapter 6). The arithmetic volumes, Fractions and Decimals and Percents, are followed by Beginning Algebra, Geometry, and more advanced topics.

In the Resources section at the end of this chapter, we have listed several resources for real-life math; we strongly recommend making use of at least one or two during grades 5, 6, and 7. The website LivingMath.net can continue to be of use as you evaluate your student’s progress.

Schedules

Schedule formal math lessons four days per week, allotting one day to real-life math or math games. Math is generally best done first thing in the morning or at the beginning of your scheduled schooltime.

Sample Schedules

Fifth Grade

45–60 minutes per day		M, T, W, TH		Math lesson
60 minutes or more		F		Real-life math

Sixth Grade

45–60 minutes per day		M, T, W, TH		Math lesson
60 minutes or more		F		Real-life math

THE SHIFT TO UPPER-LEVEL MATH

Seventh grade begins the real journey into symbolic mathematics. College-bound students (and seventh grade is too early to cut off the possibility of college) should plan on taking—as a minimum—pre-algebra, geometry, and two years of algebra. Most students should try to continue their math studies with a pre-calculus/trigonometry course. The mathematically inclined can then, in their senior year, take an advanced elective such as calculus.

A mastery of algebra has implications that go far beyond successful college admissions. Algebra, even at its most basic level, requires the student to work with unknowns, which means that she cannot memorize set answers and fill them in mechanically. Instead, she must analyze each problem, discover its central point, and then apply knowledge already acquired to its solution. Algebra, like logic, teaches the mind to think straight. It demands not only the memorization of information, but also the ability to apply that information in a number of different situations. That is higher-order thinking.

We can’t emphasize enough that higher-order thinking requires mastery of those lower-order skills. Calculators in seventh grade are fine, but only if the student has already comprehended basic mathematical operations. Again, we depart from the opinion of the National Council of Teachers of Mathematics, which recommends the use of calculators beginning in fourth grade—a standard that inevitably produces seventh graders with little intuitive understanding of mathematics.

The student who’s still shaky on fundamentals should use the summer before seventh grade to review them. Saxon Publishers offers Saxon Middle Grade Basic Facts, which should ground any student in the operations necessary for a successful move into pre-algebra.

Once the basic operations are mastered, it’s time to prepare for algebra.

Comparing Math Programs

Of the programs we’ve described, all but Calvert and Right Start extend through the twelfth grade. Since Calvert doesn’t offer a high school program, you’ll want to switch over to another program in seventh grade to assure a smooth pre-algebra-to-algebra transition. (Chapter 15 for information on Right Start.)

Of the other programs, A Beka and Saxon are time-tested and well-used, both by home schools and private schools. Developmental Math does not offer enough practice in applying concepts to stand alone as a high school preparatory program, although it may still be valuable as a reinforcement course to take alongside another program. Singapore Math, which progresses from Primary Math (grades 1 through 6) into New Elementary Math (junior and senior high school), and Teaching Textbooks, a young company, are less “traditional” (not as strongly schoolroom oriented) in their approaches and have a much shorter track record. The same is true of Chalk Dust Math (Chapter 15). That doesn’t mean that you should avoid these programs; only that you should take seriously the task of checking the student’s progress. Frequently, ask him to complete a lesson from a different program (always one covering material he already knows), to see how well he’s able to transfer his math knowledge from one context to another. And pay attention to his yearly standardized scores.

Each one of these programs prepares the student for the transition into algebra. The seventh grade A Beka text, Basic Mathematics I, reviews all arithmetic topics and provides plenty of drill in application of math to daily life. The eighth-grade book, Pre-Algebra, introduces basic algebra concepts and continues to drill the student in word problems. The student who struggles with arithmetic and isn’t ready to begin pre-algebra in seventh grade (as the Saxon program does) will do well with A Beka. As we’ve mentioned before, the A Beka program provides more drill than most students need; don’t feel that you need to do every problem in every set.

For seventh and eighth grade, Math-U-See begins its advanced sequence with pre-algebra (seventh or eighth grade) and then moves on to first-year algebra (eighth or ninth grade). The program presents concepts on video or DVD before the student completes his workbook assignments; the program also includes teacher’s manuals and manipulatives. This is a good course for a visual or hands-on learner, and the videos provide plenty of teacher support.

The Singapore Math books for seventh and eighth grades, New Elementary Mathematics 1 and New Elementary Mathematics 2, are the first two of a four-year series that continues on into grades 9 and 10. Each chapter contains explanations, a class activity which you’ll need to adapt to home use, several different sets of exercises, a chapter review, and a challenge for advanced students. Notes on the history of mathematics occasionally appear in sidebars. According to the publisher, NEM 1 is the equivalent of introductory algebra and geometry, while NEM 2 is intermediate algebra and geometry; all four books are the equivalent of American first-and second-year algebra, geometry, and trigonometry.

Saxon Math introduces algebra a year earlier than A Beka and slightly later than NEM. The pre-algebra book, Algebra 1/2, is excellent preparation for the Algebra I book.

The only problem with the Saxon program is that it doesn’t have a sep arate geometry book; the geometry is “integrated” into each year’s study. Proofs are introduced in Algebra II, and the study of geometry concludes with Advanced Mathematics, the book that comes after Algebra II. There’s nothing wrong with the geometry material itself, but to get the equivalent of a full geometry course, students must stick with Saxon all the way through and in a sequence that differs from standard geometry-separate high-school mathematics (see the table “Comparing Mathematics Programs,” below). Math-U-See follows the standard pattern.

Comparing Mathematics Programs: Saxon versus Standard

Grade	Saxon	Standard

Seventh	Algebra 1/2	Pre-algebra
Eighth	Algebra I	Algebra I
Ninth	Algebra II	Geometry
Tenth	Advanced Mathematics	Algebra II
Eleventh	Calculus	Pre-calculus
Twelve	Elective	Calculus

As you can see, Saxon advances the student an entire year by combining geometry with the rest of mathematics. If the student is able to stay with Saxon Math, she’ll have more math earlier than if she were involved with other programs. This allows mathematically gifted students to progress further in high-school math. Also, a student going through the Saxon sequence will have completed Advanced Mathematics by the time she takes her PSATs and SATs, which will give her a chance at a much higher score.

A problem arises when you deal with students who need to progress more slowly. In recognition that many seventh graders won’t be ready to start pre-algebra (Algebra 1/2), Saxon provides Math 8/7 to serve as a bridge between Math 7/6 (the sixth-grade book) and Algebra 1/2. If your sixth grader is still stumbling over the concepts in Math 7/6, don’t push her into Algebra 1/2; use Saxon’s alternate sequence (see the table “The Saxon Program,” below). Either sequence is perfectly acceptable for a middle-grade student. Pushing a child who struggles with math into pre-algebra before she’s ready is counterproductive. The child won’t “catch up” she’ll get frustrated and develop a long-lasting hatred for math.

The Saxon Program: Regular versus Alternate

	Regular Saxon	Alternate Saxon
Grade	sequence	sequence

Fifth	Math 6/5	Math 6/5
Sixth	Math 7/6	Math 7/6
Seventh	Algebra 1/2	Math 8/7
Eighth	Algebra I	Algebra 1/2

The only caution we have about the alternate sequence is that if the student sticks with Saxon throughout, she won’t finish geometry until the end of eleventh grade. Standardized tests—PSATs and SATs—are geometry-intensive. PSATs are taken in the fall of the eleventh-grade year, and a good score can bring multiple offers of financial aid—from National Merit Scholarships to in-house offers made by individual schools. It’s been our experience that students who take a focused geometry course in tenth grade and then finish Algebra II in eleventh grade test better. (See Chapter 27 for the suggested high-school sequence.) If your child isn’t comfortable with arithmetic by the end of sixth grade and isn’t ready to begin Algebra 1/2 in seventh grade, you may want to choose A Beka or Math-U-See instead.

The Singapore NEM program, like Saxon, has integrated geometry (this is more common in European and Asian countries than in America). The student who takes New Elementary Mathematics in grades 7 through 10 will have finished all geometry, algebra, and trigonometry before standardized testing. However, we suspect that many students won’t be ready for this highly accelerated pace. If you’re using Singapore and don’t plan to be finished with NEM 3 before standardized testing, you will run into the same complication: the student will not have finished a full geometry course and may score lower than average. In this case, consider switching to a program with a standard Algebra 1/Geometry/Algebra 2 progression.

Finally: Two more curriculum options, both of which offer DVD lectures, are Teaching Textbooks and Chalk Dust Math. Teaching Textbooks, described on Chapter 15, progresses from Math 5 and Math 6 to Math 7 and Pre-algebra (eighth grade). A placement test on the website, teachingtext books.com, is meant to help you decide whether the student is ready for pre-algebra; students who are just beginning the program in seventh grade should take the test to decide whether Math 7 or Pre-algebra is a better starting place.

Like Teaching Textbooks, Chalk Dust Math offers textbook-based courses accompanied by teaching DVDs, intended for students to use independently. Whereas the Teaching Textbooks texts are original, the Chalk Dust texts are derived from standard Houghton Mifflin texts. Teaching Textbooks appears to offer a more interactive process (the Chalk Dust DVD lectures are accompanied by Complete Solution Guides rather than additional solution lectures). Chalk Dust begins pre-algebra in seventh grade and does not have a course for students who need to move slightly slower, while Teaching Textbooks gives you the option of completing a seventh-grade year before moving on into pre-algebra. Samples of both programs can be viewed at the publishers’ websites, www.chalkdust.com and www.teachingtextbooks.com.

If you’re looking far ahead, Teaching Textbooks offers courses only through pre-calculus, while Chalk Dust also offers calculus and trigonometry.

One of these math programs will certainly match your student’s learning style. 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. Pick a program, try it, and if the student remains frustrated, try another instead. (Borrow from a friend before investing, if possible, or buy used: Sale and Swap boards are located at welltrainedmind.com/forums, and a Google search for used curricula will bring up more options.)

Schedules

Advanced math is best done five days per week, first thing in the morning or at the beginning of your scheduled schooltime. You can divide the number of lessons in the course (Saxon’s pre-algebra, for example, has 137 lessons) by the number of days in your school year (35 weeks or around 180 days on average). Use the extra days for extra drill, testing, or stretching a difficult lesson over two days. And use common sense. If your child flies through pre-algebra, start her on algebra—there’s no reason to make her wait.

Seventh Grade

50–60 minutes per day

M, T, W, TH, F

Selected math program

Eighth Grade

60 minutes per day

M, T, W, TH, F

Selected math program

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 will aid you in placing your child in the most appropriate level. We have listed basic curricula first and supplementary materials 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.

Arithmetic 5. $13.25.

Arithmetic 5, Teacher Edition. $22.25.

Student Speed Drills and Tests. $5.25.

Teacher Speed Drills/Test Key. $9.65.

Arithmetic 6. $13.25.

Arithmetic 6, Teacher Edition. $22.25.

Student Speed Drills and Tests. $5.25.

Teacher Speed Drills/Test Key. $9.65.

Basic Mathematics I (seventh grade). $19.00.

Basic Mathematics I, Teacher Edition. $25.00.

Solution Key. $21.75.

Student Test and Quiz Book. $6.75.

Teacher Test/Quiz Key. $9.50.

Pre-Algebra (eighth grade). $20.25.

Pre-Algebra, Teacher Key. $25.00.

Student Test and Quiz Book. $6.75.

Teacher Test/Quiz Key. $9.50.

Calvert Math. Baltimore, Md.: The Calvert School.

Order directly from Calvert. Each math course includes a textbook, a workbook, and a lesson manual to help you administer the lessons.

Calvert Math Grade 5. $115.00.

Calvert Math Grade 6. $125.00.

Chalk Dust Math. Sugar Land, Tex.: Chalk Dust Company.

Order from Chalk Dust Company.

Prealgebra Traditional. 10 DVDs, textbook, and solutions guide. $399.00.

Algebra 1. 6 DVDs, textbook, and solutions guide. $354.00.

Developmental Math: A Self-Teaching Program. Halesite, N.Y.: Mathematics Programs Associates.

Each level includes a workbook and Instruction Guide. Buy from Mathematics Programs Associates, or at a small discount from Rainbow Resource Center. A full description of the twenty levels available can be obtained by calling 631-643-9300 or visiting www.mathplace.com.

Level 11. Three-Unit Numbers: Multiplication and Division Skills. $20.00.

Level 12. Thousands and Large Numbers: Concepts and Skills. $20.00.

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.

Math-U-See.

This program has a number of different levels and workbook/video/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.

Finishing up elementary mathematics in grades 5–6:

Delta (division).

Teacher Kit. $35.00.

Student Kit. $20.00.

Manipulative Blocks. $30.00.

Epsilon (fractions).

Teacher Pack. $35.00.

Student Kit. $20.00.

Fraction Overlays. $30.00.

Zeta (decimals and percentages).

Teacher Pack. $35.00.

Student Kit. $20.00.

Algebra/Decimal Inserts. $20.00.

Beginning advanced mathematics in grades 7–8 (requires the same Manipulative Blocks and Algebra/Decimal Inserts listed above):

Pre-Algebra.

Teacher Pack. $50.00.

Student Kit. $20.00.

Algebra 1.

Teacher Pack. $50.00.

Student Kit. $20.00.

Right Start Mathematics. Hazelton, N.D.: Activities for Learning, Inc.

Order from Right Start Mathematics. Like Math-U-See, this program sells different combinations of manipulatives, worksheets, and instructor’s manuals. Visit the Right Start website at www.alabacus.com to see all options; the kit listed below contains the essentials.

Intermediate Level: RightStart Starter Kit, Geometric Approach. $115.00.

Saxon Middle and Secondary Mathematics. Orlando, Fla.: Harcourt Achieve.

Order from Saxon Homeschool. The Saxon home-study catalogs include a diagnostic test; you can also obtain the middle-grade placement test at their website. Request a catalog and test from Saxon Publishers at 800-284-7019, or visit their website at www.saxonhomeschool.harcourtachieve.com.

Homeschool Math 5/4 (fourth grade). 2005. $55.75.

Homeschool Math 6/5 (fifth grade). 2005. $56.75.

Homeschool Math 7/6 (sixth grade). 2005. $57.75.

Homeschool Math 8/7. 2005. $59.75.

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, 2000. $65.00.

Pre-algebra for seventh grade; also used by eighth graders who did the Math 87 book in seventh grade.

Algebra 1, 1998. $67.00.

For eighth grade; also used by ninth graders who did Algebra 1/2 in eighth grade.

Singapore Math, U.S. edition.

Singapore Math workbooks and textbooks can be ordered from Singapore Math (an independent dealer, not the program publisher), or from Sonlight. The U.S. edition uses American weights and money (the previous edition for sale in the U.S. does not).

Primary Mathematics 4A.

Textbook. $8.70.

Workbook. $8.70.