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Calculus Textbook Evaluations

Look for the following material or heading in the contents.

  • Limits and Derivatives
    · Limit of a function
    · Calculating limits
    · Limit laws
    · Continuity
  • Differentiation Rules
    · Rates of change
    · Derivative rules (polynomials and exponentials)
    · Product rule
    · Quotient rule
    · Derivatives of trigonometric functions
    · The chain rule
    · Implicit differentiation
    · Derivatives of logarithmic functions
  • Applications of Differentiation (subtitles may differ widely here)

Notes for use: This course should be equivalent to a Calculus One course at a local university. If you know what school your child plans to attend, find out which textbook they use and go with it. In fact, ask for a syllabus and sample tests. A good deal of information can be gleaned from the UAB website on the screening test and the UAB test bank. (See the resources at the end of the chapter.)

Also, note that these books are often written for Calculus I, II, and III. So your text will be a lot bigger than what you will teach in high school. Also, if you do go with a college textbook you will pay a hefty price. An alternative is to find the previous edition on eBay – much cheaper with probably little change in material.

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Pre-calculus (or Algebra II with Trig) Textbook Evaluations

Look for the following material or heading in the contents.

  • Review of Algebra
    · Basic operations
    · Factoring
    · Exponents
  • Equation and Inequalities
    · Linear equations
    · Absolute value
    · Complex numbers
    · Quadratic Equations
    · Polynomials
  • Graphs and Functions
    · Circles
    · Straight lines
    · Functions
    · Graphing functions
  • Polynomials
    · Finding zeroes of polynomials
  • Rational Functions
    · Graphs of rational functions
    · Partial fractions
  • Exponential and Logarithmic Functions
    · Exponential functions
    · Logarithmic functions
    · Common and natural logarithms
    · Exponential and logarithmic equations
  • Trigonometric Functions
    · Angles
    · Right triangle trigonometry (Basic trig functions)
    · Sine, Cosine, Tangent
    · Graphing
  • Analytic Trigonometry
    · Trigonometric identities
  • Additional Topics in Trigonometry
    · Law of Sines
    · Law of Cosines
    · Vectors
    · Complex numbers

Optional material included in some courses.

  • Systems of Equations
    · Solving systems of equations
    · Linear programming
  • Matrices and Determinants
    · Basic operations
    · Square matrices
    · Determinants
  • Sequences and Series
    · Arithmetic sequences
    · Geometric sequences
    · Binomial formula

Notes for use: While everything above is needed material, it is key to get through the trigonometric material – possibly leaving off systems of equations and the later material. At the Algebra II level and above college material should be used.

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Geometry Textbook Evaluations

Look for the following material or heading in the contents.

  • Reasoning and Proof
    · Proofs
    · Deductive reasoning
    · Direct and indirect proofs
  • Lines
    · Parallel and Perpendicular Lines
    · Angles
  • Triangles
    · Congruent
    · Isosceles
    · Equilateral
    · ASA and SAS
  • Quadrilaterals
    · Parallelograms
    · Rectangles
    · Squares
    · Trapezoids
  • Area
    · Squares
    · Rectangles
    · Triangles
  • Similarity
    · Ratio and proportion
    · Similar figures
  • Right Triangle Trigonometry
    · Pythagorean Theorem
    · Proportions
    · Tangent, Sine, and Cosine
  • Surface Area and Volume
    · Geometric solids
    · Rectangular solids
    · Spheres
  • Circles
    · Radius
    · Chords
    · Tangents
  • Transformations
    · Reflections

Notes for use: One of the keys of geometry is learning deductive reasoning or how to do proofs. This can be a challenge to teach, so getting a teachers manual will really help here. The second main idea of geometry is getting used to thinking in space with shapes and the relationships between them. Lots of figures should be drawn.

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Algebra Textbook Evaluations

Look for the following material or heading in the contents.

  • Variables
  • Exponents
  • Order of Operations
  • Equations and inequalities
  • Word problems (converting words into symbols)
  • Real Numbers
  • Adding
  • Subtracting
  • Multiplication
  • Division
  • Distributive property
  • Linear Equations (might be equations in one variable)
  • Graphing
  • Slope
  • Intercepts
  • Point-slope and or slope-intercept formulas
  • Systems of linear equations (or simultaneous equations)
  • Linear inequalities
  • Solving
  • Graphing
  • Absolute Values
  • Exponents
  • Products
  • Divisions
  • Scientific notation
  • Polynomials
  • Quadratic equations
  • Factoring
  • Radicals

Notes for use: All the material on the above list needs to be covered in algebra. Instead of moving fast, the students should understand these concepts pretty well.

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How to Evaluate Math Textbooks and Material

Steps to evaluate a math textbook:

  1. Make sure the basic material is covered. Look at the outlines below as a guide. The wording and chapter arrangement may be different, but you should see these key ideas in the contents.
  2. Is there a well-written table of contents?
  3. Since you will be using the material for homeschooling, look for material with lots of worked examples. Go through some yourself and determine if they are easy to read and follow.
  4. Look for plenty of problems to work.
  5. Do you have the answers? We like the textbooks that have the answers to the odd-numbered problems for the students and then have the solutions to all the problems in the teacher’s manual. If they do not have the answers to many problems, your students will never know if they are doing the problems correctly.
  6. Look for some real-world examples and problems. Do the problems tell about real situations? This is key to helping your child see the use of the material.
  7. Look at some chapters. Are the key points of the chapter outlined in boxes or color so that they stand out? This makes it easy to use as a reference now and later.
  8. Is there a teacher’s reference that tells you how to use the book? If so, is it useful to you? Does it make sense to you?
  9. Is there an index?
  10. What does your student think? If you can, let them compare a few and ask which they like better.

Math materials (such as textbooks, videos, or computer-based teaching) should all cover the same material per course title. In fact, many textbooks will have the exact same chapter titles. So you can do a pretty good job evaluating the coverage of material based on the chapter and subchapter titles. Here are some typical titles or subtitles that should be keywords in your comparison of material. Note that these topics may not match your textbook exactly or be in the same order as listed below. But, a majority of the key points listed should be found.

To use this guide go to the table of contents and look for these keywords. You should not have to search the entire text or videos to find them.

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The Homeschool Math Problem

The U.S. schools are weak in math – almost all of them regardless of public, private, or homeschool. The national weakness in math (and science – which is related) is a growing problem. Math and science are required for technological research, and research (like it or not) is required for national security. The fact is a majority of U.S. university graduate students in technology areas are from other countries – many with less than stable political systems. Congress and agencies that deal with national security are well aware of the problem – but fixing it is another challenge.

But what about homeschoolers?

We outperform everyone in everything – right? Wrong. Homeschoolers are weak in math.

“Homeschoolers need to do better in math. Our reading and language skills are excellent, even though we could always use a little improvement. But our math skills need real help. We only do slightly better than public schools here. We need to drill basic facts, teach concepts, and make sure we take our children through algebra II and geometry at a minimum.” [Mike Farris, “Aim high(er):,” World Magazine, April 28, 2001, Vol. 16. ]

Even though that quote is from years ago – not much has changed.

Besides the personal implications of future employment in a technology world, we need to consider the apologetic issues. Many Christians have been taken out of scientific debates about the origins of the universe, evolution, astronomy, etc due to their raw lack of basic knowledge about math and science. Remember – theologians of the past have been astute students of both the Bible and nature – what systematic theology calls special and general revelation from God. (Note – our courses are not faith-based – but we are Christians. We firmly believe that we do not need to force our faith into places like math – we just speak the truth. Many who have chased truth who do not believe in the God of the Bible have found the path to truth led them to their faith – including me.)

But the problem we have as parents is that we too were raised in a school system that was often inadequate to prepare us to teach our own children. Even then, most of us who did take the advanced math and science course have long since forgotten the skills we once had in the areas of math and science.

Are we preparing our children to live in an age of technology? …defend the faith against scientific attacks? …teach their children? Just as language is the way to study the written word of God or special revelation, math is the language used to study nature or general revelation.

So we have written this part to answer some other specific questions we often get about math.

What types of textbooks should I buy?

We suggest you lean toward a more college level in the Algebra II w/ Trig and Calculus courses – if not before. In fact, if you know where your child plans to go to college, find out what math they will need, get a syllabus, and use that text. (Our daughter would have to take one course in Calculus in her degree field, so we taught her the same calculus in the same book. She found the college calculus course just a review.) Note that the high school textbooks tend to be written with easier problems than the college level textbooks. See the From the Trenches below.

From the Trenches

As a fairly new member to the engineering faculty, I had learned that graduation rates in engineering and science had been down in all United States colleges and universities for the past twenty plus years. Most of this was because the high schools did not provide adequate training in math, so incoming students often got discouraged and moved into other fields. My curiosity drove me to call our math department and ask how students in general did in our calculus courses. The head of the undergraduate program explained that two local high schools outperformed all other students in math. (Both were public schools.) So I went to the math department at one of these schools and I asked what they did that made them better. The biggest issue was they used college level textbooks. They explained to me that the high school textbook publishers competed on how easy the problems were to work. The college level publishers would never survive if they watered down the material.
So the schools that use watered down material in their textbooks have built a large gulf between their math courses and the universities math courses. The few hours we spent at the local high school were convincing – and we have never turned back from college textbooks.

What about Saxon Math?

We often get asked about the Saxon math material. It seems people either love Saxon or hate it, but few are neutral. In our view, Saxon provides an excellent base in the younger years when we are starting to learn the concepts of math as well as the basic facts. However, we steer away from Saxon once we hit algebra – and the higher you go the less we like it. Now before you send us letters of how well your children did in Saxon let us say that any math material can do a great job depending on the student.

Our problem with Saxon is not that it does not teach, but it does so in a choppy manner – or as Saxon refers to it – incremental. The incremental method is big selling point of Saxon – but it falls terribly short in the later grades for two reasons

  1. Saxon textbooks are difficult to use as a reference. Good math
    textbooks (algebra and above) should be good reference material for future math courses. Since Saxon does things incrementally, it is difficult to go and find a reasonable treatment of any subject in one place, therefore making the Saxon material less than adequate for math reference. This also creates a challenge to parents trying to help their son or daughter with a concept. When we try to get to the root of an issue we often need to go back to where that subject was covered to make sure we (as the teacher) understand what the author is trying to do. In Saxon, this is very difficult to do.
  2. Another problem with Saxon is it does little in the way of application. One thing we have learned from teaching math, science, and engineering courses is that application is very important. Often I am asked questions such as “why would anyone ever use this?” (I must agree, that is how I – and probably you – always viewed math.) When I hear this question I will discuss how these elements are used in engineering, sales and marketing, construction, medicine, and yes homemaking. We have had discussions about aircraft, electronics, lighting, the space shuttle, relativity, high blood pressure, lung capacity, cooking, household cleaners and chemicals, (on and on) as part of the answers to these questions. When these discussions take place with students I see an element of excitement in the students that were not seen before. Teaching someone what a hammer is and does is nice, but showing them how it is used to build a house is an education! Saxon (and others) is weak in this area.

So, if you want to use Saxon, I encourage you to do two things. 1) Find a good math reference material. Some books such as Schaum’s outlines or related inexpensive material might suffice here. 2) Find a source to get at applications of the math. The best source of this material is to use the web and have your children find ways in which the math they are using is used in the “real” world. However, this might be a challenge at times, so a supplement of another type of math text in the same subject would be useful. We will soon be offering a series of math experiments (expected to be about $25 per course) that will be usable with any math curriculum at the high school level. Join our email list to get updates on these supplements.

How does math fit into Classical Education?

Classical education has become very popular, and we are big fans of it. But frankly, it is weak in math, and possibly weak in science. Conventional wisdom on classical homeschooling has the higher math courses as electives at best. Yet many sources for classical education recommend reading material written by Copernicus, Kepler, and Einstein as part of the science curriculum. Without knowledge of calculus, these works would be overwhelming. So if you like the classical approach we applaud you – just do not skip on the math! At the very least, your children should get through Algebra II with Trig.

What math material should I use?

Here is a big question! We have often been asked to review math material – and overall we are disappointed in what is available for homeschooling in the area of math. Not that the concepts are missing, but the method of presentation is similar to the presentation given in the public and private schools. Dry and without any application to the real world. College textbooks are (in general) richer in their treatment and application of the concepts. However, you may need some outside help with the textbooks since they are designed to be instructor based.

Any curriculum is OK as long as it meets the basic objectives of the course title. The rest is how well your child takes to it. If they love it, you have found a perfect match. The section Evaluating Math Texts will give some guidelines for picking the proper math curriculum.

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How Do I Get Help if I Get Stuck?

As a customer of AskDrCallahan (directly or if you bought from a vendor such as Rainbow Resources or Veritas Press) you can get support. Whether you need help with homework or test problems, have a question about the solutions manual, or need something more involved like video tutorials on a specific concept and ideas for at-home activities, all of these features are included, and FREE, for as long as your student -nor any of their siblings – are using our video course.

How to Get Help


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Can my son/daughter do this course on their own without my help?

Yes. We know, we homeschool too. Parent overload is common. We have designed all of our courses so that you can either do it with them or have them do it on their own.

At our home we teach our children to do everything on their own. We help them plan out a schedule for the term at first using the syllabus. Then each week we review progress. They work on their own except when they have questions, need group interaction, or need to have a test graded.

Our view is that we are preparing our kids for college and life, and we want to teach them to learn on their own, to have a love for learning, and to develop personal responsibility.



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CLEP Testing

Mary Asked:

My son just completed your course Alg II & Trig.  Is this enough preparation to pass the clep exams for Algebra or Pre-calculus?  If not, do you have any suggestions to prepare for it? Same questions for your calculus program?


Dr. Callahan Answered:


For CLEP credit after ALGEBRA II WITH TRIG, your options are:


Either would work with a little prep. I might go after precalc first.

For Calculus there is only one exam, and our calculus course would be most of (if not all) the prep needed.

But here are some guidelines for all CLEP:


  1. Check with the university you plan to attend. How do they treat CLEP? Most will accept a lot of CLEP – so make sure. If they do not accept much, I might rethink the school unless they are offering scholarships to you.
  2. Then check the department of study. For instance, if going into engineering or a sciences field, they may prefer to see one over the other – or may not accept either. Ask. In many engineering schools they will not allow you to CLEP out of math. But remember, EVERYTHING IS NEGOTIABLE. A good student who is proven is hard to pass up. So if you get a no, ask “What will you need to see to accept my CLEP credit?”
  3. Then assuming you have a direction, prepare for the test. There is USUALLY something that was not covered in your course, so you might need to go cover it quickly. In our Algebra II with Trig course, you might want to get familiar with the other chapters (at least a few of them) that we did not cover. Almost every course is that way – so just make sure you know what you need. Also, make sure you pretest and brush up on what they need. Get the study guide and see what you are missing. Key is GET SAMPLE TESTS.

Then take the test. If you do not pass it on first try, see what you got wrong and then go try again. Do not consider not passing as failure, just instead as telling you where you were weak.

For almost every college, CLEP is well worth the trouble. Costs of CLEP is less than $200 (including study material) to save over $1000 – and you will have to learn the extra material anyway. Do it now, do it fast, and do it much cheaper.
Hope this helps.

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Geometry or Algebra II withTrig First?

Question from Melissa:

I am trying to decide which course to do first?  My son is finishing 8th grade this year and has completed Alg 1.  As a freshman, he would need either Geometry or Alg/Trig.  Is there a preferred sequence?  Would the sequence matter for the SAT or ACT?

Answer from Dr. Callahan:

We usually recommend Geometry first. The book is still at high school level (mostly), has  lot of Algebra review, and introduces the student to Trig. 

The Algebra II with Trig (Barnett text we use) is actually a college book and a bit more intense. 

As for the SAT/ACT, you will see both Geometry concepts and Algebra II concepts on the tests. The ACT has small amount of Trig (actually what is done in Geometry gets you there) 

So what we would do it this:

1. Take Geometry first. 

2.Same time get prepared for or take the ACT/SAT. We would recommend a coach for this if you can because they can really help prepare and push up the score. Note we have a partner Higher Score Test Prep who helps in this and offers our customers a discount. 

3. Then I would evaluate the ACT/SAT with respect to content and work on improving it while moving into Algebra II with Trig. 


Also you might see this article on scope and sequence for the college bound

Hope this helps