Category: General High School Questions

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Scope and Sequence for College Bound or Not

Your cover school or state may require 2-4 high school maths. If you are working toward smaller scholarships or high-end private colleges, they may have requirements for specific courses in high school. Contact the colleges you are interested in and ask for their requirements. However, for the most part, colleges do not really care as long as they get money, and your student scores well enough on the ACT and SAT.  You also need to contact the college of your choice to find out the score required on these exams. Sometimes the front line admin people will say they want things – but really they want money. 😉 Know that no matter how high your student’s ACT/SAT score is, they will be required to take a math placement exam at the university and will be placed according to that score. Our courses will help with the ACT/SAT and math placement exams.

So as far as a typical math sequence goes, you can break it up in all kinds of ways. The key is how much time did you spend and what does your school (cover) require. We see some who call it

Algebra
Geometry
Algebra II With Trig
Calculus

and others

Algebra
Geometry
Algebra II (do this course over a longer period)
Trig
Calculus

or

Algebra
Geometry
Algebra II (do first few chapters of Alg II – say chapter 1-4)
Precalc (Do 5-8)
Calculus

So it really just depends.

What I would suggest is to first find out where your student plans to go to college (or the options including what specific fields of study) then contact them and ask what they would like to see. Once you get to the department level advisors, you will get some good counsel. DEPARTMENT LEVEL is key!!! Not just the admissions or Freshman counselors.  I would do this no matter what. They might have other helpful hints you can use now. Do not worry about it being too soon. You are making a plan, and rather than listen to all the common advice about what they want to hear, just hear it from them.

But trust me, even in engineering we would never run off a student for lack of high school courses or high school grades. They are going to focus on those test scores and their own entrance/placement tests. (Which are written by the same people who write the ACT.)

Not Going the College Path? Want some Math Application Studies?

Maybe you aren’t headed to the ACT/SAT or college admissions office. You just want to learn some useful, real-life math applications for entrepreneurship endeavors, trade careers, and controlling your financial life. Here is another scope and sequence that’s good education for everyone.

Algebra
Geometry
Personal Economics  (3rd-year math requirement)
Career Planning (NOT a math credit, but a great way to spend a summer or junior/senior year defining the right path for your student’s future) – a great business or elective credit.
Entrepreneurship 101 – Building Your Own Business (Not a math credit, but definitely a career path.) – a great business or elective credit.

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Calculators for High School Math and Beyond

Calculators come in four various forms (as for math)

  1. The 4 function calculator:  A few bucks and NOT what you want. These are: add, subtract, multiply, and divide.
  2. Scientific calculator:  These calculators can do square roots, exponentials, and trig functions (plus many other things) and are great for high school math courses and can get you all the way to calculus – but not as good as the more powerful calculators. A scientific calculator can be found for a little over $10. The TI-30 series is a great example. Most smartphones have this function built-in.  Turning an iPhone sideways will open up all these functions to the basic calculator. Be careful here as the touch screen is easy to punch in (or miss punching) incorrect information.
  3. Graphing calculator:  These do everything a scientific calculator does, plus it allows you to graph functions. Start around $50
  4. CAS (Computer Algebraic Calculator) with graphing: These do all of the above plus computer algebra. In other words, they will solve problems for you. VERY powerful. They will typically be over $100. the TI-89 is a popular example.

Any of the later 3 will work, but my choice for Algebra II with Trig or higher would be the TI-89 or equivalent. Not a ton of money for something you will use for a few years. A student could certainly do well with a TI-30 also, but if they will learn to use it, a TI-89 is a good deal.

The TI-30 Calculator is included in the Everything You Need Jacob’s Elementary Algebra and Jacob’s Geometry packages. This calculator is allowed on the ACT and can be used throughout your student’s academic career and college. You can use the TI-30 with Wolfram Alpha (see online calculators below) for your graphing needs if you don’t want to stick with the TI-30.

Online Calculators

My favorite tool is the wolfram website.  It is far more powerful by far than any of the above. But, you cannot use it on any tests.

Find the answer to “Do I let my students use a calculator?” here.

 

 

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Say YES to the calculator!

One of the most frequent math questions we get from homeschool parents is

“Do I let my students use a calculator?”

Is your student begging for the calculator? Have you stood firm against it all through elementary school? Do you feel like you are the “no” monster? Here’s your chance to provide a win without just giving in!

Does Using a Calculator Mean You are Not Thinking?

Somewhere along the way, we got it into our heads that using a calculator prevented thinking. Those who pulled out the calculator were lazy and were never going to remember their multiplication tables. 

While there might be some truth to that concern in the early years, once your students are in high school their math skills need to change from memorization to thinking – especially critical thinking. In courses like Algebra, Geometry, and above we are no longer trying to get them to memorize topics – but instead to understand concepts. And in understanding the concepts, the calculator can be our friend in two ways.

  1. The calculator lets our mind focus on learning the concepts and not the labor of adding or multiplying numbers together.
  2. The calculator actually makes us think harder. Whenever you plug in a series of numbers and get something you do not expect, like a negative number in a trig function, we have to think about why the calculator is giving us something we did not expect. In other words, the calculator helps us learn. 

The truth is, the calculator cannot think for us.  In all reality, we can’t even use a calculator until we understand those basic mathematical functions of addition, subtraction, multiplication, and division. Otherwise, we’re just punching buttons.  We have to be able to know what an operation is before we can accurately punch that operation into the machine. 

What if My Student Has Not Memorized their Math Facts?

OK, a reality check is in order. If your student is in high school and still does not know all their math facts by memory – are you really going to hold them back?

Time to face it – if your student is at the high school level and hasn’t memorized those pesky math facts, they aren’t going to.

I will be honest here. I have a Ph.D.. in Engineering and I still need to think about it when I need to know  9 times 7. Judge me if you like, but I just never got it. Machines are made to help me.

The more skill your student has using a calculator, the better that tool works for them. They need to learn to use it to add and multiply so that they can later be comfortable with the tool when it is time to do more complex math.

So if they have not mastered the math facts – move forward anyway. You are holding them back. It’s a bit like continuing to study the alphabet when you’re actually ready to read Shakespeare, or chopping up cabbage by hand when you need so much done that a food processor is a smart way to go.

It’s time to move onward and upward! And the power of a calculator is essential to saving time so you can focus on critical thinking.

At the high school level (Algebra 1 and above), we are exercising and building critical thinking skills rather than rote operations skills. In elementary mathematics, our students were learning the mechanics of manipulating numbers. Long-hand calculations are helpful in learning what is happening in these operations. However, in higher math, we move from rote crunching to critical thinking and analysis.

When a Calculator is Needed

We recommend using a calculator in Algebra 1 and up. Longhand is no longer needed. You will probably find a student new to the calculator using it when they don’t need to, such as for simple problems like 2×3. Don’t worry about it. Let them use it as much as they want, they will soon learn there are some operations that happen quicker in their heads than they can punch in. 

They will also learn that they can’t always trust the calculator.  A calculator is a great tool and it only serves to strengthen what they have memorized (like looking at flashcards.) Using it early builds their calculator skills and experience in when to trust it. 

You will find problems early in Algebra 1 that are simple addition, subtraction, multiplication, and division that are not intended as exercises in longhand mathematical operations. They are exercises in mathematical logic and understanding. These problems are highlighting relationships between numbers and the operations done on them. They are also great experience and skill-building in punching in operations correctly. The calculator only does what we tell it to. We must instruct it correctly and that takes practice. 

When you get to Geometry, your student will find they use the calculator much less than they’d expect. Much of Geometry is logic. They will learn to think through situations, truths, and analysis. A little computation will come into play as well, but the bigger thinking is in the logic.

Algebra 2 with Trigonometry and Calculus will bring in the big guns of the calculator. Trigonometry will make the calculator a good friend of your student. Here they will find it tricky too, as again, the calculator only does what we tell it to, and if you punch it in wrong or in the wrong setting, you get a very wrong answer. You don’t want your student to wait until this stage to become familiar with a calculator. Start in Algebra 1 so they can develop the knowledge of the many operations the calculator can do as they are introduced throughout their high school career.

If you want to know which calculators are useful for which courses, check out this post on calculator types.

Have Fun Exploring.

Learning can be a struggle, especially math. This is a chance to lighten the load without compromising the learning.

The calculator is part of the next step in your student’s learning journey. Let them be thrilled with your YES to the calculator. Let them overuse it. They will soon learn that the calculator doesn’t have all the answers. It only has the answers the student knows how to tell it to calculate.

High School math opens a whole new area to understanding God’s Creation. Have fun exploring and use the great tools that make that deeper understanding a better journey.

 

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Help with Grading

There is no wrong way to grade. It’s completely up to you. You are in charge.

But sometimes we need some help deciding where to even begin. Our advice is to remember that the goal is learning, not grades. Grades are a reflection to OTHERS of how much you’ve learned. Most people can game this system and make great grades while knowing very little. This is especially true in math. Students are great at crunching the mechanics of math and can make the grade, but then actually have little understanding. 

The ACT/SAT exams are geared to catch those deficiencies. They test for understanding rather than crunching. On those exams there isn’t enough time to crunch out the answers, there is a few seconds to guess the correct answer based on your understanding and elimination of impossible answers. 

Math is critical thinking. Logic. 

Learning and understanding is your goal.  At this point, your child is in high school – young adults. If they aren’t already, its time for them to own their education. Let them check their work. Help them understand, if they cheat, they are only cheating themselves. 

Grading Homework

Below are some options to choose from for grading. Using these methods, you may discover a combination of them or a completely different method works best for you. But here’s some ideas to start from:

Option 1:

With our children, we did not give a grade for daily work. We did have them check their work and redo the problems they missed.  The key here is to have the STUDENT check their OWN work. Then have them rework the missed problems – this is where learning happens!

Option 2:

Have the student check and rework their problems and give some credit, not for correct problems but, for the work being done as a percentage of their total grade. 

Example:

30% of the total grade for homework (100% of this if they do and check all their problems)

70% of the total grade for tests

This option takes some pressure off the tests and incentive to do homework, however, if you follow our test grading guide that gives points back for corrections (again, where real learning happens), then the tests aren’t too pressured anyway. 

We also give our tests as open book tests. Math books are good resources and everything in there can’t be memorized as you continue on into higher math. You will need the resource. If you are not comfortable with open-book testing, you could allow them a “cheat sheet” or “note sheet” where they put down the formulas, theorems, and anything from the sections the test is on to use during the test. Even in college math, we were allowed a cheat sheet of a certain size (usually a large notecard or half 8.5×11 paper) for formulas or anything we wanted to put on it.

The goal is to know how to use all the math tools, not to keep the toolbox in your head with no understanding of what they do. 

Grading Tests

In our AskDrCallahan Teacher’s Guide, you will find a test grading guide (mentioned above) that allows for regaining points for reworked problems. 

Example:

Attempt # 1

     a)  Number of problems correct ___30___

     b) Total number of problems   ____50____

     c) Grade  (100*a/b)            ____60____(round up to nearest integer)

Attempt #2            

    d)  Number of problems fixed  ____10___

    e) Points added (70*d/b) ____14____(round up to nearest integer)

 Attempt #3           

    f) Number of problems fixed  ____8____

    g) Points added  (50*f/b)        ____8____(round up to nearest integer)

  Test Grade

    h) Final Grade  (c + e +g)      ____82____(round up to nearest integer)

You can find the test grading guide for your course included with your Online Access in the Introduction Section. Just click on it to download. If you don’t have Online Access you can find our Grading Guide inside the AskDrCallahan Teacher’s Guide on the Everything You Need page for your particular course. Click on the box below for your course to purchase the guide separately. (This guide is included for free with online access so don’t buy it again if you have that access.) Scroll way down to the square that has your course name (looks just like the square below)  to find the AskDrCallahan Teacher’s Guide containing the test grading guide.

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Homework Help Complex Problems – Moving Deeper into Concepts

As you move deeper into each textbook (Algebra, Geometry, Algebra 2 w/Trig, Calculus, etc), the problems will contain several concepts you’ve learned earlier – all in one problem.  This complexity can make it difficult to find the “how-to” or examples in the book to help us understand how to work it.

Recently, we received this question from our student support page: “There doesn’t seem to be any examples or teaching that I can find that helps me solve this problem.”

This is a common question – not this particular problem, but in general, as the problems develop into including several simple concepts stacked into compound calculations. However, the explanations are there, they just may be back a few, or several, chapters.

For example, Jacobs Algebra Chapter 12 Summary and Review Problem 14h.

Concepts include but are not limited to:

  • Chapter 12: Square Roots. Simplify radical as much as possible. Example of this step page 480-481
  • Chapter 5: Equations in One Variable. Specifically for this review problem, Equivalent Equations (Lesson 3) page 162-163.
  • Chapter 12: Square Roots: Radical Equations. Page 505 has examples of squaring both sides to eliminate the radical.

Math builds on itself. As you learn more and more concepts, the problems reach back and build on concepts and analyzing skills learned earlier in the book or even in an earlier course. These complex problems can be hard to find “how-to” when we just can’t see it! The solutions manual is a good resource to help with steps, but sometimes even with those steps, we need to see where it was explained or taught.

We are here to help! Send us your homework questions and let us help. Go to our support page for help. Be sure to include the 5 points below. These points make sure you’ve told us what we need to know to help you.

Be sure to tell us:

  1. the course,
  2. the chapter,
  3. the lesson,
  4. the problem,
  5. YOUR issue as best as you can explain it.

We love to help.

 

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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.