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Geometry Ch8 Review #32

Question from Jennifer:
Chap 8 review
#32
We aren’t sure how the answer would be 24mm.
Answer:
This is not explained well in the textbook.
First – the translation is the movement of the bat from position B to position C (see chap 8, Lesson 2) A translation is two rotations around a parallel line – or in other words, we moved it over from one place to another.
So he is asking, how far did it move. Or, what is the length of the black line moving from B to C.
Since we can measure the bat to be about 42 mm (with a ruler), the length of the black line is about 24 mm using the same ruler.

 

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Geometry Ch5.1 #29

Question from Courtney

I’m having trouble with Chapter 5, Lesson 1, Problem 29.
I have worked the problem, I know what the answer should be, but given the figure, I don’t understand how that could be false.
Without the figure, I understand it.

Answer:

You can never assume anything. You only know what you are given in the definition. Don’t trust the figure. If it doesn’t tell you, you don’t know it. So for problem # 29 we know the following:

It’s a line
AB is less than BC
BC is less than CD
CD is less than DE

While the lengths in the figure look similar (or even equal), we don’t know that they really are similar or equal. Don’t trust anything but the defined statements in the problem and marked items in a figure.

Here is the question to think about. Given the definitions AB < BC < …. can you draw the figure such that the last DE is much larger.

Put some numbers in it and think inches.

AB = 1
BC = 2
CD = 3
DE = 500

Does that fit the definition of the problem? (Not the figure – the definition?)

Again – figures are often only ONE example that fits the problem – but they do not show every example.

 

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Algebra 2 with Trig Course Description for Transcript

If you are looking for details on for a transcript for Algebra 2 with Trig,  this should do the trick.

Algebra 2 and Trigonometry covers build on basic algebra and geometry to prepare the student for calculus or other college-level mathematics courses.

Students who complete Algebra 2 with Trig should take Calculus next. Concepts from this course show up on the ACT and the SAT.

Topics Covered in this Course:

  • Algebraic Operations
  • Equations and Inequalities
  • Graphs and Functions
  • Polynomial and Rational Functions
  • Exponential and Logarithmic Functions
  • Trigonometric Functions
  • Trigonometric Identities and Conditional Equations
  • Additional Topics and Applications in Trigonometry
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Algebra Ch5.1 #10x

This is an error in the Algebra Solutions Manual

Mary asked,

In Algebra 1, chapter 5 lesson 1 problem 10x, the answer key says the answer is true for all positive integers.  But isn’t it true for all integers? 1 to the power of x =1.  If it is a negative power, it is still one, right?

Mary
Dr. Callahan Answer:
You are correct. Should be for all real numbers – not just integers. I tend to test these in a calculator to make sure though 😉
So that is incorrect in the Solutions manual.
Just for completeness sake – my goto math tool is Wolfram – so here is there answer.
 
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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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When your child is actually ahead, not behind.

I am used to having parents tell me that they think their kid is behind, missed something in 3rd grade, or has a suspected learning disability. That dialogue is almost always the first thing parents say when they are outlining why they need help from a math tutor. For some students, however, they do not need to be held back or to see a doctor. What they need is to move ahead and go on to the next step.

Continue reading When your child is actually ahead, not behind.