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Understanding Basic Math – A Test

 

Understanding Math

This is the first in a series of posts that allows you to test whether you understand basic math, by which I mean the math that you learned in school, from kindergarten to Grade 12. I will assume that you are either a parent, or a student (school or college-level), and that you realize that in fact you don’t fully understand basic math. No doubt you have already made an effort to upgrade your math skills and understanding, but have found the available material (books or the web) not too helpful for some reason. My recently published book Math Overboard! (Basic Math for Adults), which reviews all of basic math (K-12), attempts to explain fully and understandably all of basic math. By skimming through several of the tests in this series you can find out what you need to study, and what Chapters of Math Overboard! are relevant.

But what does “understanding” math actually mean? I once asked a class of Honors Calculus students to write a short essay on this question, and was surprised to find that they didn’t have a clue. I soon found that I wasn’t very good at answering the question myself. Most mathematicians will say that to understand a theorem you have to understand the proof. That is true, but it’s only part of the story.  First, there are many topics in math other than theorems as such. Algorithms are one example. Applications are another. You have to know (memorize) these things, and also to understand why they are valid. For example, you need to know the long-division algorithm, and why it is valid. How are long division with remainders, and long division with decimal expansions, related? And why is it true that the decimal expansion of a fraction of whole numbers either terminates or repeats?

I am not going to provide you with the answers to the test questions, as this would take too much space. You will need to read Math Overboard! or some other book (or use the web). By reading through the tests you can find out what you need to study.

Arithmetic: Base-Ten Numbers

  1. Explain what base-ten notation, as in 2,316 for example, means, in terms of the powers of ten.
  2. Explain how the addition (or sum) of two base-ten numbers is defined, in terms of counting.
  3. Of course you know that 59 + 37 = 37 + 59 (without actually doing the additions), but how do you explain that a + b = b + a in general?
  4. Describe the addition algorithm for base-ten numbers (which are the same as whole numbers). Explain why the algorithm is valid. (An algorithm is a method of calculation that is routine and repetitive.)
  5. Repeat questions 3-5 for the multiplication of base-ten  numbers.
  6. Another basic law of arithmetic is that a + (b + c) = (a + b) + c. What is the meaning of the brackets (or parentheses)? How do we know that this law is true?
  7. Repeat question 6 for multiplication.
  8. Yet another all-important law of arithmetic is that a(b + c) = ab + ac. Give some examples. Then explain why this law is valid.
  9. What does the expression a < b mean? Give examples.
  10. State and explain the algorithm for deciding which of two given whole numbers is the larger.

Decimal numbers

  1. Explain what decimal number notation, as in 17.62, means.
  2. What is meant by the number line for decimal numbers? How is a given decimal number located on the number line?
  3. Explain the addition algorithm for decimal numbers.
  4. Show how to multiply two decimal numbers (Math Overboard! doesn’t explain why this method is correct, so perhaps you can explain that.)
  5. List the Laws of Arithmetic for decimal numbers. How do we know that these laws are valid?

Subtraction

  1. If a < b, how is the difference b – a defined?
  2. State and explain the subtraction algorithm for decimal numbers a < b.
  3. Explain both addition and subtraction in terms of the number line.

Negative Numbers

  1. Describe the system of integers. Show how to subtract two whole numbers b – a, in the event that a > b.
  2. Explain the subtraction algorithm for two decimal numbers b – a, in the event that a > b.
  3. Define multiplication for integers, and for decimal numbers (positive and negative).

Division and Fractions

  1. If a, b, and c are whole numbers, what does the equation a / b = c mean? Discuss examples. (Be careful: it may not mean anything if we insist that all three symbols refer to whole numbers.)
  2. Now explain the concept of division with remainders.
  3. State and explain the algorithm for division with remainders. Discuss examples.
  4. What is meant by a fraction m/n? Explain how to locate a fraction on the number line.
  5. Explain the cancellation law am / an = m/n.
  6. Explain addition of fractions, in the case that both fractions have the same denominator.
  7. How do you add fractions having different denominators? What is the quickest way to check such an addition?
  8. Explain how to obtain the decimal representation of a fraction.
  9. How are two fractions multiplied?
  10. What is the definition of division for fractions? What is the algorithm, and why is it valid?

If you can answer these questions confidently, you already understand the number system, at least as far as rational numbers (fractions, positive or negative) are concerned. If not, you need to brush up on basic arithmetic.