Math Logic and Vocabulary

Math is a precise practice and getting math right often relies upon a mutual understanding.  When I say "integer," it's important that everyone understand what an integer is (it's any non-decimal number, positive, negative, or 0).

So, not only is math vocabulary critical to executing math problems, a lot of math tests ask questions that rely explicitly on vocabulary.  So, make sure that you know these terms:

• Integers: Positive or negative whole numbers (including 0), but no decimals or fractions. -96, -21,  -2, -1, 0, 1, 2, 56, 121 are all integers.
• Consecutive: In a row.  1, 2, 3 are consecutive numbers.  8, 10, 12 are consecutive even numbers.  Consecutive numbers need to occur in a row but can start anywhere on the number line.
• Divisible: Can be divided by (e.g., 20 is divisible by 1, 2, 4, 5, and 10).
• Remainder: The number left over after you divide a number.  So, if you do 23 divided by 5, the answer is 4 remainder 3 (5 goes in 4 times, with 3 left over).  Most calculators will not do remainders (they convert remainders to decimals).  The easiest way to do remainder problems is to divide by hand like you did in 5th grade (answers will look like: 4 R 3)
• Even:  Divisible by 2.  2, 4, 6, 8, 10 are even numbers.
• Odd:  Not divisible by 2.  1, 3, 5, 7 are odd numbers.
• Prime: Divisible only by 1 and itself. 2, 3, 11, 13, 17, 23 are prime numbers.  2 is the only even prime; 1 and 0 are NOT prime numbers (there is some debate in the math community about this, but according to SAT and ACT, 1 and 0 are not prime, so don’t count them if you are counting prime numbers!).
• Sum: Answer to an addition problem. The sum of 3 and 4 is 7.
• Difference: Answer to a subtraction problem. The difference of 10 and 4 is 6.
• Product: Answer to a multiplication problem. The product of 2 and 4 is 8.
• Quotient: Answer to a division problem. The quotient of 18 and 9 is 2.
• Factor: Numbers in a multiplication problem. In the problem 3 x 2 = 6, 3 and 2 are factors.  Sometimes SAT uses the word factors to describe all of the numbers that multiply to make another number (or, another way to say it is, all the numbers that a number can be divided by evenly).  The factors of 10 are: 1, 2, 5, 10.
• Divisor: In division, the number divided into the other number (the “divided by” number). If the problem 10 ÷ 2 = 5, 2 is the divisor.
• Dividend: In division, the number that gets divided. If the problem 10 ÷ 2 = 5, 10 is the dividend.
• Exponent: The little number next to a base number that means to multiply the number by itself that number of times (e.g., 22 = 2 x 2 = 4; 24 = 2 x 2 x 2 x2  = 16). In 24, 4 is the exponent.
• Base: The big number next to the exponent in an exponent.  In 24, 2 is the base.
• Inclusive: Including.  If a problem says "consider the numbers 6 to 12, inclusive" consider 6, 7, 8, 9, 10, 11, and 12.
• Exclusive: Excluding. If a problem says "consider the numbers 6 to 12, exclusive" consider only 7, 8, 9, 10, and 11, not the numbers named in the problem.

• Coordinate Plane: The plane, defined by an $x$-axis (horizontal) and $y$-axis (vertical), on which we plot coordinates.

  
  • Coordinates: The (x,y) values of a point, that allow you to plot a point on a coordinate plane. These x and y values can be substituted in for x and y in an equation.
  • Axis/Axes: The x and y (or labeled with other letters) lines that define the plane. Sometimes these are in the middle of the plane (when there are negative values on the axes) and sometimes they define the bottom and left sides of the plane (more typical in a graph with only positive values). 
  • Quadrants: In a coordinate plane with positive and negative x and y values, there are 4 quadrants, named with roman numerals.
  • Intercept: The point where a line crosses an axis.  The y-intercept, where the line crosses the y-axis, is $b$ in the $y=mx+b$ linear equation.
• Minimum: The highest number in a dataset (can also be seen as the highest point on a graph).
• Maximum: The lowest number in a dataset (can also be seen as the lowest point on a graph).

Once you know these terms solidly, you'll be able to interpret even tricky math logic and vocabulary questions.  

Example: What is the greatest of three consecutive integers whose sum is 24?

First, define the words in the problem:

Consecutive = "in a row"

Sum = "answer to an addition problem"

So, you need three numbers in a row that add up to 24.

Start with an estimate:

You need three numbers, in a row that add up to 24.  So, let's figure out what number, added up three times, will make a sum of 24: $24 \div 3 = 8$

So, we know that $8+8+8=24$

What consecutive numbers add up to 24?  

How about: $7+8+9=$?

Check your estimate:

$7+8+9=24$

Remind yourself what the question wants: "the greatest of three consecutive integers."

The greatest number of 7, 8, and 9 is 9.

Overall, knowing math vocabulary well will help you quickly understand many math problems (and also help you eliminate answers -- if an answer has to be an integer, you can eliminate 2.5 right off the bat!).  Knowing math vocabulary, and applying some logic, will also help you work your way through a number of seemingly tricky math problems and puzzles.

When it doubt, follow the vocab and the rules, make estimates, and guess and check.  You may have to guess and check a few times, but do it a few times.  Either the answer or a pattern usually becomes clear pretty quickly!