MATH CHEAT SHEET (6TH – 8TH GRADE)


Unless you’re an accountant, an engineer or a math teacher, you’ve probably forgotten some math lessons from long ago. As a reminder: middle school means algebra.

Junior high math teachers attempt to bridge the gap and make a smooth transition from the basics learned in elementary school to the theory-based lessons taught at the high school level. An awkward stage for your child in general, middle-school doesn’t need not mean additional angst over math homework.

Using this worksheet as a refresher course, you can help your child feel more comfortable with these increasingly complex math sets.

Figuring Out Fractions

A fraction is a number written in the form: N/D where N is the numerator and D is the denominator. In the typical case, the numerator and denominator are whole numbers. However, the denominator cannot be zero.

A proper fraction has a numerator that is less than the denominator:
4/9

An improper fraction has a numerator greater than or equal to the denominator:
9/4

A mixed number is a whole number and a fraction.
1 ¾

The reciprocal is the inverse of a number. For a fraction, it’s obtained by “turning the fraction over.”
Fraction: 2/3
Reciprocal: 3/2

Fraction • Reciprocal = 1
2/3 • 3/2 = 1

Equality Rule: a/b = c/d if and only if a • d = b • c

When the cross products, the results of a • d and b • c, are the same values, the two fractions are equal.

Adding and Subtracting Fractions:

Like fractions have the same denominator (2/3 and 1/3 are like fractions). You can add and subtract like fractions easily—simply add or subtract the numerators and write the sum over the common denominator.
1/3 + 2/3 = 3/3
5/7 - 2/7 = 3/7

Before you can add or subtract fractions with different denominators, you must first find equivalent fractions with the same denominator, or the least common denominator (LCM). Here’s how:
  1. Find the smallest multiple (LCM) of both numbers.
  2. Rewrite the fractions as equivalent fractions with the LCM as the denominator.
1/5 + 1/3 =  1 • 3  +  1 • 5  = (3/15) + (5/15) = 8/15
5 • 3 3 • 5
The same rules apply for subtracting fractions with different denominators.

Multiplying and Dividing Fractions:

Multiplication Rule: a/b • c/d = ac/bd

Multiply the two numerators over the two denominators.

1/3 • 4/5 =  1 • 4  = 4/15
3 • 5

Division Rule: Multiply the dividend by the reciprocal of the divisor.

2/5  = 2/5 • 4/3 = 8/15
3/4

Keep It Simple!

When a problem can be simplified, you should simplify before substituting numbers for the variables. This will make your job a lot easier. Here’s How:

2(3 + x) + x(1-4x) + 5
  1. Simplify the parentheses.
    6 +2x + x – 4x2 + 5
  2. Combine like terms by adding coefficients.
    6 + 3x - 4x2 + 5
  3. Combine the constants.
    11 + 3x - 4x2
To keep it clear, here are a few rules on the order of operations.
  1. First, do all operations that lie inside parentheses.
  2. Next, do any work with exponents or radicals.
  3. Working from left to right, do all multiplication and division.
  4. Finally, working from left to right, do all addition and subtraction.
Here’s an example to work through.

8 • 22 + 7y (4 + 1) = 32+35y
  1. First add the elements in parentheses. (4 + 1) = 5.
  2. Next, carry out the exponent. 22 = 4.
  3. Multiply 8 • 22 or 8 • 4 = 32.
  4. Multiply 7y • 5 = 35y.
  5. Add the remaining elements from left to right: 32 + 35y
Graphing

The coordinate plane is determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin (0, 0). Each point in the coordinate plane can be specified by an ordered pair of numbers, or a set of two numbers in which the order has an agreed-upon meaning (x, y). The first number is the horizontal coordinate and the second is the vertical position.

Plotting straight lines

Every straight line can be represented by an equation: y = mx + b. The coordinates of every point on the line will solve the equation if you substitute them in the equation for x and y.

The slope (m) of this line—its steepness, or slant—can be calculated like this:

m = change in y-value
change in x-value

The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept of this line is the value of y at the point where the line crosses the y axis.


By Natalie Bauer

Worksheet

Algebra
1.) -6 = x – 2
2.) -3x – 5 = 7
3.) -106 = 4x + 6x + 4
4.) 4 (7 + 6x) = -260
5.) -6x – 3 = 5x -47


Fractions

6.) Are 4/5 and 8/10 equal?
7.) 4/5 – 2/3 =
8.) ½ + 3/8 =
9.) 3/10 + 4/15 =
10.) 11/56 + 3/7 =
11.) 2/7 x 3/8 =
12.) 2/9 x 4/9 =
13.) 1/8 ÷ ½ =
14.) 2/9 ÷ 6/8 =
15.) 2/3 ÷ 3/8 =

Answers
1.) x = -4
2.) x = -4
3.) x = -11
4.) x = -12
5.) x = 4
6.) Yes; 4 x 10 = 8 x 5
7.) 2/15
8.) 7/8
9.) 17/30
10.) 5/8
11.) 3/28
12.) 8/81
13.) ¼
14.) 8/27
15.) 16/9