Introduction

Mastery of basic arithmetic is essential for students to succeed and progress through Algebra I topics and beyond. Students must understand the application of math properties, operations with rational numbers, order of operation to solve equations, and the use of variables.

Math Properties

The understanding of math properties can help students to make sense of Algebra I topics. By applying the properties to basic arithmetic problems, students are more able to connect to higher level applications.

Commutative Property for Addition and Multiplication

  • $a + b = b + a$
  • $a\times b=b\times a$

Associative Property for Addition and Multiplication

  • $a+ \left(b+c\right) = \left(a + b\right) + c$
  • $ a\times \left(b\times c\right) = \left(a \times b\right) \times c$

Distributive Property

  • $ a\times \left(b+ c\right) = \left(a \times b\right) +\left(a\times c\right)$

Additive Identity Property

  • $ a+0=a$

Multiplicative Identity Property

  • $ a\times1=a$

Additive Inverse Property

  • $a+\left(-a\right)=0$

Multiplicative Inverse Property

  • $ a\times\frac{1}{a}=1$
  • $ a\neq0$

Zero Property

  • $ a\times0=0$

Rational Numbers

Rational numbers can be expressed as a fraction or ratio. Rational numbers include natural numbers, whole numbers, integers, and terminating and repeating decimals.

  • Natural numbers - numbers you use to count items: 1, 2, 3, 4, 5…
  • Whole numbers – natural numbers plus 0
  • Integers – whole numbers plus their inverses: …-3, -2, -1, 0, 1, 2, 3…
  • Fractions: an integer divided by an integer that is unequal to 0

All numbers that can be displayed on a number line are rational numbers.

numberline

All rational numbers can be displayed as a fraction.

  • $\frac{1}{3}=0.\bar{3}$
  • $\frac{16}{16}=1$
  • $\frac{4}{2}= 2$
  • $\frac{-12}{4}=-3$
  • $1.375 = \frac{11}{8}$

Integer Operations

When adding and subtracting integers, watch out for potential sign errors. One strategy for solving integer sums and differences is to first determine the sign then calculate the number.

  • $-2 + 8 + - 10 = -4$
  • $-10 + -10 = -20$
  • $8 + -24 = -16$
  • $4 - -4 = 8$

When multiplying and dividing integers, remember like signs give a positive answer, and unlike signs give a negative answer.

  • $-20\times 2= -40$
  • $16\times -2= -32$
  • $-25\times -2.2= 55$

Order of Operations (PEMDAS)

For calculations involving more than one operation, mathematicians established a standard order of operations. Use this mnemonic device to remember the standard order, PEMDAS, “Please excuse my dear Aunt Sally.”

pemdas

Follow the standard order of operations to solve equations, parentheses, exponents, multiplication and addition in order left to right, and addition and subtraction in order left to right:

$\begin{align} 2\times -3 +8\times4\div2 +2^{2} & = \\ 2\times -3 +8\times4\div2 +4 & =\\ -6 +32\div2 +4 & = \\ -6 +16 +4 & = 14\\ \end{align}$

Variables

Variables are letters or symbols that represent unknown values. The letter $x$ is the most commonly used variable. Simplifying variable expressions first can make the evaluation of variable expressions less complicated.

Simplify this variable expression for $x = 6$ and $y = 10$.

$\begin{align} 2x + 3x -4y +8y + (-9) & = 5x +4y + (-9)\\ 5\times 6+ 4\times 10+(-9) & = 30 + 40 + (-9) = 61\\ \end{align}$

This expression has only one variable. Simplify for $x = 5$.

$\begin{align} 2x^{3} +5x -1\frac{1}{2}x & = 2x^{3 } +3\frac{1}{2}x \\ 2\times5^{3} +3\frac{1}{2}\times5 & = 2\times125 +17.5 = 250 +17.5 = 267.5 \end{align}$

circle.

Use algebra equations to solve geometry problems. Find the area of the circle with $r = 4$.

$\begin{align} A & =\pi r^{2} \\ A & =\pi 4^{2} \\ A & =16\pi \end{align}$

Videos Lessons and Practice Problems for Basic Arithmetic

12 Video Lessons

Printable Worksheets for Basic Arithmetic

Us013 Simplifying Variable Expressions Preview Download
Us012 Evaluating Expressions Preview Download
Us002 Order of Operations Preview Download
Us003 Distributive Property Preview Download
Us001 Adding Integers Preview Download
Us004 Subtracting Integers Preview Download
Us005 Multiplying and Dividing Integers Preview Download
Us006 Types of Numbers Preview Download
Us007 Transforming Terminating Decimals to Fractions and Vice Versa Preview Download
Us009 Transforming Simple Repeating Decimals to Fractions and Vice Versa Preview Download
Us010 Rational Numbers on the Number Line Preview Download