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.
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.”
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 $x = 5$.
$\begin{align} 2x^{3} +5x -1\frac{1}{2}x & = 2x^{3 } +3\frac{1}{2}x \\ 2\times5^{3} +3\frac{1}{2}\times5 & = 2\times125 +17.5 = 250 +17.5 = 267.5 \end{align}$
Use algebra equations to solve geometry problems. Find the area of the circle with $r = 4$.
$\begin{align} A & =\pi r^{2} \\ A & =\pi 4^{2} \\ A & =16\pi \end{align}$