- Categorize real numbers as counting numbers, whole numbers, rational numbers, irrational numbers, or integers
- Use the properties of real numbers and the order of operations to accurately calculate and manipulate algebraic expressions.
- Evaluate and simplify algebraic expressions to enhance problem-solving efficiency in mathematics.
Defining and Identifying Real Numbers
The Main Idea
Real numbers are numbers that are either rational or irrational.
Real numbers are the backbone of our numerical universe, encompassing both rational and irrational numbers. Think of real numbers as a big family gathering where everyone from counting numbers to irrational numbers shows up.
- Natural Numbers or Counting Numbers: Start with [latex]1[/latex] and continue. [latex]1,2,3,4,5…[/latex]
- Whole Numbers: Counting numbers plus zero. [latex]0,1,2,3,4,5…[/latex]
- Integers: Whole numbers and their negative counterparts.
- Rational Numbers: Numbers that can be written in the form [latex]{\Large\frac{a}{b}}[/latex], where [latex]a[/latex] and [latex]b[/latex] are integers and [latex]b\ne o[/latex]. In decimal form, the numbers terminate or repeat.
- Irrational Numbers: Numbers that can’t be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.
The following mini-lesson provides more examples of how to classify real numbers.
You can view the transcript for “Identifying Sets of Real Numbers” here (opens in new window).
Recognizing Properties of Real Numbers
The Main Idea
Real numbers have a set of rules that they play by, and these rules are called properties. Think of these properties as the “grammar” of math, setting the stage for how numbers interact with each other. Whether it’s addition, multiplication, or even the use of parentheses, these properties ensure that numbers behave in a predictable way.
- Distributive Property: Imagine you’re sharing a pizza equally among friends. If you have [latex]5[/latex] friends and [latex]3[/latex] pizzas, each friend gets a share from each pizza. Mathematically, [latex]a×(b+c)=a×b+a×c[/latex].
- Commutative Properties: In music, a playlist on shuffle plays songs in any order but the music is still enjoyable. Similarly, in math, whether it’s addition or multiplication, the order doesn’t matter. For addition, [latex]a+b=b+a[/latex], and for multiplication, [latex]a×b=b×a[/latex].
- Associative Properties: Think of a team huddle in sports. It doesn’t matter how players are grouped; the huddle remains the same. In math, whether you’re adding or multiplying, the numbers stick together like a team. For addition, [latex]a+(b+c)=(a+b)+c[/latex], and for multiplication, [latex]a×(b×c)=(a×b)×c[/latex].
- Identity Properties: Zero and one are like the superheroes of the number world. Zero, when added to any number, doesn’t change its identity ([latex]a+0=a[/latex]). One, when multiplied with any number, keeps it the same ([latex]a×1=a[/latex]).
- Inverse Properties: These are your “undo” buttons in math. For addition, every number has a negative that will bring it back to zero ([latex]a+(−a)=0[/latex]). For multiplication, every non-zero number has a reciprocal that will bring it back to one ([latex]a×\frac{1}{a}=1[/latex], provided [latex]a \ne 0[/latex]).
- [latex]5×(6+19)=5×6+5×19[/latex]
- [latex]41.7+(−41.7)=0[/latex]
Watch the following video for more information on the properties of real numbers.
You can view the transcript for “Properties of Real Numbers” here (opens in new window).
Algebraic Expressions
The Main Idea
Algebraic expressions are mathematical statements that combine numbers or constants, variables (letters that represent unknown numbers), and operations such as addition, subtraction, multiplication, and division. They serve as the foundational language for algebra and allow us to represent and solve complex mathematical problems.
An algebraic expression can be as simple as a single variable “[latex]x[/latex]“, or as complex as a multi-term expression like “[latex]3x^2 - 2x + 5[/latex]“. The parts of the expression separated by addition or subtraction are called terms, each of which can be a combination of numbers (coefficients) and variables raised to a power.
To evaluate an algebraic expression, you substitute specific numerical values for the variables and perform the indicated operations. For instance, if we substitute [latex]x=2[/latex] into the expression “[latex]3x^2 - 2x + 5[/latex]“, we would get [latex]3*(2)^2 - 2*2 + 5 = 12 - 4 + 5 = 13[/latex].
You can view the transcript for “Algebraic Expressions (Basics)” here (opens in new window).
- [latex]x=0[/latex]
- [latex]x=1[/latex]
- [latex]x=\dfrac{1}{2}[/latex]
- [latex]x=-4[/latex]
In the following video we present more examples of how to evaluate an expression for a given value.
You can view the transcript for “Evaluate Various Algebraic Expressions” here (opens in new window).
Simplify Algebraic Expressions
The Main Idea
Simplifying algebraic expressions is the process of making an expression as concise as possible without changing its value. It involves several steps, often relying on mathematical properties and operations to combine like terms and eliminate unnecessary parts.
One fundamental process is the combination of like terms, which are terms in the expression that have the same variables and exponents. For example, in the expression [latex]3x + 2y - 5x + y[/latex], you can combine the [latex]3x[/latex] and [latex]-5x[/latex] to get [latex]-2x[/latex], and the [latex]2y[/latex] and [latex]y[/latex] to get [latex]3y[/latex]. Thus, the expression simplifies to [latex]-2x + 3y[/latex].
The distributive property is also frequently used in simplification. It allows us to remove parentheses in expressions like [latex]3(x + 2)[/latex], which becomes [latex]3x + 6[/latex] after distribution.
You can view the transcript for “Algebraic Expressions (Advanced)” here (opens in new window).
Evaluating Formulas
The Main Idea
A formula is an equation expressing a relationship between constant and variable quantities.
A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm2) is found to be [latex]A=\left(L+16\right)\left(W+16\right)-L\cdot W[/latex]. Find the area of a mat for a photograph with length [latex]32[/latex] cm and width [latex]24[/latex] cm.