Lesson 2: Equal and Equivalent

Property

An algebraic expression is the same as an arithmetic expression, except that some of the entries are letters representing numbers.
An algebraic expression, or simply an expression, is any meaningful combination of numbers, variables, and operation symbols.

To write an algebraic expression:

1. Identify the unknown quantity and write a short phrase to describe it.
2. Choose a variable to represent the unknown quantity.
3. Use mathematical symbols to represent the relationship.

Examples

• The phrase "a number x increased by 12" translates to the expression $x + 12$.
• To represent "8 times the price p", you write the expression $8p$.
• "The total cost C split among 4 friends" is written as the expression $\frac{C}{4}$.

Property

To solve percentage problems, you can use the equation:

The percent must be written as a decimal or a fraction. You can represent the unknown value with a variable.

Examples

• What is $20\%$ of $50$?

Let $x$ be the unknown part.
$x = 0.20 \times 50$
$x = 10$

• $12$ is what percent of $60$?

Let $p$ be the unknown percent.
$12 = p \times 60$
$p = \frac{12}{60} = 0.20$, which is $20\%$.

• $15$ is $30\%$ of what number?

Let $w$ be the unknown whole.
$15 = 0.30 \times w$
$w = \frac{15}{0.30} = 50$

Explanation

This skill applies your ability to write algebraic expressions to real-world percentage scenarios. By translating the words "is," "of," and "what" into mathematical symbols ($=$, $\times$, variable), you can create a solvable equation. Remember to convert the percentage to its decimal form for calculation by dividing by 100. This method allows you to find any missing part of a percentage problem: the part, the whole, or the percent itself.

Property

Two algebraic expressions are equivalent if their corresponding area models have the same total area for any positive value of the variable. For example, the area of a single rectangle representing $a(b+c)$ is visually identical to the combined area of two adjacent rectangles representing $ab + ac$.

Examples

• The expression $3(x+2)$ can be shown as a single rectangle with height $3$ and width $(x+2)$. The expression $3x+6$ can be shown as two adjacent rectangles: one with area $3x$ and one with area $6$. Since the diagrams represent the same total area, the expressions are equivalent.
• The expressions $2x+5$ and $7x$ are not equivalent. An area model for $2x+5$ would show a rectangle of area $2x$ and a separate region of area $5$. An area model for $7x$ would be a single rectangle with area $7x$. These models are visually different and only represent the same area for a specific value of $x$, not for all values.

Explanation

Area models provide a visual way to understand and verify if two expressions are equivalent. If two expressions are truly equivalent, their geometric representations as areas will be identical, regardless of the length chosen for the variable part. This method helps to confirm properties like the distributive property by showing how an expression like $a(b+c)$ represents the same area as $ab+ac$. Using diagrams can make the abstract concept of equivalence more concrete and intuitive.