Translating Words into Mathematical Expressions

Objective: Translate a phrase into an expression with correct order.

Prerequisite skills: basic arithmetic, variables, and parentheses. The phrase b less than a means a − b. The phrase b times the sum requires parentheses around the entire sum. A quantity described as per unit usually multiplies the number of units. Common error: reversing a less-than subtraction. Restrictions and interpretation: Define variables and units before translating; subtraction and division depend on order.

Objective: Build an expression for a fixed charge plus a rate.

A fixed charge is paid once, while a rate is multiplied by the quantity used. Keep units compatible: dollars per hour times hours gives dollars. A coefficient represents a per-unit change, not a starting amount. Common error: multiplying the fixed fee by the quantity. Restrictions and interpretation: Define variables and units before translating; subtraction and division depend on order.

Objective: Turn a contextual relationship into an equation and solve.

Choose a variable for the unknown, then express both compared quantities using that variable. Words such as equals, total, or is give the equality. A difference compares two entire expressions, so parentheses may be needed. Common error: dropping a fixed term when forming an equation. Restrictions and interpretation: Define variables and units before translating; subtraction and division depend on order. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.