Skill: Explain reasoning with mathematical vocabulary

A clear mathematical explanation names the quantities, operations, and relationships in a problem. It also explains what each number, letter, or symbol means.

Example:
Three notebooks cost $6. How much would five notebooks cost at the same price per notebook?

1. Name the quantities and units.
   The quantities are the number of notebooks and the cost in dollars. The ratio of cost to notebooks is

$$\frac{\$6}{3\text{ notebooks}}.$$

2. Find the unit rate.
   A unit rate tells how much corresponds to one unit. Divide the cost by the number of notebooks:

$$\$6 \div 3\text{ notebooks}=\$2\text{ per notebook}.$$

The division finds an equivalent rate with $1$ notebook.

3. Use a variable to represent the unknown cost.
   Let $c$ represent the cost of five notebooks, in dollars. The relationship can be written as the equation

$$c=5\left(\frac{\$2}{\text{notebook}}\right)=\$10.$$

The symbol $c$ is a variable because it stands for an unknown quantity. The expression 2))$shows multiplication, while the equation uses the equal sign to state that the expression has the same value as$c$.

4. State and justify the answer.
   Five notebooks cost $10 because the unit rate is $2 per notebook, and

$$5\text{ notebooks}\times\frac{\$2}{\text{notebook}}=\$10.$$

The unit “notebook” cancels, leaving dollars, which matches the question.