# Lesson 5

Changes Over Rational Intervals

### Lesson Narrative

In an earlier course, students explored how exponential functions grow by equal factors over equal intervals but with a focus on whole-number intervals. This lesson extends that work to include fractional intervals. Students see that for *any* equal intervals of input, including those for a fractional amount, the output values change by the same factor. They use their knowledge of exponential functions to calculate these factors for different fractional changes of input.

Students continue to examine exponential functions in context during this lesson, but they do so with less and less scaffolding. They observe that exponential functions grow by equal factors over equal intervals when those intervals have fractional values and apply this property to solve problems (MP7).

Technology isn't required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.

### Learning Goals

Teacher Facing

- Understand that an exponential function still changes by a constant factor over equal intervals, even if the interval is a fractional amount.

### Student Facing

- Let’s look at how an exponential function changes when the input changes by a fractional amount.

### Learning Targets

### Student Facing

- I can explain why an exponential function changes by the same factor over equal intervals, even when those intervals are not whole numbers.