### Clarifications

*Clarification 1*: Instruction includes making the connection to determining the slope of a particular line segment.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Functions

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Rate of Change
- Slope

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students determined the slope of a linear equation in two variables and analyzed graphical representations of functional relationships. In Algebra I, students calculate the average rate of change in real-world situations represented in various ways. In later courses, this concept leads to the difference quotient and differential calculus.- The purpose of this benchmark is to extend students’ understanding of rate of change to allow them to apply it in non-linear contexts.
- Instruction emphasizes a graphical context so students can see the meaning of the average
rate of change. Students can use graphing technology to help visualize this.
- Starting with the linear function $f$($x$) = 3$x$ − 2, shown below, ask students to calculate the rate of change between two points using the slope formula. Lead students to verify their calculations visually.
- Once students have successfully used the formula, transition to the graph of $f$($x$) = $x$
^{2}. - Highlight the same two points and ask students to discuss what the rate of change
might be between them
*(MTR.4.1).*Lead students realize that while there is not a constant rate of change, they can calculate an average rate of change for an interval. Show students that this is equivalent to calculating the slope of the line segment that connects the two points of interest.

- Once students have an understanding, ask them to find the average rate of change
for other intervals, such as −2 ≤ $x$ ≤ −1 or 0 ≤ $x$ ≤ 2. As each of these
calculations produce different values, reinforce the concept that non-linear
functions do not have constant rates of change
*(MTR.5.1).* - Look for opportunities to continue students’ work with function notation. Ask students to find the average rate of change between $f$(1) and $f$(4) for $f$($x$).

### Common Misconceptions or Errors

- Some graphs presented to students will only display a certain interval of data. Some students may mistakenly interpret the rate of change for that entire interval rather than a given sub-interval. In these cases, lead students to highlight the domain and range for the sub-interval requested so they can see it more clearly.
- Students may be confused if the average rate of change is 0, even though the function is
not constant.
- For example, the average rate of change of the function $f$($x$) = $x$
^{2}from $x$ = −1 to $x$ = 1 is 0.

- For example, the average rate of change of the function $f$($x$) = $x$

### Strategies to Support Tiered Instruction

- When determining an average rate of change on an interval where the function only increases or only decreases, instruction includes directions to highlight the domain and range for the interval so that the change can be seen more clearly.
- Instruction includes providing the graph to visualize the change in the $x$- and $y$-values by
drawing a line from the leftmost point on the graph within the interval to the rightmost
point on the graph within the interval. Discuss with students how the average rate of
change is the slope of the line between the two points.
- For example, for the function $f$($x$) = $x$
^{2 }, on the interval [−2, 2], the average rate of change is zero. This can be visualized by drawing a line from the point (−2, 4) to (2, 4) which has slope of zero (horizontal line).

- For example, for the function $f$($x$) = $x$
- Instruction includes assistance recognizing the connection between slope (MA.912.AR.2) and rate of change for a linear function. A linear function has a constant rate of change. Regardless of the interval, the rate of change is the same (constant). For nonlinear functions rate of change is not constant, so it is not considered slope. However, the formula for slope can be used to calculate the average rate of change over an interval.
- Teacher provides instruction on assigning the ordered pairs when calculating average rate of change. The formula for slope is the change in $y$ divided by the change in $x$. The designation of the interval points as the first or second pair of coordinates does not matter.
- Teacher co-creates an $x$-$y$ chart to organize information when solving real-world problems involving a graphical representation. The $x$ column should be labeled with the input description used for the $x$-axis. The $y$ column should be labeled with the output description used for the $y$-axis.

### Instructional Tasks

*Instructional Task 1 (MTR.3.1, MTR.4.1)*

- Jorge invests an amount of $5,000 in a money market account at an annual interest rate of 5%, compounded monthly. The function $f$($x$) = 5000(1 + $\frac{\text{0.05}}{\text{12}}$)
^{12$x$}, represents the value of the investment after $x$ years.- Part A. Find the average rate of change in the value of Jorge’s investment between year 5 and year 7, between year 10 and year 12, and between year 15 and year 17.
- Part B. What do you notice about the change in value of the investment over each interval?

### Instructional Items

*Instructional Item 1*

- The graph below represents the number of producing gas wells in Kansas from 1989 to 2017. What is the average rate of change between 2000 and 2015?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Teaching Idea

## MFAS Formative Assessments

Students are given a graph of an exponential function and are asked to calculate and compare the average rate of change over two different intervals of time.

Students are asked to estimate the average rate of change of a nonlinear function over two different intervals given its graph.

Students are asked to calculate and interpret the rate of change of a linear function given its graph.

Students are given a table of functional values in context and are asked to find the average rate of change over a specific interval.

## Original Student Tutorials Mathematics - Grades 9-12

Learn how to calculate and interpret an average rate of change over a specific interval on a graph in this interactive tutorial.

## Student Resources

## Original Student Tutorial

Learn how to calculate and interpret an average rate of change over a specific interval on a graph in this interactive tutorial.

Type: Original Student Tutorial