Lesson 10

Rate of Change

These materials, when encountered before Algebra 1, Unit 5, Lesson 10 support success in that lesson.

10.1: Growing Bamboo (10 minutes)

Warm-up

The purpose of this warm-up is to revisit the meaning of statements that use function notation, when the function is represented with a graph. Students will need to interpret such statements and read values from a graph both later in this lesson and in the associated Algebra 1 lesson.

Student Facing

The graph represents function \(h\), which gives the height in inches of a bamboo plant \(t\) months after it has been planted.

Horizontal axis, time in months. Vertical axis, height in inches. Line graphed, Y intercept = 12. Slope =9.

  1. What does this statement mean? \(h(4)=24\)
  2. What is the value of \(h(10)\)?
  3. What is \(c\) if \(h(c)=30\)?
  4. What is the value of \(h(12)-h(2)\)?
  5. How many inches does the plant grow each month? How can you see this on the graph?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display the graph and draw some slope triangles, reminding students how to calculate the slope of a line using any two points on the line.

Remind students that the rate of change of the function, which is constant since the function is linear, can be seen in the slope of the graph that represents the function.

10.2: A Growing Account Balance (15 minutes)

Activity

This activity is similar to the warm-up, but asks questions in a slightly different way.

Launch

If they understood the warm-up, students should be able to get started right away. Let students work to make sense of the expression \(\dfrac{b(7)-b(4)}{7-4}\) before explaining what it means. Encourage them to use the graph to make sense of the expression. When students interpret \(\dfrac{b(7)-b(4)}{7-4}\) and see \(b(7)-b(4)\) as an object that has a value, they have an opportunity to notice and make use of structure (MP7).

Student Facing

The balance in a savings account is defined by the function \(b\). This graph represents the function.

horizontal axis, time in months. vertical axis, account balance in dollars. line with y intercept of 300 and slope of 100 graphed.

  1. What is . . .
    1. \(b(3)\)
    2. \(b(7)\)
    3. \(b(7)-b(3)\)
    4. \(7-3\)
    5. \(\dfrac{b(7)-b(3)}{7-3}\)
  2. Also calculate \(\dfrac{b(11)-b(1)}{11-1}\)
  3. You should have gotten the same value, twice. What does this value have to do with this situation?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Either reveal the correct responses or invite students to share their responses. Focus on the last question, drawing slope triangles on the graph as needed. This can be connected back to the grade 8 understanding of slope based on the side lengths of similar right triangles. The important point to emphasize is that for a given line, you can choose any two points to calculate its slope.

10.3: The Temperature Outside (15 minutes)

Activity

The purpose of this activity is to practice interpreting function notation, reading values from a table or a graph, and making sense of what quantities mean in a situation.

Since this function is not perfectly linear, we’d typically refer to the rate of change between any two points as an average rate of change. In the associated Algebra 1 lesson, students will need to recognize this term.

Launch

Arrange students in groups of 2. Explain that students will take turns matching each expression to a value, and explaining to their partner how they know it’s a match, and explaining what the value means in this situation. If necessary, choose a student to act as your partner and demonstrate the protocol before students start working.

Student Facing

Here is a graph and a table that represent the same function. The function relates the hour of day to the outside air temperature in degrees Fahrenheit at a specific location.

\(t\) \(p(t)\) \(t\) \(p(t)\)
0 48 6 57
1 50 7 56
2 55 8 55
3 53 9 50
4 51.5 10 52
5 52.5    

Scatterplot. Horizontal axis, time in hours. vertical axis, temperature in degrees fahrenheit.

Match each expression to a value. Then, explain what the expression means in this situation.

  1. \(p(12)\)
  2. \(p(8)\)
  3. \(p(12)-p(8)\)
  4. \(12-8\)
  5. \(\frac{p(12)-p(8)}{12-8}\)
  6. \(p(10)\)
  7. \(p(20)\)
  8. \(p(10)-p(20)\)
  9. \(10-20\)
  10. \(\frac{p(10)-p(20)}{10-20}\)
  • 4
  • -2.75
  • 44
  • -1.4
  • 55
  • 14
  • -11
  • 38
  • -10
  • 52

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Reintroduce the term average rate of change, which is what we’d typically call the rate of change between any two points in a function that is not perfectly linear.