Lesson 20

Evaluating Functions over Equal Intervals

These materials, when encountered before Algebra 1, Unit 5, Lesson 20 support success in that lesson.

20.1: Finding Slopes (10 minutes)

Warm-up

The purpose of this activity is to recall that the slope of a line is the change in \(y\) every time \(x\) increases by 1.

Student Facing

  1. Find the slope of each line.
    1. The line that passes through \((2,2)\) and \((3,6)\).
    2. The graph of \(f(x)=\text-2+\frac13x\).
  2. Show on the graph where each slope can be seen.
2 comma 2 and 3 comma 6 on a graph of a line 
Graph of line f. y intercept = -2. Slope = the fraction 1 over 3.

 

Student Response

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

Activity Synthesis

Use the first question to review finding the slope of a line given the coordinates of two points on the line, and the slope of a graph of a linear function given an expression that defines the function.

Display the graphs for all to see, and select a few students to describe how they saw the slope on each graph. By drawing slope triangles, ensure students see that the slope of each line is the vertical change when \(x\) increases by 1.

20.2: Incrementing by One (20 minutes)

Activity

In this activity, students practice some of the mechanics needed for the associated Algebra 1 lesson, but stop short of analyzing the implications. When students repeatedly perform calculations and then generalize, they are expressing regularity in repeated reasoning (MP8). If students decide to use technology to efficiently evaluate the same function for different input values, they are using appropriate tools strategically (MP5).

Launch

Display the task statement for all to see, and point out that two functions are given, along with several pairs of input values. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Student Facing

  1. For the function \(f(x)=3x+4\), evaluate:
    1. \(f(0)\) and \(f(1)\)
    2. \(f(100)\) and \(f(101)\)
    3. \(f(\text-10)\) and \(f(\text-9)\)
    4. \(f(0.5)\) and \(f(1.5)\)
  2. What do all those pairs of numbers you found have in common?
  3. Write an expression for \(f(w)\) and \(f(w+1)\).
  4. What would you expect to be the result of subtracting \(f(w)\) from \(f(w+1)\)?
  5. Subtract \(f(w)\) from \(f(w+1)\). If you don’t get the answer you predicted, work with a partner to check your algebra.
  6. For the function \(g(x)=2^x\), evaluate:
    1. \(g(3)\) and \(g(4)\)
    2. \(g(0)\) and \(g(1)\)
    3. \(g(\text-1)\) and \(g(\text-2)\)
    4. \(g(10)\) and \(g(11)\)
  7. What do all those pairs of numbers you found have in common?
  8. Write an expression for \(g(u)\) and \(g(u+1)\).
  9. What would you expect to be the result of dividing \(g(u+1)\) by \(g(u)\)?
  10. Divide \(g(u+1)\) by \(g(u)\). If you don’t get the answer you predicted, work with a partner to check your algebra.

Student Response

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

Activity Synthesis

Display a graph of \(f\) and ask students where you can see the “3”. Draw a bunch of slope triangles on it with a horizontal side of length 1 and vertical side of length 3, corresponding to the inputs and outputs in the task.

Then, display a graph of \(g\) and ask students where you can see the “2”. Show that when \(x\) increases by 1, \(y\) doubles for a few different inputs and outputs in the task.

Ask a student to demonstrate, or demonstrate yourself, working through writing \(3(w+1)+4-(3w+4)\) and \(\dfrac{2^{u+1}}{2^u}\) with fewer terms.

20.3: Rewriting Expressions (10 minutes)

Activity

In this practice activity, students prepare for the associated Algebra 1 lesson by practicing applying some of the properties they will need to use.

Student Facing

  1. Evaluate:
    1. \(\dfrac{3^5}{3^4}\)
    2. \(\dfrac{3^1}{3^0}\)
    3. \(\dfrac{3^{\text-1}}{3^{\text-2}}\)
    4. \(\dfrac{3^{100}}{3^{99}}\)
    5. \(\dfrac{3^{x+1}}{3^x}\)
  2. Solve for \(m\):
    1. \(\dfrac{2^m}{2^7}=2\)
    2. \(\dfrac{2^{100}}{2^m}=2\)
    3. \(\dfrac{2^m}{2^x}=2\)
  3. Write an equivalent expression using as few terms as possible:
    1. \(3(x+1) + 4 - (3x + 4)\)
    2. \(2(x+1) + 5 - (2x + 5)\)
    3. \(2(x+2) + 5 - (2(x+1) + 5)\)
    4. \(\text-5(x+1) + 3 - (\text-5x + 3)\)
    5. \(\dfrac{5^{x+1}}{5^x}\)
    6. \(\dfrac{7^{x+4}}{7^x}\)

Student Response

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

Activity Synthesis

Either display the correct solutions or invite students check with another student to see if their answers are the same. If there are any differences between solutions, invite students to ask questions or discuss whole group.