EDITED Resources. The resources on this page have been aligned with the revised K-12 mathematics TEKS. Necessary updates to the resources are in progress and will be completed Fall 2006. These revised TEKS were adopted by the Texas State Board of Education in 2005–06, with full implementation scheduled for 2006–07.

Clarifying Lessons

Algebra II: Crickets—Nature's Thermometer

activity under revision

Lesson Overview

Students determine and use functions to examine the relationship between the number of chirps of three different types of crickets and the temperature.

Mathematics Overview

Students translate functional relationships into equations to answer questions. Students answer questions using different representations of the relationships: function rules, graphs of the function, tables of values, and equations and graphs of the inverse functions.

Set-up (to set the stage and motivate the students to participate)

  1. Have students read about crickets on Worksheet A (or do a little research on their own about crickets).
  2. Have students work in pairs to complete the tasks on Worksheet B.
  3. Have students come back together as a whole group to share their strategies and solutions.

Teacher Notes (to personalize the lesson for your classroom)

Guiding Questions (to engage students in mathematical thinking during the lesson)

  • What are some ways in Part I that you can determine a symbolic representation that describes the temperature in terms of the number of chirps per minute for each type of cricket? (2A.1.B)
  • Based on what you did in Part I, what are some ways in Part III that you can determine a symbolic representation that describes the number of chirps per minute to expect from a given type of cricket at a given temperature? (2A.4.C)

Teacher Notes (to personalize the lesson for your classroom)

Summary Questions (to direct students' attention to the key mathematics in the lesson)

  • Compare and contrast the function for each type of cricket with its inverse function. (Possible responses: The slopes of each function and its inverse are reciprocals; if (a,b) is an ordered pair in one function, then (b,a) is an ordered pair in its inverse function.) (2A.4.C)
  • How could you use the functions and graphs to answer the question, "How many times will a field cricket chirp when the temperature is 70ƒF?" (Possible responses: substitute 70 for T in the function n/4 + 38 = T and solve for n; graph T1 = n/4 + 38 and T2 = 70 and determine the point of intersection of the two graphs; use a table of values for the function T = n/4 + 38 to look for the value of n when T = 70; use any of the above processes with the inverse function 4T - 152 = n) (2A.1.B)
  • For the functions in the form T = f(n), temperature expressed in terms of chirps per minute, explain the physical meaning of the slopes of their graphs. (The slope is the increase in the temperature for each increase of 1 chirp per minute, described by the value (change in temperature) — (change in number of chirps per minute).) (2A.1.B)
  • For the functions in the form n = f(T), chirps per minute expressed in terms of temperature, explain the physical meaning of the slopes of their graphs. (The slope is the increase in the number of chirps per minute for each increase of 1 degree in temperature, described by the value (change in number of chirps per minute) — (change in temperature).) (2A.1.B)

Teacher Notes (to personalize the lesson for your classroom)

Assessment Task(s) (to identify the mathematics students have learned in the lesson)

Teacher Notes (to personalize the lesson for your classroom)

Extension(s) (to lead students to connect the mathematics learned to other situations, both within and outside the classroom)

Students can rewrite the function for the field cricket as a relationship of the number of chirps per minute to temperature in degrees Celsius instead of degrees Fahrenheit.

equation

Students can then convert this function to a verbal rule, such as one of those originally given, that can easily be used to convert chirps to temperature in degrees Celsius. The rule should be written in the following form: Count the number of chirps in ____ seconds and add ____. [Note: In order for this to be an easily used rule, it is necessary to round quantities to the nearest whole number.] (To determine the "rule," we need to know the number of seconds to use when counting chirps. To find this, we need to convert 5/36 to x/60. TC = 5n/36 + 10/3 is approximately equal to 8.33n/60 + 3.33. For a field cricket, you need to count the number of chirps in 8 seconds and add 3 to obtain the temperature in degrees Celsius.)

Teacher Notes (to personalize the lesson for your classroom)