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

Grade 8: Making Connections

activity under revision

Lesson Overview

Formalize the input/output model for function and connect multiple representations: tables, function rules, equations, and graphs.

Mathematics Overview

Students will investigate the input/output model for building function tables. Then they will connect tables, graphs, function rules, and equations in one variable. Finally, they will work backwards to determine function rules for given data sets or graphs.

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

  1. The teacher discusses the use of a function machine and input/output variables using the illustration in Activity 1, number 1. The teacher reviews the notion with a simple example: input 3, output 8; input 4, output 9; etc. The rule is to add 5.

    The teacher then tries a more complicated example with a two-step rule: input 1, output 3; input 4, output 9; etc. The rule is to multiply by 2 and add 1.

  2. Students complete Activity 1 in small groups and discuss. This activity will help move students from tables to function rules. Some groups may do number 1 and other groups do number 2.
  3. In small groups, students plot their data (from number 1 or 2) from Activity 1 on 1" graph paper using markers or peel-and-stick dots. The graphs should be displayed and discussed as a whole class.
  4. Students complete Activity 2, transferring their group's graph to their activity sheet.
  5. Students complete Activity 3, using tables to develop rules. Students should work independently at first, and then discuss their strategies with their small group.
  6. Students discuss Activity 3 as a whole class.
  7. Students complete Activity 4, moving from graphs back to tables, function rules, and equations. Students should work independently at first, and then discuss their strategies with their group.
  8. Students discuss Activity 4 as a whole class.

Teacher Notes (to personalize the lesson for your classroom)

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

Activity 1:

  • What is the input and output of the function machine?
  • How did you use mental math to complete the table?
  • What variables did you investigate when completing your table?
  • What process did you use to complete the table?
  • What patterns do you see in the table?
  • By looking at your table, can you tell if this illustrates a proportional relationship? How?
  • By looking at your table, can you tell if the relationship is linear? How? (Students may use coffee stirrers or flat spaghetti to help.)
  • What equation did you write?
  • What does the variable in the equation represent?
  • What does the coefficient of the variable represent?
  • When given the output, what process strategy did you use? (For example, in number 1 some students may use an undoing of the process such as 775 - 100 = 675, then 675 — 15 = 45. This numerical approach is an important first step to solving equations.)
  • What sentence did you use to describe the total cost?
  • Did you use your sentence to write your equation?
  • What similarities did you see between the sentence and the equation?
  • By looking at your equation, can you tell if this illustrates a proportional relationship? How?
  • By looking at your equation, can you tell if the relationship is linear? How?
  • How would you find the nth term?
  • How can you use your graphing calculator to find an input value if you are given the output value?
  • In your graphing calculator table, which variable is the input variable and which is the output variable?
  • By looking at your equation, can you tell if this illustrates a proportional relationship? How?
  • By looking at your equation, can you tell if the relationship is linear? How?
  • Can you create a situation in which one quantity is dependent upon another and identify the input and output variables?
  • What would be some reasonable input and output values for your situation?

Activity 2:

  • When your graphed your data in Activity 2, how did you label the graph?
  • How did you determine the scales in your graph?
  • Do you see any patterns in your graph?
  • By looking at your graph, can you tell if this illustrates a proportional relationship? How?
  • By looking at your graph, can you tell if the relationship is linear? How?
  • How can you determine the rate of change from the graph?
  • Using your graph can you predict an input value given an output value? Predict an output value given an input?

Activity 3:

  • In Activity 3, what strategies did you use to find the output for 270?
  • What do the variables in the equation represent?
  • How can you use your graphing calculator to confirm your process?
  • What kind of relationship is illustrated in Activity 3, number 2?
  • Explain how you can use a table to find the solution to an equation? Give an example.
  • Describe situations in which the data in a table represents a linear function.
  • Compare the relationships found in numbers 1, 2, and 3 of Activity 3. (One relationship teachers should note is that number 1 is proportional, and numbers 2 and 3 are nonproportional.)
  • Could you have written a function rule directly from the verbal description? What would it be?
  • Did the table values help you find the function rule? How?

Activity 4:

  • Describe your strategies for using the graph in Activity 4 to find the rule describing the relationship.
  • How can you find the rate of change from the graph?
  • How does the starting point (y-intercept) help you find the rule?

Teacher Notes (to personalize the lesson for your classroom)

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

  • Which representation, table, equation, or graph do you find most helpful? Why?
  • Which representation would you use to find the nth term? Why?
  • What types of relationships did we investigate?
  • What does it mean for a relationship to be proportional? Nonproportional?
  • What are some characteristics of a linear relationship?
  • Can a linear relationship also be proportional? Nonproportional? When?
  • Are there characteristics of the graphs that are similar?
  • Which representation would you prefer to use to predict information?
  • What generalizations can you make from the table, equation, and graph?
  • Can you find the constant rate of change in the table, equation, and graph? Explain.
  • How can you tell whether the set of data is proportional by examining a table? An equation? A graph?
  • How can you tell whether the set of data is linear by examining a table? An equation? A graph?
  • Given an equation, can you identify the input, output, and process of the function machine?

Teacher Notes (to personalize the lesson for your classroom)

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

Students create a real-world situation that can be investigated by this process and then model the situation using several representations.

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 generate tables or graphs and have a partner guess the rule.

Teacher Notes (to personalize the lesson for your classroom)