Math Background

Lesson: Functions
Introducing the Concept

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In Grade 5, students first work with functions that involve addition and subtraction and those that relate to patterns.

Materials: overhead projector and blank transparency; Function Cards (PDF file) for each pair of students; Fishing for Functions game sheets for each student; a set of number cards (5 copies of numbers 0-9)

Preparation: Make copies of the Fishing for Functions (PDF file) game sheets for each student and sets of number cards (5 copies of 0-9) and Function Cards (PDF file) for each pair of students.

Prerequisite Skills and Concepts: Students should be able to write and solve equations. They should be able to find the rule or the missing values in an input-output table.

  • Say: I'm going to show some of you how to make a pattern. I will also draw this pattern on the overhead.
    Have three students come to the front of the room. Have them stand in the following arrangement.
    figures
  • Say: This is the first term of our pattern. Draw the pattern on the overhead and label it 1. Call another student to the front of the room.
    Add that student to the pattern in this way.
    figures
  • Say: This is the second term of our pattern.
    Draw the pattern on the overhead, next to pattern 1, and label it 2.
    Continue bringing students one at a time to the front of the room and adding them clockwise around the others until you have the following pattern (which is labeled 4 on the overhead).
    figures
  • Ask: How many students would I need for the tenth term?
    Give students time to determine the number. Eventually they should say “12.”
  • Ask: How did you find the answer?
    Answers may vary. Students may say they drew 10 versions of the pattern or that they determined that each time the number of people was 2 more than the number of the term.
  • Ask: Some of you drew 10 versions of the pattern and some of you figured out the relationship between the term and the number of people. Which way would be easier if you wanted to find the number of people needed for the fiftieth term?
    Students should say that it would be easier to figure out the relationship between the term and the number of people.
  • Say: Write an equation that shows the relationship between the term and the number of people. Let x stand for the term and y stand for the number of people.
    Students should generate the equation y = x + 2. Write this on the overhead.
  • Ask: The equation y = x + 2 describes the relationship between the variables x and y. It tells us that y is always 2 more than x. A rule that explains how two variables relate is called a function. Is this the type of equation in which there is only one right answer?(no) How many values are there for each variable?
    Students should say that there are many possible values for x and y.
  • Ask: There are an infinite number of values for each variable. Suppose x equals 12; what is the value of y?
    Students should say “14.”
  • Ask: When x equals 12, can y equal anything other than 14?
    Students should say no.
  • Say: That's right. If x = 12, y must equal 14. For every value of x, there is only one related value for y.
    Draw a function table on the overhead like the one shown below and have the students copy it.  y = x + 2
    x y
    1   
    2  
    3  
    4  
    5  
    6  
    7  
    8  
    9  
    10  
    11  
    12  
  • Ask: One way to show some of the pairs of values for a function is in a function table. If x = 1, what must y equal? (3) In the function y = x + 2, can y equal anything other than 3 when x equals 1? (no)

    Complete the function table together, or start it together and then have the students complete it themselves.

  • Ask: Where have you seen a table like this before?
    Elicit input/output tables.
  • Say: Function tables are the same as input-output tables. Whenever you had to find the rule for an input-output table, what you were really doing was finding the function for that table.
    To assess student understanding, have them write a function for x and y using addition or subtraction. Then have students, on a separate sheet of paper, describe the relationship between x and y in word form and create a function table with the first 15 values filled in. Finally, have them trade the sheet with only the function on it with another student. Each student then has to describe the other's function in word form and then create a function table with the first 15 values filled in. You may also wish to have each student draw the pattern described by the other student's function.

    For extra practice, have students pair up to play the Fishing for Functions game. Each pair of players needs a set of number cards (5 copies of each number, 0 - 9) and Function Cards (PDF file). Each player needs a Fishing For Functions (PDF file) game sheet. To play, each student picks a function card and 5 number cards. The function card tells students the rules for their tables. They may place any of their number cards in either column of their function tables or hold them until later. Players take turns either drawing from the number pile or asking for a number from the other player. If the other player does not have the number, the first player loses a turn. Number cards may be paired up to make double-digit numbers, and numbers may be placed in the function table at any time. The object is to correctly complete all 5 rows of the function table.


Houghton Mifflin Math Grade 5