Illuminations: The Variable Machine

The Variable Machine
This lesson provides an introduction to the use of variables, and develops students' algebraic thinking. Students create variable machines to discover the value of words.

This lesson was adapted from a lesson by Gilbert Cuevas and Karol Yeatts, as found in Navigating Through Algebra in Grades 3 ‑ 5, from the Navigations Series, NCTM 2001.

Learning Objectives
 
Students will:
  • explore the idea of variable as a symbol that can stand for any member of a set of numbers
  • substitute numbers for variables (letters) to discover unknown values

Materials
 
Two strips of lined notebook paper per student: one 5 cm wide and one 3 cm wide
Transparent Tape
Cracking the Code Activity Sheet

Instructional Plan

You may wish to group your students in pairs to carry out the activity. Later on students may choose to work independently to create codes for their friends to solve.

Tell the students that they are going to create a variable machine to discover the value of words. On the three-centimeter-wide strip of lined notebook paper, have them write the letters of the alphabet in order down the left side of the paper. 

Down the right side of the five-centimeter strip of notebook paper, have them write the numbers from 0 to 25. They should then attach the ends of the number strip together with a piece of tape; wrap the letter strip around the number wheel, matching the letters to the corresponding numbers:

A to 0
B to 1
C to 2
and so on....

and tape the ends of the letter strip together, as shown below:

 

A Completed Variable Machine

 

Explore

Have the students find the value of their first names, using their variable machines; for example, the value of Amy's name is 36: 

A = 0, M = 12, Y = 24
0 + 12 + 24 = 36

Have the students find the value of their last names, and ask the following questions:

  • Which name has the higher value - your first name or last name? 
  • What is the difference in the values of your first and last names?

 

Students should explore the values of various words. For example, ask students to find words:

  • whose values are equal to 25, 36, or 100.

 

Ask, "Do most of the words you checked have a value that is even or odd?" Students should discuss strategies they might use to answer this question.

These questions and activities are also found on the Cracking the Code activity sheet. Students can record their findings on the activity sheet.

Cracking the Code Activity Sheet Cracking the Code Activity Sheet

As students find the values of various words, ask questions such these:

  • What is the three-letter word with the greatest value?
  • Are the greatest values always associated with words that contain the most letters?

Questions for Students
 

What is a variable?

[Students may suggest that a variable is a letter which can stand for a numeric value. Accept other definitions which are mathematically accurate.]

How did you use your Variable Machine to determine the value of your first name?

[Students should state that they used the Variable Machine to identify the value of each of the letters in their first name, and then they added those values together to find a sum.]

Is it possible to change the values of each of the letters in a Variable Machine, or are they always the same value?

[It is possible to change the values. Students should participate in the Extension to discover another way of assigning values.]

Extensions
 
  1. Challenge students to find words of more than ten letters whose values are less than the values of words having only three letters.
  2. Realign the number strips to let A equal 7, as shown in the figure below. Doing so changes the assigned values. The number strips can be altered also by writing different values, such as decimal or fractional numbers, on different strips.

     

Teacher Reflection
 
  • Did the context of the problems provide a high level of enthusiasm? If not, what context might be better?
  • What adjustments could be made to extend these problems for students who quickly understood the concept of variables and a Variable Machine?
  • What additional support could be given to students who had difficulty understanding the concept of variables and Variable Machines?

NCTM Standards and Expectations
 
Algebra 3-5
  1. Represent the idea of a variable as an unknown quantity using a letter or a symbol.
  2. Express mathematical relationships using equations.
  
1 period   

NCTM Resources

Navigating through Algebra in Grades 3‑5

Web Sites
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
© 2000 National Council of Teachers of Mathematics
Use of this Web site constitutes acceptance of the Terms of Use

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.