Making and Using x-y (Input-Output) Tables

Input-Output tables are an excellent way to introduce students to linear, proportional, and non-linear algebraic reasoning. When set up correctly, students can use the tables to describe relationships between the independent and dependent variables; how the output is changing between each row; and what transformations take place between the input and the output.


Questions to Ask about Input-Output Tables

Real world situations

• Unit price (specials, 2 cans/$5)
• Constant speed (time vs. distance traveled)
• Year vs. age

What is the rule inside the machine?

• Is it a one-step or multi-step transformation?
• Is the rule constant or changing?
• Is the rule describable, either in words as y = f(x)?

What is the pattern?

• As x increases, what happens to y?
• How can you predict the next row on the table?
• If you skip a few rows on the table, what would the 100th row be?

Important numbers

• What happens when you plug in 0?
• What do you have to plug in to output 0?
• By how much do the inputs and outputs increment each time?

Translate table to other forms

• Rows into (x, y), graph on coordinate plane
• Come up with situation or graphic that the table could describe
• Predict what characteristics the graph will have based on the table.

 In this problem, students can make a connection between the term in the sequence and how many squares form that picture. They can see that as the numbers in the Input column increase by 1 each time, the number of squares grows by 2 every time. You can (literally) highlight that change visually by shading the two new squares that are added on to form the new shape. As an extension, you can ask students to analyze other characteristics of the shape by gathering data and making conjectures- for example, finding the perimeter of those shapes, or redrawing it to be a 3-d figure and finding surface area or volume. Or, students may choose to do away with the original independent variable and find the relationship between area and perimeter, or any other two quantities.

The utility of Input-Output tables lies in their flexibility to span grade levels and depths of content, from elementary through calculus and beyond. Students should be able to use tables as a tool to organize and interpret many instances of data and numerical relations.