Linear Equations: Step by Step

Ralph Pantozzi, a 2024-2025 Albert Einstein Distinguished Educator Fellow at the Library of Congress, wrote this post. 

Consider the equation y = mx + b. Where does it come from? Examining stairs is one way to understand the origins of this equation, which is the equation of a line.  

Before recorded history, humans encountered stair-like formations in the natural environment, such as the Giants’ Stairs in Maine. Geological formations sometimes exhibit striking regularity. Humans mimicked such regularity in their own structures. 

Ask your students to examine this image of a set of rustic stairs leading to a bridge. 

• What do they observe in this photo? 
• What hypotheses do they make?
• What do they wonder? 

Students may not at first realize that the stairs have been cut out of a log. Direct them to examine the steps for clues about how the stairs were made, such as the rounded edges and striations. Then, ask students to think about what measurements, and what calculations, they would need to create such a set of steps.  

Make this investigation tactile by providing students with a strip of paper to represent the log. Select a particular overall horizontal distance for the log to span, and a vertical distance for it to rise. Students can then draw the steps and cut them out with scissors. No matter the size of the steps students draw, the ratio of the vertical height to the horizontal length will be the same.   

There are many images of stairs in the Library’s collections to inspire questions about design, mathematical and otherwise. Consider the image below, from 1896. 

For any staircase, the overall rise in height, the overall horizontal distance, and the slope of the incline are all related measurements. If, for example, one step is 12 inches in length, and 9 inches in height, the ratio is 0.75 inches of height for every 1 inch of horizontal distance (9 divided by 12)

The repetition of steps gives rise (pun intended) to the multiplication in the linear equation y = mx + b. If x is 240 units of horizontal distance, 240 times 0.75 results in 180 inches of change in height over that span. The b? That’s the initial height of the staircase, measured from an arbitrary point, as in the image below.

What might the measurements of each step of this staircase be? Students can suggest a variety of possible answers.  Which of them might be practical for such a staircase? 

Students may also enjoy determining the measurement of each stair behind the slide in this 1907 postcard from Glenwood Springs, Colorado. Is this a good slope for a slide?   

This slide (ramp?) has no stairs, but the same principles apply. The slope of the incline can be determined by drawing a small step aligned to the side, and dividing the vertical height of the step by the horizontal length.

To repeat, y = mx + b is an equation that multiplies a value x by another value m. Multiplication can represent repetition. It’s a step-by-step process that is a route to the equation of a line. 

Additional resources

Here are some other stairs to examine. What makes each distinctive?    

Congressional Library, grand stairway from vestibule

Stairway. U.S. Custom House, Portland, Maine

Stairway in hall, Sturdivant Hall, Selma, Alabama

Jewett Arts Center, Wellesley College, Wellesley, Massachusetts. Stairway with sculpture

An earlier version of this blog misidentified some of the measurements.