This post was co-written by Trey Smith, the Library of Congress 2015-16 Science Teacher in Residence, and Max Ray-Riek, a Project Manager from The Math Forum, now on staff at the National Council of Teachers of Mathematics.
Last August, Julie Miller in the Library’s Manuscript Division wrote about Thomas Jefferson’s quest for an odometer. Jefferson’s search for a tool to measure distances he traveled in a horse-drawn carriage was just one of his many efforts to quantify and logically describe the natural world. He also wrote a report on weights and measures, kept copious weather records, and created a chart detailing the fruits and vegetables sold at a vegetable market throughout the year. A closer look at Jefferson’s notes about odometers presents a range of possibilities for engaging students in mathematical reasoning and problem solving.
Before digging into Jefferson’s notes, engage students in an investigation of circumference using wheels or discs. They may:
- use markers, paper, pencils, ribbon, or even trays of sand to explore lengths and revolutions of wheels or discs; and
- consider the relationships among circumference of a wheel, number of revolutions of that wheel, and the total distance one travels in a carriage.
Through their investigations, students might arrive at a formula in which circumference times the number of revolutions is equal to total distance traveled.
Jefferson included a note on the September 30, 1807, itinerary of his travel from Monticello to Washington, DC, that his odometer had been built for a wheel that was 16 feet in circumference. He then noted that the actual circumference of the wheel on his carriage was 16 feet and 5/8 of an inch.
Teachers might ask students if and when such a small difference in measurement would matter. Students may or may not agree that a difference of 5/8 of an inch makes much difference.
Jefferson recorded more notes about the distance between locations along his journey, including:
- an undated itinerary of a journey from the President’s House (today known as the White House) to Monticello; and
- an undated itinerary and map of travel from Washington, DC, to Stevensburg, VA.
Teachers might ask students: If Jefferson’s odometer had been built with a specific wheel circumference in mind, would it be possible to create a mathematical formula and update the measurements Jefferson recorded in his itineraries?
While considering Jefferson’s notes, students might:
- discover they can work backwards from Jefferson’s initial distances and incorrectly measured circumference to calculate the number of revolutions and then multiply number of revolutions by the correct circumference;
- use the tables in each of Jefferson’s itineraries to come up with a revision factor to multiply each of Jefferson’s recorded distances by; or
- practice unit conversions and express their work in miles and feet, just feet, or even inches.
Ask students to find the percent error in Jefferson’s measurements and then describe how far off he would be on journeys of 1 mile, 10 miles, or 100 miles. Students should revisit their opinions about whether 5/8 of an inch really matters in measurement.
What other mathematical questions and investigations might students pursue?
These investigations are great ideas for developing critical thinking skills in students.
Asking students how to determine distance traveled on a bicycle (prior to the lesson) might be a good introduction.
Another connection: Distances on trails, such as the Appalachian Trail, are determined with odometers when maps are prepared.
Just sharing some thoughts,
It’s great that someone decided to relate this article to maps and trails!!! Some of us can relate to that better.
I live close to Monticello and have taken all my out of town guests to Monticello many, many times. The above story about Jefferson is new to me. I enjoyed reading every word of it. Even at 70 plus years I can still learn new things. Thanks for the article.