Distance Over Time
copyright 2011 by Tim Griffin

It was twelve o’clock in Vegas when the car club headed West
They’d all agreed to put their speedy sports cars to the test
They really made their engines scream with radar detectors glowing green
Drove slowly into Barstow at one-fifteen… and were placed under arrest

The drivers put their righteous indignation on display
We never saw a single lawman anywhere today
I tell you we were driving slow, it could be true for all you know
You can’t show how fast we go while driving on the way

The lawman laughed and told them what they said was true indeed
But any child could prove it with a clock and map to read
He’d called Las Vegas: they left at noon and got to Barstow way too soon
A radar detector won’t make you immune to the laws of time and speed

It’s basic math to get your speed, he told them with a smile
The distance that you traveled was one hundred fifty miles
Divide that by the time you spent, that’s how I know how fast you went
He took them all without consent to jail for a while

The reckless drivers thought it through and knew he’d proved the crime
‘Cause all you need for average speed is distance over time
This formula will never fail, the answer was they went to jail
Sold their cars to make their bail and pay off all their fines

Now unless you want to ride a bus to Vegas all your lives
Obey the laws of physics and of men if you are wise
Remember this and you won’t be a former driver’s licensee
‘Cause all you need for average speed is distance over time

All you need for average speed is distance over time
(spoken) Sir, do you know how fast you were going?

Notes

Writing about math presents a special challenge since a math procedure seldom comes with its own story, as do most history or science standards. In this case the story happens to be true, with a few changes in details to make the calculations cleaner. The answer is intentionally not included in the song so you can have the fun of doing it yourself. The solution is down at the bottom of these notes.

Here are some standards from the Common Core and the state of California addressed by this song:

Fourth Grade:
(note that most 4th graders will need more guidance on selecting an approach; consider solving in teams)

• 4.OA. Use the four operations with whole numbers to solve problems.
• 4.MD. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. Note from Tim: discuss implications of hours & minutes using base-sixty (that’s sexagesimal, if you can say it without laughing) rather than base-ten notation.
• MP.1. Make sense of problems and persevere in solving them.
• MP.2. Reason abstractly and quantitatively.
• MP.4. Model with mathematics.

Fifth Grade:

• 5.OA. Analyze patterns and relationships.
• 5.NF. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
• 5.NF.6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
• 5.MD. Convert like measurement units within a given measurement system.
• MP.1. Make sense of problems and persevere in solving them.
• MP.2. Reason abstractly and quantitatively.
• MP.4. Model with mathematics.

Sixth Grade:

• 6.RP. Understand ratio concepts and use ratio reasoning to solve problems.
• 6.EE.9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.
• MP.1. Make sense of problems and persevere in solving them.
• MP.2. Reason abstractly and quantitatively.
• MP.4. Model with mathematics.

Seventh Grade:

• 7.RP. Analyze proportional relationships and use them to solve real-world and mathematical problems.
• 7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.
• MP.1. Make sense of problems and persevere in solving them.
• MP.2. Reason abstractly and quantitatively.
• MP.4. Model with mathematics.

Eighth Grade:

• MP.1. Make sense of problems and persevere in solving them.
• MP.2. Reason abstractly and quantitatively.
• MP.4. Model with mathematics.
• CA.PS.8.1.b. Students know that average speed is the total distance traveled divided by the total time elapsed and that the speed of an object along the path traveled can vary.
• CA.PS.8.1.c. 8.1.c: Students know how to solve problems involving distance, time, and average speed.

Guitar Chords:
C, Am, F, G, G7

Solutions:
Okay, stop reading now unless you want the answer. Still reading? Okay, either you’ve already done the problem or you tried and got stuck, or maybe you’re just too lazy to do it yourself so I’ll explain. As with most tricky story problems there are multiple paths to the solution. One way is to literally divide 150 (the distance in miles) by 1.25 (the elapsed time in hours, converted into decimal form). That’s pretty hard, you’ll probably want a calculator if you do it that way.

Another option is to use a ratio comparing miles to minutes rather than miles per hour. As it happens, 150 (miles) is exactly twice as much as 75 (minutes), so now it’s easy to see that you’re going 2 miles per minute. Multiply that by 60 (minutes per hour) to get 120 miles per hour. Man, that’s fast!

If you like fractions, another way to do the problem would be to think of the elapsed time as 5/4 of an hour. Dividing 150 by 5/4 is a huge pain, but if you love fractions as much as I do then you probably know you can just multiply by 4/5 instead of dividing by 5/4. So to get 4/5 of 150, just multiply by 4 to get 600, then divide by 5 to get 120 miles per hour. Not as easy as the previous solution, but pretty cool nonetheless.

Other feasible approaches might include a timeline, drawing a map, or whatever floats your math boat. If you came up with a creative solution, please let me know!