# A Graph to Visualize Average Velocity

Note: This is an expansion of the today’s Noschese 180 post. I thought it was too good not to share here.

We started Constant Acceleration in college-prep today. Rather than dive right in with carts and motion detectors, I propped up one end of a lab table with textbooks (best use ever) and let a C-battery roll down. (Batteries accelerate more slowly than marbles and hot wheels cars. They also roll much straighter.)

“What do you observe?” I asked

“It rolls down and gets faster.” they said.

“Prove it. You have 10 minutes.” I challenged them. I hate prescribing directions for activites like this. I want to see how my students approach these tasks.

They wanted stopwatches and metersticks. Some wanted tape.

One group wisely rolled the battery down a whiteboard and left marks at one second intervals. They were done in 2 minutes.

The other groups marked out equal intervals of distance to time with a stopwatch. Most groups made data tables to show that it takes less time to travel each successive distance interval, thereby showing it continously increases in speed.

Many groups added a velocity column and calculted the “velocity” for each interval to show it changes. (But velocity when? where? average? I didn’t want to go down that road just yet. I just let it be.)

Some groups went further and also made distance-time graphs of their data to show the slope increases.

Two groups went even further and added an average-velocity step graph like this one:

It was beautiful. And something I had never considered doing.

***

You see, over the years, I’ve tried a variety of acceleration labs. Kids would collect position-time data and make position-time and velocity-time graphs. And getting the velocity-time graph was always laborious. Here are some methods I’ve tried in years past…

Method 1: Manually draw tangent lines on the position-time graphs. Calculate and graph the slopes of the tangent lines. (Tedious)

Method 2: Use the slope tool in Logger Pro to get the slope of the tangent at each data point. Graph the slopes of the tangent lines. (Computer issues)

Method 3: Kids calculate the average velocity for each distance/time interval. Tell them to graph it at the midpoint in time. This typically involved a lot of hand waving b/c kids didn’t quite understand why at the midtime rather than the end time. And I’d still have groups that would incorrectly graph the average velocity at the end time. One time I made a data table worksheet to avoid this issue — but it was scary table with rows in between rows for midtime data.

Method 4: Method 3, but using Excel (OMG, what was I thinking?)

***

The average velocity step-graph method is perfect. It doesn’t matter how the students took the data. They calculate the average velocity for each interval, then graph each average velocity as a step that is as long as the interval. No need to handwave about midtimes. No need to assume the acceleration is constant.

The board pictured above inspired me, so I had all groups make their own average velocity step graph as well, just to see if it would work.

“Is this how the velocity-time graph really looks?” I asked.

“No. There wouldn’t be any steps. It would be a line. Or a curve.” they said.

They made the leap on their own to draw a line through the steps. And, lo and behold, the “best fit line” cuts through the middle of each step — the midtime.

You can’t miss it. A great visualization.

Kids who took data at equal time intervals had equal sized step-widths and step-heights. Kids who took data at equal distance intervals had unqual step-widths and step-heights (the steps got narrower and shorter over time — which in a data table looks like non-constant acceleration). But the line still cut through the midtime of each step. Now we can talk about why that happened and what that means AFTER, rather than all the handwaving and number crunching first.

Several graphs also got a y-intercept, which we chalked up to reaction time error.

I love it when I learn from kids!

UPDATE: There’s a mistake in the step-graphs here. Read my follow-up post “A Mistake Made in Haste.” Sorry!

### 12 responses to “A Graph to Visualize Average Velocity”

1. “Prove it. You have 10 minutes.” – I love it. I’d never thought of looking at average velocity as a step graph before either. Thanks for sharing.

2. Peter Hoh

Love that line about the textbooks!

3. This is AWESOME! Already thinking of connections with chemical Kinetics. Thanks!

4. Rafael

I absolutely love the “You have 10 minutes to prove it” idea. My students get a similar idea to your step graph when we use the very old-fashioned ticker tape timers. We make them cut the tape every 5 or 10 dots and stick their strips of tape side by side. They get a bar chart which is basically the same as your average velocity graph.

5. Totally agree, Frank. My students have been doing the step graph for a few years now — it arises naturally from the CVPM unit — and it is very effective for them.

6. Great post. The line about the use of textbooks reminded me of this even better description:

“Returning to teaching after raising a family, [a teacher] said, she had her husband and son bringing in some of her favorite equipment to the new classroom, including a sand table and a water table. Seeing this, her principal spoke to her nervously: “You know, Mrs. ______, our school has a strong academic emphasis.” “Oh yes,” she replied, “I couldn’t teach physics without this equipment!” She later found piles of fill-in-the-blanks workbooks in her closet. Asking if she would be required to use them, she was told indeed she must. “So,” she said, “we used them. We weighed them singly and by twos and threes. We weighed them dry and we weighed them wet.”” (Hawkins, 1990)

Hawkins, D. (1990). Defining and Bridging the Gap. In SCIENCE EDUCATION: A Minds-On Approach for the Elementary Years (pp. 97-139). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

7. Helene Dauerty

Awesome. We used our AP books last spring as the counterwieght for our trebuchet. It very helpfully saved on materials costs.

8. That’s also all I used the textbook for in my class last year. :oP Is there ever the issue of science literacy in your class or school? As much as I’d like to be proud of not using the textbook, it’s mostly because my students’ literacy levels aren’t quite at that level, and ideally, I would like to help them get to a level where they can learn *something* from the book. I know physics is not learned well from a book, even if you can comprehend the paragraphs (best case, “clear and easy to understand” like a KA video, rite? :oP heh), so it’s not like I’d try to teach them Newton’s 2nd law with that, but maybe just read and understand a passage on how Newton’s laws apply in the engineering industry, or one of those other tidbits in the book. I know you do a lot of the “sexy” stuff in your class (which is ironic, because to outsiders, modeling is mundane and KA is sexy), but is there also “mundane” stuff that goes on in your class that helps build literacy around science? I’m not saying whether you should or not, but just wondering.

• Hi Frank,

I’ll admit that I haven’t done much to promote literacy directly in my classes. But now that NY adopted the Common Core Standards, we have been taking science department meeting time to explore the reading and writing standards and to see how well our courses align with them.

* Current events reading and responses in writing (LHC, Astronomy, Medical Physics, etc.)
* Socratic seminars around pieces from these 4 great books about experiments and model building in science:
– The 10 Most Beautiful Experiments (Johnson)
– The Prism and the Pendulum (Crease)
– Great Experiments in Physics (Shamos)
– What’s the Matter (Whitfield)
* Pieces from popular science writing:
– A Brief History of Everything (Bryson)
– The Physics of Superheroes (Kakalios)
– The Physics of the Impossible (Kaku)
– Why Does E=mc^2 (Cox)
– How to Teach Physics to Your Dog (Orzel)
– Six Easy Pieces (Feynman)
– Demon Haunted World (Sagan)
– The Physics of ____________ (there are so many of these books now).

But now how do I assess those assignments? Feedback only, no grade? Do I add reading and writing standards to my SBG system? (I can’t just assign point values, you know :))

And what’s the purpose of reading/writing? To get information we’ll use later? To broaden our horizons? To deepen our backgroud? To write better?
Who’s going to read their responses? Me? Other peers? Public on the web?

So overwhelming….

• Ooh, that’s quite the list. I’m also considering having them do shorter readings like current events or articles, or “how stuff works” type of articles. The only attempt at writing I did so far this semester was have them write their own experiment procedures for finding out if heavy things fall faster. If I remember, I’ll post my rubric and some student work samples, but it’s definitely far from perfect, as it’s my debut to any sort of writing and one of few examples of creative work. My attempt at making it more authentic was to give their procedures to another period to follow. If it’s unclear, it may be misinterpreted by another group, giving real consequences to their miserable attempt at communication. The irony and difficulty was asking my students to create a clear cut cookbook lab when I’ve never modeled such a thing for them because I half ask them to come up with their own procedures all the time. Oh, as for grading, I tried to take from your experimentation standards, and make one of them “I can create an experiment procedure that can be repeated consistently by other parties” or some similar (or way different heh) wording. I only gave them credit for perfect papers. However, they can revise it as many times as they want.