Tag Archives: Stopping Distance

Stopping Distance Reflection – Where did it take us?

We are studying quadratic functions.  We started with the Stopping Distances data to look at quadratic data visually.  Our hypothesis was that the distance required to stop while braking is proportional to the square of speed, d=k·v².  Many of our learners had trouble fitting the curve; they were hesitant to take a swing.

What if we read Peter Reynold’s The Dot in class?  What if we encouraged our learners to just try to find an equation, to see where it takes them?  What if we used the TI-Nspire Navigator to “frame” their first marks, to celebrate that they took a swing?

Here’s what @fencersz, a learner in DD’s 3rd period, said in a couple of tweets to me about the class:

@jgough I thought it was really interesting to look at math in a way that involved creative problem solving as opposed to just applying a set model to the problem, was surprised b/c I’ve never though about math like that b4 and would like 2 get better at this.

What if we did this for each other too?  DD just happened to come for part of my class while I was teaching Stopping Distances.  She liked it, but said she thought she could never teach it.  We agreed to team-teach the lesson in one of her classes.  The following is her reflection on teaching it alone for her other class.

I had the best time teaching my 5th period today.  I know it was not as smooth as yours, but I think it was pretty good for my first go-round.  I remembered things to say that you said; there were even things I sort of got lost in, yesterday, that I wasn’t planning to do, but understood what you did when I got to it and was able to show the kids.  I don’t think they had any clue that I was just learning what I was talking about when I gave my spiel on velocity being speed with direction to explain one reason it would be quadratic rather than exp.  The kids were asking great questions and making good connections.  They were also engaged.  Thank you for teaching this to my third period so I could learn it.  I know you felt horrible yesterday, but you trooped on over anyway.  What about you and me team teaching this lesson in 4th period PLC as our first lesson study (so we can improve it for next year) with the new TI-Nspire software?

Model learning.  Encourage others.  Try new things.  Collaborate.  See where it takes you.

Swing Even If You Miss; Stepping Up To The Plate Is Not Enough

Is a swing and a miss better than no swing at all?  I used to coach my learners to step up to the plate, just step up.  You can do it.  But now I realize that just stepping up is not what it takes to learn.  It is a great first step, but it is just a first step.  Will you ever hit it out of the park if you don’t swing?

Today I got to co-teach Algebra I with D. Dietrich with her learners.  It was day two of Stopping Distances.  After reviewing what we accomplished during the last class, we stopped to ask them to write a complete sentence about what the graph of reaction distance vs. velocity represented. 

When I asked for volunteers, no one responded.  Yikes!  Now, they all had written something.  Finally, one sweet, brave learner volunteered.  It was a sentence that needed some coaching.  We talked about the structure of the sentence as well as the need to be more specific about units.  Since the distance is in feet, what are the units of velocity?

We collected this sentence with our TI-Nspire Navigator system.  The screenshot above allowed the entire class to have multiple representations of the sentence.  They could see it as it was being read.  The screenshot is evidence of a swing.

I asked for another volunteer…  …  …  …  So, I finally asked the one child that was making eye-contact with me why he didn’t want to read.  “Because I know it is not perfect.” Isn’t this what learning should be about?  Can’t we have a culture where it is okay to not be perfect?  How will I ever learn if I am not brave enough to share my thoughts with others?  Being in school should be about drafting and making revisions, shouldn’t it?  Have we built a culture where we are afraid to try because the only correction I get is criticism?  (double-Yikes!!)

We edited and coached and then tried again for the second graph, braking distance traveled vs. velocity.

We got, without question, the best sentences of the day for the second writing. The power of feedback and revision was demonstrated.  Unknown to me, the learner that volunteered for the second writing rarely speaks in class and never volunteers.  Isn’t that great?  She found the confidence to write and share.

But, as the facilitator of the learning, do I know how many of these learners actually took a swing?  Not really.  I love that one student wrote his sentence on the screen of his calculator.  But it is the only one I could monitor.

Which makes me wonder…how many times are my learners telling me they understand when they have not actually taken a swing?  Are they so conditioned to tell us that they understand that they are not testing their understanding and seeking feedback?  WOW!  That is a problem!

We stopped and talked about trust.  We talked about risking being wrong to grow and learn.  But, again, were they just agreeing with us because they know it is what we want to hear?  Would they take action…to swing and possibly miss?  Would we celebrate the swings and give quality, safe feedback for a miss?  Risky for everyone, right?  Trust is the key.

Yesterday we wrote the equations for both of the graphs above.  y = 1.1x and y = 0.05x², where x is velocity in miles per hour and y is the distance traveled in feet.

New learning for today…How do we find the model, the equation for the data, of total distance traveled in feet based on velocity in miles per hour.  Important questions must be asked.

  1. How will we calcuate the total distance traveled based on the given data?
  2. What pattern does this data follow?
  3. Is the pattern exponential or quadratic?Okay, so let me stop here and say that the question exponential or quadratic is an important question.  If you can’t answer it, how will you know what type of model to write?  This is Algebra I; we are not using regressions.  We are hand fitting data.Aren’t the results interesting?  Q for quadratic, E for exponential.

Before you boo, a bunch of learners came without their technology or had login issues.

But notice, even when anonymous, one learner did not take a swing. It was a 50-50 shot at being right and this learner would not take a swing.

New and MUCH more interesting questions are now possible.

  1. Is the majority always right?
  2. Can we listen to an opposing view and try to understand their reasoning?
    (Shouldn’t Democrats and Republicans try this occasionally?)
  3. Can each side make a reasonable argument for why they made their choice?
  4. Are you willing to consider that the other side might be right?

The minority view, quadratic functions, explained first.  Then a member of the majority party raised her hand and said “I voted exponential, but I can now give another reason why it is, in fact, quadratic.  Is that okay?”  WOW!  We stopped and voted again.

Progress!  But, what do we do about the 2 remaining e’s?  Remember, the majority is not always right.  More discussion ensued.  The two e’s identified themselves, AB and SS, and explained why they were now in the quadratic function camp.  Also notice, they everyone took a swing this time around.

Let me stop here and say that this is where and why I prefer using my TI-Nspire Navigator over Poll Everywhere and the SMARTBoard clickers.  While it appears to be anonymous, I can click on Poll Details and see which learners need intervention if they don’t self-identify during the lesson.  I can also see what is on every learner’s graph which I will show if you keep reading.

Now that we established that we are pretty sure the function to write is quadratic, we again returned to the data.

How was the data in the total column computed?  “Everyone knows, Ms. Gough, that you just add the reaction distance to the braking distance, duh!” Eyes rolling as only 14 year-olds can do.

So, if that is true, write the equation that fits the total distance traveled in feet based on the velocity in miles per hour.  You have all that you need.  You can do it.  Take a swing.

No one got it right the first time. Some took a swing and missed; some got close; some just stood at the plate.  At least two students were still in the dugout – maybe the parking lot; Caswell and St. Cloud were absent the day before.  They did not ask for the file.  St. Cloud was working to catch up; Caswell was just observing.  And we absolutely would not have known if we didn’t see their calculator screens.

Vuckovic looks like he took a swing, but no.  He connected his scatter plot; he had not attempted to construct a function.  Gibson has tested a function that is close, but does not fit very well; he took a great swing which we celebrated.

So we asked again, how was the data in the total column calculated?  Can you use this information to write an equation that will fit the data?

Before class was over, everyone took a swing and got a hit.  It was very collaborative.  We then took the model and began to interpolate and extrapolate.  If I was in a car going 65 miles per hour, how much distance would I need to not crash?  Easy, but we needed to talk about how to document our work and thinking so that future physics teacher would know how we arrived at our answer WITH units.

If I am a stunt car driver and know that I have a football field’s length to stop my car, what should my top speed be when I hit the brakes?  This question was not so easy.  Learners fell in to three camps.

  • Camp 1:  I substitute 100 in for y.  They got feedback quickly from their peers that 100 was in yards and the units were feet!
  • Camp 2:  I substitute 300 in for x because that is where I substituted 65 before.  They also got quick peer feedback that x was in mile per hour and to pay attention to your units!!
  • Camp 3:  I substitute 300 in for y, but what do I do now?  Unfortunately, in this class, they got no peer feedback.  We facilitators were informed to say the least!  Not one child connected this to using the quadratic formula to solve.  (triple YIKES!) Good formative assessment that they are not connecting the skill to the application.

Once they realized that they could use the quadratic formula to solve, they were successful.  We could monitor the swings, and lack of swings, using our Navigator.  By now, everyone was swinging because they knew we were supporting their effort and learning.

The big take-away for me:  Formative assessment that offers feedback and support in the moments of learning are critical for success and confidence.

The big take-away for my learners:  You can do this; we will help!

You will not hit it out of the ballpark if you do not swing.  A swing and a miss is so much better than no swing.  Step up to the plate; dare to swing.  Miss; swing again.  You can do this; we will help!

Stopping Distances

… written in collaboration with Ruth Casey and Sam Gough.

Distracted driving is any non-driving activity a person engages in that has the potential to distract him or her from the primary task of driving and increase the risk of crashing.
From D!straction.gov
The Official US Government Website for Distracted Driving

A typical rule for the distance you should follow behind a car is given by the “three second rule.” To determine the right following distance, select a fixed object (a tree, a sign, an overpass ..) on the road ahead. When the vehicle ahead of you passes the object, begin counting “one one thousand, two one thousand, three one thousand.” If you reach the object before you complete the counting, you’re following too closely.

  • When you see an object in your path, can you stop your car instantly?
  • What happens between the time you realize that something is in your path and when the car actually stops?
  • How much distance has been covered before the car has stopped?
  • How does your reaction time affect these distances?

As an introduction, watch Vehicle Stopping Distance from teacher’sdomain.org or Think! – Slow Down which is embedded below.  (Warning…it is tough to watch.  A dummy is used, but you should preview before you show it to students.  I like it because you can see the screeching tires and the struggle to stop.)

If you are interested in the physics, check out the Vehicle Stopping Distance Calculator from Computer Support Group and their online division, csgnetwork.com.

For an experiment of calculating your reaction time, do the math.

Let’s look at the data.

Suppose you want to visualize the pattern in the distance traveled while reacting versus the speed of your car.  Do I travel the same distance while I’m reacting no matter the speed or does the speed influence the distance traveled just while reacting?


  • What does this pattern tell us about reaction distance traveled vs. speed?
  • Can you find the mathematical model for these data?
  • What is the slope? What is the meaning of the slope?
  • Is this direct variation?
  • Which of our learners can find success with this?

How about the pattern in the distance traveled while braking versus the speed of your car?

  • What does this pattern tell us about braking distance traveled vs. speed?
  • Can you find the mathematical model for these data?
  • What is happening with the slope?
  • Which of our learners can find success with this?

Now, how about the pattern or relationship between the total distance traveled while stopping the vehicle vs. the speed?

  • What does this pattern tell you about the total braking distance vs. speed?
  • Can you find the mathematical model for these data?
  • What is happening with the slope?
  • Which of our learners can be successful with this?

I don’t want to give away the mathematical models; I want you to have time to consider and think about the mathematical models.  If you need or want a hint, please leave a comment below and I’ll write you back.