# 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.

# Phases of the Moon…Middle School Connections with Trigonometry and Science

My learners struggle to read and interpret graphs for meaning.  It makes me wonder…How are we teaching them to read and interpret graphs? When our learners get to precalculus, are they adept at reading graphs for meaning so that they can concentrate on mathematical modeling?  Wouldn’t it be advantageous for a new-to-precalculus or new-to-physics learner to already have a context with which to identify when presented with periodic data?

What if we integrated the ideas of plotting points and interpreting graphs with some earth science?  We are not going to have middle school students model this data, but we are going to have them interpret the data and label the graph.  We are going to expect them to connect the math and the science.  I’m pretty sure that there’s a great connection to periodic poetry too.

I believe that somewhere in 6th or 7th grade science we teach our learners about the phases of the moon.  The geometry is awsome.  (Take the Lunar Cycle Challenge.)  In pre-algebra and algebra, we work on plotting points on the Cartesian coordinate plane.  What if we practiced plotting points and pattern-finding by plotting real data?

In terms of diagonstic assessment, ask:

1. Can you name the 8 traditionally recognized phases of the moon?
Then wait…6th and 7th graders know this, which means that older learners will need a little time for recall.  Wait time is critical.  Full moon and blue moon almost always occur.  Generally the vocabulary will come back to any group of learners.  This is a great place to integrate teachnology.  Let them find the answers.
2. Is there an order to these phases?  Are there any patterns?
Ask your learners to sketch a graph of what they have described.  You might want to check their understanding of the vocabulary and the images.
3. Are the phases identifiable when not in order?
To reinforce the geometry, you can use the Lunar Phase Quizzer.

1. Can we find data for the percent of the moon showing every day?
2. If we plot this data, will there be a pattern?
3. How much data should we plot to see the pattern?

Let’s look at the data for January and February 2011 from the United States Naval Observatory.

Questions to ask to check for numeracy, geometry, and understanding of the vocabulary:

1. Is the moon waxing or waning on January 1, 2011?  How do you know?
2. Is the moon crescent or gibbous on January 1, 2011?  How do you know?
3. Sketch the moon’s illumination on January 1, 2011.  Check using Today’s Moon Phase.

Don’t you think that these are great TI-Navigator formative assessment questions?  You can repeat those three questions about any date in January and/or February until you have consensus.

Let’s discover if there is a visual pattern in this data.

This graph always gets a big WOW! if you are using TI-Nspire and can see the points animate into place.  Questions to ask to check for graphical interpretation and connection between the graph and the earth science:

• When was the full moon in January of 2011?  How do you know?
• When was the new moon in January of 2011?  How do you know?
• When was the first quarter moon in January of 2011? … the third quarter moon? How do you know?

Again, you can check using Today’s Moon Phase.

Deeper questions connecting to writing and interpreting inequalities with connections to more vocabulary:

• Name one day in January of 2011 that the moon was waxing?… waning? … crescent? … gibbous?  How do you know?
• Over what days in January of 2011 was the moon a waxing crescent moon? … waning gibbous?  etc.

Pretty good stuff if your students are struggling with writing inequalities, particularly compound inequalities.

Now for patterning…

• Will February’s data look like January’s data?
• How are these data similar?  How are these data different?
• What would the graph look like if we graphed the percent of the moon showing vs. the number of days since December 21, 2010?  (In other words, February 1 would be day 32.)  Would the pattern continue?

How cool is that?

• Can you identify all of the above questions for February?
• Can you predict the day of the full moon for March?

Please use Today’s Moon Phase to check!

Here are the burning questions for me…

• Can middle school students plot these points?
• Will they find connections between the math and the science?
• Will these connections help them understand how to interpret graphs?
• Will these connections help them understand the moon and its phases?

If  our students would start off in trigonometry and physics understanding the connections between the math and science and could interpret what they see, would they be more likely to find success modeling the data?

If you teach trigonometry or physics, there is a clear path from the graphical interpretation to finding a function that models this data.

TI-Nspire Resource Files
• Phases of the Moon Diagnostic Assessment
• Phases of the Moon PublishView document
• Phases of the Moon Data .tns file

From my webpage….

During every month, the moon seems to “change” its shape and size from a slim crescent to a full circle. When the moon is almost on a line between the earth and sun, its dark side is turned toward the earthThe moon’s cycle is a continuous process, there are eight distinct, traditionally recognized stages, called phases, which are ordinarily adequate to designate both the degree to which the Moon is illuminated and the geometric appearance of the illuminated part, to the extent that Moon visibility has relevance to everyday human activities.

In this activity, we will investigate the fraction of the moon seen each day for a month and then for a year.

1. Identify the eight traditionally recognized stages of the moon’s cycle.
2. Find the approximate period of the moon’s cycle.
3. Extract the fraction of the moon showing from the United States Naval Observatory, for the days of the year.
4. Set up a scatter plot of the fraction of the Moon showing in January versus the day of the year.
5. From the data or the graph, determine the amplitude and the vertical translation.
6. Find the cosine function that fits this data.  Identify the phase shift for this function.
7. Find the sine function that fits this data.  Identify the phase shift for this function.
8. Edit the data to graph the fraction of the moon showing in January and February versus the day of the year.
9. How well do your functions fit this extended data?
10. Determine which day of the year corresponds to today’s date.  Predict the phase of tonight’s moon.
11. Check the accuracy of your prediction using Today’s Moon Phase.
12. Take the Lunar Cycle Challenge.