Wednesday, July 29, 2015

Technology as educational motivator, formative assessment tool, and documentation of both

The voice on my arm chirped "Four kilometers. Twenty-four minutes and ten seconds. Five minutes and fifty-three seconds per kilometer."

With that data I knew I could run a sub-30 minute five kilometer time, but I would have to push harder.

That is formative assessment at its absolute best: exactly the data the student needs at the exact moment the student needs that data and delivered in a way that was completely accessible to a student with his hands full. When I run, I juggle, a sport called "joggling." Joggling means my hands are literally full and constantly in motion, I have no ability to check a wrist mounted pace monitor
~ ~ ~
"How are you doing in that class?" I asked Bill as part of a advisor-advisee conversation just after early warning.

"I don't know," he responded.

"Why not?" I asked.

"The teacher has not marked any of our papers, we don't have a grade yet," he replied.

At midterm I asked Bill about the same class, "How is that class now? Do you know how you are doing?"

"Yes, I have a C. But I do not know why. The teacher has not yet handed back any papers."
~ ~ ~
"I ran a mile at the high school track tonight," posted Newleen to her friends on social media. "#6/41!" she added, indicating she was on the sixth day of a 41 day challenge to run one mile every day for 41 consecutive days.

"Post a screenshot of your app," admonished Lynda in a reply to the post. The screenshot is proof of the accomplishment, documentation, evidence. The app refers to software on a smart phone that the running group is using to share proof of their accomplishing the daily one mile running challenge. With group members scattered across ten time zones around the planet, only screenshots of running apps can provide evidence.
~ ~ ~
"How are you doing in statistics class?" I asked Tom.

A puzzled look briefly flashed across his face and then he smiled, "I doing OK," Tom responded.

"What's your grade?" I asked.

"'B', I just checked Schoology this morning." Tom replied. "But if I get the missing homework done, it will go up."
~ ~ ~
When I reached home and finished the run, a new voice piped up, that of professional runner Shalane Flanagan, announcing that my five kilometer time was a personal best for me.

From inside the house my wife asked, "Who's that woman out there with you?"

"Shalane Flanagan," I responded.
~ ~ ~
Immediate feedback is the motivating and the critical enabling factor is the technology that makes this possible. The running app, enables immediate formative feedback during the process of running.

The technology also provides the ability to share the experience as an image. Anyone could post "I ran a mile today" but the image documents the accomplishment, provides proof. One is motivated to run in order to get that image to share with friends who are waiting to see the image. The technology is central to the motivation to get up and go out for a run.

In the classroom too, the key is the technology. My student knew their grade even when I did not because of technology, my use of the Schoology learning management system. And they had the information they needed on how to improve that mark. I used on line tests that provided instant feedback upon submission of right and wrong answers. The students reacted very positively to this feature of the course. Yet only technology made that possible.

There is also Lynda's comment to post a screenshot. Provide evidence. Not just for the benefit of the student, but for those who supervise iinstructors. Technology can enable an academic supervisor to complete the supervision loop. The chair for the division in which Bill was taking that class had no idea that their instructor was not providing any feedback, and in the eighth week could only provide a grade without a "screenshot" showing how that grade was obtained. Supervisor's need to have real time insight into their runners - the employees they supervise.

Technology can again play a pivotal role. If instructors utilize an on line learning management system (and here I intentionally mean a learning management system replete with assignment submission and rubric marking capacity, not merely an on line grade book), then by purchasing an institutional license, authorized administrators can have real time insight into whether formative assessment is occurring.

The starting point for motivating the runners above is the enabling, connected, networked technologies. And that also has to be the starting point for a 21st century classroom. Which is why I cannot be a supervisor: if you are not using a learning management system and other enabling technologies in your classes, then it is time for you to move on.

Note that this article was inspired in part by What Fitness Bands Can Teach Us About Classroom Assessment. The names used are pseudonyms to protect privacy. The author is a 55 year old community college instructor in the Pacific. The running app pictured is the Android version of the Nike+ running app

Wednesday, July 22, 2015

Cubic bezier sine wave in SVG

In the past I generated the Scalable Vector Graphics path for a sine wave by using a sine function and inputting x values.

The difficulty with this approach is that the crests and troughs become unsmooth at combinations of high frequencies and high amplitudes. I had to increase the sampling density along the x-axis to smooth out the crests and troughs, which led to very large blocks of coordinates in the path command. 

This summer I found that a cubic bezier could generate a cosine curve using just the points at the crest and trough, with control points at 0.1875*wavelength and 0.3125*wavelength. Those values were visually determined, not mathematically obtained.

The result is a smooth cosine wave of any combination of wavelength, frequency, and amplitude. A transform="translate(kπ,0)"can then move the cosine curve horizontally to generate a sine wave. Masks can be used to hide unwanted portions.

The cubic bezier routine can be seen in the code underneath question number one on my test five for the summer session. A transformed version that appears to be a sine wave was deployed in a similar question, question 21, on the final.

Sunday, July 19, 2015

Assessing learning in physical science

SC 130 Physical Science proposes to serve two institutional learning outcomes (ILO) through four general education program learning outcomes (GE PLO) addressed by four course level student learning outcomes (CLO). Not listed are proposed specific student learning outcomes that in turn serve the course level learning outcomes.  This report assesses learning under the proposed course level learning outcomes which in turn supports program and institutional learning outcomes.

ILO 8. Quantitative Reasoning: ability to reason and solve quantitative problems from a wide array of authentic contexts and everyday life situations; comprehends and can create sophisticated arguments supported by quantitative evidence and can clearly communicate those arguments in a variety of formats.

3.5 Perform experiments that use scientific methods as part of the inquiry process. 1. Explore physical science systems through experimentally based laboratories using scientific methodologies
3.4 Define and explain scientific concepts, principles, and theories of a field of science. 2. Define and explain concepts, theories, and laws in physical science.
3.2 Present and interpret numeric information in graphic forms. 3. Generate mathematical models for physical science systems and use appropriate mathematical techniques and concepts to obtain quantitative solutions to problems in physical science.

ILO 2. Effective written communication: development and expression of ideas in writing through work in many genres and styles, utilizing different writing technologies, and mixing texts, data, and images through iterative experiences across the curriculum.

1.1 Write a clear, well-organized paper using documentation and quantitative tools when appropriate. 4. Demonstrate basic communication skills by working in groups on laboratory experiments and by writing up the result of experiments, including thoughtful discussion and interpretation of data, in a formal format using spreadsheet and word processing software.


Explore physical science systems through experimentally based laboratories using scientific methodologies

Laboratory fourteen in the penultimate week of the term provided a vehicle for assessing this course level outcome. The students were given a system to explore and two questions to answer. The students were provided with string and weight pendulums and asked to determine the relationship between the length and the period. The class started in the classroom working with lengths from 25 cm to 150 cm.

Using the gathered data and spreadsheets, the students fit linear regressions to the data. Over the lengths involved the non-linearity of the data is not apparent. The students then used their linear regressions to predict the period for a 700 cm pendulum.

After posting predictions on the board, the class walked down to the gym where a 700 cm pendulum can be swung. The predictions were up around 10 seconds, but the actual period was around 5 seconds (4.85 seconds and 5.38 seconds were the recorded values). The data did not fit the model. The students were then instructed to write up the laboratory taking into account the errant prediction.

The laboratory reports were assessed to determine whether students properly recorded data in a labeled table, generated an xy scatter graph, added a regression equation to their data, made a determination as to the non-linear nature of the relationship, and then discussed their analysis and results in a reasonably meaningful manner.

Student performance was assessed using a simple binary rubric based on the metrics noted above. Ten of the fourteen students completed this term end laboratory.

The ten students demonstrated mastery the core mechanics of the laboratory report style used in SC 130 Physical Science. The students gathered data and reported results in a properly formatted and labeled table. The students used spreadsheet software to generate xy scatter graphs also properly formatted and labeled. All of the students added a trend line to the graph, but only seven specifically discussed their choice of mathematical relationship. All ten students chose a linear regression. Only a single student noted that the data was "not really linear." That one student went on to note that there existed the possibility of the "math model used being wrong."

During the fall and spring terms, laboratory submission rates tend to fall as the term progresses. This summer saw no fall off in laboratory submission rates. One student simply failed to complete laboratories in the later half of the term, but the remaining thirteen consistently turned in laboratories after midterm with the only exception being the last laboratory. Being at the end of the term, the final laboratory (laboratory fourteen) had a narrower window for submission.

Summer term is short and intense. Only two students were taking classes other than physical science, thus this was the only course for the majority of the students. These factors may have contributed to the turn-in rate remaining high during the summer session.


2. Define and explain concepts, theories, and laws in physical science.

Twenty-nine items on the 66 item final examination asked students to define and explain concepts, theories, and laws in physical science. The average for fourteen students on these 29 items was 80%. This is significantly higher than the 61% success rate seen in the spring and the 54% success rate fall 2014. The strong performance is also likely a unique effect of the brevity and intensity of the summer term.


3. Generate mathematical models for physical science systems and use appropriate mathematical techniques and concepts to obtain quantitative solutions to problems in physical science.

While the laboratory fourteen assessment above provides some data on this course learning outcome, this outcome supports the general education program learning outcome "3.2 Present and interpret numeric information in graphic forms." With this focus in mind, a pretest and post-test was included in the course. The post-test was embedded in the final examination.

Student performance on the pretest can only be characterized as abysmal.

Nine of the fourteen students completed a 100 or higher level college mathematics course prior to enrolling in the SC 130. The mathematics background for five of the students was not determined. Given that background the general inability of the students to calculate linear slopes and intercepts is disappointing. The only skill the students consistently appear to bring into physical science is the ability to plot points on an xy scattergraph.

SC 130 Physical Science includes a focus on the mathematical models that underlie physical science systems. Laboratories one, two, three, five, seven, nine, eleven, and twelve have linear relationships. A number of assignments in the course also have linear relationships. The students also encounter a quadratic relationship in laboratory three. A plot of height versus velocity generates a power relationship, specifically a square root relationship. By the end of the course students have repeatedly worked with linear relations.One relationship at a time, not "problems one to thirty even problems only." Every equation is built from data that the students have gathered. From the concrete to the abstract, repeated throughout the term, providing cognitive hooks on which to "hang" their mathematical learning.

By term end, the students who were present on the day of the pretest had all improved significantly as measured by the post-test. Note that ten students sat the preassessment, fourteen completed the post-assessment, hence the use of percentage success rate in the following chart.

In the chart above, the left end of the line marks the percent of students answering an item correctly on the pretest, the right end the percent answering correctly on the post-test. Note that items such as the slope and y-intercept assessment improved on the first two items even though the problems were made more difficult by a non-zero y-intercept on the post-test.

With eleven items on the pre-test and post-test, scores for students could range from zero to eleven. The median score on the pre-test was one, on the post-test items was eight. The rise in the mean score was from 1.43 to 7.5. The score distributions as box plots provide some insight into the lift in scores from the pretest to the post-test. Note that ten students sat the pre-assessment and fourteen sat the post-assessment.

The post-test does not answer whether there will be long term student retention of the ability to present and interpret numeric information in graphic forms. The course has had, at least in the short run, a strong positive impact on the students' ability to work with numeric information in graphical forms.


4. Demonstrate basic communication skills by working in groups on laboratory experiments and by writing up the result of experiments, including thoughtful discussion and interpretation of data, in a formal format using spreadsheet and word processing software.

Course level learning outcome four focuses on communication, specifically writing. In the late 1990s assessment data suggested some students were graduating with limited writing communication skills. As noted by the languages and literature division at that time, two college level writing courses in the general education core cannot by themselves produce collegiate level writers. Writing must occur across the curriculum, across disciplines. In 2007 SC 130 Physical Science at the national campus was redesigned to put an emphasis on writing. A "fill-in the blank" cook book style laboratory manual was replaced by laboratories which led to laboratory reports constructed using spreadsheet and word processing software.

By the end of the term students could produce a laboratory report with tables and charts integrated from a spreadsheet package. The students could produce reports that included the use of quantitative tools.

As reported above, student ability to include thoughtful discussion and interpretation of data supported by their quantitative evidence was not accomplished by all students as measured by laboratory fourteen. In fact only one student questioned the whether the data was linear (the data was not linear).

Producing sophisticated scientific arguments was a bridge too far for a number of students. The analysis of laboratory fourteen suggested that only three of fourteen students (21%) could engage in a meaningful discussion of their data. Spring 2015 seven of eighteen students (39%) showed an ability to reasonably and meaningfully discuss their results. The compressed summer term may leave less time for reflection, contemplation, and insight.

Inherent in supporting institutional learning outcome two, which course learning outcome four serves, is proper mechanics. Physcial Science laboratory report marking rubrics at the national campus include evaluation of four broad metrics: syntax (grammar), vocabulary and spelling, organization, cohesion and coherence. Each of these four metrics is measured on a five point scale yielding a total possible of twenty points. In general, students enter the course with writing skills. Errors of tense and agreement tend to mirror areas in students' first language that do not have similar tense or agreement structures. All students in the class are working in English as a second language.

In general there is no significant change in mechanics measurable from laboratory one to laboratory fourteen. The sample size is small and the change in individual scores is also small. Measured across all four metrics, the median score went unchanged. On the first laboratory the median was the maximum possible of 20 points, the median at term end was 20 (n = 10). The students in the class generally had basic grammar skills when they arrived in the class. Over the duration of the summer only two students had frequent grammar and syntax issues.

The mean score at term start was 17.57 out of 20, at term end the mean score was 19.5. The course may be more beneficial to the weakest writers.

Considering the four course level outcomes, there are areas where improvement can be sought. A challenging area for improving performance would be in the students ability to engage in more sophisticated discussions of their experimental results. The students do not have a rich and varied background in science, and the course is serving principally non-science majors.

The course also has the intent to communicate affective domain messages to the students. One is the idea that doing science is fun, the other is that the students can do science. Science is not a subject in which the students have all experienced success. This term these impacts were not measured, these are subtle and difficult to quantify.

Over the years, however, students will at times comment on an image posted by a student from the class. The comments are always positive, of the nature that the course was interesting and fun. Learning happens only where there is motivation to learn, and when an activity is enjoyable, a learner will engage more fully with that activity.

Physics, mathematics, and fun are three words that do not often co-exist in a single sentence, a single class, or in the mind of a student. Yet they should. The mathematical nature of the world is fascinating and fun. My hope is that those three words can now co-exist in the minds of the physical science students.

Friday, July 17, 2015

Assessment in algebra and trigonometry

MS 101 Algebra and Trigonometry now includes a course wide student learning outcome common math assessment (CMA) along with a more traditional final examination (FX). This provides the opportunity to compare performance on the common assessment versus the final examination.

Summer 2015 fourteen students enrolled in and completed the course. As with any course, the focus and parameters vary by instructor. My own approach includes an open book evaluation, the elimination of all rote memorization, a de-emphasis on trigonometric manipulations, the inclusion of exponential and logarithmic regressions using spreadsheets, the use of trigonometry in contexts such as scalable vector graphics programming, and the use of technologies such as the WolframAlpha engine to aid in solving problems.

Student using WolframAlpha app on Android 

The common math assessment is intended to evaluate the following student learning outcomes:

1.1 Solve and evaluate exponential equations and functions.
1.2 Solve and evaluate logarithmic equations and functions.
1.3 Interpret the inverse relationship between exponentials and logarithms.
1.4 Use exponentials and logarithms to model and solve real-life problems.
2.1  Sketch and recognize angles using degree mode.
2.2  Sketch and recognize angles using radian mode.
2.3  Convert angles between degree and radian mode
3.1  Use right triangles to evaluate trigonometric ratios of acute angles.
3.2  Use reference angles to evaluate trigonometric ratios of nonacute angles.
3.3  Use the unit circle to evaluate trigonometric ratios of quadrantal angles
3.4 Use a calculator to evaluate trigonometric ratios of angles.
3.5  Use right to triangles to model and solve real life problems.
4.1  Recognize and sketch the basic sine and cosine curves.
4.2   Find the period, amplitude and midline of sinusoidal curves
4.3  Use sinusoidal functions to model real-life phenomena.

While students in the course could and did use computer based technologies to assist in solving problems, on the common math assessment the students were restricted to the use of scientific calculators. While in class tests were open book, the common math assessme
nt was closed book.

The common math assessment is multiple choice, the final examination was paper based with all answers in a "fill-in-the-blank" format. The common math assessment was programmed into Schoology for delivery to the students. This provided automated tabulation of the results.

Multiple choice provides a random chance of being right even when the student does not know how to answer a question. The answers also provide information to the student on the form of answer which in turn may assist in solving the problem.

Rendering the common math assessment in Schoology meant generating diagrams. To gain full control over a diagram's design, the diagrams were done in Scalable Vector Graphics, imported into GIMP, and then saved as PNG files for uploading in Schoology. Some answers, as seen above, were done using the preformatted element in HTML. Schoology provides support for HTML in multiple choice answers.

Schoology provided detailed analysis of the student's responses including how many chose which particular right or wrong answer.

Due to an absence, thirteen students sat the common math assessment. All fourteen students completed the final examination. Note that the shift in emphasis of the course and the coverage choice on the final meant that the final examination did not cover all of the same outcomes as the common math assessment.

The average correct on the common math assessment was 51%, on the final examination the average correct based on an item analysis was 73%, a difference of 21%. Noting that the two instruments are not fully aligned, performance on outcomes on the common math assessment which were also on the final examination saw a 59% performance, a difference of 14% against the final examination. Bear in mind that the final examination permitted the students the use of the full array of tools encountered during the course and was open book, the common math assessment restricted the students to scientific calculators and was closed book.

For those outcomes which were evaluated by both the common math assessment and the final examination, the correlation was 0.76. The correlation is significant (p-value: 0.047), or to return closer to the language used by Fisher, the correlation would be surprising for someone who presumed that the two data sets are unrelated.

Source SLO CMA% Source SLO FX% Diff zDiff
CMA 1.1 0.31 FX 1.1 0.75 0.44 1.54
CMA 1.2 0.54 FX 1.2 0.68 0.14 -0.49
CMA 1.3 0.77 FX 1.3 0.77 0.00 -1.44
CMA 1.4 0.23 FX

CMA 2.1 0.77 FX

CMA 2.2 0.69 FX

CMA 2.3 0.77 FX 2.3 0.95 0.18 -0.20
CMA 3.1 0.54 FX

CMA 3.2 0.46 FX

CMA 3.3 0.31 FX

CMA 3.4 0.85 FX 3.4 0.98 0.14 -0.51
CMA 3.5 0.31 FX 3.5 0.38 0.08 -0.91
CMA 4.1 0.08 FX

CMA 4.2 0.56 FX 4.2 0.57 0.01 -1.33
CMA 4.3 0.54 FX


0.73 0.2125
Stdev (diff) 0.23

0.21 0.1490

0.73 0.1415

Diff = FX - CMA
zDiff = z-score for the difference = (Diff-0.73)/0.1490
StDev (diff) is the standard deviation of the difference values, not the difference in the standard deviations of the CMA and FX
MeanSelct is the mean for selected outcomes, those found on both the CMA and FX

As a side note, the final examination asked the students to perform a series of routine sine and cosine calculations. The students answered these correctly at 98% rate on the final examination.

When asked, as in 19A and 19B seen above, to make essentially the same calculations as presented in a context involving SVG graphics, the success rate fell to 39%. Question 19h* then asked the student to plot the heptagon vertices on their computer using SVG. 36% of the students were able to generate the heptagon on their computer with the vertices correctly placed. For students in the Computer Information Systems major, SVG is a language they could conceivably have to work with in the future. There is an application to the knowledge in the course, I view this as mathematics in a context. These problems are hard and all of the tools of WolframAlpha or Mathematica do not help a student solve such a "real world" problem. For a CIS major, I would argue that this constitutes a form of authentic assessment.

Measuring pi out on the lawn

*Note that 19h is a complex task.The students only had to plot the vertices using circle elements, some worked on adding the outline of the heptagon using the path element.

The code to generate the heptagon 

The course is open book, open note, allowing the students to use pre-existing class work as a starting point for the diagram. Thus some portions of the SVG code are boiler plate that the students already know to leave alone. Note too that a transform moves the center to (0,0) which means that the calculations such as 100 sin (39) generate the correct SVG grid values except for the flipped y-axis. The students were also working in SVG from within HTML using the HTML5 doctype to avoid the problems that full SVG (XML) bring (yellow screens of death). Notes for the students covers how to set up these files.

Wednesday, July 8, 2015

Mathematics only in a context

I tried a slight twist on the last laboratory of the summer session. I walked into the room empty handed and then began an abstract lecture on site swap mathematics.

I deliberately avoided placing the notation into a context, introducing the topic as an abstract system of symbolic notation and manipulation. Note that I labeled the sites as alpha and beta, referring only to objects by color names. "The purple will move three sites down from alpha to beta, then another three sites from beta back to alpha."


I initially put up a 3 and then showed that one could swap two site to yield the 42 seen in the middle of the pattern above. I noted that sums and averages were invariant under a site swap. The students dutifully took notes and worked on drawing the diagram.


With the above on the board, I went around the room to orally check for understanding. Seven of the fourteen students smiled and nodded yes when I asked if they felt they basically understood the site swap system. Three were non-committal, but a fourth student simply said, "That does not make any sense. Too confusing." A recheck found that the initial three non-commits were now willing to admit that they had no idea what the markings on the board meant.

Julie-Ann, one tennis ball out-of-shot high above

Part of the above effect is something I have come to see as part of the way of interacting out here - smile, nod, shake your head in agreement, even if you have no idea what the other person is really trying to say or talking about.

Cherish, three ball juggle under control

I then explained that mathematical models, mathematical systems, should never be introduced as abstract systems without a context. I noted my own position that mathematics as a subject should never be taught in isolation. Mathematics should be taught within the contexts within which mathematics arises. Such as physical science class engenders. Data that is related leads to graphs with linear regressions which are expressed as linear equations.

Barry masters three ball juggling, site swap pattern 3

I then revealed the context within which site swap mathematics arises, that the alphas and betas are right and left hands, landing sites for juggled balls, and then illustrated the meaning of the system. The notation now had a context, a meaning. While still confusing, the idea of a site swap could now be physically demonstrated.

Mathematics without context is meaningless. I realize that I am attacking the ivory tower of pure mathematics, a field that will surely point to meaningless abstractions that paved the way for scientific advances. Or argue that their abstractions are beauty in the purest form. There will always be a place for the 0.01432% who professionally engage in mathematics. For the other 99.98568% of the population, mathematics without context is meaningless holds.

The class wraps up with the students attempting to learn a 3, and those who master a 3 attempting a 342 site swap. Juggling is not useful per se from a survival perspective, I would not argue mathematics has to be useful, only that math should be taught from within the contexts in which mathematics is useful.

Tuesday, July 7, 2015

Story time and predictions with mathematical models

I had the students read from 13 Planets as a follow-on activity to the Journey to the Edge of the Universe video yesterday. Although somewhat unorthodox, I noted that the exercise would clarify some of facts that are so colorfully tossed around in that video.

Perdania reads, covering the sun.

Joemar practices his reading technique

Rofino demonstrates good reading out loud technique. I had noted the importance of reading to children and that this exercise also exemplified that skill.

Cherish faces the class, but then the listeners cannot see the picture.

Sharon reads.

Pendulum data predictions.

For the lab, the students measured length versus period for the pendulum in the classroom and then used a linear regression to predict the period to predict periods at longer lengths. In class the students topped out at about 150 centimeters. Predictions were on a linear model.

Eddie with the in class rig.

Then the class moved to the gym where a 700 cm length can be tested. The 700 cm pendulum had a period of around five seconds, with students obtaining slightly different times on their stopwatches. The prediction were all too high. The lab is then trying to figure out what went wrong. First time I have taken the pendulum lab to where the length refutes the linear model. Well worth doing, the students expressed disappointment at being wrong - exactly what I had hoped. The new data throwing their predictions into disarray means a new theory is needed.

RipStik for vector exposition and addition

Vectors are fun. I sat in the computer laboratory staring at section 8.3 of Larson's Algebra and Trigonometry text. Nothing I saw on the page looked like fun. My mind wandered to my RipStik. And then in a flash I knew I wanted to fling a ball at a right angle from a moving RipStik and use the radar gun to get the ball launch speed. The RipStik could be timed across a distance with a stopwatch. 

Kenneth used the radar gun to obtain the ball launch speed.

Marla timed the RipStik over an 800 cm distance. She obtained a time of 3.28 seconds for a
RipStik velocity of 243.9 cm/s.

Maggie handled the RipStik riding and ball launching, freeing me up to photo-document the experiment.

Lecture then proceeded on the sidewalk. Note that the radar gun measured in kilometers per hour. 31 kph was 861 cm/s. The ball was thrown perpendicular to the path of the RipStik directly at the radra gun, but the path over the ground, and the net velocity vector, carried the ball towards the Araucaria pine. The i, j notation is upside down so that it would be right side up for the students who had settled onto the faculty building stairs.

The class then returned to the classroom to provide some definitions for magnitude, direction angle, and the unit vectors. 

The day before I sneaked a couple vectors onto the board when covering a trajectory equation in section 7.3.

Vectors are fun, especially when is able to play with them outside!