Vectors for Everyday Life
Author
Bartley Richardson
Subject
Mechanical Physics
Grade Level
11th and 12th
Duration
70 minutes (one class period)
Rationale (How this relates to engineering)
Engineers use vectors to represent aspects acting upon certain systems. For example,
if multiple forces are applied on an object at varying angles, the mechanical or civil
engineer would use vector addition to find the total force acting on the object. Civil
engineers often deal with multiple forces acting on a bridge. In addition to forces, vector
addition can be used to find distances where a straight line-of-sight may not exist.
Regardless of the application, the ability to manipulate vectors and understand their
meaning is essential in engineering.
Activity Summary
General Description
This lab experiment utilizes two methods to reinforce the concept of vector addition. In
the first activity, students use surveying equipment to find the straight-line distance
between two points where a direct line-of-sight between the points does not exist. They
also use the equipment to find the angle between the distance vector and the horizontal.
In the second activity, students apply vector addition to forces acting on a bolt. The
objective is to find the total force acting on the bolt and the angle at which this force acts.
Teaching Philosophy
This lesson was created at the request of a high school physics teacher. Since the
students had already been introduced to vector addition, completed multiple homework
assignments on the concept, and finished one experimental lab dealing with vector
addition, I designed this lesson to mainly reinforce concepts they were already familiar
with. I took the opportunity to incorporate real-world engineering applications throughout
the lesson, and I designed the calculations so that the students would need to take their
work a few steps further than they do in the homework questions. The lesson was
designed to be self-guided, and different groups of students can work at difference
speeds. Minimal initial instruction is required (assuming the students are already familiar
with vector addition), and the only verbal instructions given include a brief overview of
the lesson and a quick tutorial on how to use the surveying equipment.
Objectives
1. Students will be able to accurately measure distances and angles (to proper
significant digits).
2. Students will be able to maintain proper significant digits throughout the
calculations.
3. Students will be able to create an accurate diagram of their measurements.
4. Students will be able to apply right triangle trigonometric rules to calculate the
hypotenuse of the triangle and a given interior angle.
5. Students will be able to use the Law of Sines and Law of Cosines to calculate the
hypotenuse of a triangle and a given interior angle.
6. Students will be able to identify sources of error in the lab and explain their
significance.
7. Students will be able to use a total station or theodolite to measure distances and
angles.
8. Students will be able to calculate the percent error of a given calculation.
9. Students will be able to apply the principle of vector addition to a real-world
engineering problem involving forces.
10. Students will be able to accurately and effectively report their results in scientific
language.
11. Students will be able to identify when vector addition is necessary and what
steps need to be taken before the vectors can be added.
Standards
Science
 Standard: Science and Technology
Students recognize that science and technology are interconnected and that using
technology involves assessment of the benefits, risks and costs. Students should
build scientific and technological knowledge, as well as the skill required to design
and construct devices. In addition, they should develop the processes to solve
problems and understand that problems may be solved in several ways.
o
Benchmark A: Predict how human choices today will determine the
quality and quantity of life on Earth.
 Indicator 1 (Grade 12): Explain how science often advances with
the introduction of new technologies and how solving
technological problems often results in new scientific knowledge.
Mathematics
 Standard: Number, Number Sense and Operations
Students demonstrate number sense, including an understanding of number systems
and operations and how they relate to one another. Students compute fluently and
make reasonable estimates using paper and pencil, technology-supported and mental
methods.
o
o
Benchmark B: Develop an understanding of properties of and
representations for addition and multiplication of vectors and matrices.
Indicator 2 (Grade 11): Determine what properties hold for vector
addition and multiplication, and for scalar multiplication.
o

Indicator 5 (Grade 11): Model, using the coordinate plane, vector
addition and scalar multiplication.
o Indicator 9 (Grade 11): Use vector addition and scalar multiplication to
solve problems.
Standard: Measurement
Students estimate and measure to a required degree of accuracy and precision by
selecting and using appropriate units, tools and technologies.
o

Benchmark A: Explain differences among accuracy, precision and error,
and describe how each of those can affect solutions in measurement
situations.
o Indicator 1 (Grade 11): Determine the number of significant digits in a
measurement.
Standard: Geometry and Spatial Sense
Students identify, classify, compare and analyze characteristics, properties and
relationships of one-, two-, and three-dimensional geometric figures and objects.
Students use spatial reasoning, properties of geometric objects and transformations
to analyze mathematical situations and solve problems.
o

Benchmark A: Use trigonometric relationships to verify and determine
solutions in problem situations.
o Indicator 2 (Grade 11): Represent translations using vectors.
o Indicator 4 (Grade 11): Use trigonometric relationships to determine
lengths and angle measures; i.e., Law of Sines and Law of Cosines.
o Indicator 2 (Grade 12): Derive and apply the basic trigonometric
identities; i.e., angle addition, angle subtraction and double angle.
Standard: Patterns, Functions and Algebra
Students use patterns, relations and functions to model, represent and analyze
problem situations that involve variable quantities. Students analyze, model and solve
problems using various representations such as tables, graphs and equations.
o
o
Benchmark D: Apply algebraic methods to represent and generalize
problem situations involving vectors and matrices.
Indicator 7 (Grade 11): Model and solve problems with matrices and
vectors.
Technology
 Standard: Technology and Society Interaction
Students recognize interactions among society, the environment and technology, and
understand technology's relationship with history. Consideration of these concepts
forms a foundation for engaging in responsible and ethical use of technology.
o
Benchmark A: Interpret and practice responsible citizenship relative to
technology.
 Indicator 3 (Grade 11): Compare and evaluate the advantages
and disadvantages of widespread use and reliance on technology
in the workplace and in society as a whole.
 Indicator 1 (Grade 12): Make informed choices among
technology systems, resources and services.
o
Benchmark C: Interpret and evaluate the influence of technology
throughout history, and predict its impact on the future.
 Indicator 2 (Grade 11): Understand the basic elements of
evolution of technological tools and systems throughout history.
Background Knowledge
 Previous exposure to vector addition (even if it is only a few homework problems)
 Knowledge of significant digits
 Exposure to right-triangle trigonometric rules
o Pythagorean theorem
o Angle equations (SOH-CAH-TOA)
 Ability to rearrange algebraic equations to solve for a given variable
Materials Required
 Vectors for Everyday Life Packet (1 per student)
o Step-by-step instructions
o Worksheets / data tables
 Total station or theodolite (1 total)
 100 ft. tape measure (4 total)
 Ruler or other straight-edge (1 per student)
 Data sheet (if real data cannot be measured or otherwise obtained)
 Grading rubric
Activities
1. Introduction to the activity (summary by the teacher)
2. Quick tutorial on how to use the total station (or theodolite)
3. Vectors for Everyday Life Packet
a. Students are taken outside to measure the required distances and angles
(as shown in the packet).
b. Students work in groups (3 to 4) to complete the lab. While some groups
are measuring, one group is working at the total station.
c. Once all the measurements are taken, students go back inside to analyze
the data and complete the calculations.
d. Students are given a grading rubric that explains the expectations of the
lab.
e. An activity feedback form is attached to the back of the packet (to be
completed at the end of the lesson).
Assessment of Student Learning
The lab packets will be graded and the results compiled. These grades will be
compared to the grades for the other vector addition lab.
Assessment of the Activity
A student feedback form is provided with the activity. Students complete the form at the
end of the lab and give it back to the teacher. The results are compiled and used to
determine the effectiveness of the activity. In addition, the teacher will be watching the
students while they complete the activity in order to gauge their interest.
Reflection
The following reflection refers to a lesson on vector addition taught in Sharon Bachman’s
11th and 12th grade Mechanical Physics classes on Monday, 11/22/04.
General (Setup)
I thought I was going to have a total station for use in this lesson. As it turned out, the
surveying class was using them, and the instructor keeps them in his car. Dr. Baseheart
helped me to get a theodolite to use instead. Amy Dimmerling graciously agreed to go
over to Hughes with me to help with the measurements/surveying. When we got there, it
was just starting to rain. Amy was able to set-up the theodolite, and we proceeded to
take all the necessary measurements (in the rain). My secondary location (the gym) was
in use, and they wouldn’t let me in there. I slashed open my thumb on the tape measure,
but it’s better now.
12th Grade Mechanical Physics (Block 2)
I explained to the seniors that we couldn’t go outside due to the rain. They seemed okay
with it. I then explained the lab, and I gave them the measurements they needed to do
the lab. One of the biggest problems was them not knowing which triangle to use for
which calculations (since I had them calculate the same distance twice – using different
triangles). They also wanted to use their normal right triangle rules to calculate distance
and angles. I explained to them that these equations only apply to right triangles and
that they only could use these with the first triangle (since the second triangle was
obtuse). They worked quietly and efficiently in their groups. Another big problem was
finding the direction of segment AC from the horizontal (segment AB). I’m not sure they
ever understood why they had to add the two angles in order to find this.
The bolt problem (second part of the lab) went okay, but the big problem was the
students not being able to move Q to the head of P. They wanted to overlay the vectors
instead of simply sliding Q along P. Also, finding angle B proved to give some (about
25% of the students) problems.
Most of the seniors were at least half-way through the bolt problem when they had to
leave. I assigned the bolt problem for homework (to be turned in on Tuesday, 11/23/04).
What I noticed the most was the seniors not being able to solve equations. For
example, given the equation
sin( A) sin( B)

a
b ,
they were unable to solve for sin( A) , much less actually solving for A . I went around to
all of the groups to explain how to do this, but it’s simply something they have done over
and over again but still don’t understand how to rearrange equations. I wonder if this
has something to do with their math curriculum.
11th Grade Mechanical Physics (Block 3)
Although it had stopped raining after lunch, it had turned very cold outside. On top of
that, there was a truck parked in the “No Parking” zone – right where I was going to take
the students to measure. To that end, I decided to keep the juniors inside as well. One
change I did make to their lab was to change angle A of the second triangle (Δ ACD) to
force more error in their second calculation of length AD. There is a question in the
packet that asks them to identify sources of error in their lab, and the measurements
given to the seniors didn’t yield any variance (since Amy and I measured very
accurately).
In order to let the students experience the theodolite (after all, I did carry it over there),
Amy set it up at the front of the classroom and measured the angle from one back corner
to the other.
31°
theodolite
While they were working on their lab in groups, I had the students come up two at a time
to measure the angle with the theodolite (Amy explained it to them and helped them
measure).
For the juniors, I went step-by-step through the directions. I would read the directions to
them, give them the measurement, and then tell them to “record, draw, and label.” They
were able to follow along, and it only took about 10 minutes of the class. After this I had
them get into groups of 3-4 and start working. They immediately just started working (on
their diagram sheet) and didn’t read the directions. This resulted in them doing the right
thing but in the wrong place. I stopped the class from working to tell them to make sure
to show their work in the appropriate place (just follow along with the directions). I told
them I’d be grading on how well the followed the directions (in addition to the correct
answers).
Since the juniors hadn’t had as much math and physics as the seniors, I gave them the
bolt problem as extra credit (with Sharon’s approval). My fear was that they wouldn’t
understand the Law of Cosines and the Law of Sines (both required for the bolt
problem). In the past, even when I give them the formula with the correct variables (all
they have to do is plug-and-chug), they still have problems. Much to my surprise, the
juniors were better at the calculations than the seniors. In fact, one student went so far
as to look-up the Laws in his SHAB. So much for following directions, but I was
impressed with the initiative.
All of them started on the bolt problem before they left. The problem the seniors had
(where to move Q along P) was very minimal with this class. Most of them completely
understood what they needed to do. I normally use the seniors as a pseudo “litmus test”
for the juniors (since I have the seniors first). In this instance, my theories about how the
juniors would do proved to be wrong.
Overall, I would say that the lesson went well. The students enjoyed working with the
theodolite, and they were very well-behaved during the lab. I’m disappointed that they
weren’t able to actually measure the distances and angles on their own, and I learned
that I need better contingency plans for future outside activities.
Things to Consider
What Worked Well
 Having the students work in groups was a necessity because they were going
outside to measure. However, having them work together in groups in an inclass setting proved to be effective. Those who understood were able (and
willing) to help those who were struggling.
 Including two different real-life applications of vector addition showed the
students how their classroom activities can be applied in the real world. Both
activities held the interest of the students.
 By giving the students problems requiring them to use equations they may not
have seen before (such as the Law of Cosines and Law of Sines), they had to do
a little bit of thinking past their normal classroom knowledge. This worked well
for both sets of students, but the juniors in particular did extremely well working
with concepts they hadn’t explicitly covered in class.
 Using the same packet but allowing different classes to have different objectives
allows both classes to learn the material but cuts-down on planning time (and
grading time).
What Needs Improvement
 I need to have better (more) contingency plans in case of inclement weather. I
thought a Plan B would be sufficient, but there needs to be a Plan C.
 These students refuse to read directions. If possible, I would like to better
integrate the calculations into the directions (but I’m not sure how).
 Adding another hands-on activity (using forces) would be helpful. This way the
students would get hands-on experience with vector addition using forces, and it
could be used in addition to or in place of the bolt problem.