Virtual Bridge Design: Distance and Midpoint Formula
Grade Level:
Duration:
9
3 Days
Subject:
Algebra 1 (Distance and Midpoint Formula)
Prepared By:
Nick Hanlon
Day 1
Materials Needed
Map of the Philippines with an XY-coordinate system overlaid on map.
Calculators
Analyze Learners
Overview & Purpose (STEMcinnati theme)
Education Standards Addressed
The purpose of the lesson is to introduce the concept of design in engineering
practice. Any engineer can create a strong bridge but engineers must adhere to
production design, costs and other constraints. Students are given a project in
which they will be competing among themselves for the best design. The bridges
the students can select from represent three of the Cincinnati bridges (Brent
Spence, Big Mac, and Purple People Bridge)
(Math) Patterns, Functions and Algebra Standard
13. Compute and interpret slope, midpoint and distance
given a set of ordered pairs.
(Science) Scientific Inquiry Standard
5. Develop oral and written presentations using clear
language, accurate data, appropriate graphs, tables, maps
and available technology.
The first day introduces the formulas for calculating the distance and midpoint
between two data points on an X-Y coordinate graph. The students are then given
(Technology) Optimization and Trade-offs Standard
the task of designing the bridges to connect the Philippines by minimizing cost or
distance.
1. Describe situations in which the selection of resources
involves trade-offs between competing values.
Overview:
A: Bridge building
C: Civil and Materials Engineering (see description of each field at the end of the
lesson plan)
S: Transportation
Select Goals and
Objectives
Teacher Guide
Student Guide
Assessment
Goals and
Objectives
(Specify
skills/information that
will be learned.)
Select
Instructional
Strategies –
Information
(Catch, give and/or
demonstrate
necessary
information,
misconceptions,
etc…)
Goals:
1. Students should understand how to find the distance
between two data points on an XY-coordinate graph.
2. Students should understand how to find the midpoint
between two date points on an XY-coordinate graph.
Objectives:
1. Students will be able to calculate the distance
between two data points on an XY-coordinate graph.
2. Students will be able to calculate the midpoint
between two data points on an XY-coordinate graph.
Catch (10 mins)
A video of bridge support beam under an increasing load until
the support beam factures is shown. The purpose of the
catch is to illustrate the need to design bridges to meet
specific guidelines for safety.
Formative:
Summative:
Students should have
minimum 2 bridge designs
completed. The must
show their design work as
a check to ensure the
students understand the
material and are
progressing forward.
Direct Lesson (20 mins)
Students should be knowledgeable of the Pythagoras
Theorem prior to this lesson.
Given two data points on an XY-coordinate graph, the formula
for the distance and the midpoint between the points and will
be shown by applying the Pythagoras Theorem.
See Appendix A for in class example for the distance and
midpoint formula.
Activity (30 mins)
The activity is introduced to the classroom. The activity will
take the remainder of the day1 and all of day 2.
Utilize Technology
Calculators
Other Resources
(e.g. Web, books, etc.)
Require
Learner
Participation
Activity
(Describe the
independent activity
to reinforce this
lesson)
Task: Design a way to connect strategic islands in the
Philippines
Teacher: Governmental agency accepting bids
Students: Independent civil engineer contractors making a
bid for the job.
Students (in teams of 2) are to design bridges based on the
map of the Philippines (an XY-coordinate grid is overlaid on
the map). The bid goes to one of two teams: first, to the team
that finds the shortest distance of the bridges to connect the
islands; second, to the team that has the lowest cost.
Students must select one of three bridges at each location
(Brent Spence bridge, Big Mac bridge, Purple People bridge).
Each bridge has an associate cost per distance. In addition,
each bridge has constraints regarding the distance it can
traverse. See Appendix B for specifics of activity.
Students will select data
points on the map and
calculate the distance and
midpoint of between the
islands. The students will
then select the appropriate
bridge that meets the
constraints.
By the end of the first day, students are to have at least two
bridges completed with the math work to support their
findings.
Evaluate
(Assessment)
(Steps to check for
student
understanding) – See
Objectives above
Pre-assessment
See Appendix C for pre- and post-assessment
N/A
Additional Notes
Day 2
Materials Needed
Map of the Philippines with an XY-coordinate system overlaid on map.
Calculators
Analyze Learners
Overview & Purpose (STEMcinnati theme)
Education Standards Addressed
The purpose of the lesson is to introduce the concept of design in engineering
practice. Any engineer can create a strong bridge but engineers must adhere to
production design, costs and other constraints. Students are given a project in
which they will be competing among themselves for the best design. The bridges
the students can select from represent three of the Cincinnati bridges (Brent
Spence, Big Mac, and Purple People Bridge)
(Math) Patterns, Functions and Algebra Standard
13. Compute and interpret slope, midpoint and distance
given a set of ordered pairs.
The second day is a continuation of a first day. Since this is a competition among
the groups, enough bridges must be built to see a difference in the results. Thus,
the second day allows the students to experiment with different routes or bridges
to fine-tune their project.
(Science) Scientific Inquiry Standard
5. Develop oral and written presentations using clear
language, accurate data, appropriate graphs, tables, maps
and available technology.
(Technology) Optimization and Trade-offs Standard
1. Describe situations in which the selection of resources
involves trade-offs between competing values.
Overview:
A: Bridge building
C: Civil and Materials Engineering (see description of each field at the end of the
lesson plan)
S: Transportation
Select Goals and
Objectives
Teacher Guide
Student Guide
Assessment
Goals and
Objectives
(Specify
skills/information that
will be learned.)
Select
Instructional
Strategies –
Information
(Catch, give and/or
demonstrate
necessary
information,
misconceptions,
etc…)
Goals:
1. Students should understand how to find the distance
between two data points on an XY-coordinate graph.
2. Students should understand how to find the midpoint
between two date points on an XY-coordinate graph.
Objectives:
1. Students will be able to calculate the distance
between two data points on an XY-coordinate graph.
2. Students will be able to calculate the midpoint
between two data points on an XY-coordinate graph.
Catch (5 mins)
Uncooked spaghetti is passed to each student. The students
will break the spaghetti by only holding onto the ends of the
spaghetti. This process is repeated with each broken half until
the student can no longer break the piece of spaghetti. This
idea demonstrates the need for supports in bridges as length
of the bridge increases.
Activity (55 mins)
Day 2 is a continuation of day 1. The students are to wrap-up
their bridge design by the end of the day.
Have the students give a progress update (gives an idea of
their progress and their understanding). Students must be
able to state
a) Reason for selecting a particular bridge location
b) How the length of the bridge was calculated
c) Are the constraints met for their bridge
Groups are randomly selected from a hat to bring them up for
the progress report.
Utilize Technology
Calculators
Formative:
Formative assessments
are built into the activity
section
Summative:
Students will attempt to break
the spaghetti by only holding
onto the ends. They will
continue until they can no
longer break the piece of
spaghetti.
Students will continue
working in their groups
designing the bridges that
meet the constraints of the
contract.
Other Resources
(e.g. Web, books, etc.)
Require
Continuation of activity from day 1.
Continuation of activity from
day 1.
During the progress updates of each group, students must
show their mathematical calculations of at least one bridge,
explain why they chose their specific design, and demonstrate
that the constraints have been met.
N/A
Learner
Participation
Activity
(Describe the
independent activity
to reinforce this
lesson)
Evaluate
(Assessment)
(Steps to check for
student
understanding) – See
Objectives above
Additional Notes
Day 3
Materials Needed
Map of the Philippines with an XY-coordinate system overlaid on map. Bridges should be laid
out on the map at this point.
Analyze Learners
Overview & Purpose (STEMcinnati theme)
Education Standards Addressed
The purpose of the lesson is to introduce the concept of design in engineering
practice. Any engineer can create a strong bridge but engineers must adhere to
production design, costs and other constraints. Students are given a project in
which they will be competing among themselves for the best design. The bridges
the students can select from represent three of the Cincinnati bridges (Brent
Spence, Big Mac, and Purple People Bridge)
(Math) Patterns, Functions and Algebra Standard
13. Compute and interpret slope, midpoint and distance
given a set of ordered pairs.
The third day is the presentation of the design project. The presentation has
stringent rules so the students stay on course but it also allows the student to
practice public speaking skills.
Overview:
(Science) Scientific Inquiry Standard
5. Develop oral and written presentations using clear
language, accurate data, appropriate graphs, tables, maps
and available technology.
(Technology) Optimization and Trade-offs Standard
1. Describe situations in which the selection of resources
involves trade-offs between competing values.
A: Bridge building
C: Civil and Materials Engineering (see description of each field at the end of the
lesson plan)
S: Transportation
Select Goals and
Objectives
Teacher Guide
Student Guide
Assessment
Goals and
Objectives
(Specify
skills/information that
will be learned.)
Select
Instructional
Strategies –
Information
(Catch, give and/or
demonstrate
necessary
information,
misconceptions,
etc…)
Goals:
1. Students should understand the relationship between
distance and cost.
Objectives:
1. Students will be able to justify their choices made in
the bridge design by presenting their work.
Catch (10 mins)
A mock presentation by the teacher is performed to show
what is expected for the presentation. This includes the
objectives that must be met and good presentation skills such
as speaking clearly, making eye contact with the audience,
etc.
Presentations (30-40 mins)
Students present their design projects. Each member of the
group must participate in the presentation. Students must
show:
 The layout of the bridges on their map
 The bridges meet the constraints of the project
 The total length of their bridge design
 The total cost of their bridge design
 Prove one case of a bridge. Provide distance
formula, cost calculation, and bridge selection.
Utilize Technology
None
Require
Students will be presenting their design projects that they
have been working on for the past couple days.
Learner
Participation
Students will present their
design project and ensure
that all aspects of the
presentation are met.
Other Resources
(e.g. Web, books, etc.)
Activity
(Describe the
independent activity
to reinforce this
lesson)
Evaluate
(Assessment)
(Steps to check for
student
understanding) – See
Objectives above
Post-Assessment – See Appendix C
N/A
Additional Notes
Important Attachments:
1. Pre-Post Assessment
2. Worksheets
3. PowerPoint
4. Reflection after lesson
Summary of Engineering Fields
Aerospace engineers design, develop, and test aircraft, spacecraft, and missiles and supervise the manufacture of these products. Those
who work with aircraft are called aeronautical engineers, and those working specifically with spacecraft are astronautical engineers. Aerospace
engineers develop new technologies for use in aviation, defense systems, and space exploration, often specializing in areas such as structural
design, guidance, navigation and control, instrumentation and communication, or production methods. They also may specialize in a
particular type of aerospace product, such as commercial aircraft, military fighter jets, helicopters, spacecraft, or missiles and rockets, and
may become experts in aerodynamics, thermodynamics, celestial mechanics, propulsion, acoustics, or guidance and control systems.
Biomedical engineers develop devices and procedures that solve medical and health-related problems by combining their knowledge of
biology and medicine with engineering principles and practices. Many do research, along with life scientists, chemists, and medical scientists,
to develop and evaluate systems and products such as artificial organs, prostheses (artificial devices that replace missing body parts),
instrumentation, medical information systems, and health management and care delivery systems. Biomedical engineers may also design
devices used in various medical procedures, imaging systems such as magnetic resonance imaging (MRI), and devices for automating insulin
injections or controlling body functions. Most engineers in this specialty need a sound background in another engineering specialty, such as
mechanical or electronics engineering, in addition to specialized biomedical training. Some specialties within biomedical engineering include
biomaterials, biomechanics, medical imaging, rehabilitation engineering, and orthopedic engineering.
Chemical engineers apply the principles of chemistry to solve problems involving the production or use of chemicals and biochemicals. They
design equipment and processes for large-scale chemical manufacturing, plan and test methods of manufacturing products and treating
byproducts, and supervise production. Chemical engineers also work in a variety of manufacturing industries other than chemical
manufacturing, such as those producing energy, electronics, food, clothing, and paper. They also work in health care, biotechnology, and
business services. Chemical engineers apply principles of physics, mathematics, and mechanical and electrical engineering, as well as
chemistry. Some may specialize in a particular chemical process, such as oxidation or polymerization. Others specialize in a particular field,
such as nanomaterials, or in the development of specific products. They must be aware of all aspects of chemicals manufacturing and how the
manufacturing process affects the environment and the safety of workers and consumers.
Civil engineers design and supervise the construction of roads, buildings, airports, tunnels, dams, bridges, and water supply and sewage
systems. They must consider many factors in the design process, from the construction costs and expected lifetime of a project to
government regulations and potential environmental hazards such as earthquakes and hurricanes. Civil engineering, considered one of the
oldest engineering disciplines, encompasses many specialties. The major ones are structural, water resources, construction, environmental,
transportation, and geotechnical engineering. Many civil engineers hold supervisory or administrative positions, from supervisor of a
construction site to city engineer. Others may work in design, construction, research, and teaching.
Computer hardware engineers research, design, develop, test, and oversee the manufacture and installation of computer hardware.
Hardware includes computer chips, circuit boards, computer systems, and related equipment such as keyboards, modems, and printers.
(Computer software engineers—often simply called computer engineers—design and develop the software systems that control computers.
These workers are covered elsewhere in the Handbook.) The work of computer hardware engineers is very similar to that of electronics
engineers in that they may design and test circuits and other electronic components, but computer hardware engineers do that work only as it
relates to computers and computer-related equipment. The rapid advances in computer technology are largely a result of the research,
development, and design efforts of these engineers.
Electrical engineers design, develop, test, and supervise the manufacture of electrical equipment. Some of this equipment includes electric
motors; machinery controls, lighting, and wiring in buildings; automobiles; aircraft; radar and navigation systems; and power generation,
control, and transmission devices used by electric utilities. Although the terms electrical and electronics engineering often are used
interchangeably in academia and industry, electrical engineers have traditionally focused on the generation and supply of power, whereas
electronics engineers have worked on applications of electricity to control systems or signal processing. Electrical engineers specialize in areas
such as power systems engineering or electrical equipment manufacturing.
Environmental engineers develop solutions to environmental problems using the principles of biology and chemistry. They are involved in
water and air pollution control, recycling, waste disposal, and public health issues. Environmental engineers conduct hazardous-waste
management studies in which they evaluate the significance of the hazard, advise on treatment and containment, and develop regulations to
prevent mishaps. They design municipal water supply and industrial wastewater treatment systems. They conduct research on the
environmental impact of proposed construction projects, analyze scientific data, and perform quality-control checks. Environmental engineers
are concerned with local and worldwide environmental issues. They study and attempt to minimize the effects of acid rain, global warming,
automobile emissions, and ozone depletion. They may also be involved in the protection of wildlife. Many environmental engineers work as
consultants, helping their clients to comply with regulations, to prevent environmental damage, and to clean up hazardous sites.
Materials engineers are involved in the development, processing, and testing of the materials used to create a range of products, from
computer chips and aircraft wings to golf clubs and snow skis. They work with metals, ceramics, plastics, semiconductors, and composites to
create new materials that meet certain mechanical, electrical, and chemical requirements. They also are involved in selecting materials for
new applications. Materials engineers have developed the ability to create and then study materials at an atomic level, using advanced
processes to replicate the characteristics of materials and their components with computers. Most materials engineers specialize in a
particular material. For example, metallurgical engineers specialize in metals such as steel, and ceramic engineers develop ceramic materials
and the processes for making them into useful products such as glassware or fiber optic communication lines.
Mechanical engineers research, design, develop, manufacture, and test tools, engines, machines, and other mechanical devices. Mechanical
engineering is one of the broadest engineering disciplines. Engineers in this discipline work on power-producing machines such as electric
generators, internal combustion engines, and steam and gas turbines. They also work on power-using machines such as refrigeration and airconditioning equipment, machine tools, material handling systems, elevators and escalators, industrial production equipment, and robots used
in manufacturing. Mechanical engineers also design tools that other engineers need for their work. In addition, mechanical engineers work in
manufacturing or agriculture production, maintenance, or technical sales; many become administrators or managers.
Table 2: Earnings distribution by engineering specialty, May 2006
Lowest
10%
Specialty
Lowest
25%
Median
Highest
25%
Highest
10%
Aerospace engineers
59,610
71,360
87,610
106,450
124,550
Biomedical engineers
44,930
56,420
73,930
93,420
116,330
Chemical engineers
50,060
62,410
78,860
98,100
118,670
Civil engineers
44,810
54,520
68,600
86,260
104,420
Computer hardware
engineers
53,910
69,500
88,470
111,030
135,260
Electrical engineers
49,120
60,640
75,930
94,050
115,240
Environmental engineers
43,180
54,150
69,940
88,480
106,230
Materials engineers
46,120
57,850
73,990
92,210
112,140
Mechanical engineers
45,170
55,420
69,850
87,550
104,900
Table 3: Average starting salary by engineering specialty and degree , 2007
Curriculum
Bachelor's
Aerospace/aeronautical/astronautical
Master's
Ph.D.
$53,408
$62,459
Bioengineering and biomedical
51,356
59,240
$73,814
Chemical
59,361
68,561
73,667
Civil
48,509
48,280
62,275
Computer
56,201
60,000
92,500
Electrical/electronics and communications
55,292
66,309
75,982
Environmental/environmental health
47,960
Materials
56,233
Mechanical
54,128
62,798
72,763
Footnotes:
(NOTE) Source: National Association of Colleges and Employers
Bureau of Labor Statistics, U.S. Department of Labor, Occupational Outlook Handbook, 2008-09 Edition, Engineers, on the Internet
at http://www.bls.gov/oco/ocos027.htm(visited November 20, 2009).
Reflection
Overall, the lesson was successful. A large majority of the students were highly involved in the competition, trying to outwit their competitors with
a better score. The most successful part of the lesson was the activity itself, since it was designed as a competition with a reward at the end.
I would reinforce to the students that they do not have to have three separate routes to each desired island. The point of the activity is for an
individual to reach each of the three islands from the initial start island. Thus, if one bridge has already been designed, the student may use that
bridge again for another route.
The activity was designed such that a midpoint was needed whenever a Brent Spence bridge was used. However, I found at that the bridge
design could be completed without using a Brent Spence bridge. Therefore, I would suggest a few ways to overcome this issue. Shorten the
constraints of the bridges thus requiring a Brent Spence bridge to be used. Or require that a midpoint support beam be required for all the bridges
(or for the Big Mac and Brent Spence only). From my observations, the students performed the distance equation multiple times and became
quite efficient at its calculation. However, they still lacked a full understanding of the midpoint due to the setup of the activity. This was evident is
the results of the post-assessment.
I was very pleased with the presentations of the bridge designs by the students. The combination of a strict guidelines and a mock presentation
helped the students better understand what was required. However, some groups still struggled with covering all aspects that were requested.
These were typically the students who did not always pay attention to directions. I do not see a reason to make any changes to this section of the
lesson but feel free to make any modifications that you may feel would help the presentations.
The incentive I used to entice the students to strive for the best design was a free lunch at Frisch’s (which was located on the campus of the
school). Although this may not be possible, feel free to use any incentive or none at all. The results from my lesson showed that usually the group
that won one of the categories (shortest path or lowest cost), won at both categories.
Appendix A
Class Example
Given two points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ), what is the distance and midpoint between those two points?
(x2,y2)
(x1,y1)
First, create a right triangle
(x2,y2)
d
b
a
(x1,y1)
(x2,y1)
a
Pythagoras Theorem tells us that
𝑑 2 = 𝑎2 + 𝑏 2
Solve for d:
𝑑 = √𝑎 2 + 𝑏 2
We find that
𝑎 = (𝑥2 − 𝑥1 )
𝑏 = (𝑦2 − 𝑦1 )
Therefore, we can write the Pythagoras Theorem as:
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
The midpoint between the two points is similar to finding the average of two numbers
The midpoint of ‘a’ and ‘b’ will tell us the midpoint:
𝑎=
(𝑥2 +𝑥1 )
2
and 𝑏 =
(𝑦2 +𝑦1 )
Therefore, the midpoint is:
𝑥1 + 𝑥2 𝑦1 + 𝑦2
(
,
)
2
2
2
Example: Find the distance and midpoint of the two data points (1,1) and (4,5)
Distance
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑑 = √(4 − 1)2 + (5 − 1)2
𝑑 = √(3)2 + (4)2
𝑑 = √9 + 16
𝑑 = √25
𝑑=5
Midpoint
1+4 5+1
(
,
)
2
2
5 6
( , )
2 2
5
( , 3)
2
Appendix B
Activity Directions
Philippines Bridge Design Contract
Funded by the W&H Agency
The Philippines is the 12th most populous country in the world
with an estimated 92 million people, covering about 116,000
square miles. The Philippines has been an ally of the United
States since World War II. Recently, a tsunami caused vast
destruction and the country is relying on the US to help rebuild
its bridges between the islands.
The W&H Agency is accepting bids to complete the bridge
design project.
Your goal is to meet one of the following two objectives:
1. Design bridges such that the total bridge length of your
project is less than any other group.
2. Design bridges such that the total bridge cost of your project
is less than any other group.
You may choose from 3 different bridges:
1. Brent Spence Bridge
a. Cost ......................................................... $1000 per 1 km
b. Minimum bridge distance ........................ 12 km
c. Maximum bridge distance ....................... N/A
d. Must contain a midpoint support beam
2. Big Mac Bridge
a. Cost ......................................................... $500 per 1 km
b. Minimum bridge distance ........................ 3 km
c. Maximum bridge distance ....................... 20 km
3. Purple People Bridge
a. Cost ......................................................... $300 per 1 km
b. Minimum bridge distance ........................ N/A
c. Maximum bridge distance ....................... 5 km
Constraints




All bridges must start from island H.
Your objective is to build bridges starting from island H and reach the designated islands as indicated on the map
(islands E, G, and C).
You may design your bridge to have a direct route OR you may choose to design bridges that go through other
islands. In some cases, it will be less distance and cost to design a bridge to go through other islands to reach
your goal.
All bridges must start and end at an intersection of XY coordinates. Thus, the XY coordinates must clearly be on
the island, otherwise, the residents would have no way of reaching the bridge if it starts in the water. If an
intersection is not clear whether it’s on the island, ask Mr. Ward or Mr. Hanlon.
Judging




Each group must present their bid to the W&H Agency and the rest of the class on the last day.
Each group is responsible to give a 2-3 minute presentation of their bid.
The presentation must include:
o The layout of the bridges on the map
o The total length of their bridges
o The total cost of their bridges
o One distance and one midpoint calculation
Each member of the group must speak during the presentation.
What to submit



Along with the presentation, each group must submit their bridge design with their map.
Each bridge on the map must be labeled 1 through n bridges
On a separate sheet of paper, show the calculation of each bridge labeled 1 through n. This includes
o The two data points,
o Distance of bridge,
o Bridge selection,
o Cost of bridge,
o Midpoint (if required).

The winning group of each category (minimum bridge length and minimum bridge cost) will receive a free lunch
from Frisch’s.
If a group is able to win both categories, the group will receive a free lunch and desert from Frisch’s.
Winning Groups

Brent Spence Bridge
Big Mac Bridge
Purple People Bridge
Appendix C
Pre- and Post-Assessment
Virtual Bridge Design: Distance and Midpoint Formulas
Pre-Assessment
1. Solve for ‘d’ using Pythagoras Theorem
d
3
2
2. Find the distance between -3 and 6
-3
0
3. Find the distance between -2 and 10
6
2
0
-8
4. Find the average of 4, 9, 2, and 5
Virtual Bridge Design: Distance and Midpoint Formulas
Pre-Assessment Key
1. Solve for ‘d’ using Pythagoras Theorem
d
𝑑 2 = 𝑎2 + 𝑏 2
𝑑 = √𝑎2 + 𝑏 2
𝑑 = √22 + 32
𝑑 = √4 + 9
𝑑 = √13
3
2
2. Find the distance between -3 and 6
-3
0
6
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 6 − (−3)
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 9
3. Find the distance between -2 and 10
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 2 − (−8)
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 10
2
0
-8
4. Find the average of 4, 9, 2, and 5
𝑎𝑣𝑔 =
4+9+2+5
4
𝑎𝑣𝑔 =
20
4
𝑎𝑣𝑔 = 5
Virtual Bridge Design: Distance and Midpoint Formulas
Post-Assessment
Given the two data points (1,2) and (4,3):
1. Find the distance between the two data points.
2. Find the midpoint between the two data points.
Virtual Bridge Design: Distance and Midpoint Formulas
Post-Assessment Key
Given the two data points (1,2) and (4,3):
1. Find the distance between the two data points.
𝑑 2 = 𝑎2 + b2
𝑑 = √𝑎 2 + b 2
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
𝑑 = √(4 − 1)2 + (3 − 2)2
𝑑 = √ 32 + 22
𝑑 = √9 + 1
𝑑 = √10
2. Find the midpoint between the two data points.
(
(𝑥1 + 𝑥2 ) (𝑦1 + 𝑦2 )
,
)
2
2
(1 + 4) (2 + 3)
(
,
)
2
2
5 5
( , )
2 2
Appendix D
Philippines Map with XY Grid Overlay
The map should be printed from a large format printer with 36”x50” dimensions.
DO NOT use the image below for printing due to the low resolution quality, it is only displayed below
for a visual of the file. Find attached with the lesson plan files a high resolution image titled
‘Philippines Map High Res.pdf’ for printing.