7. You must also keep track of the directions from point to point so that another military troop can go back to those locations and cover up any evidence.

8. At the last clue, you will be able to find the orders which tells you the President’s location!

Each group will create clues using transformations from each of the categories below. The answer to the clue will lead them to the next coordinate on the grid.

- Translation (one that changes left/right and also up/down)
- Reflection (one that reflects over the x axis AND the y-axis)
- Rotation (one that rotates at least twice)
- Dilation (include your scale factor)
- Translation and Dilation
- Reflection and Rotation
- Translation and Rotation
- A series of transformations that will get you back to a previous waypoint and then translates from that location by 5 units).

Make sure that you specify the ending location and reveal where the President is hidden. You can even do that with a latitude/longitude location!

__Constraints:__

- You can go no further than the football field in the back
- You can go no further than the first tree in the front of the school

__Materials Given:__

- A map of the school’s front and back yard
- A compass and satellite map (use an app from a smartphone)
- Tools for transformation – miras and patty paper
- Paper to write down your clues

__Obstacles (Redesign):__

- Option 1: Reporters and paparazzi have taken over (Insert an area from the coordinate grid) and you must redesign your map avoiding all of these areas.
- Option 2 – Just let the students test out their waypoints/map and make adjustments depending on their satellite map or how easy/hard these locations are to get to.

**The Hook**

- Option 1 – Students will participate in a Geocache hunt at the park next door or around the school. Students will participate in a Geocache hunt on school grounds. The hunt will teach students three overarching goals: (1) learn how to use GPS receivers/Cell phone apps, (2) learn to geocache, gather real-world data, and problem-solve authentic and personally meaningful challenges and (3) explore ideas in the fields of math and science.
- Option 2 – Show students how triangulation works by using people in the classroom (someone can be lost and someone can play the part of a mountain and someone can play the part of a giant tree. Have a GPS on hand to show the students how a satellite can find exactly where you are located at this very second.
- Here is a lesson that also explains triangulation

Lesson: Where Am I: Navigation and Satellites

**Teacher's Guiding Questions**

- Do you know how a GPS system works?
- Why are satellites involved with the locating of specific locations?
- Do you know why the latitude and longitude system was created?
- Would you know how to get yourself home if you were lost using a map or other means?
- Is it necessary to use team work to solve the problem? If so, would it be helpful to give each team member their own job?
- How can a compass be useful in navigations?
- How does a GPS system relate to Mathematics and the coordinate grid?
- What changes do line segments in the coordinate plane undergo that have been rotated, reflected, or translated?
- How do similar figures compare?
- What transformations can figures undergo that enable them to remain congruent to the original image?
- What are the differences between rigid and non-rigid transformations?
- How can we prove something is congruent to another figure by a series of translations, rotations, and reflections?
- How can we prove two objects are similar by a series of translations, rotations, reflections, and dilations?

**ACS (Real world applications; career connections; societal impact)**

__Application:__ A GPS system has many applications. It was developed by the military and used to navigate unknown regions, but has progressed so it can be used individually by drivers, hikers, hunters, and boaters. The system is also utilized for a variety of engineering uses including road construction, structural analyzing, earth moving, fleet management projects, and agricultural needs.

__Career:__ A variety of careers incorporate the use of GPS including engineers such Civil, Environmental, Aerospace, Earth Science, Agriculture, Meteorology, Cartography, Oceanographers, and Marketing.

__Societal Impact:__ The advances in technology have made GPS easily accessible to anyone with internet access, a smart phone, or a GPS device and it is free. Therefore, the impact on our society has been huge because it can be used for anything from recreation to very elaborate military operations. It is used to make our environment a safer and easier place to live. We can track and locate people if needed such as missing children or even criminals. We can quickly and easily access maps and directions. It is used by the military, agriculture, and industry to find destinations, locate people and buildings, and to predict the weather.

##### Where the CBL and EDP appear in the Unit

__Day 1:__

Part 1:

1. Reserve a computer lab or obtain ipads for the day.

2. Students will have already been given an overview of what a gps system is and how the latitude/longitude system works (Introduction Powerpoint).

3. Students will be instructed to go to the website http://itouchmap.com/latlong.html

4. Students should type in the school’s address and an aerial view of the school will pop up.

5. Students should record the latitude and longitude of the school.

6. Give students about 10 minutes to play around with the zoom function tool and to explore the school grounds.

7. Students will then be instructed to find the latitude/longitude of the following locations:

a. The football field

b. The baseball field

c. The elementary school

d. The high school

e. The tennis court

f. Their choice of 2 places

8. Students will then be instructed to use the “area” tool that will help them calculate the area of selected locations. Students should click on “Land Plat Size”

9. By dropping the pins, students should be able to draw lines around the selected land plots. Students will be instructed to find the area of: (in square feet.)

a. The football field

b. The tennis court

c. The baseball field

d. The school itself

e. The whole campus

10. It should be mentioned the importance of being able to find the measurements of land plot sizes. Surveying Engineers use this tool to find the distances and areas of certain locations to plan, design, and build upcoming projects.

__Part 2: Planning a Route__

1. Students will then be given this scenario: A new student has just moved in to the city of Goshen. She has never been here before. Using the following website: Planning Map - Click Here!

Pin 5 locations that is a must see attraction that will be helpful for this student to get around town for the essentials. Make Goshen Middle School the starting location. Some examples might include:

a. The grocery store

b. The library

c. A restaurant etc.

2. Use the site and “drop pin” in the places of interest. You should also find the latitude and longitude of these locations.

3. Students should “screen shot” the map at the end of the lesson and turn it in. A summary paragraph describing the locations will also be required.

__Part 3: Google Earth- __

1. Students will be instructed to open up the app “Google Earth”. There are more functions available if a desktop version is used over an ipad or iphone app.

2. Explain to students that the point of Google Earth is to let the user explore places in the entire world by using technology. Places that a person may never get to see in real life! The “part 2” activity can be used in collaboration with this activity as long as the computer desktop version is available. If the teacher decides to use Google Earth for the part two lesson, the instructions should be followed the same (the data sheet will be fine to use as well) except, the “pin” that will be dropped to mark the location can be seen in the Google Earth toolbar and it looks like a thumb tack.

3. Students will be given a couple different locations to view on Google Earth (ex: their house, downtown Cincinnati, The Grand Canyon, etc.), they will be instructed to write down the latitude and longitude of these points.

4. To change it up a bit, students will then be given the latitude/longitude of a few different places (ex: The Great Wall of China, The Eiffel Tower, The Pyramids of Giza) and they will have to name the point of interest.

5. As an extension, students can explore Google Earth and find the coordinates of 5 points of interest and give those coordinates to another student to navigate to.

6. At the end of the activity, the teacher might want to show other tools that students might find interesting, such as the

a. planet tool – will show what it looks like on the Moon and Mars

b. The clock tool – will show what a location looks like throughout time

c. Ruler tool – will allow students to measure from one point to another.

__Day 2:__

1. Continue viewing the Geocaching Powerpoint to familiarize students with what Geocaching is and how it is used in the real world (a real life treasure hunt for grown-ups using latitude/longitude coordinate points!) If there is a nearby park next to your school, take the students geocaching! My students will be rewarded with a Geocache at the end of the unit!

2. The teacher needs to download the app “X Note”. This app will allow teachers to drop clues/caches at any destination on the school yard. Prior to this lesson, the teacher will need to find out how many students have a smartphone and divide groups based on that (see materials).

3. Teacher will need to have the geocache links already made into QR codes (or use the students email address to send to them). Students will be put into groups of 3 or 4. The group members will need to decide among these jobs:

a. Navigator (holds the GPS unit and voices direction)

b. Scribe – writes down necessary information

c. Reader – will read any of the clues

d. Illustrator or photographer – will take pictures of the journey or draw out anything needed

4. Students will receive the clues to the “virtual” cache by the teacher handing them a QR code. They will use a QR code reader to see the message waiting for them. This message will lead them to the “virtual hidden message”.

5. Basically, the teacher will view a map of the school through the app. The teacher will “drop virtual caches” by touching the screen of the phone on the point on the map in which the students will only be able to open the message once they physically follow the map and are standing at the point. Example: Teacher touches the map where the concessions stand is on the football field. The students will follow a satellite map to get to that location. Once they arrive, a hidden message will pop up. The messages can be anything related to the unit of study. For this particular lesson, I will be using the virtual geocache as a way to take notes. At each of the locations, I will have a fact about transformations that they will have to take notes on. I will be introducing transformations tomorrow in class. So this will serve as a sneaky way to take notes and a way to generate interest!

6. When arriving at a location, students should use the compass app to determine the latitude and longitude of the “way point” (the point at which they are standing).

7. See the data sheet for the clues, QR codes (but they will only work for my school yard), and notes topic.

8. At the end of the lesson, students will have had the opportunity to use a navigational tool, waypoints which are latitude and longitude, maps, a compass, take their “transformation notes” and be prepared for the next day.

9. In addition to this, at the end of the activity, the students will be given an exit ticket to research or write about the type of engineering that deals with navigation.

__Day 3:__

1. Part 1: Introduction/Review to transformations warm-up lesson. The focus of this lesson will just be to refresh their memories on transformations because they have been taught this concept already. See the worksheet (4.2.1c). Also, distribute patty paper and miras to let students use as a tool.

2. Quick Notes – see the Powerpoint for a few notes of the symbols and vocabulary Transformation Powerpoint

3. Part 2: Students will begin the geocache. There will be 4 Geocache locations to set up. The teacher should put the clues in a manila envelope and put it outside the classroom in which they will travel to. Each location will have directions of what transformation they will have to perform to move the 1^{st} pre-image. Once the transformations have been performed, the triangle (pre-image) will land on a room number in the school. This is the location the students must travel to get their next clue.

4. Students should work in groups of 4. Because there will be too many students traveling to one location at a time, the teacher should request that only one student from the group travel to the hidden cache. Since there are 4 locations, they can take turns hunting for the cache!

5. At the end of the activity, students will have solved 8 different transformation problems. The teacher should give the students some sort of prize at the end of the geocache hunt such as erasers or a certificate!

6. While students are performing the problems, they should write down the coordinate points on the data sheet. At the end of the activity, the class discussion will be examining or reviewing? if there is any kind of pattern that they notice with the coordinate points themselves. This will lead into transformation notation and is added in the Powerpoint (students will add into their notes)

__Day 4-5:__

1. Warm-up – give students two congruent triangles to hold in their hands. Tell them to hold one straight up and down and to hold the other upside down. Ask them to explain how they know the two triangles are congruent.

2. Go to the website www.geogebra.org

3. Give students a few minutes to explore with the functions of Geogebra and what they can do with it before you give them a tutorial.

4. Give the students a tutorial of the web applet. Really focus on showing them the transformation buttons and symbols. Especially how rotations rotate around a point, so they will have to create a point for the object to rotate around. Don’t just pick the origin, show them some other points to rotate around as well.

5. Intro: show them the following situation: http://www.geogebratube.org/student/m135736

This is to get them thinking about how you can PROVE something is congruent. You can have them draw the triangles at their locations and then have them perform transformations that will allow them to get triangle EFG on top of triangle ABC – which will prove congruence. Students will look at you like you are crazy when you ask them – how do you know they are congruent?” They will probably say something like “well look at them, they are the same”. In mathematics, we must never “assume” something is exactly alike just by looking at it… we must perform a series of rotations, reflections, and translations to prove that they are congruent.

6. Have students write down the steps that they took to prove the congruence. Answer is:

a. Rotate triangle EFG 180° counter-clockwise

b. Reflect EFG over the y-axis

c. Translate EFG 4 units up

7. Step 2 – show the students the following example: http://nlvm.usu.edu/en/nav/frames_asid_295_g_3_t_3.html?open.instructions

8. Then have them prove similarity by creating this situation by a series of rotations, translations, reflections, and dilations. http://www.geogebratube.org/student/m135993 answers:

a. Reflect over the x-axis

b. Rotate 90°

c. Reflect over x-axis

d. Dilate by 2

9. Step 3:

After the introduction part is over, give the students a choice between two activities:

__Activity 1: (see student handout)__

Design a poster using a polygon of your choice that has a variety of transformations (with Geogebra)

- There must be a total of 10 images, labeled A-J in the key below.
- The original image (A) must appear in the upper left corner, and all other transformations will be made relative to that original.
- Each type of transformation should occur at least once: clockwise rotation, counter-clockwise rotation, horizontal reflection, vertical reflection, translation (left, right, up, down), a dilation that enlarges, and a dilation that reduces the original image.

__Activity 2: (see student handout)__

Design a new layout to our school ground with a variety of transformations.

- There must be a total of 10 images, labeled A-J in the key below.
- Each type of transformation should occur at least once: clockwise rotation, counter-clockwise rotation, horizontal reflection, vertical reflection, translation (left, right, up, down), a dilation that enlarges, and a dilation that reduces the original image.
- This would be the template to start with: http://www.geogebratube.org/student/m136026

10. Circulate to monitor students’ discussions. Encourage students to use the correct vocabulary and terminology when describing the transformations. (Look for evidence of MP3.) What changed from one floor plan to the other? What stayed the same? (Look for evidence of MP3.)

11. When finished you will give your directions to another classmate to see if they can produce the same picture as you (so your directions should be very specific including which point your rotated around and which line you reflected over)

12. Step 4: Students will be given homework that will focus on proving similarity and congruence.

__Day 6-7:__

1. Present to students the big idea and reflect from the “hook” in the earlier part of this unit. (start the CBL process – use powerpoint to guide: CBL Guided Powerpoint

2. Generate ideas for the essential question (CBL Guided Notes

3. Students should work in their groups and come up with 5 things/questions that they would like to know more about dealing with the big idea and the transformations.

4. Students should agree upon 1 idea for their group and submit it to the google doc.

5. List the ideas on a google doc or google form and share with the class.

6. Teacher should use these ideas as a guideline and design the challenge accordingly.

7. Present the challenge with the criteria and constraints. (you may want to do this the next day so that you can incorporate some of their ideas into the challenge).

__Day 8-10:__

1. Design, build, and explore your troops map to find the president!

##### Misconceptions

Students can perform transformations but they often have trouble explaining/writing how perform the transformation. They also do not know how to use transformations to prove similarity and congruence.

- Geogebra web applet
- Google Earth (web application)
- GPS receiver or a smartphone app that will show a map
- Latitude/longitude finder (use a smartphone app)
- Compass (use a smartphone app)
- X Note App – (one subscription is $2.99 on the apple store and only the teacher needs it)
- Become a member to http://www.geocaching.com/ and you can take your students to the park if you have one nearby. This will allow you to “find” hidden geocaches and track “travel bugs” that your class has left. Then you as a class can follow this travel bug for years to come!

##### Pre-Unit Assessment Instrument

Pre-Test

##### Post-Unit Assessment Instrument

Post-Test

**How to Make This a Hierarchical Unit**

This unit would be very easy to adapt to high school Geometry. Transformations continue at the high school level. I would keep all of the same concepts and activities, as a teacher I would just make sure that the content was a little more rigorous and fit the level of instruction at the high school level. ** **

##### Reflection

In the future when I do this project again, I would make sure that every student had a GPS receiver. I had them use their smart phones, which was fine, but some of them ran out of power before the activity was over. Or I would warn them to make sure that their phones were fully charged. I found out that you can check out GPS receivers from the Clermont County Educational Services Center, so this would be very helpful in the activity.

I would also modify the number of “waypoints” that each group had to create. I would require only 5 points to decode and create instead of 8. This was definitely lengthier than anticipated.

I also might change the scenario from being the president is visiting – to a scenario that might be a little more realistic such as a military troop is hiding clues for another military troop. Or just use a geocaching scenario. Using a geocache scenario would allow the theme to be more relevant. Another scenario that would work would be to use the map as a “missing persons” or “criminal hiding” and the student would have to use the clues to find and pinpoint an appropriate search area.

At the end of the unit, I took my students next door to the park. There were 3 real “geocache” boxes that we discovered using a GPS unit and the app “Geocaching 101.” This is a registered site. We were able to put something called a “travel bug”at each location and we are watching as a class to see where these “travel bugs” travel to all over the state (you can watch them travel as people track them and upload their story on the internet).

Lastly, if I were to use the Geogebra program again to teach them transformations, then I would make sure I used a computer lab. Using the ipads was a little confusing and students weren’t able to completely manipulate the shapes on the software.

I think this unit took so long because the students were being taught for the first time “Transformations” and I was teaching it as they were learning it. This project may go faster as a culminating project.

Overall, the students really enjoyed this project. They learned a lot of ways that transformations and geometry could be connected to the real world. Students increased their scores from the pre-test to the post-test (see pre and post-test grades below).