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# Hiking Proportions: Shenandoah National Park

### Summary

Students will compute unit rates and solve word problems associated with ratios of fractions, including ratios of lengths. Students will analyze proportional relationships and use them to solve real-world mathematical problems. Students will develop and produce creative or informational media messages using technology as a tool to research, organize, evaluate and communicate information.

### Coin Type(s)

- Quarter

### Coin Program(s)

- America The Beautiful Quarters

### Objectives

- Students will compute unit rates and solve word problems associated with ratios of fractions, including ratios of lengths.
- Students will analyze proportional relationships and use them to solve real-world mathematical problems.
- Students will develop and produce creative or informational media messages using technology as a tool to research, organize, evaluate and communicate information.

### Major Subject Area Connections

- Language Arts
- Math
- Technology

### Grades

- Seventh grade
- Eighth grade

### Class Time

**Sessions**: Six

**Session Length**:
45-60 minutes

**Total Length**:
151-500 minutes

### Groupings

- Whole group
- Small groups
- Individual work

### Background Knowledge

Students should have a basic knowledge of:

- Ratios
- Setting up and solving proportions
- Multimedia presentation software
- Hiking
- Conversion between miles and kilometers
- Conversion between minutes and hours
- Conversion between fractions and decimals

### Terms and Concepts

- Quarter
- Obverse (front)
- Reverse (back)
- Circuit trail
- Round trip trail
- Pace
- Strenuous
- Ridge
- Panorama
- Terrain
- Avid

### Materials

- 1 overhead projector or equivalent technology (optional)
- 1 overhead transparency (or photocopy) of each of the following:
- "Shenandoah National Park Quarter" page
- "Shenandoah Trails Chart"

- Copies of the following:
- "Shenandoah Trails Chart"
- "Shenandoah Trails Multimedia Rubric"
- "Hiking Proportions" worksheet (2 pages)
- "Hiking Proportions Video/Presentation Rubric"

- 1 class map of the United States
- Texts that contain information on hiking and Shenandoah National Park, such as:
*75 Hikes in Virginia’s Shenandoah National Park*by Russ Manning*101 Essential Tips: Hiking*by Hugh McManners*Day and Overnight Hikes: Shenandoah National Park*by Johnny Molloy*The Hiking Companion*by Michael Robbins

- Chart paper, whiteboard or interactive whiteboard
- Computers with Internet access and presentation software

### Preparations

- Make an overhead transparency or equivalent of each of the following:
- "Shenandoah National Park Quarter" page
- "Shenandoah Trails Chart"

- Make copies of each of the following:
- "Shenandoah Trails Chart" (½ sheet per student)
- "Shenandoah Trails Multimedia Rubric" (½ sheet per student)
- "Hiking Proportions" worksheet (2 pages, 1 each per student)
- "Hiking Proportions Video/Presentation Rubric" (1 per student)

- Locate texts that contain information on hiking and Shenandoah National Park (see examples under "Materials").
- Arrange to use the school computer lab for one or two sessions.
- Bookmark Internet sites that contain information about Shenandoah National Park and hiking, such as:
- Prepare a graphic organizer in the format of your choice on chart paper.
- Find the hiking map for the Overall Run Falls Trail in the Mathew Arm area. Display the map or make a copy for each student.
- Prepare a multimedia presentation template on the Overall Run Falls Trail that includes a slide for each of the following:
- Introduction (area, name)
- Information (trail length, circuit length)
- Information (difficulty)
- Summary (other interesting information)
- Conclusion (why visit this site or trail)

- Grade the "Hiking Proportions" worksheets after Session 4.
- Find some examples of instructional videos for sessions 5 and 6.

### Worksheets and Files

### Sessions 1 and 2

- Display and examine the "Shenandoah National Park Quarter" page. Locate this site on a class map. Note its position in relation to your school’s location.
- As background information, explain to the students that the United States Mint began to issue the quarters in the America the Beautiful Quarters® Program in 2010. By the time the program ends in 2021, there will be a total of 56 quarter designs. Each design will focus on a different national site—one from each state, territory and the District of Columbia.
- Tell the students that the front of a coin is called the "obverse" and the back is called the "reverse." Have the students identify the images on the quarter’s reverse. Tell the students that the image depicts a day hiker taking in the view from Little Stony Man summit. Skyline Drive is visible in the distance.
- Have the students share what they know about Shenandoah National Park and hiking.
- Display or have the students search the Shenandoah National Park Web site at www.nps.gov/shen/planyourvisit/mapshiking.htm. Find the hiking map for Overall Run Falls Trail in the Mathew Arm area. Display the map or give a copy to each student. Explain that this trail is being used as an example for their own work later.
- Record facts about the Overall Run Falls Trail trail on a web, concept map or topic map on chart paper. Ask the students what they should consider and prepare for when planning to hike the trails in Shenandoah National Park. Some examples may include type of shoes, plant life to see and avoid, wildlife to look for and to avoid, water and food. Have them include the trail length, the distance and the difficulty of the hike.
- Have the students visit bookmarked Web sites or use selected texts to find more information to add to the web or graphic organizer.
- Have the students define the terms "ridge" (a long area of land on top of a mountain or hill), "panorama" (a full and wide view or something), "terrain" (the physical features of a tract of land) and "avid" (characterized by enthusiasm and vigorous pursuit) during their research and write the definitions on the graphic organizer, chart paper or note paper.
- Distribute a copy of the "Shenandoah Trails Chart" to the students. Discuss the difference between a round trip hike and a circuit hike. Explain to the students that a round trip hike goes to a certain place and returns along the same trail. In a circuit hike, hikers begin and end at the same location but do not retrace their steps along the trail. Allow students time to complete the blank columns on the "Shenandoah Trails Chart" using the park Web site or selected texts.
- Review the students’ research and be sure their figures match those on the "Shenandoah Trails Chart."
- Using the prepared presentation template, demonstrate to the students how to set up a multimedia presentation about the Overall Run Falls Trail trail to encourage park visitors to hike the trail. Include information from the chart generated in Step 6, focusing the presentation on the trail length, the distance and the difficulty of the hike.
- Have the students select a trail from the "Shenandoah Trails Chart" and research more information on the trail. Have the students work in groups of two or three to create some type of multimedia presentation on the trail, to include trail information (area, name, length, distance, difficulty), preparation, safety tips and persuasive reasons to hike the trail.
- Distribute the "Shenandoah Trails Multimedia Rubric" and review it with the students.
- Allow time for the students to research and put together their presentation.

### Session 3

- Have the students present their multimedia presentations to the class or set up a center in the room where students can watch the presentations.
- Have the students complete the rubric for their presentation.

### Session 4

- Review the process of setting up and solving proportions with the class. Distribute a copy of the "Hiking Proportions" worksheet to each student.
- Discuss the word "pace" with the students. Explain that it is a ratio of the distance hiked to the time it takes to hike.
- Read the sample problem with the students. Have the students set up the proportion and solve it. Review the conversion of fractions to decimals, decimals to fractions, minutes to hours and hours to minutes.
- Assign the problems on page 2 of the "Hiking Proportions" worksheet to the students. Have the students use the "Shenandoah Trails Chart" to complete the problems. Allow the students time to complete the worksheet.
- Collect the "Hiking Proportions" worksheet.

### Sessions 5 and 6

- Return and review the graded "Hiking Proportions" worksheet with the students.
- Tell the students that they will be working in groups of three and creating their own problem from the chart. They will create a one-minute instructional video, oral presentation or skit demonstrating the problem and how to solve it.
- Distribute a copy of the "Hiking Proportions Video/Presentation Rubric" to the students. Review the rubric. Review and discuss tips for creating a video/ presentation with the students. Show the students some examples of instructional videos.
- Allow students time to create their problem and presentation of their choice.
- Allow time for students to view the completed presentations.
- Have students complete and hand in the rubric.

### Differentiated Learning Options

- Allow students to work in pairs or small groups on the research and on solving the problems on the worksheets.
- Simplify the measurements in the problems.
- Allow additional time for students to complete the problems.

### Enrichments/Extensions

- Have the students interview someone who has hiked in a national park.
- Present the videos to the rest of the school in a gallery format.
- Have students use personal electronic devices for research.
- Have students complete a Web quest on hiking.
- Instead of having students make their presentations, have them let the class solve the problems before showing the instructional presentations.

- Use the worksheets and rubrics to evaluate whether the students have met the lesson objectives.
- Use the "Hiking Proportions" worksheet (both pages) to assess the students’ understanding of solving proportion problems.

**Discipline**: Math

**Domain**: 7.RP Ratios and Proportional Relationships

**Grade(s)**:
Grade 7

**Cluster**: Analyze proportional relationships and use them to solve real-world and mathematical problems

**Standards**:

**7.RP.1.**Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.- For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

**7.RP.2.**Recognize and represent proportional relationships between quantities.- Decide whether two quantities are in a proportional relationship, eg, by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
- Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships.
- Represent proportional relationships by equations.
- For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn

- Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

**7.RP.3.**Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

**Discipline**: Mathematics

**Domain**: 6-8 Geometry

**Cluster**: Analyze characteristics and properties of 2 and 3D geometric shapes and develop mathematical arguments about geometric relationships.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
- understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects; and
- create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

**Discipline**: Mathematics

**Domain**: 6-8 Algebra

**Cluster**: Use mathematical models to represent and understand quantitative relationships.

**Grade(s)**:
Grades 6–8

**Standards**:

In grades 6–8 all students should

- model and solve contextualized problems using various representations, such as graphs, tables, and equations.