## Summary

## Coin Type(s)

- Quarter

## Coin Program(s)

- America the Beautiful Quarters

## Objectives

- Students will solve problems involving the circumference of a circle.
- Students will solve problems involving scale drawings of geometric figures.
- Students will use research methods and create models to investigate practical problems and questions.

## Major Subject Area Connections

- Math
- Language Arts
- Science

## Grades

- 7th
- 8th

## Class Time

- Sessions: Four
- Session Length: 45-60 minutes
- Total Length: 151-500 minutes

## Groupings

- Whole group
- Pairs
- Individual work

## Background Knowledge

Students should have a basic knowledge of:

- Ratios
- Diameter
- Scale
- Circles
- Pi
- Formulas
- Hypothesis
- Proportions
- Fraction/decimal conversion

## Terms and Concepts

- Quarter
- Obverse (front)
- Reverse (back)
- Circumference
- Span
- Semicircle
- Arch

## Materials

- 1 overhead projector or equivalent technology (optional)
- 1 overhead transparency (or photocopy) of each of the following:
- "Arches National Park Quarter" page
- "Arches Formation Sequence" graphic organizer

- Copies of the following:
- "Arches Formation Sequence" graphic organizer
- "Arches Exit Slip"
- "Arches and Semicircles" worksheet (2 pages)
- "Arches and Scale" worksheet
- "Arches Competition Score" worksheet

- 1 class map of the United States
- Locate texts that contain information on arches and Arches National Park, such as:
*Arches National Park by Day and Night*by Grant Collier*The National Parks: America's Best Idea*by Dayton Duncan and Ken Burns*Arches and Canyonlands National Parks Pocket Guide*by Damian Fagan*Guide to National Parks of the United States*by National Geographic*Moon Spotlight Arches and Canyonlands National Parks: Including Moab*by W.C. McRae, Judy Jewell*Our National Parks*by David Mensch*Our National Parks*by Readers Digest*National Geographic Complete National Parks of the United States*by Mel White

- Locate texts that contain information on bridges and arch bridges, such as:
*How We Build Bridges*by Neil Ardley*Bridges: A History of the World's Most Famous and Important*by Judith Dupre*Encyclopedia of Bridges and Tunnels*by Stephen Johnson

- Chart paper, whiteboard or interactive whiteboard
- Computers with Internet access
- Assorted weights such as weights from a balance scale or weight set
- Pictures of arch bridges and arches from Arches National Park
- Calculators
- Rulers
- Graph paper
- Clay
- Clay modeling tools such as tooth picks, plastic spoons or ice pop sticks
- Index cards for numbering arches
- Parchment paper

## Preparations

- Make an overhead transparency or equivalent of each of the following:
- "Arches National Park Quarter" page
- "Arches Formation Sequence" graphic organizer

- Make copies of each of the following:
- "Arches Formation Sequence" graphic organizer (1 per student)
- "Arches Exit Slip" (1
*/*2 sheet per student) - "Arches and Scale" worksheet (1 per student)
- "Arches and Semicircles" worksheets (2 pages, 1 each per student)
- "Arches Competition Score" worksheet (1 per student)

- Locate texts that contain information on arches and the Arches National Park (see examples under "Materials").
- Locate texts that contain information on bridges and arch bridges (see examples under "Materials").
- Arrange to use the school computer lab for one or two sessions.
- Bookmark Internet sites that contain information about Arches National Park and arches, such as:
- Locate pictures of arches in Arches National Park and examples of arch bridges.
- Check the "Arches Exit Slip" after Session 1 for clarification.
- Grade the "Arches and Semicircles" worksheet after Session 2.

## Worksheets

Worksheets and files (PDF)

## Lesson Steps

### Session 1

- Display and examine the "Arches 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 in the design. This design depicts Delicate Arch, a 65-foot freestanding natural arch. It is a widely recognizable Utah landmark, depicted on postage stamps and license plates. The Olympic torch relay for the 2002 Winter Olympics passed through this arch. The Lasal Mountains are visible in the background.
- Have the students define the word "arch" (a curved formation with an opening formed by erosion and weathering). Have the students share what they know about Arches National Park. Take notes on chart paper if desired.
- Display or have the students search the Arches National Park Web site at www.nps.gov/arch/index.htm. View some of the pictures of arches found on the site or in selected texts. Ask the students if they know how the arches were formed. Record answers on chart paper.
- Distribute the "Arches Formation Sequence" graphic organizer to the students. Explain that they will be watching a short video on the formation of the arches and writing the steps in the formation of the arches on their "Arches Formation Sequence" graphic organizer. Review the definition of "sequence" and the sequence words on the graphic organizer.
- Play the video at www.nps.gov/arch/photosmultimedia/geologyvideo.htm for the students or have them watch the video on their own device. Play the video more than once if necessary. Allow time for the students to complete the "Arches Formation Sequence" graphic organizer. Answers may include (especially the last five steps):
- Techtonic plates move, raising the Rocky Mountains
- Faulting and subsidence create the Paradox Basin
- Salt water floods the basin repeatedly
- Evaporation leaves salt layer (over 5,000 feet thick)
- Rock from mountain erosion covers the salt
- Pressure forces the salt to flow westward
- Flow is blocked by faults, forming salt wall 2 miles high
- Loose rock covers wall 1 mile thick
- Techtonics force the deeper rock to bend and form a dome
- Joints (cracks) form in the bent rock
- Colorado rivers erode the loose rock covering
- Water seeps through cracks and dissolves some salt
- Rock cover collapses, forming Salt Valley
- Rock erodes around the valley's edge, forming fins
- Fins erode, leaving sand between fins
- Carbonic acid wears away the rock, causing openings in weaker rock
- Weaker rock falls away, enlarging the openings
- Arches continue to erode (and collapse).

- When the students have finished, complete the displayed "Arches Formation Sequence" graphic organizer.
- Ask the students where else they have seen arches. Record their responses on chart paper or the board. Guide the students if necessary to include arch bridges.
- Display some pictures from selected Web sites or texts of arch bridges. Emphasize the fact that arch bridges were one of the first bridge designs widely used. The Romans, for example, were well known for their use of arches in their bridge and building construction.
- Have the students complete the "Arches Exit Slip."

### Session 2

- Review the Web sites, exit slips and charts from the previous session.
- Distribute the "Arches and Semicircles" worksheets. Explain to the students that they will be treating all the arches as if they were semicircles rather than ellipses. Ask the students for the definition of "semicircle" (a half of a circle). Have the students record the definition and draw an example on page 1 of the "Arches and Semicircles" worksheet.
- Review the definitions of "diameter," "radius" and "pi" with the students (see the answer key). Have the students write the definitions and draw the example on page 1 of the "Arches and Semicircles" worksheet.
- Define "circumference" and give the formula for finding the circumference of a circle. Have the students write the definition and the formula on the worksheet.
- Using the park Web site or a selected text, display some pictures of Tower Arch.
- Read the sample problem. Have the students set up the equation and solve it.
- Assign the problems on page 2 of the "Arches and Semicircles" worksheet to the students. Allow time for the students to solve the problems. When they finish, collect the worksheets so you can grade them.

### Session 3

- Return the graded "Arches and Semicircles" worksheets to the students. Review the pictures, definitions and formulas from the previous sessions.
- Discuss the arches from Arches National Park. Explain to the students that they are going to construct arches from clay. Distribute the "Arches and Scale" worksheet and graph paper.
- Review scale with the students. Explain to the students that they will use scale from the size of actual arches for their construction. Complete the "Arches and Scale" worksheet sample problems with the students.
- Allow time to research the arches and calculate the scale for the clay arches.
- Allow the students time to create the arches from clay.

### Session 4

- Take pictures of the completed arches to display on a bulletin board or space in the classroom. Explain to the students that they will have a voluntary competition to determine whose arch can hold the most weight.
- Assign numbers and display the arches that will be part of the competition. Distribute the "Arches Competition Score" worksheet. Have each student write their hypothesis and share it with a partner. Have the students record the necessary information in the first 3 columns on the "Arches Competition Score" sheet.
- Use weights and determine which arch can hold the most weight. Have the students record the results on the "Arches Competition Score" sheet. Have the students complete the reflection on the score sheet.
- Display the winner of the competition. Have the students discuss their reflections and why the winning arch was able to hold more weight.
- Have the students discuss reasons why the National Park Service would not want people to climb on the arches

## Differentiated Learning Options

- Allow students to work in pairs or small groups on the building of the arches.
- Install the arches as a diorama with an illustrated background.
- Simplify the measurements in the problems.

## Enrichments/Extensions

- Have students discuss variables that can increase the load-bearing capacity of arches, such as different densities of clay, using an armature or arch thickness.
- Have students build other types of bridges from other materials and compare the strength of the different bridges.
- Have students create a multimedia presentation about the arches in Arches National Park or about arch bridges. Include the dimensions of the arch or the bridge.
- Have students work backwards by measuring the arch and its span and dividing the two to determine if pi is always used to find the answer.
- Have students set up ratios of distance under the arch and the span (diameter). Compare the different ratios to determine if they are equivalent.
- Have students calculate length of arches based on their actual elliptical shape.

## Assess

- Use the "Arches and Semicircles" worksheets to assess the students' ability to find the circumference of a circle.
- Use the "Arches Competition Score" sheet to assess the students' understanding of using the scientific process.
- Use the "Arches Exit Slip" to assess the students' understanding of the formation of the arches.

## Common Core Standards

**Discipline**: Math

**Domain**: 7.G Geometry

**Grade(s)**: Grade 7

**Cluster**: Draw, construct, and describe geometrical figures and describe the relationships between them

**Standards**:

**7.G.1.**Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.**7.G.2.**Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.**7.G.3.**Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

**Discipline**: Math

**Domain**: 7.G Geometry

**Grade(s)**: Grade 7

**Cluster**: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume

**Standards**:

**7.G.4.**Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.**7.G.5.**Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.**7.G.6.**Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms

## National Standards

**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**: Science

**Domain**: 5-8 Content Standards

**Cluster**: Physical Science

**Grade(s)**: Grades 6–8

**Standards**:

- Properties and changes of properties in matter
- Motions and forces
- Transfer of energy