Hands-on Activity: Topographic Maps and Ratios: A Study of Denali

Contributed by: University of Wyoming

The image shows a satellite image overlaid by a topographic map of Denali, the tallest peak in Alaska. Glaciers, snow, and rock cover the majority of the mountain, while topographic (or contour) lines show elevation. The area is characterized by steep areas and large basins formed by glaciers.
Topographic maps offer users many uses, and engineers must know how to navigate them. This image was taken from Google Earth and shows a topographic map with satellite imagery of Denali, a peak in Alaska.
copyright
Copyright © 2017 Jake Schell, University of Wyoming

Summary

In this activity, students overlay USGS topographic maps into Google Earth’s satellite imagery. By analyzing Denali, a mountain in Alaska, students discover how to use map scales as ratios to navigate a map, and use rates to make sense of contour lines and elevation changes in an integrated GIS software program. Students also problem solve to find potential pathways up a mountain by calculating gradients.
This engineering curriculum meets Next Generation Science Standards (NGSS).

Engineering Connection

Engineers use ratios, rates, and GIS with great frequency and must be able to work with them in various forms. Often, engineers combine physical documents with technology to make the data easier to understand and manipulate. In addition, technology is helping engineers increase efficiency and problem solve while simultaneously globalizing businesses and services. The concept of slope or gradient is ubiquitous in mathematics and applies to all fields in STEM. For example, when designing a road in mountainous terrain civil engineers must analyze slope to make the road passable while considering costs while environmental engineers simultaneously assess the slope and terrain type above and below the road for safety purposes.

Pre-Req Knowledge

Students should have knowledge of ratios, rates, and proportional relationships prior to this activity. Also, students need to be able to convert from fractions to percentages.

Learning Objectives

After this activity, students should be able to:

  • Compute unit rates of contour lines
  • Use rates to understand the growth of mountains
  • Determine whether contour lines are proportional
  • Calculate a grade (gradient/slope)

More Curriculum Like This

Topo Map Mania!

Students learn to identify the common features of a map. Through the associated activities, students learn how to use a compass to find bearing to an object on a map and in the classroom.

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What Is GIS?

In this lesson, students learn the value of maps, how to use maps, and the basic components of a GIS. They are also introduced to numerous GIS applications.

Middle School Lesson
Navigational Techniques by Land, Sea, Air and Space

Students learn that navigational techniques change when people travel to different places — land, sea, air and space. For example, an explorer traveling by land uses different navigation methods and tools than a sailor or an astronaut.

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Students learn about projections and coordinates in the geographic sciences that help us to better understand the nature of the Earth and how to describe location.

Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

  • Construct an explanation based on evidence for how geoscience processes have changed Earth's surface at varying time and spatial scales. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use appropriate tools strategically. (Grades K - 12) Details... View more aligned curriculum... Do you agree with this alignment?
  • Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve real-world and mathematical problems involving the four operations with rational numbers. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Decide whether two quantities are in a proportional relationship, e.g., 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. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Analyze proportional relationships and use them to solve real-world and mathematical problems. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • New products and systems can be developed to solve problems or to help do things that could not be done without the help of technology. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • Technological systems often interact with one another. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • A product, system, or environment developed for one setting may be applied to another setting. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use appropriate tools strategically. (Grades K - 12) Details... View more aligned curriculum... Do you agree with this alignment?
  • Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. (Grade 6) Details... View more aligned curriculum... Do you agree with this alignment?
  • Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve real-world and mathematical problems involving the four operations with rational numbers. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Analyze proportional relationships and use them to solve real-world and mathematical problems. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Decide whether two quantities are in a proportional relationship, e.g., 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. (Grade 7) Details... View more aligned curriculum... Do you agree with this alignment?
  • Construct an explanation based on evidence for how geoscience processes have changed Earth's surface at varying time and spatial scales. (Grades 6 - 8) Details... View more aligned curriculum... Do you agree with this alignment?
Suggest an alignment not listed above

Materials List

Each group needs:

Introduction/Motivation

Consider what it takes to construct an aerial tram up the side of a mountain. What does it take to plan, design, integrate, and execute such a monumental feat of engineering? Construction of any infrastructure requires the collaboration of a variety of engineers, especially if, for example, they are working on something like an aerial tramway in a steep, rocky area like Jackson Hole, Wyoming. [Show video on the construction of the Jackson Hole Aerial Tram to engage students.]

The image shows an aerial tram as it ascends above snow a snow covered, rocky mountain. Far below in the distance the docking building can be seen.
All over the world, aerial tramways are used for people to travel up steep peaks. What would you need to know if you were an engineer tasked with building one to take you to the top of Denali?
copyright
Copyright © 2015 Akiry, CC BY-SA 4.0, Wikimedia Commons, https://commons.wikimedia.org/wiki/File:Les_Arcs_-_telepherique_aiguille_rouge.jpg

The image shows a winding road ascending through a valley and up a mountain. The road is made of several switchbacks which climb on top of each other, with large mountains in the background.
Steep passes are an impressive engineering feat. What kinds of problems would an engineer have to solve to build a pass like this one, from southern France?
copyright
Copyright © 2012 xuuxuu, Public Domain, Pixabay. https://pixabay.com/p-69363/?no_redirect

In order to think like an engineer and to wrap our minds around some of the phenomenal civil engineering that happens every day, we are going to use Google Earth along with some digital maps from the United States Geological Survey to think about steepness and size of features on earth such as mountains. Think about our aerial tram example, and then imagine yourself driving over a mountain or a mountain pass. What kinds of engineering had to go into the design of that road? How would people know where to build or where the easiest path lies? When you plan a trip and use a map to navigate, how do you know how far away two points on the map are? Have you ever seen two maps of the same area and compared the detail that goes into each map?

Part 2:

In this image, the lines crossing the landscape are called “contour lines” and show surfaces of the earth that all lie at the same elevation, or distance above sea level. Lines above a given contour represent higher elevations and lines below represent lower elevations. Based on this concept, would you expect that steeper parts of the mountain would have lines spaced closer together or further apart? Why?

[Show image of Google Earth with a topographic map overlay, either from this document (Image 1) on your electronic device]

[Have students discuss the question in groups and share with the class. Students should determine that steeper parts of the mountain have contour lines spaced closer together. Next, review what students will be doing in the activity by going over the Student Activity Sheet.]

In this activity, you will be taking on the role of an engineer who needs to build a tram up Denali, the highest peak in North America at 6190 m (20,310 ft) above sea level. What would your main engineering challenges look like? How would you start problem solving for solutions? What kind of information would you need to know about an area to build a tramway there?

As the engineer, you will be analyzing the steepness of Denali using satellite imagery and contour maps. You will need to use ratio and proportional reasoning skills to calculate slope (steepness) and analyze elevation. As you work through the activity, you should be thinking about everything you do from the lens of an engineer. You will need to use your calculations throughout the activity to suggest a location for the tram based on the topography of the mountain. To summarize the activity:

  • First, you will need to familiarize yourself with Google Earth and set up the maps. An engineer will have more sophisticated software to work with, but we can do a lot with free programs!
  • Next, we will calculate and analyze how scale affects what we see on the map. As you work through the activity, consider how scale would be important to an engineer.
  • Next, you will calculate gradients (slope) and determine the unit rate of each contour line interval.
  • You will then use your newfound knowledge to analyze the ascent of two climbing groups, taking into account their locations and the topography they need to traverse to get to the top. Think about how this relates to your mission as an engineer designing the tramway.
  • Next, you will use unit rates to think about how the mountain grows over time. This is the work that a geologist might undertake, but would be important information to an engineer, as well.
  • Finally, you will draw the path that you think a tram should take up the mountain, and explain why you chose the location you did. You will need to print the screen with the map showing your tram’s route. You may draw the route by hand or use the path tool in Google Earth.

Vocabulary/Definitions

contour lines: Lines on a topography map that designate areas that have the same elevation, or vertical distance above sea level. Different maps have different unit rates per contour line, such as 200 m between each contour line.

geographic information system: A way to store, edit, and manipulate data or information, usually using maps; commonly referred to as GIS.

grade: The equivalent percent of a given slope.

rate: Ratio that compares numbers with different units, such as miles per hour.

ratio: Comparison of two or more numbers, commonly written as a fraction. Ratios can be written in three ways: a to b, a/b, or a:b.

scale: Ratio (typically shown as 1:x) describing a proportional relationship between the dimensions of an actual object and some model representation of the object.

slope: Value of the ratio when comparing the vertical change to the horizontal change of two quantities. Slope coincides with the steepness of the mountain between any two points on the map, and can be shown as a fraction or decimal equivalent.

topography map: Sometimes referred to as a “topo” map; shows different elevations, or relief, by way of contour lines or other means. These maps may also show significant geographic points such as roads, peaks, and rivers.

Procedure

Background

Review the following concepts with students. Depending on where your class is relative to these terms, you may want to spend more or less time here. You will want your students to have their handouts and display Google Earth to the class for reference.

Ratio:  A comparison of two of more numbers. Can be shown as a:b, a to b, or a/b. The value of a ratio is the number we get when we divide a by b.

Proportion/Proportional Relationship: A relationship or equation showing that two or more ratios are equivalent, or have the same value. All of the following ratios are equivalent (equal to 0.5 as a decimal), so the relationship between them is proportional. Example: 1/2 = 2/4 = 5/10.

Scale: A scale is simply a ratio (typically shown as 1:x) describing a proportional relationship between the dimensions of an actual object and some model representation of the object. Typically, we write the ratio as model:actual. On a map of the United States, you may see a scale such as 1 in: 300 mi. This means that each inch on the map represents 300 miles on the surface of the earth.

Refer to Part 2 of Student Activity Sheet called “Determining Scale,” for information about scale as it pertains to Google Earth. Google Earth, like all maps, uses scale to represent distances in real life as distances on the map. Different maps have different scales. Google Earth gives its scale in the form of a bar on the lower left of the screen. (You can display this bar in Google Earth settings.)  The scale changes when you move the screen around, or when you zoom. You can write scale as a ratio comparing the distance on the map to the distance represented in real life.

Note that the scale does not have units because the units cancel out when written as a fraction. This scale works because it uses the same units on the map as in real life. If the scale was 1:4, then one inch on the map would equal four inches in real life and one centimeter on the map would equal four centimeters in real life. For this activity, you will not include units in your scale.

Rate: A rate is a ratio comparing quantities with different units. We often use the word “per” instead of “to” when describing rates. For example, “there is one inch per 300 miles on this map” or “there are 300 miles per inch.”

Contour Lines: Refer to Part 3 of Student Activity Sheet, “Contour Lines and Gradient (Slope).” Topography maps allow users to determine elevation on the map. Elevation means the vertical distance a point is above sea level, and is represented by the small numbers on the lines in the map. These lines are called contour lines. Using unit rates allows users to know the elevation change between each line. Unit rates are determined by division, with the small numbers representing elevation in feet above sea level.

Slope: A slope is the value of the ratio of the vertical change to the horizontal change of two quantities. Slope coincides with the steepness of the mountain between any two points on the map, and can be shown as a fraction or decimal equivalent. A slope of 4/5 means that for every 4 meters of elevation you go up the mountain, you are moving horizontally 5 meters.

Grade (or Gradient): This is the equivalent percent of a given slope. If the slope between two points on the mountain is 4/5, the decimal equivalent is 0.8. To convert between a decimal and a percent, you multiply by 100, so the gradient is 80%. The steepest mountain passes for vehicles are typically around 10%.

Before the Activity

You need to familiarize yourself with the Google Earth GIS platform and experience the USGS topographic map enhancement feature with the KMZ file. You can display data in Google Earth in metric units from the view panel in the top of the screen. Click “Google Earth” (labeled as “Google Earth Pro” in the Figure 1), click preferences, and choose Meters, Kilometers under “Units of Measurement.” Secure electronic devices with internet connections and make copies of the Student Activity Sheet, one per student.

Determine whether students will need to download Google Earth onto the devices they will be using. Your admin or tech department can determine if the devices already have the software or if students will be able to download the software if they need (many downloads are password protected).

This is a screenshot of the preferences tab in Google Earth, showing how to change the units of measurement in the program.
Figure 1. How to change the units of measurement to metric in Google Earth.
copyright
Copyright © 2017 Jake Schell, University of Wyoming

With the Students

  • Break students into groups of two or three (two is preferable but the number of devices available to you may dictate your group size.) Hand out the Student Activity Sheet to each student and a ruler for each group.
  • All students should download the KMZ file NGS Topo 2d from http://services.arcgisonline.com/arcgis/rest/services/NGS_Topo_US_2D/MapServer
  • Once students have opened the site, click Google Earth under the heading “View In” at the very top of the page. A KMZ file should begin to download.
  • Once the KMZ file is downloaded, it should load in Google Earth. If not, look for a pane on the left-hand side of the program called “Places.” This allows you to toggle the new file on and off with a checkmark. When your file downloads, it may be located under the tab “Temporary Places.” See Figure 2.
  • Students should set Google Earth to display information in metric units (kilometers) and show the scale bar. To set this preference, click on the Google Earth heading (shown as Google Earth Pro in Figure 1) and then click on Preferences. From within the preferences dialogue box, students can set the units to kilometers. To show the scale bar, click on “View” the top pane and make sure there is a checkmark by “Scale Legend.”
  • Familiarize students with maneuvering through Google Earth by discussing the first section of the Student Activity Sheet as a class and work through the rest of the activity as a class.

This is an image of Denali and its surrounding area with USGS topographic map overlays. Most of the area is colored in white and brown indicating snow and steep terrain. The image highlights the pane viewers see in Google Earth showing the places, layers, and search box with standard navigation tools across the top such as File, Edit, View, etc.
Figure 2. Make sure the checkmark next to the file NGS_Topo_US_2D is marked for the file to load. It may be located under Temporary Places on your device.
copyright
Copyright © 2017 Jake Schell, University of Wyoming

Attachments

Troubleshooting Tips

  • If the USGS topographic map overlay file is not displaying in Google Earth, try double clicking on it in the “Places” pane on the left-hand side of the screen. This will then zoom out and load the file. From here, you may need to locate Alaska and slowly zoom in on it and locate Denali manually instead of zooming in on it from the search tool. As you slowly zoom in, stop frequently to let the topo map overlay load before zooming in further. 
  • If the map overlay is not displaying the “Places” pane, you can manually click and drag the file from the save location into Google Earth. Release the mouse over the map portion of the program.

Assessment

Pre-Activity Assessment

Discussion Questions: Take note of responses to the discussion questions in the introduction. This could provide data on grouping.

Activity Embedded Assessment

Activity Sheet: Have students complete the Student Activity Sheet. You can grade one per group or each student’s work. 

Post-Activity Assessment

Reflection Questions: Ask students the following questions:

  1. What did you learn in the activity about engineering as it relates to using topographic maps and contour lines?
  2. In what situations would an engineer building a tram want to use a large-scale map (more detail) and in what situations would they want to use a smaller scale (see less detail)?
  3. What, if anything, is different about the engineer’s route up the mountain from those of the climbers? How does this relate to slope?
  4. What factors might the engineer consider, besides slope, that would influence the placement of the tram?

Activity Extensions

Emphasize the connections of this activity to engineering, a variety of extensions are possible. Consider the following ideas:

Different groups could be assigned to take on the role of different types of engineers to put forth more effort into the design of a tram (or road) in a rocky, mountainous area. Instead of just using this idea as an opening or short write up as seen in part 5 of the activity sheet, consider spending a few days doing more research on real world engineering feats and have students problem solve. They could create a model of a mountain pass or tramway with different groups researching the different engineering aspects of the project.

Activity Scaling

  • For lower grades, modify the ratios to involve fewer conversions of units.
  • For higher grades, focus more on slope. You could potentially have students graph a cross section of various portions of the mountain to get an idea of the different slopes of the peak, perhaps following popular climbing routes.

Contributors

Jake Schell; Andrea Burrows

Copyright

© 2018 by Regents of the University of Colorado; original © 2017 University of Wyoming

Supporting Program

University of Wyoming

Acknowledgements

This digital library curriculum was developed under the guidance of Andrea Burrows, Linda Hutchinson, and Michele Chamberlin at the University of Wyoming.

Last modified: October 6, 2018

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