SCALE ME DOWN

Master Teacher: Karen Mapes

Overview

Scale drawings are a real-world application of classroom learning on ratio and proportion. Students can see the utility of the mathematics they are learning. These lessons are designed to be taught after students have done work with ratios, proportions and cross products. Through the activities presented in this series of lessons, students make progressively more complicated scale drawings and, finally, a scale model.

Learning Objectives

Students will be able to:

• Demonstrate understanding of ratio and proportion.
• Use a scaling factor to scale down a physical object into a scale drawing - plan view and elevation view.
• Analyze the concept of scaling by comparing the scale drawing to the physical object.
• Apply the usefulness of scaled drawings to real situations.

Oregon Standards Available at:

http://www.ode.state.or.us/cifs

Mathematics - Algebraic Relationships

• Recognize, create, describe and analyze patterns and sequences (arithmetic and geometric).

Mathematics - Statistics and Probability

• Design and carry out simulations to compare experimental and theoretical probability and to make predictions.

National Standards from the National Council of Teachers of Mathematics

(http://www.nctm.org/)

The National Council of Teachers of Mathematics has recommended the following standards for students in grades 5-8:

• The mathematics curriculum should include the continued development of number and number relationships so that students can understand and apply ratios, proportions and percents in a wide variety of situations.
• The mathematics curriculum should develop the concepts underlying computation and estimation in various contexts so that students can:
  • Develop, analyze and explain methods for solving proportions.
  • Use computation, estimation and proportions to solve problems.

Media Components

Video

Check the link at http://www.opb.org/edmedia/trs/ to find access to the video(s) from unitedstreaming™ referenced in this lesson plan.

• "Mathematical Eye: Ratio and Scale" (20:08)
  o Clip: "Defining Scale" (04:05)

Web

• Balloonist Tom Deering's Web Site
  A balloonist, Tom Deering, has drawn several objects in his life superimposed on his balloon but to the same scale - a semi truck, his apartment and himself.
  http://www.deering.org/balloon/
• NOVA: Pyramids
  This is a link to the NOVA site on pyramids. This site provides information and math problems for students to solve related to the size of the Great Pyramid at Giza. The main Web site for the show has other features such as an interactive tour of the passages and information on excavations at the Pyramids.
  http://www.pbs.org/wgbh/nova/pyramid/geometry/../index.html

Materials

• Computer hooked up to TV or projector with speakers working
• Brown construction paper
• Graph paper: 1/4-inch grid, 1-cm grid

Prep for Teachers

When using media, provide students with a Focus for Media Interaction, a specific task to complete and/or information to identify during or after viewing of video segments, Web sites or other multimedia elements.

Download the video clip listed above from OPB's TRS Video On Demand. You can obtain a free copy of the Windows Media Player on the Web that will make viewing the downloaded clip much more convenient and versatile. QuickTime Player can also be downloaded and used for viewing on most of the TRS Video On Demand clips. Cue the video clip to 01:24 and pause.

Bookmark the Web sites you will be using. Explore the NOVA: Pyramids Web site. Become familiar with the links and which ones you want to use.

Find a photo of yourself or another object available in the classroom to use in the Introductory Activity.

Introductory Activity

Talking About Scale

Show students a photo of you or a photo of an object that you have in the classroom. Ask them to discuss with a neighbor what is the same and different about the two things (same thing, different sizes). Ask some pairs to share with all. Ask them to estimate how much bigger or smaller the object or person is compared to the photo. Pairs share. Explain that the comparison they are making about size is called "scale."

Provide students with a Focus for Media Interaction by asking them how much bigger they think a hot air balloon is compared to them. Show students the drawings of objects compared to a hot air balloon on the Web site, http://www.deering.org/balloon/. Ask them to try to figure out how much bigger the objects are compared to one another and to an adult person 6 feet tall. Ask students to write it down in terms of scale. (The drawing of the balloon is about 8 times as tall as the drawing of Tom, so the scale would be 8 to 1; or from Tom's point of view, he is 1/8 the size of the balloon.)

Explain that when they do this, students are looking at scaled drawings and trying to figure out a scaling factor (8 times or 1/8 times). The scaling factor is the amount by which the drawing has been enlarged or reduced compared to the actual object.

Learning Activities

Drawing a Table

Step 1: Students work in groups to draw a plan view and elevation view of a desk or table in the classroom. Students use a given scaling factor, measure the table's dimensions, create a list of scaled parts and draw the table as seen from the top (plan view) and the side (elevation view). You will begin this part of the lesson with a video clip, "Defining Scale" (04:05), from the video, "Mathematical Eye: Ratio and Scale" (20:08).

Tell the students that they will be working in groups to make a scale drawing of a table in the room. They will be drawing two views of the table that are proportional to the real table but smaller. They will be drawing a plan view, as if they were looking down at the table from the ceiling, and an elevation view, as if they were sitting on the floor looking directly at the side of the table. They will be scaling the table down to 1/12th of its original size, so that 1 inch in the drawing equals 12 inches (1 foot) of real table. The ratio 1 to 12, written as a fraction, 1/12, gives the scaling factor of the drawing.

Explain that first they will watch a short piece of a video of some students in England working with a scaled drawing. Provide students with a Focus for Media Interaction by asking them to pay close attention and try to figure out the difference between what these students are doing and what their groups will be doing. (English students are enlarging the drawing to make the table; your students are reducing the dimensions of the table to make the drawing.)

Start the video clip at approximately 01:47, right after the studio segment when the narrator says, "… So they have a better chance of making a good program." Show the clip of the students looking at the drawing and then holding the ruler up to the screen (approximately 01:57). Pause and ask for answers to the focus question. Point out that the elevation view was called a front view in the video.

(Older students can also determine the scaling factor for the enlarged camera lens projected on the screen. The student measuring it has it at 8 cm wide. Measure it across your screen and set up a proportion that will come up with the scaling factor for the enlargement. You can also talk, at this point, about how the English students are using the metric system and your students will be using the customary system of measurement.)

Step 2: Let the groups work. Give them 1/4-inch-grid graph paper. Depending on the grade level, have them measure the table's dimensions to the nearest 1/2 inch, 1/4 inch or 1/8 inch. Information should be organized in a table that includes the actual dimensions and the scaled dimensions of various table parts. For instance:

Make sure students know that every part must be scaled down by the same factor. If they are using a scale (ratio) of 1 inch to 12 inches, then all measurements must be divided by 12 (or multiplied by 1/12) so each part is 12 times smaller than the original. This will create proportional sizes.

Step 3: The proportions can be checked by creating a length-to-width ratio for both the actual measurements and the scaled measurements and making sure they are equal. For instance, if the actual table top is 48 feet by 24 feet then its length-to-width ratio is 48 in/24 in. In the drawing that should translate to 4 in/2 in, since they are both divided by 12. Since 48/24 = 4/2, the sizes are proportional.

Ask students to finish the activity by calculating some proportions from their measurement charts. You might consider assigning each group member a different proportion comparison. Ask them to write a summary of how they know their drawing was proportional by using their proportion calculation as evidence.

A Scale Model of a TV Studio

Review what we did yesterday by asking: "What do you remember about how to make a scale drawing?"

Ask: "What other uses can you think of for scale drawings in the real world?" Make a list on the overhead, chalkboard or chart paper of ideas the students have. (These could include architectural plans, maps, manufacturing ideas, building, etc.)

Step 4: You are going to start this part of the lesson with another video clip from the very beginning of "Defining Scale" (04:05), from the video, "Mathematical Eye: Ratio and Scale" (20:08). Provide students with a Focus for Media Interaction by asking them if they can figure out the scale being used in this video by the TV studio designers. Play the video from the beginning until " … The distance along the side of the studio is measured in feet" (approximately 00:38). Pause and let students absorb that information. Ask them what they noticed (each square equals 1 foot - 1/4 inch equals 1 foot. This would make a scale factor of 1/48 of the real size).

Step 5: Before you continue playing the video, provide students with a Focus for Media Interaction by asking them, "How wide do you think a real studio camera is?"

Play the video and pause after the narrator asks that question (approximately 00:59) and ask for student input. If you have a clear pause function, they should be able to figure out the size of a real camera from the scale from what they calculated before (1/4 inch = 1 foot; studio camera would be about 41/2 feet).

After this in the video, there is a nice segment where the scale model turns into a real studio. Provide students with a Focus for Media Interaction by asking them to notice this change. Pause at " … better chance of making a good program" (approximately 01:12). Rewind it and play again as needed so students can discuss the ways the scale model is true to the original and ways in which it differs from the set.

The Great Pyramid

Step 6: Collecting Data

Tell students that, now that they have seen how useful scale drawings can be and how drawings can be turned into scale models, they are going to make a scale drawing and model of the Great Pyramid in Egypt.

This activity uses the Web site, http://www.pbs.org/wgbh/nova/pyramid/geometry/../index.html.

Provide students with a Focus for Media Interaction by telling them they need to collect data about the Great Pyramid so that they will be able to make a scale drawing. You will be slowly moving through the links provided and they will need to write down the measurements they find for various parts of the Pyramid.

Access the Web site and focus on the picture of the Pyramid. The data students will need to find as you follow the links are:

Length of one side of the base: Click on the word "base" on the pyramid picture. Each side of the Great Pyramid measures 230 m (756 ft). A measurement for the area of the base (the footprint of the pyramid) is given in hectares and compared to the size of a football field. A multiple-choice question helps students understand the scale and involves the concept of perimeter and multiplying 230 m x 4.

Height: Click on "height" at the main pyramid site and students will find that the Great Pyramid is 137 m (449.5 ft) tall. It used to be taller and that information is also included at this point. A graphic shows the height compared to other famous structures.

These are the two measurements students will need to make a scale drawing of the Great Pyramid with a plan view and an elevation view.

Extensions can be made by exploring the other key words on the picture of the pyramid at the main site. Students can practice angle measurements, talk about the weight and size of each block, and take a virtual tour of a passageway. It's all good stuff and just depends on the time you have and direction you want to take the lesson.

Step 7: Making a Scale Drawing

You need to choose (or have students choose) an appropriate scale for the drawing. Calculations for different scales follow - you may want to have students, or groups of students, do these calculations and choose their own scale.

The pyramid site uses 30 m = 1 cm as a scale, creating a drawing that is 3,000 times smaller. Using that scale, the sides of the base are 7.7 cm and the height is 4.6 cm. (On the paper pattern provided by the Web site, a different height is given - 6.2 cm - but doing calculations of 137/30 will give you 4.6 cm, so that is the height I suggest using.)

A scale that is larger would also fit an 8.5" x 11" sheet of paper: 10 m = 1 cm is a bit too large.
20 m = 1 cm gives a base side of 11.5 cm and a height of 6.9 cm.

Once the scale is chosen, the drawings are made on 1-cm-grid graph paper. The plan view will show the area of the base. The elevation view will show the height and shape of the side. The point of the top of the pyramid should be directly over the middle of the base in the elevation view. This should result in an angle of 51 degrees for the slope of the sides.

Culminating Activity

Step 8: The Scale Model

Students will use their scale drawings to build a scale model of the Great Pyramid. Four sides and a base can be cut out of brown construction paper and glued/taped together. If students include tabs along the sides of their cutouts, those can help with the assembly since the tabs can be folded and pasted inside (see diagram).

More capable students could model the pyramid in clay. Less capable students could be provided with the outline available through a link on the Web site.

Cross-Curricular Extensions

History

• Explore the Great Pyramid Web site further to learn more about ancient Egypt and its culture and excavations of the Pyramids.
• Watch "This Old Pyramid," a video that talks about how the pyramids were built and tries to reconstruct some of the ancient methods.
• Work with hieroglyphics. Some related resources on the Internet are located at:
  http://www.newton.mec.edu/oakhill/sixtwo/egypt_lesson.html
  
• Explore Egypt lessons on the Web. Some related resources on the Internet are:
  http://www.mrdowling.com/604print.html
  http://members.aol.com/donnandlee/ - EGYPT
  http://members.aol.com/Donnclass/Egyptlife.html
  http://www.newton.mec.edu/oakhill/sixtwo/egypt_lesson.html

Language Arts

• Read books about Egypt. Some related resources on the Internet that might help you find titles are:

http://www.libsci.sc.edu/miller/Egypt.htm

http://infolynx.ci.tucson.az.us:90/kids/10,173,181/search/degypt-ivilization/degypt+civilization/1,9,74,E/2browse

http://ed-web3.educ.msu.edu/literacy/newsltr2.htm

http://www.ptahhotep.com/categories.html

Math

• Diane Coates, a former high school and middle school math teacher, has written a lesson in which students choose a picture or cartoon and increase its size by using a scaled-up grid.
  http://www.frontiernet.net/~dcoates/scale.htm
• A Web site created by Net Resources International for the aerospace technology industry shows a scale drawing of a helicopter complete with measurements.
  http://www.aerospace-technology.com/projects/model333/model3332.html
• Artificial Reefs of the Keys is a nonprofit group in Key West, Florida working to bring the de-commissioned ex-USAFS Gen. Hoyt S. Vandenberg to their area to become an artificial reef. At this Web site, they provide a scale drawing of the Vandenberg - a very large ship - compared to drawings of the Statue of Liberty, football fields and a person.
  http://www.bigshipwrecks.com/../../images/mediakit/vandenberg_liberty.htm

Art

• A site with links to all kinds of modeling sites; many paper modeling sites with free patterns can be found. Search over 5,729 hobby-related Web sites.
  http://scalemodel.net/
• A Japanese Web site (in English) with paper scale models of spaceships (real and fictional).
  http://member.nifty.ne.jp/papermodel/htmls/Links/Link45e.html
  
  Science

• A site by Mitchell N. Charity with links to many ways of scaling down the solar system. The idea: Making scale models of the solar system is a useful way to learn about it.
  http://www.vendian.org/mncharity/dir3/solarsystem/
• An interactive scaling site for sizes of a scale model of the solar system. This site is created by San Francisco's Exploratorium Museum. The online museum now contains over 12,000 Web pages exploring hundreds of different topics.
  http://www.exploratorium.edu/ronh/solar_system/

Community Connections

• Invite an architect to talk about blueprints and scale models.
• Invite a product designer to talk about how a product goes from an idea into actual production.
• Students can create a math trail around the school or community with a scale drawing of the route. Stations along the route would have math problems related to something found there. This is a good activity for a Family Math Night at school, a field trip or a community-based project at a local park or wild area. Maps and problem sheets can be distributed to the community as appropriate.
• Invite someone who has lived in or traveled to Egypt to talk about modern-day Egypt and the influence of the past on the present.