During this lesson and its extensions, the students will have the opportunity to apply scales on maps to the real world, construct their own scale models of objects, and use the concepts of scale and ratio to draw their own maps, drawings or models of real life objects. The students should be very familiar with the concept of equivalent fractions, and using multiplication and division by common factors or multiples to obtain equivalent fractions. The initial lesson could be covered in one day, but follow-up activities and extensions could take from two days to a week or more, depending on the extent to which other subjects are integrated.

Math Talk #114: Sizing Things Up: Exploring Scale and Ratio
Students will be able to:
1. explain how equivalent ratios are used to represent real world objects and distances on scale drawings and map scales.
2. use measurement tools to make a scale drawing of a room.
3. research science concepts such as the solar system or ocean life to create scale models of real objects.

Math Objectives:
#11: Determine solution strategies and analyze or solve problems.
#12: Express or solve problems using mathematical representation.
Science Objectives:
#8: Relate and apply scientific and technological information to daily life.

Standard 4: Mathematical Connections
Standard 13: Measurement
Mapping A Room
Per group of 2-4 students:
• a 12 inch ruler
• several yards of string
• 4 copies of centimeter or 1" grid paper taped together

Science Projects
Per group of 2 students:
• map of solar system
• encyclopedia or CD ROM for information gathering
• books on life in the oceans, especially large sea creatures

Review with the class the concept of equivalent ratios by putting a few examples on the board and calling on volunteers to complete them. Hopefully, the students will be familiar with metric measurements. If not, some review and or instruction will be necessary on using metric units of length or distance. It might be helpful to conduct an activity that would help make a kilometer more real to the students. Ask them to complete the statement, "A kilometer is about the distance between _____ and ______," filling in the blanks with familiar landmarks. These can be discussed as a group. Have the students bring in pieces of string that can be tied together to make a piece that is 100 meters long. This could take several days. When you have a total of 100 m of string, a small group can use it to measure out a full kilometer--10 of the 100 m lengths. It does not have to be in a single straight line; it might be five times around the playground. Let other groups use the string to find other paths that are exactly 1 km long, so that the students will have a mental picture of what a kilometer really is.
The focus for viewing is a specific responsibility given to students to focus and engage their viewing attention. Tell students this is an explanation of scales and ratios used in those scales. As they watch this video, they are responsible for finding and recording the scales used in each of the two segments, and for explaining why maps and models must be made to scale and not actual size.

BEGIN the Math Talk video at the beginning. PAUSE when Maria says, "Buster, look at the scale." The screen is showing a map with the scale "1 cm : 20 km" in the right corner. Ask for students to offer their definitions of scale. Summarize correct answers and write a suitable definition on board. RESUME the video. PAUSE the video when the navigator places the meter stick between the X on the map and the triangle representing the iceberg. He tells the captain, "Take a look for yourself." Ask the students, "What is wrong here? Can the iceberg be only 10 cm away?" Discuss their responses. Summarize by stressing to the class that the navigator does not realize that the map is not a 1-1 portrayal of reality. RESUME the video. PAUSE when the navigator finishes the captain's statement: "10 cm on map equals 10 x 10 or...100 km." Check for students' understanding of the calculations involved. Relate to previous lessons on equivalent fractions. RESUME the video.
PAUSE when Maria and Buster return to the screen. Have the students practice their understanding of the map scale introduced in the video by first asking them to reproduce it on their papers. After checking that everyone has written "1 cm : 10 km", give them the following problems to solve with a partner:
1. The navigator said that the giant squid was 240 km away from the ship. How many centimeters would this be on the map? (24 cm)
2. He also said the home port was 25 km away. What distance was this on the map? (2.5 cm)
3. If his rubber ducky Melvin was 2466 km away, how would you find what this distance would be on the map? (divide 2466 by 10 to get 246.6 cm)
4. If another ship was 132 cm away from their ship on the map, how far would it be from them in km? (1320 km)
RESUME the video as Maria says, "Now do you understand how a scale on a map works, Buster?" PAUSE after Buster says, "But each cm stands for 20 km in the real world." Ask the students to figure out how far the bird sanctuary is from Sylvia's house. RESUME the video to check answers. FAST FORWARD to the segment concerning architect Frank Lloyd Wrong. The opening scene portrays a red-headed architect at his drawing board. He is sneezing into his handkerchief as his dark-haired assistant looks at some plans on the drawing board. PAUSE the video when Eegore the assistant says, "Mr. Wrong, we did not build a tower 600 ft. tall." Ask the students what they think has happened. RESUME the video. PAUSE video when Eegore says, "I thought a scale was something you weighed yourself on." Ask the students for further predictions on what has gone wrong. Say, "What do you think the architect has built? Continue watching to find out." RESUME the video. STOP the video when the segment is finished. Have the students record the scale the architect was supposed to use to complete the building for his client. Check to see that everyone has written "1 inch : 20 feet". Ask the students to write a few sentences explaining why maps and models must use a scale and cannot be drawn or built to actual size. Allow willing students to share their sentences.

Have the students work in groups of 2-4. Provide each group with a 12 inch ruler, several yards of string, and four copies of centimeter or 1-inch grid paper taped together to make a large graph. Ask volunteers to measure specific distances and length using a piece of string. Then wrap the string around the 12 inch ruler the long way to find the number of feet and inches. If using metrics, meter sticks may be used or centimeters added together to get meters. Discuss how to best show the dimensions of the classroom and of the objects in it on the large graph paper. Students should consider the size and number of squares on the graph paper as they think about the scale. Once a scale has been decided upon, it is time to make and record the necessary calculations. For each measurement of the real classroom, students will have to calculate to figure out how that measurement will be shown on their drawings. Groups can then make their drawings based upon their calculations. Have them show the sizes and positions of major pieces of furniture and other features.
The next activity depends on the course of study in science for each particular grade level involved. Studies of the solar system and ocean life are particularly well suited. Provide the students with encyclopedias and/or CD ROM and allow them to research the solar system or particular species of large ocean creatures. Have them work in groups to complete either a scale model of the solar system using materials of their own choosing, or a model of a particular ocean animal, especially very large ones. Groups will provide information on the actual sizes of the objects of which they have made models, as well as the scale used in making the models. Models can be displayed in the school library or other suitable place for viewing by other grade levels.
Scale and ratio play important roles in many occupations. Architects, surveyors, designers, and navigators create maps or models every day, while people in many other occupations must read and interpret maps, charts, blueprints, or scale drawings. Have each student interview, write a letter to, or e-mail a person having one of these jobs and find out what the job really entails. The students should be encouraged to identify professionals with characteristics similar to their own. The students should write a report of their interview or a summary of their findings. They can share work samples collected with the class.
Math: Students can choose areas or buildings that are particularly relevant to them, such as basketball court, football field, band hall, soccer field, etc. They should then make a map or model of what they chose. They should describe how they chose the scale factor and what calculations they had to make in creating the drawing or model.
Science: Use the idea of scale and ratio to make models of the human body systems.
Social Studies: Create scale models of mummies studied in world cultures.
Math/Art: Students can make scale models of a room in their homes.
Math: Use knowledge of scale and ratio to draw similar plane figures in the study of geometry.

1995-1996 National Teacher Training Institute / Austin

### Master Teacher: Jan Nutt

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