 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 2, Part B:
The Role of Ratio (45 minutes)

In This Part: Ratio and Scale | Constant Ratios | Using the Pythagorean Theorem

 Measurement is the process of quantifying properties of an object by comparing them to some standard unit. Thus, a measure is a ratio. When we state that an object is 8 in. long, this is in comparison to the unit of 1 in. Likewise, stating that a bag of sugar weighs 5 lb. implies that the 5 lb. are being compared to the unit of 1 lb., even though we don't state this explicitly. We use proportional reasoning in other ways in measurement situations. For example, we are all familiar with map scales. If 1 cm on a map represents a distance of 250 km, what is the approximate distance of a length represented by 2.7 cm? We can set up a proportion to show that the distance is 675 km: Solving the equation for x, we get x = 250 • 2.7 = 675 km. One unit of measurement on a scale drawing corresponds to n units of measurement in reality. The units can be anything -- centimeters, meters, etc. In fact, they don't even have to be the same units; the example above used centimeters and kilometers. That scale could have used the same units (1 cm on the map representing a specific number of centimeters in reality), but converting the centimeters to kilometers makes it easier for the user.  Problem B1 The 1 cm:250 km scale compares centimeters to kilometers. Rewrite the scale to show the same relationship comparing centimeters to centimeters (1 cm:x cm, or simply 1:x).  Remember that 1 km = 1,000 m, and 1 m = 100 cm.   Close Tip Remember that 1 km = 1,000 m, and 1 m = 100 cm. Scale drawings and models are another way that ratio is used in measurement. Usually, a scale compares linear measures. Examine the scale drawings below. A scale of 1:1 implies that the drawing of the grasshopper is the same as the actual object. The scale 1:2 implies that the drawing is smaller (half the size) than the actual object (in other words, the dimensions are multiplied by a scale factor of 0.5). The scale 2:1 suggests that the drawing is larger than the actual grasshopper -- twice as long and twice as high (we say the dimensions are multiplied by a scale factor of 2). If no units are listed in the scale, then you can assume that the drawing and the object are measured using the same units. For example, the scale 1:2 might represent 1 cm:2 cm or 1 in.:2 in.  Problem B2 A mural of a dog was painted on a wall. The enlarged dog was 45 ft. tall. If the average height for this breed of dog is 3 ft., what is the scale factor of this enlargement? Can you express this scale in more than one way? Problem B3 Imagine that you need to make a drawing of yourself (standing) to fit completely on an 8.5-by-11-in. sheet of paper. Determine the scale factor, allowing no more than an inch of border at the top and bottom of the page. How long will your arms be in the drawing? Note 8  Try a scale of 1:10 (i.e., your drawing would be one-tenth your actual size) or 1:8. Measure different body parts, such as the length of your head, arms, torso, and legs, and then use a ratio to determine the size of that body part in your drawing.   Close Tip Try a scale of 1:10 (i.e., your drawing would be one-tenth your actual size) or 1:8. Measure different body parts, such as the length of your head, arms, torso, and legs, and then use a ratio to determine the size of that body part in your drawing. Scale drawings are especially useful when comparing the relative magnitudes of objects that are very large. Science museums often have a scale model of our solar system to help us grasp the enormous distances between the Sun and each planet. Imagine that you had to design a model of the solar system for your school.

Below is a table with some relevant data. Notice that the distance from the Sun is given in scientific notation:   Diameter (in km) Distance from Sun in Scientific Notation (in km)      Sun 1,392,000 Mercury 4,900 5.8 • 107 Venus 12,100 1.08 • 108 Earth 12,760 1.5 • 108 Mars 6,790 2.28 • 108 Jupiter 143,000 7.78 • 108 Saturn 121,000 1.43 • 109 Uranus 51,000 2.87 • 109 Neptune 50,000 4.5 • 109 Pluto 2,300 5.9 • 109       Problem B4 a. What scale would you use if you wanted to show students how far the planets are from the Sun? b. What scale would you use if you wanted to help students understand the differences in diameter size among the planets? Can the same scale be used for both goals? Note 9 Problem B5 A science park in Westerbork, Holland, uses the scale of 1:3.7 • 109 for a scale model of the solar system. What units do you think the park chose to use?    Session 2: Index | Notes | Solutions | Video