 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

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

Ratio plays an important role in measurement and can be used to make predictions. If the ratio of inches to centimeters is 1 to 2.54 (1:2.54), then we can assume that a length of 12 in. is approximately 30 cm (30.48).

Not all ratios in nature are constant, though. According to mathematician Ernest Zebrowski Jr., "Most ratios, in fact, are not constant. If, for instance, it took 24 rowers to row a galley at 15 mi/h, this does not mean that 48 rowers would get the boat up to 30 mi/h and that with 144 rowers the boat would hit 90 mi/h. (In fact, this line of reasoning would suggest that the ancients could have broken the sound barrier just by getting together enough rowers.) .... Although it's a simple matter for an accountant or mathematician to assert that a particular ratio is constant, the laws of nature are the final arbiter. Clearly, before making predictions on the basis of an assumed constant ratio, we need to get someone to check out the reality of the situation."

While Zebrowski states that many ratios are not constant, there are some constant ratios found in measurement situations. One constant ratio that we use regularly is . We will explore this ratio further in Session 7, which focuses on circles.

Another common measurement situation involves right triangles. We will now look more closely at right triangles, beginning with a number of right triangles of different sizes.   Problem B6 Print the shapes from the PDF file (be sure to print this document full scale). With a centimeter ruler, measure the hypotenuses of these triangles. We will explore whether there is a constant of proportionality.

Complete the chart. (Notice that these are all isosceles right triangles.):  Side Lengths (S) in cm Hypotenuse Length (H) in cm Ratio S:S Ratio H:S        1   2 3 4 5 6    Side Lengths (S) in cm Hypotenuse Length (H) in cm. Ratio S:S Ratio H:S        1 1.4 1:1 1.4:1 2 2.8 2:2 2.8:2 3 4.2 3:3 4.2:3 4 5.7 4:4 5.7:4 5 7.1 5:5 7.1:5 6 8.5 6:6 8.5:6 Answers may vary due to measurement. Here, answers are given to the nearest tenth of centimeter. Problem B7 a. What constant ratios did you find in the isosceles right triangles (45°-45°-90°)? b. Sometimes you can't measure something directly (e.g., by using a ruler), but you still can determine its measure. Measures found indirectly using mathematics are often referred to as "derived" measures. For example, if we know the lengths of the legs of an isosceles right triangle, how can we determine the measure of its hypotenuse?   Video Segment Watch this video segment to see how one of the participants, David Cellucci, reasoned about the results obtained from finding the hypotenuse-to-side ratios of the right isosceles triangles. Watch this segment after you've completed Problem B6. Did you arrive at a similar conclusion? If you are using a VCR, you can find this segment on the session video approximately 14 minutes and 33 seconds after the Annenberg Media logo.   Quote taken from Zebrowski, Ernest, Jr. A History of the Circle. pp. 3. © 1999 by Ernest Zebrowski Jr. Reprinted by permission of Rutgers University Press.   Session 2: Index | Notes | Solutions | Video