Global Measurements
Recalling the experience of working through the ambiguity of deciding on the constant for the fire hydrant activity will encourage students to persevere in solving problems and make use of structure when they reach the IM Measuring Circles unit and must derive that pi is the constant of proportionality between the diameter and circumference of any circle. In IM lesson 8.3, given a coordinate plane with the x-axis labeled diameter and the y-axis labeled circumference, students are tasked with plotting the data for various circles and determining that there is a constant even though the points do not lie on a perfectly straight line. Students are encouraged to wonder why and conclude that there could be slight variation due to rounding to the nearest tenth rather than hundredth or thousandth. Students will come up with a working value, by finding the average between the constants they found, arriving at 3.14 as an approximation for pi.
Having been introduced to circles, students will begin working with spherical maps, simply called globes. Presented with measuring tapes and globes of various size, students in triads will measure their circumference and diameter. If students become too frustrated with measuring the diameter of a sphere, the teacher can suggest that 2 of the three partners hold either side of the globe while the third partner measures across the widest part. The teacher should ask students to consider a way of finding the diameter when only the circumference is known. Students should deduce that they can divide the circumference by the constant of proportionality they previously found, pi or as an approximation. In creating a table and graph of their findings, they may observe that their points do not lie on a straight line this time either, but they must realize that a line can be drawn that passes through (0,0) and closely hits all points. This inquiry and conclusion set the groundwork for later line of best fit statistical work.
As the module on measuring circles is the shortest, the recommendations in this section are brief but impactful for transfer. Students are not expected to memorize circumference and area formulas and will have access to formula sheets during activities and testing but they are expected to apply the appropriate formula. To foster contextualization and practice appropriate application, students will continue working with maps to increase their fluency. Given circular flat maps such as the Nautical Atlas of the World, students will consider what the total circumference and area would be if placed in a circular frame that is 1.75 inches all the way around. Different groups should investigate different frame widths, and the teacher will collect and display results, promoting discussion. Students should have access to actual frames or frame-shaped cardboard to physically manipulate and help them visualize the situation. The nautical atlas, though round, is imposed on a rectangular background with surrounding cartouche. Students will determine the area of the background that is exposed by subtracting the area of the circle from the area of the rectangle. They can transfer this learning to word problems that require them to find the area by composition and decomposition.
Comments: