 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Solutions for Session 2, Part A

See solutions for Problems: A1 | A2 | A3 | A4 | A5 | A6| A7    Problem A1 Counts are exact; they are not on a scale, nor are they ratios. In a count, the unit is absolute. In contrast, measurements are not exact; the units are relative, and typically they don't directly match what we're measuring. For example, a person's height, measured in centimeters, is very unlikely to be an exact number of centimeters, so we approximate. A measurement is continuous, not discrete; someone can be 180 cm tall, 181 cm tall, or any number in between.   Problem A2

 a. It is not possible to just "count" inches or centimeters, since the result of a measurement may not be an exact number in those units. Also, depending on the measuring device used, the unit of the measurement can change; for example, the same measurement could be expressed as 6 (in.), 0.5 (ft.) or 1/6 (of a yard). b. Again, it's a question of relative vs. absolute: When we hear that the temperature is 63 degrees, this means that the temperature has been rounded off to the nearest whole number. When we count that we have 63 pennies, there is no rounding off; we have exactly 63 pennies.   Problem A3 Transitivity is used in many places -- in parallelism, for example. If lines A and B are parallel, and lines B and C are parallel, then lines A and C are parallel.   Problem A4 Answers will vary. One possible answer is that in part-whole interpretation, the number of parts that the whole is divided into is predetermined, whereas in measurement, you can vary the number of equal parts according to whatever is most appropriate for your measurement situation.   Problem A5 Partitioning is important in measurement, because the measurements taken depend entirely on the partitioning. The example of timing a swim meet is relevant here, since the partitioning of time determines the measured times in the event (to the nearest second, 100th of a second, and so on). There are an infinite number of possible partitions of the number line, since we can always break any partition into a smaller one.   Problem A6

 a. Since the number line between 0 and 1 is already partitioned into 12 equal parts, we will need to partition the 12ths into two equal parts so that each is 1/24. Then, since 1/3 = 8/24, count one partition to the left of 1/3. b. The number line between 0 and 1 is partitioned into 18 equal parts now (since 1/6 is three partitions over). To locate 3/8, partition each of the 18 parts into four equal parts so that each is 1/72 (so 3/8 = 27/72). Since 1/6 = 12/72, count over to 2/6 (i.e., 24/72), then three partitions beyond it.    Problem A7 Answers will vary. In either case, it is also possible to start with a new number line and make partitions different from the ones you made before. For example, to locate 7/24, you could partition the number line into thirds and then partition those into eighths, which would also result in 24ths (as 3 and 8 are both factors of 24). Other combinations with other factors of 24 are also possible.   Problem A8 Write and reflect.     