# What Does It Mean To Measure? Homework

## Session 1, Homework

How Large Is It?
This table shows the height to the shoulder and the weight of several species of buffalo. Which is the largest (i.e., one that is both tall and heavy)? The guar is the tallest, and the water buffalo and the American bison are the heaviest, but how might we determine which is the largest?

 Name (Country/Continent) Height (cm) Weight (kg) Water Buffalo (Asia) 165 1,000 African Buffalo 135 560 Yak (Tibet) 200 550 Guar (Asia) 220 850 American Bison (North America) 180 1,000 European Bison 200 900

Problem H1

Maria decides to add the height and the weight as a measure of the total size. The animal with the greatest sum is the largest.

Which three animals, in order from largest to smallest, are the largest, according to Maria’s criterion?

Problem H2

Jacob decides to rank the animals from 1 to 6 for both height and weight. The smallest in each category is given a rank of 1, the largest a rank of 6. If two animals have the same height or weight, each receives the average of the ranks they would have received. He then adds the height and weight ranks. The largest sum indicates the largest animal.

Which three animals, in order from largest to smallest, are largest according to Jacob’s system?

Problem H3

Quentin decides to find the product of height and weight as a measure of the total size.

Which three animals, in order from largest to smallest, does Quentin think are the largest?

Problem H4

Is Quentin’s method a good measure of how large an animal is? Do you think it is better or worse than Jacob’s method?

Problem H5

Greg argues that the question would be easier to decide if one of the buffalo were 220 cm tall to the shoulder and weighed 1,000 kg. That animal would clearly be the largest, because it would be both the tallest and the heaviest.

a. Plot the height and weight of each buffalo as an ordered pair. You may want to print out the image (PDF) of the axes below to plot the points.

 Heights and Weights of Buffalo

b. How far from the point (220,1000) does each buffalo fall? Which point is the shortest distance away? Complete this table:

 Name (Country/Continent) Coordinates Distance from (220,1000) Water Buffalo (Asia) (165,1000) African Buffalo (135,560) Yak (Tibet) (200,550) Guar (Asia) (220,850) American Bison (N. America) (180,1000) European Bison (200,900)

Use the Pythagorean theorem to find distances between two points. Imagine a right triangle whose hypotenuse lies between the two points and whose legs are parallel to the x- and y-axes (the right angle between them). The shortest distance between the points will be the length of the hypotenuse. How can you use the coordinates of the two points to figure out the lengths of the legs you will need in order to calculate the length of the hypotenuse?

Problem H6

Which three species, in order from largest to smallest, are closest to the point (220,1000)?

Marcy alters Greg’s argument somewhat. She argues that the smallest animal would be (0,0), or no animal at all. Since largest is the opposite of smallest, the largest animal is the one farthest from (0,0).

Problem H7

Which three buffalo, in order from largest to smallest, are largest by Marcy’s rule? Complete the table below.

 Name (Country/Continent) Coordinates Distance from (0,0) Water Buffalo (Asia) (165,1000) African Buffalo (135,560) Yak (Tibet) (200,550) Guar (Asia) (220,850) American Bison (N. America) (180,1000) European Bison (200,900)

In all the criteria used so far, the students equate 1 cm in height to 1 kg in weight. It is not always reasonable to treat all units of measure as being equally important. For example, Maurice thinks that a 1 cm increase in height is more noticeable than a 1 kg increase in weight. In fact, he believes that a 1 cm change in height is twice as important as a 1 kg change in weight, so he uses the value of the expression 2(height) + weight as his measure. According to Maurice’s rule, the largest animal is the American bison, followed by the water buffalo and the European bison.

Problem H8

How would you have solved this problem? Can you think of an additional rule that can be used to determine which three animals are the largest?

Problem H9

Which species of buffalo would you say is the smallest? Why?

### Solutions

Problem H1

According to Maria’s criterion, here are the values:

 Name (Country/Continent) Height (cm) + Weight (kg) Water Buffalo (Asia) 1,165 African Buffalo 695 Yak (Tibet) 750 Guar (Asia) 1,070 American Bison (North America) 1,180 European Bison 1,100

The three largest animals by this criterion are the American bison, the water buffalo, and the European bison.

Note that this method is somewhat problematic since the answers could change drastically if we change the units of measurement. For example, the results would look quite different if we measured weight in tons (1 metric ton = 1,000 kg) instead of in kilograms. Clearly, this can lead to potential errors.

Problem H2

Here are the ranks according to Jacob’s criterion:

 Name (Country/Continent) Height Rank Weight Rank Total Water Buffalo (Asia) 2 5.5 (tie) 7.5 African Buffalo 1 2 3 Yak (Tibet) 4.5 (tie) 1 5.5 Guar (Asia) 6 3 9 American Bison (North America) 3 5.5 (tie) 8.5 European Bison 4.5 (tie) 4 8.5

The three largest animals by this criterion are the guar, the American bison, and the European bison.

Problem H3

Here are the products for each animal:

 Name (Country/Continent) Height (cm) • Weight (kg) Water Buffalo (Asia) 165,000 African Buffalo 75,600 Yak (Tibet) 110,000 Guar (Asia) 187,000 American Bison (North America) 180,000 European Bison 180,000

The three largest animals by Quentin’s criterion are the guar, the American bison, and the European bison.

Problem H4

These two methods result in the same ordering of the animals, but some may argue that Quentin’s method is more capricious than Jacob’s, since there is no natural meaning to multiplying height and weight.

Problem H5

1.
2. Using the Pythagorean theorem, we can derive a formula for the distance between two points A (x1,y1) and B (x2,y2) as Next, we can calculate the distances. So for the Asian water buffalo Here is the completed table:
 Name (Country/Continent) Coordinates Distance from (220,1000) Water Buffalo (Asia) (165,1000) 55 African Buffalo (135,560) 448.13 Yak (Tibet) (200,550) 450.44 Guar (Asia) (220,850) 150 American Bison (North America) (180,1000) 40 European Bison (200,900) 101.98

To learn more about the Pythagorean theorem and distance formula, go to Learning Math: Geometry, Session 6.

Problem H6

The three species closest to (220,1000) are the American bison, the water buffalo, and the European bison.

Problem H7

Here is the completed table:

 Name (Country/Continent) Coordinates Distance from (0,0) Water Buffalo (Asia) (165,1000) 1,013.52 African Buffalo (135,560) 576.04 Yak (Tibet) (200,550) 585.23 Guar (Asia) (220,850) 878.01 American Bison (North America) (180,1000) 1,016.07 European Bison (200,900) 921.95

According to Marcy’s criterion, the three largest animals are the American bison, the water buffalo, and the European bison.

Problem H8

Answers will vary. Another possible method is to use weight as the first determining factor and to use height only in case of a tie.

Problem H9

Answers will vary, but the most likely answer is the African buffalo, which is the shortest and nearly the lightest. It wins easily on Jacob’s criterion and is the smallest for many others.