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)

Distance from (220,1000)

 55 448.13 450.44 150 40 101.98

 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?   Close Tip 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)

Distance from (0,0)

 1,013.52 576.04 585.23 878.01 1,016.07 921.95

 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?

Duke, C. (May, 1998). Tangrams. Mathematics Teaching in the Middle School, 485-487.
Reproduced with permission from Mathematics Teaching in the Middle School. © 1998 by the National Council of Teachers of Mathematics. Used with permission. All rights reserved.

 Homework problems taken from Teague, Dan. How Large Is It? Student Math Notes. © by the National Council of Teachers of Mathematics. Used with permission. All rights reserved.
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