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Private: Learning Math: Measurement
Fundamentals of Measurement Part B: The Role of Ratio (45 minutes)
Session 2, Part B
In This Part
- Ratio and Scale
- Constant Ratios
- Using the Pythagorean Theorem
Ratio and Scale
Measurement is the process of quantifying properties of an object by comparing them to some standard unit. Thus, a measure is a ratio. When we state that an object is 8 in. long, this is in comparison to the unit of 1 in. Likewise, stating that a bag of sugar weighs 5 lb. implies that the 5 lb. are being compared to the unit of 1 lb., even though we don’t state this explicitly.
We use proportional reasoning in other ways in measurement situations. For example, we are all familiar with map scales. If 1 cm on a map represents a distance of 250 km, what is the approximate distance of a length represented by 2.7 cm? We can set up a proportion to show that the distance is 675 km:
1 cm = 2.7 cm
250 km x km
Solving the equation for x, we get x = 250 • 2.7 = 675 km.
One unit of measurement on a scale drawing corresponds to n units of measurement in reality. The units can be anything — centimeters, meters, etc. In fact, they don’t even have to be the same units; the example above used centimeters and kilometers. That scale could have used the same units (1 cm on the map representing a specific number of centimeters in reality), but converting the centimeters to kilometers makes it easier for the user.
Problem B1
he 1 cm:250 km scale compares centimeters to kilometers. Rewrite the scale to show the same relationship comparing centimeters to centimeters (1 cm:x cm, or simply 1:x).
Remember that 1 km = 1,000 m, and 1 m = 100 cm.
Scale drawings and models are another way that ratio is used in measurement. Usually, a scale compares linear measures. Examine the scale drawings below. A scale of 1:1 implies that the drawing of the grasshopper is the same as the actual object. The scale 1:2 implies that the drawing is smaller (half the size) than the actual object (in other words, the dimensions are multiplied by a scale factor of 0.5). The scale 2:1 suggests that the drawing is larger than the actual grasshopper — twice as long and twice as high (we say the dimensions are multiplied by a scale factor of 2). If no units are listed in the scale, then you can assume that the drawing and the object are measured using the same units. For example, the scale 1:2 might represent 1 cm:2 cm or 1 in.:2 in.
Problem B2
A mural of a dog was painted on a wall. The enlarged dog was 45 ft. tall. If the average height for this breed of dog is 3 ft., what is the scale factor of this enlargement? Can you express this scale in more than one way?
Problem B3
Imagine that you need to make a drawing of yourself (standing) to fit completely on an 8.5-by-11-in. sheet of paper. Determine the scale factor, allowing no more than an inch of border at the top and bottom of the page. How long will your arms be in the drawing? Note 8
Try a scale of 1:10 (i.e., your drawing would be one-tenth your actual size) or 1:8. Measure different body parts, such as the length of your head, arms, torso, and legs, and then use a ratio to determine the size of that body part in your drawing.
Scale drawings are especially useful when comparing the relative magnitudes of objects that are very large. Science museums often have a scale model of our solar system to help us grasp the enormous distances between the Sun and each planet. Imagine that you had to design a model of the solar system for your school.
Below is a table with some relevant data. Notice that the distance from the Sun is given in scientific notation:
Constant Ratios
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 π. 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.):
|
Hypotenuse Length (H) in cm |
Ratio S:S |
Ratio H:S |
|
| 1 | |||
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 |
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? |
Using the Pythagorean theorem
Remember that the Pythagorean theorem states that in right triangles with leg lengths a and b, and hypotenuse length c, the following relationship holds: a2 + b2 = c2. Note 10When you use the Pythagorean theorem, your answer may not reduce easily from radical form (as a square root). Rather than using a calculator to take the square root, you can instead express the answer in reduced radical form. Here’s how: Express the number as a product of factors, where one of the factors (if possible) is a square number. Then take the square root of just the square number and leave the answer as a product of the square root and the radical:
Problem B8
- Use the Pythagorean theorem to find the lengths of the hypotenuses for all the triangles from Problem B6. Leaving the length in radical form, fill in the blank columns in the chart below. Do this on paper by printing this page if it’s not possible to type in the square root symbol on your computer. Notice the patterns in the ratio of H:S.
Side Lengths (S) in cm
Hypotenuse Length (H) in cm
Ratio S:S
Pythagorean Ratio H:S
1 1:1 2 2:2 3 3:3 4 4:4 5 5:5 6 6:6 - Which measures are more accurate — those done with a ruler or those determined using the Pythagorean theorem? Explain.
Problem B9
In most right triangles, one or more of the side-length values is irrational. Note 11
In terms of measurement, what are the implications of one or more of the values being irrational?
Notes
Note 8
If you are working in a group, work in pairs on Problem B3. Practice setting up proportions (two ratios that are equal to each other) to determine the length of the different body parts in your drawing.
Note 9
This problem may take some time, especially if you are trying to use one scale for both the diameter of the planets and their distances from the Sun. Often, models are created that focus on one or the other (size vs. distance). If you choose a scale that allows distances from the Sun to fit into a large room, you will find that the models of some of the planets are very, very small. If you choose a scale that allows the models of the planets to be big enough that you can observe them, you will find that the distances between planets in the model must be very large.
Note 10
To learn more about the Pythagorean theorem, go to Learning Math: Geometry, Session 6.
Note 11
The √2 is an irrational number, because it cannot be expressed as a fraction a/b, where a and b are integers and b 
0. In other words, this value can’t be written as a fraction or as a repeating or terminating decimal. If we expressed it as a decimal, it would have an infinite number of digits to the right of the decimal point in a non-repeating pattern. The real-number system is made up of an infinite number of rational numbers (those that fit the fraction property above) and an infinite number of irrational numbers. There are many situations where a length is actually an irrational number (such as the hypotenuses of isosceles right triangles), so we cannot measure the length exactly. The idea that a measure is always an approximate value is a hard one to grasp, since in everyday life we treat measures as exact quantities.
Solutions
Problem B1
Since 1 km = 100,000 cm, the scale 1 cm:250 km is equivalent to 1 cm:250 • 100,000 cm, or 1:25,000,000.
Problem B2
The scale factor is 45:3. This can be simplified to 15:1 or expressed in other ways, such as 7.5:0.5 or 150:10.
Problem B3
Answers will vary. Here’s one example: Suppose that someone is exactly 6 ft. tall, with arms 3 ft. long. The scale factor will be 9 in.:6 ft. in order to leave 1 in. of border at the top and bottom. To simplify this, remember that 6 ft. = 72 in. The scale factor can then be expressed as 9:72 or 1:8. This person’s arms would be 1/8 as long in the scale drawing; 1/8 of 3 ft. (36 in.) is 4.5 in.
Problem B4
- The farthest planet from the Sun, Pluto, has an average distance from the Sun of about 5.9 billion km. To fit on school grounds (say, within 100 m), this would require a scale factor of 100 m:5,900,000,000 km. Since 1 km = 1,000 m, the scale factor can be expressed as 100 m:5,900,000,000,000 m, or 1 m:59 billion m:
100 m 1 • 102 m 1 5.9 • 109 km = 5.9 • 1012 km = 5.9 • 1010m - This scale would be hopelessly large for visualizing the difference in diameter sizes among planets, since the largest diameter (Jupiter’s) is only about 143,000 km. A scale of 1 m:59 billion m would make Jupiter’s diameter roughly 2.4 mm, extremely small. A better scale might be 1 m:590 million m, which would make Jupiter’s diameter roughly 24 cm. The smallest planet, Pluto, would have a diameter of 3.8 mm on this scale, which is still small but certainly visible.
Problem B5
Most likely, they chose miles. The longest distance from the Sun, 5.9 • 109, becomes 1.6 km. This is just under one mile, so the scale was likely chosen to let the entire model fit within one mile. Using this scale, the smallest piece of data (Pluto’s diameter) becomes 0.62 mm, which is very small but still visible.
Problem B6
Answers may vary due to measurement. Here, answers are given to the nearest tenth of a centimeter:
|
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 |
Problem B7
- Not surprisingly, the ratio between the sides is constant at 1:1, since we worked exclusively with isosceles right triangles. The ratios between the hypotenuse and a side also seem to be about the same (this is evident if you divide the H:S ratios and write them as decimals). So there may be a constant ratio involved there as well. All the observations are between 1.4 and 1.425, so the constant ratio (if there is one) may be between these values.
- We could multiply the length of the side by 1.41 (the average ratio) to get an approximate answer. We could also use the Pythagorean theorem (covered in the next section) to derive the measure of the hypotenuse.
Problem B8
|
Side Lengths (S) in cm |
Hypotenuse Length (H) in cm |
Ratio S:S |
Pythagorean Ratio H:S |
| 1 | √2 | 1:1 | √2:1 |
| 2 | 2√2 | 2:2 | 2√2:2 |
| 3 | 3√2 | 3:3 | 3√2:3 |
| 4 | 4√2 | 4:4 | 4√2:4 |
| 5 | 5√2 | 5:5 | 5√2:5 |
| 6 | 6√2 | 6:6 | 6√2:6 |
b.The measures using the Pythagorean Theorem are more accurate. They show the constant ratio of √2:1 in all six cases.
Problem B9
An irrational number is a number that cannot be written as an exact ratio of two integers; √2 is an example. In terms of measurement, the important implication (and one that the Greeks missed for centuries) is that there is no possible exact unit conversion between a whole number and an irrational number. If one measure is rational and another is irrational, they are incommensurate; that is, we can never say “A of these make B of these,” where A and B are integers.
