Session 2 Fundamentals of Measurement Key Terms in This Session Previously Introduced • measurement

• precision

New in This Session • accuracy

• compensatory principle

• conservation

• partitioning

• ratio

• scale

• scale factor

• transitivity

• unit iteration

Introduction In Session 1, you began to explore what it means to measure. In this session, you will investigate the difference between a count and a measure and examine some of the important ideas essential to meaningful measurement. You will learn about the many uses of ratio in measurement and how scale models can help us understand relative sizes. Finally, you will investigate the constant of proportionality in isosceles right triangles and learn about precision and accuracy in measurement. For the list of materials that are required and/or optional in this session, see Note 1.

Learning Objectives In this session, you will do the following: • Understand the three components of measuring: conservation, transitivity, and unit iteration • Consider the effects of partitioning a unit into smaller subunits • Understand the use of proportional reasoning and ratio in measurement situations • Use accuracy and precision to determine how much error is involved in any given measurement

Note 1. Materials Needed: •

Centimeter ruler

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Part A: Measuring Accurately (45 min.) Conservation, Transitivity, and Unit Iteration In Session 1, we established that in order to measure something, we have to (1) select an attribute of the thing to be measured; (2) choose an appropriate unit of measure; and (3) determine the number of units. In conjunction with these three steps, many educators have noted that there are three components of measuring that contribute to students’ ability to make meaningful and accurate measurements: conservation, transitivity, and unit iteration. [See Note 2]

Conservation Conservation is the principle that an object maintains the same size and shape even if it is repositioned or divided in certain ways. If you understand this principle, you realize that a pencil’s length remains constant when it is placed in different orientations. For example, two pencils that are the same length remain equal in length when one pencil is placed ahead of the other:

You also realize that two differently shaped figures have the same area if they have the same component pieces. For instance, a jigsaw puzzle covers the same amount of space whether the puzzle is completed or in separate pieces.

Transitivity When you can’t compare two objects directly, you must compare them by means of a third object. To do this, you must intuitively understand the mathematical notion of transitivity (if A = B and B = C, then A = C; if A < B and B < C, then A < C; if A > B and B > C, then A > C). For example, to compare the length of a bookshelf in one room with the length of a desk in another room, you might cut a string that is the same length as the bookshelf. You can then compare the piece of string with the desk. If the string is the same length as the desk, then you know that the desk is the same length as the bookshelf. Developmentally, conservation precedes the understanding of transitivity, because you must be sure that a tool’s length (area, volume, etc.) will stay the same when moved in the process of measuring.

Unit Iteration In order to determine the correct unit for measurement, you must understand the attribute you are measuring. For instance, when measuring distance, a linear measurement is appropriate. When measuring area, you need two-dimensional units, such as squares, to cover the surface. When measuring volume, you need a three-dimensional unit. Note 2. Take a few minutes to read the information about conservation, transitivity, and unit iteration. Whereas adults conserve measures, we can sometimes become confused (as with the tangram activity in Session 1) by a visual image. Transitivity is used in algebra and geometry (for example, as justification for steps in a proof ) as well as in measurement, when comparing the equality of a number of measures. Examining the concept of units leads us to consider the kind of units that are used when we count versus when we measure. “Conservation, Transitivity, and Unit Iteration” adapted from Chapin, S. and Johnson, A. Math Matters: Understanding the Math You Teach, Grades K–6. pp. 178–180. © 2000 by Math Solutions Publications. Used with permission. All rights reserved.

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Part A, cont’d. Another key point to grasp is that the chosen unit influences the number of units. For example, weighing a package in grams results in a larger number of units (2,000 g) than weighing it in kilograms (2 kg). This inverse relationship—a larger number of smaller units—is a conceptually difficult idea. Unit iteration is the repetition of a single unit. If you are measuring the length of a desk with straws, it is easy enough to lay out straws across the desk and then count them. But if only one straw is available, then you must iterate (repeat) the unit (straw). You first have to visualize the total length in terms of the single unit and then reposition the unit repeatedly. Problem A1. Counts of a number of objects are exact (e.g., you can have either three chairs or four chairs around the table, not between three and four chairs), yet measurements cannot be made exactly. Why is that so? What makes a count different from a measure? [See Note 3] Problem A2. The units on measurement instruments, such as rulers and thermometers, run together; they are not distinct as are, for example, the number of books on a shelf. a. Why might this aspect of measurement cause confusion? b. How is understanding a length of 7 in. or a temperature of 63 degrees Fahrenheit different from understanding that you have seven balloons or 63 pennies? Problem A3. Where else in mathematics is the concept of transitivity used? Give an example other than measurement.

Partitioning Let’s look more closely at the idea of a unit and how one goes about partitioning that unit into subunits. How are rational numbers (fractions and decimals) interpreted in measurement situations? Imagine that you are timing a swim meet. If you timed a 100 m backstroke race to the nearest hour, you would not be able to distinguish one swimmer’s time from another’s. If you refined your timing by using minutes, you still might not be able to tell the swimmers apart. If the swimmers were all well trained, you might not be able to decide on a winner even if you measured in seconds. In high-stakes competitions among well-trained athletes (the Olympics, for example), it is necessary to measure in tenths and 100ths of a second. Now suppose that you are working on a project that requires some precision. You need to determine the exact length of a strip of metal in inches. Holding the strip up to your ruler, with one end at 0, you see that the other end lies between 4 and 5 in. Note that only the right end of the metal strip is shown here. What would you say its length is?

You might think to yourself, “The length is between 4 9/16 and 4 10/16, so I’ll call it 4 19/32.”

Note 3. If you are working in a group, discuss Problems A1–A3 together. When discussing Problem A2, consider the fact that young children first learn about numbers using discrete quantities. How does that differ from measurement, which is never exact (discrete), as we can infinitely divide continuous quantities? “Partitioning” adapted from Lamon, Susan J. Teaching Fractions and Ratios for Understanding, pp. 113–121, © 1999 by Lawrence Erlbaum Publishers. Used with permission. All rights reserved.

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Part A, cont’d. These situations illustrate the nature of measurement interpretation of rational numbers. A unit of measure can always be divided into finer and finer subunits so that you can take as accurate a reading as you need. On a number line; on a graduated beaker; on a ruler, yardstick, or meterstick; on a measuring cup; on a dial; on a thermometer—some subdivisions of the unit are marked. The marks on these common measuring tools allow readings that are accurate enough for most general purposes, but if the amount of the object you are measuring doesn’t exactly meet one of the provided hash marks, it certainly doesn’t mean that you can’t measure it. Rational numbers provide us with a means to measure any amount of stuff. [See Note 4] If meters will not do, we can partition into decimeters; when decimeters will not do, we can partition into centimeters or millimeters—and so on. When we talk about rational numbers as measures, the focus is on successively partitioning the unit. Certainly partitioning plays an important role in other models and interpretations of rational numbers, but there is a difference. In measurement, there is a dynamic aspect; instead of comparing the number of equal parts you have to a fixed number of equal parts in a unit, the number of equal parts in the unit can vary, and what you name your fractional amount depends on how many times you are willing to keep up the partitioning process. In the above example, you’ve seen how the units were first divided into 16 equal parts and then into 32 equal parts (the fractional amount was thus expressed in 16ths or 32nds, respectively). If necessary, you could further partition the unit into 64 or more equal parts, each time refining the precision of your measurement. Problem A4. In your own words, clarify the difference between the measurement interpretation of rational numbers and the part–whole interpretation of rational numbers. [See Note 5] [See Tip A4, page 46] Problem A5. Why is the concept of partitioning so important in measurement?

Partitioning on a Number Line How many partitions of a number line are possible? To use a rational number to describe how far a point on the number line is from 0, you can begin by partitioning the unit interval into an arbitrary number of equal parts. Each of those parts can then be partitioned into an arbitrary number of equal parts, and those, in turn, can be partitioned again. This process is actually a composition of operations. You can use arrow notation to keep a record of your partitioning actions, as well as the size of the subintervals being produced.

Note 4. Rational numbers are what is known as a dense set: A dense set is such that for any two elements you choose, you can always find another element of the same type between the two. To learn more about the concept of density, go to the Learning Math: Number and Operations Web site at www.learner.org/learningmath and find Session 2. Note 5. To learn more about rational numbers and the part-whole interpretation of fractions, go to the Learning Math: Number and Operations Web site at www.learner.org/learningmath and find Session 8. “Partitioning on a Number Line” adapted from Lamon, Susan J. Teaching Fractions and Ratios for Understanding, pp. 113–121. © 1999 by Lawrence Erlbaum Publishers. Used with permission. All rights reserved.

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Part A, cont’d. For example, what if you wanted to locate 17/48 on a number line from 0 to 1? You would start by drawing the number line on a piece of paper and repeatedly folding it, making sure to mark the locations of 0 and 1 before you start folding:

Here’s one set of partitioning actions to find 17/48:

Video Segment (approximate times: 2:40–5:15): You can find this segment on the session video approximately 2 minutes and 40 seconds after the Annenberg/CPB logo. Use the video image to locate where to begin viewing. In this video segment, the participants place a fractional value on a number line using the method of partitioning. They explore the reciprocal relationship that exists between partitioning and the number of units in a measure.

Is there more than one way to do the partitioning to arrive at a particular fraction?

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Part A, cont’d. Problem A6. Try these partitioning tasks: [See Note 6] a. Locate 7/24:

b. Locate 3/8:

Take It Further Problem A7. Find another way (or ways) to locate the fractions in Problem A6 (a) and (b). [See Tip A7, page 46] The compensatory principle states that the smaller the subunit you use to measure the distance, the more of those subunits you will need; the larger the subunit, the fewer you will need. When multiples of two different subunits cover the same distance, different fraction names result. There is only one rational number associated with a specific distance from 0, so these fractions are equivalent. [See Note 7] For example, when measuring the diameter of a pencil using two different subunits, we would have the following:

But 1/4 and 2/8 are equivalent fractions, so these are the same measurements.

Write and Reflect Problem A8. State the compensatory principle in your own words. What type of relationship exists between the size of a measuring unit and the number of that unit needed to measure a property?

Note 6. If you are working in a group, work in pairs on both parts of Problem A6. First use the fraction given to find one unit; then consider how you can use partitioning and equivalence to locate the desired fraction. Note 7. The compensatory principle is an important mathematical idea. The idea of an inverse relationship between the size of a unit and the number of units can be examined numerically (e.g., the area of a surface that is 1 m2 can also be expressed as 10,000 cm2). An inverse relationship can also be shown graphically. A linear inverse relationship produces a straight line that is drawn diagonally from the upper left to the lower right in the first quadrant. Be sure to reflect on or discuss other inverse relationships when working on Problem A8.

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Part B: The Role of Ratio (45 min.) 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 pounds 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. The 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). [See Tip B1, page 46] 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.

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Part B, cont’d. 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-11in. 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? [See Note 8] [See Tip B3, page 46]

Take It Further 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: Diameter (in km) Distance from Sun in Scientific Notation (in km) Sun

1,392,000

Mercury

4,900

5.8 • 107

Venus

12,100

1.08 • 108

Earth

12,760

1.5 • 108

Mars

6,790

2.28 • 108

Jupiter

143,000

7.78 • 108

Saturn

121,000

1.43 • 109

Uranus

51,000

2.87 • 109

Neptune

50,000

4.5 • 109

Pluto

2,300

5.9 • 109

Problem B4. a. What scale would you use if you wanted to show students how far the planets are from the Sun? b. What scale would you use if you wanted to help students understand the differences in diameter size among the planets? Can the same scale be used for both goals? [See Note 9]

Take It Further Problem B5. A science park in Westerbork, Holland, uses the scale of 1:3.7 • 109 for a scale model of the solar system. What units do you think the park chose to use?

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.

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Part B, cont’d. 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 miles/h, this does not mean that 48 rowers would get the boat up to 30 miles/h and that with 144 rowers the boat would hit 90 miles/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. Use the triangle images on page 45. 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): Side Lengths (S) in cm

Hypotenuse Length Ratio S:S (H) in cm

Ratio H:S

1 2 3 4 5 6 Problem B7. a. What constant ratios did you find in the isosceles right triangles (45º-45º-90º)? 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? Quote taken from Zebrowski, Ernest, Jr. A History of the Circle. p. 3. ©1999 by Ernest Zebrowski, Jr. Reprinted by permission of Rutgers University

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Part B, cont’d. Video Segment (approximate time: 14:33–16:14): If you are using a VCR, you can find this segment on the session video approximately 14 minutes and 33 seconds after the Annenberg/CPB logo. Use the video image to locate where to begin viewing. Watch this video segment to see how one of the participants, David Cellucci, reasoned about the results obtained from finding the hypotenuse-to-side ratios of the right isosceles triangles. Watch this segment after you’ve completed Problem B6. Did you arrive at a similar conclusion?

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. [See Note 10] When 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: √162 = √81•2 = √81 • √2 = 9√2 Problem B8. a. 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 you notice the patterns in the ratio of H:S? Side Lengths (S) in cm

Hypotenuse Ratio S:S Length (H) in cm

1

1:1

2

2:2

3

3:3

4

4:4

5

5:5

6

6:6

Pythagorean Ratio H:S

b. 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. [See Note 11] In terms of measurement, what are the implications of one or more of the values being irrational?

Note 10. To learn more about the Pythagorean theorem, go to the Learning Math: Geometry Web site at www.learner.org/learningmath and find 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.

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Part C: Precision and Accuracy (30 min.) Measuring With Precision We have learned that physical measurement involves error and that every physical measurement is an approximation. This leads us to a new question: How much error is involved in any given measurement? The terms precision and accuracy relate to how good an approximation is. For example, how precise were our measurements of the sides of the right triangles, and how accurate were our measurements of the distance from Mars to the Sun? Since measurements are approximate, the most meaningful way of interpreting a measurement is as an interval with a lower bound and an upper bound. Imagine that we have measured a line segment, using a ruler divided into centimeters, and found the length to be 5 cm. To be more precise, we can state the measure as an interval— either in words, 5 cm to the nearest 0.5 cm, or using notation, such as 5 cm ± 0.5 cm (read “5 cm plus or minus 0.5 cm”). Either presentation gives the center of the interval and the distance of the upper and lower bounds from this center (5 ± 0.5 implies a lower bound of 4.5 and an upper bound of 5.5). We can also state that the maximum possible error for this measure is 0.5 cm (which is half the size of the measurement unit). [See Note 12] In summary, the precision of a measurement depends on the size of the smallest measuring unit—whether the measurement is, for example, to the nearest 10 ft., to the nearest foot, or to the nearest tenth of a foot. The smaller the interval, the more we have “narrowed it down,” and thus the more precise the measurement. Video Segment (approximate time: 6:04–8:04): If you are using a VCR, you can find this segment on the session video approximately 6 minutes and 4 seconds after the Annenberg/CPB logo. Use the video image to locate where to begin viewing. Watch this video segment to see the participants discuss whether a measurement is an approximate or an exact value. They also discuss the role partitioning plays in answering this question.

Can you think of any other reasons that would support their conjecture? Problem C1. In Part A, we discussed how a unit can be partitioned into smaller subunits. How are partitioning and precision related? [See Tip C1, page 46] Problem C2. When measuring length, the precision unit is determined by the smallest unit being repeated on the measuring tool (the smallest hash mark). Examine the rulers below (not drawn to scale), and identify the precision unit:

Problem C3. The maximum possible error of a measurement is always half the size of the precision unit. For example, if the precision unit is 1 cm, the maximum possible error is 0.5 cm; if the precision unit is 4 cm, the maximum possible error is 2 cm, etc. What is the maximum possible error for each ruler above?

Note 12. Another way to think of an interval is as a boundary; namely, that the measurement is going to fall within a range given by an upper and lower boundary. The range is determined first by the smallest measuring unit on the instruments being used (the precision unit) and then by the maximum possible error for that precision unit. Discuss or reflect on why the maximum possible error cannot be more than half the precision unit.

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Part C, cont’d. Problem C4. In Problem B6, you measured isosceles right triangles and found that the hypotenuse of the triangle with legs of 3.0 cm was 4.2 cm. What is the precision unit? Give the interval that shows a more accurate measure of the hypotenuse.

Accuracy vs. Precision In everyday language, we use the terms accuracy and precision interchangeably. In mathematical terminology, however, the accuracy of a measure (an approximate number) is defined as the ratio of the size of the maximum possible error to the size of the number. This ratio is called the relative error. We express the accuracy as a percent, by converting the relative error to a decimal and subtracting it from 1 (and writing the resulting decimal as a percent). The smaller the relative error, the more accurate the measure.

Try It Online!

www.learner.org

This relationship can be explored online as an Interactive Activity. Go to the Measurement course Web site at www.learningmath.org/learningmath and find Part C.

Here’s how it works. We can measure two different items to the nearest centimeter: a desk and a notepad. The desk is 120 cm long, and the notepad is 12 cm long. The maximum possible error in each case is 0.5 cm. Both measures are to the same level of precision, but the relative error differs: Relative Error

=

Maximum Error Size of Number

So the relative errors for the desk and the notepad are: 0.5 120

=

0.0042

0.5 12

=

0.042

Therefore, though the measurements of 120 cm and 12 cm are equally precise, they are not equally accurate. The measurement of 120 cm is more accurate because it has the smaller relative error. You can record the accuracy as a percentage by subtracting the relative error from one and writing the resulting decimal as a percentage. So, in the first instance, the accuracy is 1 - 0.0042 = 0.9958, or 99.58%, and in the second instance, it is 1 - 0.042 = 0.958, or 95.8%. Clearly, the accuracy of the two measurements differs significantly. [See Note 13]

Take It Further Problem C5. Calculate the relative error for the measurement you made on the isosceles right triangle with side 3, and then one or two others.

Note 13. If you are working in a group, be sure to discuss the idea of accuracy. How do maximum possible error and relative error differ? Why do you think relative error is calculated as a ratio? How might we improve our approximations when measuring?

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Homework Problem H1. Imagine that you are playing the Between game with a partner. Player A picks a decimal number between 5 and 6—say, 5.7—crosses out the number 5, and writes down 5.7 in its place. Player B picks a number between 5.7 and 6, such as 5.9, crosses out the number 6, and writes down 5.9. Now Player A must pick a number between 5.7 and 5.9—i.e., 5.8—and replace Player A’s previous number (5.7) with it. Play continues for a total of 10 rounds (the above describes three rounds). Imagine that the Between game was a measuring task where you were trying to become more accurate with each measurement. What does this game tell you about the nature of measurement? Problem H2. Suppose that a scale of 1 in.:20 ft. was used for building a model train. What does this mean? If a railcar is 40 ft. long, how long is the scale model? Problem H3. A science class wants to create insects that are larger than life. They found that a queen ant is 0.5 in. long. The large model they plan to create is 5 ft. long. What is the scale factor for the ant model? Problem H4. a. If a full gas tank holds 16 gallons, put an arrow on the following gas gauge to show how much gas you would have left in your tank if you filled it up and then took a drive that used 6 gallons:

b. Your gas tank said “Empty,” but you were low on cash. You used your last $4 to buy gas, paying $1.139/gallon. If a full tank holds 14 gallons, put an arrow on the gas gauge to show how much gas you had in your tank after your purchase:

c. You filled up your tank this morning, then took a drive in the country to enjoy the fall colors. Your odometer said that you had gone 340 miles, and you have been averaging about 31 miles/gallon. If your gas gauge looked like this when you got home, how much does your gas tank hold when it is full?

Problem H4 adapted from Lamon, Susan J. Teaching Fractions and Ratios for Understanding, pp. 121–123, © 1999 by Lawrence Erlbaum Publishers. Used with permission. All rights reserved.

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Homework, cont’d. Problem H5. Suppose that you had the following 10 measurements (in centimeters) of the same object: 31.9

32.0

31.9

32.1

32.0

32.2

32.4

32.3

32.5

32.4

a. With these data, what would you give as the best approximation? Explain why you give that approximation. What would be the precision unit for your best estimate? [See Tip H5, page 46] Suppose that you made five measurements in addition to the 10 listed above: 32.1

32.2

32.3

32.3

32.4

b. What is your best approximation now? Suppose that you made a total of 20 measurements—the 15 above and the following five: 32.0

32.9

32.4

32.2

32.1

c. What is your best approximation now? Did it change? d. In general, what effect on a best approximation do you expect if the results of more and more measurements are reported? Problem H6. Take a piece of paper and measure its length and width. What level of precision will you use to measure? What is the accuracy of your measure in terms of the relative error? [See Tip H6, page 46]

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Triangles for Problems B6-B8

3 2 1 2

1

3

5 4

5

4

6

6

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Tips Part A: Measuring Accurately Tip A4. Part-whole interpretation of rational numbers refers to dividing one or more units into equal-sized parts. You can think of it as pieces of a pie—3/4 would mean three equal-sized slices from a total of four. Tip A7. Start with a new number line.

Part B: The Role of Ratio Tip B1. Remember that 1 km = 1,000 m, and 1 m = 100 cm. Tip B3. 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.

Part C: Precision and Accuracy Tip C1. Think about what you would do to a measuring instrument if you wanted to measure something more precisely.

Homework Tip H5. Think about finding the mean or an average data value of the set. Tip H6. Remember, the level of precision depends on the measuring instrument and its unit of measure. Accuracy depends on relative error. You can record the accuracy as a percent by subtracting the relative error from 1 and writing the resulting decimal as a percentage.

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Measurement

Solutions Part A: Measuring Accurately 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.), .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.

Measurement

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Session 2: Solutions

Solutions, cont’d. 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.

Part B: The Role of Ratio 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, use 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. a. 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 meters), 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 9

5.9 • 10 km

=

1 • 102 m 12

5.9 • 10 m

=

1 5.9 • 1010 m .

b. 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. Of course, at this scale, you couldn’t place the planets at their relative distances on the school ground. Problem B5. Most likely, they chose kilometers. The longest distance from the Sun, 5.9 • 109, becomes approximately 1.6 km. So the scale was likely chosen to let the entire model fit within less than 2 km. 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:

Session 2: Solutions

Side Lengths (S) in cm Hypotenuse Ratio S:S Length (H) in cm

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

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Measurement

Solutions, cont’d. Problem B7. a. 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. b. 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. a. Here is the completed table: Side Lengths (S) in cm

Hypotenuse Ratio S:S Length (H) in cm

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.

Part C: Precision and Accuracy Problem C1. The precision of a measurement is directly related to the partitioning of the measuring instrument. We can more precisely measure something when the partitions are closer together. For example, a beaker with milliliter partitions gives a more precise liquid measure than a measuring cup with 100-milliliter partitions. Problem C2. a. The precision unit is 1 ft. b. The precision unit is 1 mm. c. The precision unit is 1 cm. d. The precision unit is 1 in.

Measurement

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Session 2: Solutions

Solutions, cont’d. Problem C3. a. The maximum possible error is 0.5 ft. b. The maximum possible error is 0.5 mm. c. The maximum possible error is 0.5 cm. d. The maximum possible error is 0.5 in. Problem C4. The length of the hypotenuse mentioned in this problem is 4.2 cm, so the precision unit is 0.1 cm, because each centimeter is divided into tenths. The maximum possible error is 0.05 cm. The side length of the hypotenuse could be written as 4.2 cm ± 0.05 cm, or between 4.15 and 4.25 cm. If you expressed the length of the hypotenuse as 42 mm, then the precision unit would be 1 mm, and the maximum possible error would be 0.5 mm (in either direction). The side length of the hypotenuse could be written as 42 mm ± 0.5 mm, or between 41.5 and 42.5 mm. Problem C5. Since each centimeter is divided into tenths, the precision unit is 0.1 cm, and the absolute error for the calculations is ±0.05 cm; therefore, the relative error is 0.05 divided by the measurement. For example, when the hypotenuse was measured as 4.2 cm, the relative error was 0.05/4.2, which gives you 0.012 or 1.2/100 (about 1.2%). Larger measurements with the same absolute error will have lower relative error.

Homework Problem H1. The fact that this game does not reach an “ending” (with no moves available) shows that there can always be a more accurate measurement, and therefore the act of measuring can never be exact. Problem H2. This means that all lengths in the model are 1 in. long for every 20 ft. in the original model. If the original railcar is 40 ft. long, we can set up the following proportion to calculate the model’s length: 1 in. x in. Solving the equation for x would give us x = 2. So the model’s length will be 2 in. = 20 ft. 40 ft. Problem H3. The scale factor is 5 ft.:1/2 in. For easier calculations, we can convert the ratio into inches. Since 1 ft. = 12 in., the scale factor would be 60:1/2, or 120:1. Problem H4. a. One way to do this is to partition the gas gauge into 16 equal parts by dividing it in half four times (2 • 2 • 2 • 2 = 16). You can then count 6 gallons. Another way is to note that 6/16 = 3/8, so partitioning the gauge into eighths is sufficient.

Session 2: Solutions

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b. One way to solve this problem would be as follows: Filling up the entire tank would cost $1.139 • 14 = $15.94. Four dollars is one-fourth of $16, so it is just a little more than one-fourth of $15.94. The arrow should be just slightly above one-fourth full.

Measurement

Solutions, cont’d. Problem H4, cont’d. c. First you need to find out how much gas you used for 340 miles: 340 ÷ 31 = 10.97 So approximately 11 gallons were used. Next, look at the needle position: It is approximately one-third of the way from “Empty” to the first quarter mark. Further subdividing the gauge into 12ths, we see that the needle is about 1/12 of the way from “Empty.” Since about 11 gallons were used, and 11/12 of the tank is empty, the tank therefore holds about 12 gallons when full.

Problem H5. a. The best approximation might be about 32.2 cm, which is in the center of the data set. The average (mean) of the 10 measurements is 32.17 cm. The precision unit might be 0.1 cm (or 1 mm), as all 10 measurements are recorded to the nearest tenth. b. The mean of the 15 measurements is 32.2 cm. Again, the precision unit might be 0.1 cm. Our approximation, however, is better now since we have more measurements. c. The mean is now 32.23 cm. Even though the measurement errors are pretty high, this gives us even more confidence that the actual measure is close to 32.2 cm. d. With more measurements we will get an increasingly better approximation. Problem H6. The level of precision depends entirely on the measuring instrument. If we use a measuring stick with 1-in. precision, the maximum error is ±0.5 in. The accuracy can be obtained by subtracting the relative error from 1 and writing the resulting decimal as a percent. For the longer side of the paper, the relative error is 0.5/11, or about 0.045 (so the accuracy is 1 - 0.045 = 0.95, or 95%). The relative error for the shorter side is 0.5/8.5, or about 0.059 (so the accuracy is 1 - 0.059 = 0.94, or about 94%), which is less accurate. A measuring stick with more partitions will give more precise and more accurate measurements.

Measurement

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Session 2: Solutions

Notes

Session 2

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Measurement