Field Day Long Jump: Comparing Decimals Recorded at Different Precisions
At the fifth-grade field day, four students' best long jumps were recorded, but the judges did not all measure the same way. Ana's jump is recorded as 3.4 meters, Ben's as 3.45 meters, Cara's as 3.405 meters, and Dev's as 3.5 meters. First-, second-, and third-place ribbons will be awarded. A parent volunteer says, "Cara should get second, because 405 is bigger than 45." Another volunteer points out that one judge read the tape to the nearest tenth, one to the nearest hundredth, and one to the nearest thousandth, so the four numbers may not be directly comparable as written. Before any ribbons are handed out, the physical education teacher wants the jumps ordered from greatest to least, and she wants to know whether rounding all four jumps to the nearest tenth would change who stands on the podium.
1. Order the four jumps from greatest to least and explain what is wrong with the parent's reasoning about 3.405 and 3.45.
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Write all four with the same number of decimal places so place values line up: 3.400, 3.450, 3.405, 3.500.
Order: Dev 3.500 > Ben 3.450 > Cara 3.405 > Ana 3.400.
The parent compared 405 with 45 as though the digits after the decimal point were whole numbers. The correct comparison is in thousandths: 3.405 = 3 + 405/1000, and 3.45 = 3 + 450/1000. Since 450 thousandths is greater than 405 thousandths, Ben's jump is longer than Cara's.
A second way to see it: 3.45 has a 5 in the hundredths place while 3.405 has a 0 in the hundredths place, so 3.45 is larger regardless of what follows. Comparing decimals means comparing matching place values from left to right - never comparing the length of the digit string.
2. Round each jump to the nearest tenth and determine whether the podium changes. Justify your conclusion.
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Rounding each to the nearest tenth:
- Ana 3.4 → 3.4
- Ben 3.45 → 3.5 (the 5 in the hundredths place rounds the tenths digit up)
- Cara 3.405 → 3.4 (the 0 in the hundredths place leaves the tenths digit alone; the 5 further right does not matter)
- Dev 3.5 → 3.5
Rounded standings: Dev 3.5 and Ben 3.5 tied for first; Ana 3.4 and Cara 3.4 tied for third.
Yes, the podium changes. In the exact standings Cara is a clear third. After rounding she is tied with Ana and no longer holds third alone - a tiebreaker would have to be invented, or Cara could lose the ribbon. The rounding threw away the 0.045-meter difference that decided third place, which is why unrounded measurements should be used for awards.
3. Recommend how the meet should handle measurement precision in the future, and explain the reasoning in terms of what a recorded value like 3.4 actually represents.
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The meet should require one precision for everyone - thousandths of a meter - read the same way from the same type of tape, with each jump measured by two judges.
The reason is that a recorded value carries an implied range. A jump recorded as 3.4 to the nearest tenth means the true distance is somewhere from 3.35 to 3.45. That range overlaps Ben's 3.45, so the record cannot prove Ana's jump was shorter by the amount the digits suggest; the rounding hid up to 0.05 meter, which is larger than the gaps separating several of these jumpers.
With matched precision, differences like 3.405 versus 3.450 reflect real measurement rather than differences in how the tape was read.