Ordering Decimals: Three Errors, One Underlying Idea
Ms. Alvarez gives her fourth graders four cards - 0.6, 0.45, 0.405, and 0.5 - and asks them to order the decimals from least to greatest. Three responses surface during the share-out. Jalen writes 0.45 > 0.5 "because 45 is bigger than 5." Priya writes 0.405 > 0.45 "because 405 has more digits than 45." Diego produces the correct order but explains that 0.405 and 0.45 "are the same, since zeros don't count." Ms. Alvarez has 20 minutes left, a set of laminated hundredths grids, and a 0-to-1 number line taped to the floor. She wants one follow-up task that surfaces the shared misconception rather than three separate corrections.
1. Give the correct ordering and state the single conceptual issue that all three responses share. Then explain what each representation (hundredths grid, number line) makes visible about that issue.
Read the worked answer
Correct order: 0.405 < 0.45 < 0.5 < 0.6.
Shared issue: all three students compare numerals as whole numbers instead of attending to place value, and none of them treats the decimals as occupying the same interval. Jalen compares 45 to 5 without asking "5 what?" Priya uses length of numeral as a size cue. Diego reads a trailing zero as meaningless rather than as an unfilled place, so he cannot see that 0.45 = 0.450 and 0.405 = 0.405, which differ by 45 thousandths.
- Hundredths grid: 0.6 and 0.5 each shade 60 and 50 whole squares; 0.45 shades 45; 0.405 shades 40 full squares plus half of one. The grid forces the comparison "how many hundredths?" instead of "which numeral looks bigger."
- Number line: all four decimals sit between 0 and 1 on the same scale, so 0.405 landing just past 0.4 and 0.45 landing near the midpoint of 0.4 and 0.5 makes the ordering visual and contradicts Jalen and Priya simultaneously.
2. Design the 15-minute follow-up task, including the exact question you would pose and the numbers used. State what student evidence would show the misconception is resolved.
Read the worked answer
Task: Each pair places 0.4, 0.405, 0.45, 0.5, and 0.6 on the floor number line and then writes each as a decimal with the same number of decimal places (to thousandths) on a sticky note placed above the point.
Key question: "Rewrite every number in thousandths, then tell me which two are closest together and how far apart they are."
Anticipated work: 0.4 = 0.400; 0.405 = 0.405; 0.45 = 0.450; 0.5 = 0.500; 0.6 = 0.600. Gap between 0.405 and 0.450 is 0.045 (45 thousandths); gap between 0.450 and 0.500 is 0.050. So 0.405 and 0.45 are the closest pair - a result no student in the transcript would have predicted.
Evidence of resolution:
- Students volunteer that 0.5 must be read as 0.500 before comparing with 0.45.
- Jalen-type reasoning disappears: the student says "50 hundredths is more than 45 hundredths" rather than "5 is less than 45."
- Diego-type reasoning is replaced by "the zero in 0.405 is holding a place; it is not the same as 0.45."
- Students can place 0.405 slightly left of 0.45 on the line without counting grid lines.
3. For students who finish early, pose: "How many decimals are between 0.45 and 0.46?" What is the mathematically correct response, and what does a student's answer reveal?
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There are infinitely many. A student can always insert another decimal: 0.451, then 0.4501, then 0.45001, and so on, or take midpoints such as 0.455 and then 0.4525.
A student who says "none, because 0.45 and 0.46 are next to each other" is still using a whole-number model in which consecutive numerals are adjacent. A student who says "ten, because the hundredths place only goes up to 9" correctly sees 0.451–0.459 but has not yet recognized thousandths. A student who says "as many as you want - I can keep adding digits" has connected decimal place value to the density of the number line, which is the target idea. Follow up by asking the student to name a decimal between 0.45 and 0.451, which forces the thousandths-and-beyond reasoning.