Density and Decimal Equality in a Number-Systems Warm-Up
During a Grade 10 number-systems lesson, a teacher poses a warm-up: 'Is there a rational number between 0.33 and 1/3?' Of 28 students, 19 answer no. Their explanations cluster around two claims: (a) 0.33 and 0.333… are consecutive decimals, so nothing fits between them; (b) the gap is only about 0.00333…, too small to contain a fraction. One student also insists that 0.999… is 'just below 1' and that 22/7 is exactly π. The teacher has one 50-minute period before the unit quiz and wants to use the responses to build a conceptual lesson on density of the rationals, decimal representation, and exact equality. The department requires the lesson to include a written justification task and a check for whether students can distinguish 'close' from 'equal.' Calculators are available, but no graphing technology. The teacher plans to group the 19 responses by explanation type.
1. Determine whether a rational number exists strictly between 0.33 and 1/3, and justify your answer by constructing one. Then state whether the same reasoning applies to 0.999… and 1.
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0.33 = 33/100 = 99/300, and 1/3 = 100/300. Their average is (99/300 + 100/300)/2 = 199/600 ≈ 0.331666…, which lies strictly between them. A simpler construction is 331/1000 = 0.331, since 0.330 < 0.331 < 0.333… . This works because the rationals are dense: given distinct rationals a < b, (a+b)/2 is rational and satisfies a < (a+b)/2 < b. The case 0.999… and 1 is different: they are the same real number. If x = 0.999…, then 10x = 9.999…, so 10x − x = 9, giving 9x = 9 and x = 1. Equivalently, 0.999… = 3 × 0.333… = 3 × (1/3) = 1. There is no positive gap, so density does not produce a number between them; they are not two distinct points.
2. Diagnose the misconception in claim (a) and describe a number-line or decimal-expansion representation that would confront it.
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The misconception treats a terminating decimal and a repeating decimal as adjacent labels on a number line. It implicitly assumes a 'next' decimal after 0.33 and before 0.333…, which is false. Use a zoomed number line: mark 0.33 and 1/3, then mark 0.331 and 0.3317; students see room for more numbers. Convert to fractions with a common denominator to remove the decimal camouflage: 33/100 and 1/3 become 99/300 and 100/300; there are many rationals between, for example 199/600. Also ask students to find the midpoint repeatedly, showing there is no smallest gap. For 0.999…, the same representation can show equality: 1/3 = 0.333…, and multiplying by 3 gives 1 = 0.999…, so the number line has one point, not two.
3. Design a brief follow-up item that distinguishes procedural answer from conceptual understanding, and state what response would count as evidence.
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Item: 'Find two different rational numbers strictly between 0.49 and 0.50. Then explain why your method will always produce another rational between any two distinct rationals you choose.' Also include: 'True or false: 0.4999… = 0.5. Justify without using a calculator.' Evidence of understanding: the student constructs, say, 0.495 and 0.4975, or uses fractions 49/100 and 1/2 with common denominator 200 and picks 99/200, and explains averaging or common denominators. A rote response only gives one decimal and cannot say why the method generalizes, or claims 0.4999… is less than 0.5 because it 'looks smaller.' The teacher should look for an explicit statement that distinct rationals have a midpoint and that repeating 9s represent the same real number.