Third-Grade Fraction Misconceptions on the Number Line
A third-grade teacher has 24 students and five 45-minute mathematics lessons before the fractions unit assessment. Pre-assessment results show 14 students can identify unit fractions, 9 can order fractions with the same denominator, and 5 can compare fractions with unlike denominators. Three work samples stand out. Student A writes, '1/4 > 1/3 because 4 is bigger than 3.' Student B writes, '2/3 = 2/4 because both have 2.' Student C marks 3/4 at the third tick on a number line divided into four equal parts and labels it 3. The teacher has fraction strips, blank number lines, and picture cards but no additional aide. The district expects Tier 1 instruction to use multiple representations and to check for conceptual understanding, not just procedures. The teacher must decide what to reteach, how to sequence the five lessons, and how to regroup students without labeling them.
1. Diagnose each work sample and explain what each reveals about the student's fraction understanding.
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A treats numerator and denominator as separate whole numbers and applies a whole-number rule ('4 > 3') to fractions; she lacks the idea that a larger denominator means smaller equal parts. B sees the common numerator and assumes equality, ignoring the denominator; this suggests she is matching surface features rather than comparing the size of parts. C can partition and count ticks but confuses the tick label with the fraction value; marking 3/4 at the third tick and labeling it 3 suggests the student does not yet treat the number line as a whole from 0 to 1 with 3/4 as a point. None of these is a careless error; each points to a different missing concept: equal parts, unit fraction size, and the whole/point relationship.
2. Design a five-lesson sequence using representations to address the misconceptions, with specific teacher moves.
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Sequence: Lesson 1, equal parts and unit fractions with fraction strips; students compare 1/2, 1/3, and 1/4 by covering the same whole. Lesson 2, build fractions on a number line from 0 to 1, defining the whole first and iterating each unit fraction. Lesson 3, compare same-numerator and same-denominator fractions using strips and number lines; explicitly confront A and B. Lesson 4, compare unlike denominators by reasoning about unit fractions and the benchmark 1/2. Lesson 5, mixed practice with error analysis and a short explanation task. Teacher moves include asking 'What is the whole?', having students justify with a representation, and using contrasts such as 1/3 versus 1/4. This addresses misconceptions rather than just reteaching rules.
3. Describe a formative assessment and how the teacher should regroup students based on the results.
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Formative check: give four items-place 3/4 on a number line, compare 2/3 and 2/4 with a drawing, explain why 1/4 is not greater than 1/3, and identify a false statement. Sort responses into three groups: needs equal-parts reteaching, needs number-line/whole support, and ready for unlike-denominator comparison. Regroup for the next lesson based on the sort, not on the pre-assessment alone. A decision rule: if fewer than 70% of students correctly place 3/4 and justify one comparison, reteach with strips before moving to the assessment. Keep the grouping temporary and public as 'what we need next,' not as fixed ability labels.