Number-Line Exit Tickets in Grade 4
Mr. Chen's 24 fourth graders completed an exit ticket asking them to place 5/4 and 2/3 on a number line and explain which fraction is greater. Eighteen students placed 5/4 correctly between 1 and 2; six placed it between 0 and 1. In follow-up interviews, five of those six said 'a fraction is always less than 1,' and one said '5/4 is less than 1 because 5 is small.' When comparing 2/3 and 5/4, nine students compared numerators only, eight compared denominators only, and seven gave a correct comparison. Mr. Chen has 35 minutes for the next lesson and fraction strips, number lines, and linking cubes available. He wants to avoid jumping straight to a common-denominator procedure because several students still seem unsure what a fraction's size means.
1. Diagnose the patterns in the exit-ticket and interview data. Which misconception is most important to address first, and why?
Read the worked answer
The most consequential pattern is the idea that a fraction cannot exceed 1. Six students placed 5/4 below 1, and five stated that rule directly. That misconception blocks all later work with improper fractions, benchmarks, and number-line placement. The student who said '5 is small' is using whole-number reasoning about the numerator without attending to the unit. The comparison errors show two related habits: comparing numerators only or denominators only. Those are not identical to the 'less than 1' error, but they share the same root: students are treating fraction symbols as separate whole numbers rather than as quantities on a line.
The first instructional target should therefore be fraction magnitude with 1 as a benchmark. Common-denominator rules would let some students produce correct answers without repairing the underlying quantity idea. The data also suggest keeping 5/4 visible as a greater-than-1 example, because it directly contradicts the mistaken rule. Mr. Chen should not re-teach all fraction concepts; he should focus on locating fractions relative to 0, 1/2, 1, and beyond 1, then use those benchmarks to compare.
2. Outline a 35-minute sequence using representations that addresses the misconception. Justify the order of the representations.
Read the worked answer
Begin with a 5-minute number talk: 'Is 5/4 more or less than 1? How do you know?' Have students justify with words before symbols. Next, 10 minutes with fraction strips: students build 4/4 and then add a fourth strip to see 5/4 as one whole plus 1/4. This makes the unit and the 'more than 1' idea concrete. Then 10 minutes on the number line: students place 1/2, 1, 3/4, 5/4, and 7/4, attending to equal intervals and the whole. The number line is the bridge because it shows magnitude rather than just part-whole shading. Next, 7 minutes comparing 2/3 and 5/4 using the benchmark 1: 2/3 is less than 1 and 5/4 is greater than 1, so 5/4 is greater. Finally, 3 minutes for a new exit ticket.
The order moves from concrete strips to a linear model to a benchmark argument. It avoids the common-denominator algorithm until students can explain why the comparison makes sense. Linking cubes can support students who need to build wholes, but the number line should remain central because that is the representation on the assessment.
3. What should Mr. Chen look for on the new formative check, and how should he follow up with the six students who placed 5/4 below 1?
Read the worked answer
The check should include three quick items: place 7/4 on a number line, decide whether 3/4 or 7/4 is greater and explain with 1 as a benchmark, and write one sentence explaining why 5/4 is greater than 1. Mr. Chen should look for placement beyond 1, equal spacing, and language such as 'one whole and one fourth' or 'more than one whole.' He should also watch for students who produce the right comparison but still describe 5/4 as 'five and four,' which would suggest the symbol meaning is still fragile.
For the six students, a brief follow-up interview is better than a full reread of the lesson. Ask each to build 4/4 with strips, add one more fourth, and then place the result on a number line. If they can do it with the strip but not the line, the next step is number-line practice with visible whole markers. If they still say fractions cannot exceed 1, use a repeated-halves or fourths sequence beyond 1 and have them explain the pattern. The goal is to replace the rule with a quantity-based explanation, not just to correct one exit-ticket answer.