Elapsed Time and a Scaled Picture Graph
Mrs. Chen's third grade is finishing the measurement unit. Friday's quiz had two items. Item 1: a bus leaves at 8:45 a.m. and arrives at 9:20 a.m.; how long is the ride? Theo wrote 1 hour 25 minutes and explained, '45 minus 20 is 25, and the hour changed.' Item 2: a picture graph of favorite field-day events uses a key where one star stands for 4 students. The tug-of-war row shows 6 full stars and one half star. Omar wrote 7 students and said, 'six and a half stars.' Nia wrote 26 students and explained, '6 × 4 = 24, and half of 4 is 2.' The unit test blueprint has 8 elapsed-time items and 2 scaled-graph items. Mrs. Chen has two 30-minute review blocks before the test and wants to spend them where the evidence points.
1. Diagnose Theo's elapsed-time error and show two correct methods he could use.
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Theo subtracted the minute numbers without respecting the hour boundary; 9:20 is not 45 + 20 minutes after 8:45. The ride crosses 9:00, so the minutes before and after the hour must be counted separately. Number-line method: 8:45 to 9:00 is 15 minutes, and 9:00 to 9:20 is 20 minutes, so the total is 35 minutes. Counting-back method: from 9:20 back 20 minutes is 9:00, and back another 15 minutes is 8:45, so again 35 minutes. A third check is to add forward: 8:45 plus 35 minutes equals 9:20. Theo's 1 hour 25 minutes would put arrival at 10:10, which a clock model shows immediately. The repair is to anchor on the hour, not to memorize another subtraction rule.
2. Explain Omar's scale error and how Nia's reasoning should be used with him.
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Omar counted symbols and treated the half star as half a student, ignoring the key. The graph's unit is 4 students per star, so each half star is 4 ÷ 2 = 2 students. Nia's explanation shows the right structure: 6 stars × 4 = 24, plus 2 = 26. With Omar, cover the key and ask what one star means, then have him write '1 star = 4 students' above the graph and relabel each star with 4, 8, 12, and so on before recounting. Then ask him to find the total for 3½ stars and explain why the answer is not 3.5. A quick check is a row with 4½ stars, where the answer is 18, an integer that cannot be reached by treating the half symbol as half a student.
3. The blueprint has 8 elapsed-time items and 2 graph items. How should Mrs. Chen allocate the two review blocks, and what evidence would show readiness?
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Allocate roughly 45 of the 60 minutes to elapsed time and 15 to scaled graphs, matching the 8:2 weighting. Block 1: number-line routines with problems that cross the hour (8:40 to 9:15, 11:50 to 12:25) and problems that cross noon, where a.m. and p.m. labels matter; include both counting back and adding on. Block 2: 10 minutes of mixed elapsed-time review, then 15 minutes on keys, including half symbols and a bar graph with a scale of 5. Readiness evidence: Theo solves two crossing-the-hour items correctly with a number line and verifies by adding forward; Omar reads a 3½-symbol row as 14 and explains the key in his own words. If Theo still subtracts minute digits, keep the clock and number line rather than moving to abstract subtraction.