Diagnosing Decimal Place Value in Grade 5
Ms. Chen's Grade 5 class of 24 has just finished a short diagnostic on decimals. Item 1: 'Which is greater, 0.4 or 0.15?' Eleven students chose 0.15, eight of them writing 'because 15 is bigger than 4.' Item 2: 'Order 0.6, 0.06, 0.66.' Nine students ordered all three correctly. Item 3: 'Write a decimal between 0.3 and 0.4.' Seventeen students wrote 0.4 or left it blank; four wrote 0.35. The unit test is in eight school days, and the Grade 6 teacher has asked that students arrive able to compare and order decimals to thousandths.
1. Do the three items together indicate a problem with a rule or with place value? Cite the specific pattern in the data that supports your reading.
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Place value, not a missing rule. The giveaway is the contrast between Item 2 and Item 3.
- On Item 2, nine students ordered correctly - the same students who can apply 'line up the place values' or 'add zeros and compare.' That rule needs no quantity sense and produces correct answers on comparing and ordering given decimals.
- On Item 3, seventeen of twenty-four could not generate a decimal between 0.3 and 0.4. No comparison rule can do this; the student has to know that the interval between tenths can be subdivided into hundredths. Four students writing 0.35 is the only direct evidence of that structure in the whole diagnostic.
- Item 1's failures cluster on digit length ('15 is bigger than 4'), which is the same underlying issue seen from the other side: the digits are being read as whole numbers, not as tenths and hundredths.
So the class holds a comparison procedure without a quantity sense of tenths. Teaching the rule harder would raise Item 1 and 2 scores and leave Item 3 - and rounding, and Grade 6 work - untouched.
2. Write one follow-up question that would separate students who think 0.6 means 'just 6' from those who think of it as six tenths, and say what each response would lead you to do.
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Question: 'Which is closer to 1: 0.6 or 0.06? Place both on a number line from 0 to 1 and explain without using a rule.'
This is a discriminator because a comparison rule alone does not produce it. A student who reads 0.6 as 'just 6' places it near the end or off the line, or reasons '6 is bigger, so 0.6 is closer to 1' - right answer, wrong reason - and typically cannot place 0.06 at all. A student thinking in tenths places 0.6 in the sixth interval and 0.06 just past zero, and can say 0.06 is six hundredths, so it is nearly at 0.
Responses:
- Student places 0.6 accurately and 0.06 only approximately: the tenths structure is there; move them to subdividing a tenth strip into hundredths and straight into Item 3-type tasks.
- Student places 0.6 by counting six marks on the line: they have the linear model but not place value; go to the hundredths grid, build 0.6 and 0.06, and compare areas.
- Student places both correctly but says 'six is bigger than one, so closer to 1': keep them in the small group on hundredths grids - correct placement is not yet correct reasoning.
3. Explain why Item 3's results are the most concerning of the three, and outline two or three activities with materials that target it in the time available.
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Why Item 3 matters most: it is the only item that cannot be answered by a comparison procedure. It probes whether the student's number line has structure between the marked tenths - the idea that decimals are dense, that there is always another number in between. That structure is the prerequisite for rounding, for ordering to thousandths, and for the Grade 6 expectation the receiving teacher has named. It also predicts the failure mode that survives the unit test: students who pass a comparison test by rule and collapse on estimation or on placing a decimal in a context.
Sequence, roughly two periods:
- Build and place. Students build 0.3 and 0.4 with hundredths grids, cut a tenths strip into ten equal parts, and place both on a 0–1 line. Language target: 'three tenths is thirty hundredths.'
- Subdivide. In pairs: 'Find three different decimals between 0.3 and 0.4. Record each with a grid.' Circulate and select 0.31, 0.35, 0.39 for the consolidation, then ask for one between 0.35 and 0.36.
- Extend to thousandths. Subdivide a single hundredths square into ten to reach a decimal between 0.35 and 0.36, then between 2.45 and 2.46. Close with the original Item 3 as an exit card.
I would not use 'add a zero to compare' as the fix; it is the rule whose success is already masking the gap.