Diagnose decimal comparison errors using equal place-value units.
Teaching companion · Version 2.0.0
Compare decimal values in shared place-value units rather than digit strings.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
Compare 0.9 and 0.12 after naming both in hundredths.
Look for: 0.9 = 90 hundredths, so 0.9 > 0.12.
If you observe: Uses 9 < 12 to claim 0.9 is smaller.
Respond: Repartition nine tenths into ninety hundredths and place both counts on the same grid.
Compare 0.4, 0.40, and 0.04. Which two are equal?
Look for: 0.4 = 0.40; both exceed 0.04.
If you observe: Treats every inserted zero as value-preserving.
Respond: Write the fraction names 4/10, 40/100, and 4/100; match equal areas before comparing the written forms.
Check independently: Compare 0.8 with 0.75 and state the common unit.
0.80 > 0.75; compare eighty and seventy-five hundredths.
Use the equivalent factors short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
e1 meets its criterion: e1 gives > for 0.9 compared with 0.12. e2 also meets its criterion: e2 names hundredths and compares ninety hundredths with twelve hundredths.
Continue to the linked next lesson. Apply the same shared-unit idea to addition of unlike fractions.
Next lesson: Common Denominator Kitchene1 does not yet meet this criterion: e1 gives > for 0.9 compared with 0.12.
Ask the learner to explain the attempt, then use the matching diagnostic probe to identify the difficulty. Work through the focused reteach below and its check. Use the linked short intervention if the difficulty persists; retry e1 and e2 before moving on.
Open the linked short interventione1 meets its criterion, but e2 does not yet show this evidence: e2 names hundredths and compares ninety hundredths with twelve hundredths.
Use equivalent-factors to rebuild tenths-to-hundredths renaming, then ask the learner to apply it directly to 0.9 and explain the common unit in e2.
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Confuses tenths and hundredths or drops the zero that fixes a digit’s place.
Try: Use a 100-cell square. Shade three complete columns for 0.3 and three cells for 0.03. Name these 30 hundredths and 3 hundredths before writing the decimals.
Watch for: Thinks a final decimal zero changes value, or inserts a zero into a different place.
Try: Repartition three tenths into thirty hundredths on the same square. Write 0.3 = 0.30, then shade 0.03 separately to show why moving the 3 into the hundredths place changes its value.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Thinks a final decimal zero changes value, or inserts a zero into a different place.
Try: Repartition three tenths into thirty hundredths on the same square. Write 0.3 = 0.30, then shade 0.03 separately to show why moving the 3 into the hundredths place changes its value.
Watch for: Confuses tenths and hundredths or drops the zero that fixes a digit’s place.
Try: Use a 100-cell square. Shade three complete columns for 0.3 and three cells for 0.03. Name these 30 hundredths and 3 hundredths before writing the decimals.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Look for in the response:
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Look for in the response:
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Look for in the response:
Watch for: Thinks a final decimal zero changes value, or inserts a zero into a different place.
Try: Repartition three tenths into thirty hundredths on the same square. Write 0.3 = 0.30, then shade 0.03 separately to show why moving the 3 into the hundredths place changes its value.
Look for in the response:
Watch for: Orders decimals by their digit strings or written length rather than place value.
Try: Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Watch for: Confuses tenths and hundredths or drops the zero that fixes a digit’s place.
Try: Use a 100-cell square. Shade three complete columns for 0.3 and three cells for 0.03. Name these 30 hundredths and 3 hundredths before writing the decimals.
Look for in the response:
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Eight core questions and a two-question exit ticket. Print one set per student.
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Read the claim that 0.12 is larger than 0.9 because twelve exceeds nine.
Rename both in hundredths and check on a number line.
Partners write and repair a different misleading decimal claim.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Compare 0.9 and 0.12. Choose <, =, or >. The wholes are the same size. What unit makes this comparison simple?
Orders decimals by their digit strings or written length rather than place value.
Rename 0.9 as 90 hundredths and 0.12 as 12 hundredths. Mark both on one 0–1 line, then compare the counts of the same hundredth-sized unit.
Teacher answer key. Open responses include a rubric and one valid sample.
0.9 > 0.12. Rename them as equal-sized parts or place them on the same unit number line.
0.8 > 0.75. Rename them as equal-sized parts or place them on the same unit number line.
0.04 < 0.4. Rename them as equal-sized parts or place them on the same unit number line.
0.30 = 0.3. Rename them as equal-sized parts or place them on the same unit number line.
0.62 < 0.7. Rename them as equal-sized parts or place them on the same unit number line.
0.5 > 0.49. Rename them as equal-sized parts or place them on the same unit number line.
0.99 > 0.9. Rename them as equal-sized parts or place them on the same unit number line.
0.07 > 0.06. Rename them as equal-sized parts or place them on the same unit number line.
0.2 < 0.3. Rename them as equal-sized parts or place them on the same unit number line.
0.5 = 0.50. Rename them as equal-sized parts or place them on the same unit number line.
0.1 > 0.01. Rename them as equal-sized parts or place them on the same unit number line.
0.7 > 0.6. Rename them as equal-sized parts or place them on the same unit number line.
Twelve hundredths is less than ninety hundredths.
0.04 has more decimal digits than 0.4 but is smaller.
Eighty hundredths exceeds seventy-five hundredths.
0.4 = 0.40, but 0.04 names four hundredths rather than four tenths.
0.9 > 0.12. Rename them as equal-sized parts or place them on the same unit number line.
Hundredths: compare 90 with 12 equal hundredth units.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Designed for Grade 4. Diagnose decimal comparison errors using equal place-value units.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.