Decimal Detective: teaching companion

prepare. Before starting: Recognize a whole and count equal parts.

materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).

activity-1. Read the claim that 0.12 is larger than 0.9 because twelve exceeds nine.

activity-2. Rename both in hundredths and check on a number line.

activity-3. Partners write and repair a different misleading decimal claim.

phase-1. 5 min · warmup: Look at the first model or task. What is the whole? What do you notice?

Answer: Students: Sketch an initial prediction and explain it to a partner.

phase-2. 8 min · model: Read the claim that 0.12 is larger than 0.9 because twelve exceeds nine.

Answer: Students: Follow the model; label the whole and the equal parts.

phase-3. 10 min · partner: Rename both in hundredths and check on a number line.

Answer: Students: Work in pairs. One models, the other checks; switch roles.

phase-4. 7 min · discussion: Partners write and repair a different misleading decimal claim.

Answer: Students: Compare two solutions and explain the relationship.

phase-5. 10 min · practice: Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.

Answer: Students: Solve independently, then compare the reasoning.

phase-6. 5 min · exit: Compare 0.9 and 0.12. Choose <, =, or >. The wholes are the same size. What unit makes this comparison simple?

Answer: Students: Complete the two exit questions independently.

d1. Compare 0.9 and 0.12 after naming both in hundredths.

Answer: 0.9 = 90 hundredths, so 0.9 > 0.12.

Teacher response: If the learner uses 9 < 12 to claim 0.9 is smaller. Repartition nine tenths into ninety hundredths and place both counts on the same grid.

d2. Compare 0.4, 0.40, and 0.04. Which two are equal?

Answer: 0.4 = 0.40; both exceed 0.04.

Teacher response: If the learner treats every inserted zero as value-preserving. Write the fraction names 4/10, 40/100, and 4/100; match equal areas before comparing the written forms.

r1. Compare 0.8 with 0.75 and state the common unit.

Answer: 0.80 > 0.75; compare eighty and seventy-five hundredths.

e1. Compare 0.9 and 0.12. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

e2. What unit makes this comparison simple?

Answer: Hundredths: compare 90 with 12 equal hundredth units.

Look for:

• Name hundredths as the common unit for e1.
• Rewrite 0.9 as 90 hundredths and 0.12 as 12 hundredths.
• Use the shared unit to explain why 90 > 12 gives 0.9 > 0.12.

Teacher response: Confuses tenths and hundredths or drops the zero that fixes a digit’s place. 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.

teach-1. Read each decimal aloud with tenths or hundredths.

teach-2. Rename tenths in hundredths using the factor ten, preserving the selected area.

teach-3. Compare equal-unit counts and test a counterexample to “more digits means larger.”

next-1. 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.

Teacher response: Continue to the linked next lesson. Apply the same shared-unit idea to addition of unlike fractions. Next original lesson: /lessons/common-denominator-kitchen/.

next-2. e1 does not yet meet this criterion: e1 gives > for 0.9 compared with 0.12.

Teacher response: 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. Short intervention: /interventions/equivalent-factors/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 names hundredths and compares ninety hundredths with twelve hundredths.

Teacher response: 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. Short intervention: /interventions/equivalent-factors/.

c1. Compare 0.9 and 0.12. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

c2. Compare 0.8 and 0.75. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

c3. Compare 0.04 and 0.4. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: Confuses tenths and hundredths or drops the zero that fixes a digit’s place. 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.

c4. Compare 0.30 and 0.3. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Thinks a final decimal zero changes value, or inserts a zero into a different place. 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.

c5. Compare 0.62 and 0.7. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

c6. Compare 0.5 and 0.49. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

c7. Compare 0.99 and 0.9. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

c8. Compare 0.07 and 0.06. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

s1. Compare 0.2 and 0.3. Choose <, =, or >. The wholes are the same size.

Answer: <

Teacher response: 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.

s2. Compare 0.5 and 0.50. Choose <, =, or >. The wholes are the same size.

Answer: =

Teacher response: Thinks a final decimal zero changes value, or inserts a zero into a different place. 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.

s3. Compare 0.1 and 0.01. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: Confuses tenths and hundredths or drops the zero that fixes a digit’s place. 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.

s4. Compare 0.7 and 0.6. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

h1. Correct 0.12 > 0.9 with place-value names.

Answer: Twelve hundredths is less than ninety hundredths.

Look for:

• Rename 0.12 as twelve hundredths and 0.9 as ninety hundredths.
• Compare 12 with 90 using the shared unit.
• Correct the relation to 0.12 < 0.9.

Teacher response: 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.

h2. Give a counterexample to more decimal digits means a larger number.

Answer: 0.04 has more decimal digits than 0.4 but is smaller.

Look for:

• Give two decimals with different written lengths.
• Make the longer decimal smaller, such as 0.04 < 0.4.
• Justify the actual values using place value rather than digit count.

Teacher response: 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.

h3. Write a convincing explanation for 0.8 > 0.75.

Answer: Eighty hundredths exceeds seventy-five hundredths.

Look for:

• Rename 0.8 as 0.80 or eighty hundredths.
• Identify 0.75 as seventy-five hundredths.
• Conclude 0.8 > 0.75 because eighty equal units exceed seventy-five.

Teacher response: 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.

h4. Explain why adding a final zero is different from adding a zero after the point.

Answer: 0.4 = 0.40, but 0.04 names four hundredths rather than four tenths.

Look for:

• Show a final-zero equality such as 0.4 = 0.40.
• Contrast it with inserting a zero before the nonzero digit, such as 0.04.
• Explain that the second move shifts four tenths to four hundredths.

Teacher response: Thinks a final decimal zero changes value, or inserts a zero into a different place. 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.

e1. Compare 0.9 and 0.12. Choose <, =, or >. The wholes are the same size.

Answer: >

Teacher response: 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.

e2. What unit makes this comparison simple?

Answer: Hundredths: compare 90 with 12 equal hundredth units.

Look for:

• Name hundredths as the common unit for e1.
• Rewrite 0.9 as 90 hundredths and 0.12 as 12 hundredths.
• Use the shared unit to explain why 90 > 12 gives 0.9 > 0.12.

Teacher response: Confuses tenths and hundredths or drops the zero that fixes a digit’s place. 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.