Fraction Classroom · 2.0.0 · Teacher answers & responses
Tenths to Hundredths: teaching companion
Rename tenths as hundredths while preserving value and digit place. Student pages include two diagnostic probes, a reteaching check, and the original exit ticket. The teacher copy includes the complete lesson preparation, teaching sequence, diagnostic responses, all original question guides, and updated rubrics.
Complete lesson preparation
Grade 4 · 45 minutes. Connect tenths to equivalent hundredths and decimal notation.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Partition a strip into tenths and shade three.
activity-2. Split each tenth into ten hundredths without moving the shaded endpoint.
activity-3. Write 3/10, 30/100, 0.3 and 0.30; explain why each names the same value.
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: Partition a strip into tenths and shade three.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Split each tenth into ten hundredths without moving the shaded endpoint.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Write 3/10, 30/100, 0.3 and 0.30; explain why each names the same value.
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 0.3 and 0.30. Choose <, =, or >. The wholes are the same size. Why can a zero be added at the end of 0.3?
Answer: Students: Complete the two exit questions independently.
Diagnostic probes
Explain or draw your reasoning. Your teacher will use these responses to choose what to revisit.
d1. Use hundredths to explain whether 0.3 and 0.30 are equal.
Answer: They are equal: three tenths are thirty hundredths.
Teacher response: If the learner says 0.30 is larger because 30 exceeds 3. Shade three columns of a hundred-grid and count the thirty cells without changing the shaded area.
d2. Does 0.03 equal 0.3? Name the value of the digit three in each.
Answer: No. It represents three hundredths in 0.03 and three tenths in 0.3.
Teacher response: If the learner drops the zero after the decimal point as though it were a final zero. Shade three cells beside three columns and name 3/100 versus 30/100.
Check after reteaching
Complete this check independently after discussing the model.
r1. Rename 0.7 in hundredths, then compare it with 0.07.
Answer: 0.7 = 70/100 = 0.70, which is greater than 0.07 = 7/100.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Compare 0.3 and 0.30. 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.
e2. Why can a zero be added at the end of 0.3?
Answer: It expresses tenths as hundredths without adding any value.
Look for:
- Name 0.3 as three tenths and 0.30 as thirty hundredths.
- Explain the ten-hundredths-per-tenth relationship.
- State that the final zero changes the written form while preserving the amount selected.
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.
Focused reteach
Rename tenths as hundredths while preserving value and digit place.
teach-1. Label a whole hundred-grid, one tenth column, and one hundredth cell.
teach-2. Repartition three tenths into thirty hundredths and record 3/10 = 30/100 and 0.3 = 0.30.
teach-3. Contrast 0.03 on a separate copy, asking whether the selected amount has stayed fixed.
Use the exit evidence
A wrong answer invites a follow-up probe; it does not by itself establish a particular misconception. Check e1 and e2 separately.
next-1. e1 meets its criterion: e1 gives = for 0.3 compared with 0.30. e2 also meets its criterion: e2 explains that a final zero renames three tenths as thirty hundredths without changing the value.
Teacher response: Continue to the linked next lesson. Locate tenths and hundredths as distances on the same unit line. Next original lesson: /lessons/decimal-number-line/.
next-2. e1 does not yet meet this criterion: e1 gives = for 0.3 compared with 0.30.
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 explains that a final zero renames three tenths as thirty hundredths without changing the value.
Teacher response: Use the equivalent-factors intervention for the repartitioning idea, then return to the decimal grid and have the learner explain the specific final zero in 0.30. Short intervention: /interventions/equivalent-factors/.
Original question guide and updated rubrics
Use these question-specific teaching responses alongside the original lesson. Accept other mathematically valid open responses that meet the criteria. The original question wording and mathematical answers are retained; model descriptions below make their references readable offline.
c1. Write 0.3 as a fraction. Strip model: 1 same-sized whole, each divided into 10 equal parts; 3 parts are shaded altogether.
Answer: 3/10
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.
c2. Write 30/100 as a decimal.
Answer: 0.30
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.
c3. Compare 0.4 and 0.40. 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.
c4. 7/10 = ?/100. Enter the numerator.
Answer: 70
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c5. Write 0.09 as a fraction.
Answer: 9/100
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.
c6. Compare 0.6 and 0.60. 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.
c7. Calculate 3/10 + 4/100. Show a model or your reasoning.
Answer: 17/50
Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.
c8. Write 0.25 as a fraction.
Answer: 25/100
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.
s1. Write one tenth as a decimal.
Answer: 0.1
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.
s2. Write two tenths as a decimal.
Answer: 0.2
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.
s3. Write 0.10 as a fraction.
Answer: 10/100
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 0.2 and 0.20. 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.
h1. Explain why 0.3 = 0.30 with repartitioning.
Answer: Each tenth is ten hundredths, so three tenths is thirty hundredths.
Look for:
- Represent three tenths of one fixed whole.
- Split each tenth into ten hundredths to obtain thirty hundredths.
- Conclude 0.3 = 0.30 because the selected amount is unchanged.
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.
h2. Does 0.03 equal 0.3? Use place value.
Answer: No. Three hundredths is a tenth of three tenths.
Look for:
- State that 0.03 and 0.3 are unequal.
- Name them as three hundredths and three tenths.
- Explain that three tenths equals thirty hundredths, ten times three hundredths.
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.
h3. Show 0.47 as tenths plus hundredths.
Answer: Four tenths plus seven hundredths equals forty-seven hundredths.
Look for:
- Decompose 0.47 into 0.4 + 0.07.
- Name four tenths and seven hundredths using the correct places.
- Check that 40/100 + 7/100 = 47/100.
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.
h4. Invent a pair of decimals with different written lengths and equal value.
Answer: 0.5 and 0.50 both mean one half.
Look for:
- Give a pair of decimals with different numbers of written decimal places.
- Make the values equal, such as 0.5 and 0.50.
- Justify equality by renaming tenths as hundredths or appending only a final zero.
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 0.3 and 0.30. 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.
e2. Why can a zero be added at the end of 0.3?
Answer: It expresses tenths as hundredths without adding any value.
Look for:
- Name 0.3 as three tenths and 0.30 as thirty hundredths.
- Explain the ten-hundredths-per-tenth relationship.
- State that the final zero changes the written form while preserving the amount selected.
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.
Teaching notes
- Teaching supplement version 2.0.0. Original lesson questions and PDF materials remain version 1.0.0. Original PDFs contain their original teaching notes and rubrics; use this supplement for the updated diagnostic guidance.
- Short intervention: /interventions/equivalent-factors/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/decimal-number-line/. Locate tenths and hundredths as distances on the same unit line.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.