Fraction Classroom · 2.0.0 · Teacher answers & responses
Decimal Number Line: teaching companion
Locate and compare hundredth distances using a fixed decimal unit. 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. Locate hundredths on a unit line and compare decimal distances.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Draw a line from zero to one with tenth marks. Zoom the segment from 0.4 to 0.5.
activity-2. Divide that segment into ten equal hundredth steps and locate 0.45.
activity-3. Partners locate a second decimal and compare distances from zero.
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: Draw a line from zero to one with tenth marks. Zoom the segment from 0.4 to 0.5.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Divide that segment into ten equal hundredth steps and locate 0.45.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners locate a second decimal and compare distances from zero.
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 0.45 and 0.5. Choose <, =, or >. The wholes are the same size. How many hundredth steps equal one tenth?
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. A line segment runs from 0.4 to 0.5 in ten equal intervals. What is each interval worth?
Answer: 0.01, or one hundredth of the original whole.
Teacher response: If the learner calls each interval 0.1 because there are ten intervals. Write the segment length as 0.1, then divide that length into ten. Keep the original 0–1 unit visible beside the zoom.
d2. Which is farther right on one increasing number line: 0.45 or 0.5?
Answer: 0.5, because 50 hundredths exceeds 45 hundredths.
Teacher response: If the learner places 0.45 farther right because 45 exceeds 5. Rename the right endpoint 0.50 and count the five hundredth-steps from 0.45 to it.
Check after reteaching
Complete this check independently after discussing the model.
r1. Name a point seven hundredth-steps after 0.6, and compare it with 0.7.
Answer: 0.67; it is less than 0.70.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Compare 0.45 and 0.5. 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. How many hundredth steps equal one tenth?
Answer: Ten hundredth steps.
Look for:
- State ten hundredth steps.
- Connect 10/100 with 1/10.
- Explain that the ten steps cover the same distance as one tenth of the same unit.
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.
Focused reteach
Locate and compare hundredth distances using a fixed decimal unit.
teach-1. Mark 0, 0.4, 0.5, and 1 on a unit line.
teach-2. Enlarge only the 0.4–0.5 segment, keeping both endpoint labels, and divide it into ten hundredth intervals.
teach-3. Locate 0.45 and compare its distance from zero with 0.50.
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 0.45 compared with 0.5. e2 also meets its criterion: e2 states that ten hundredth-steps equal one tenth.
Teacher response: Continue to the linked next lesson. Use the number-line positions to challenge decimal digit-count rules. Next original lesson: /lessons/decimal-detective/.
next-2. e1 does not yet meet this criterion: e1 gives < for 0.45 compared with 0.5.
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/intervals-not-points/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 states that ten hundredth-steps equal one tenth.
Teacher response: Ask the learner to move ten 0.01 steps from 0.4 to 0.5 and connect that movement to the ten-hundredths-per-tenth statement. Short intervention: /interventions/intervals-not-points/.
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 the marked position as a decimal. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 100 equal intervals. The marked point is 45 intervals from zero.
Answer: 0.45
Teacher response: Counts tick marks, including zero, instead of equal intervals from zero. Mark 0 and 1, then make four equal spaces. Draw one jump over each space and count four jumps despite five marks. Count jumps from zero to the target point, keeping zero as the starting position. 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. Compare 0.45 and 0.5. 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 0.67 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.
c4. Compare 0.25 and 0.2. 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.
c5. Compare 0.09 and 0.1. 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.
c6. Compare 0.80 and 0.8. 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. Compare 0.56 and 0.65. 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. Write the marked position as a decimal. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 100 equal intervals. The marked point is 32 intervals from zero.
Answer: 0.32
Teacher response: Counts tick marks, including zero, instead of equal intervals from zero. Mark 0 and 1, then make four equal spaces. Draw one jump over each space and count four jumps despite five marks. Count jumps from zero to the target point, keeping zero as the starting position. 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. Name the point in decimal notation. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 10 equal intervals. The marked point is 1 intervals from zero.
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. Name the point in decimal notation. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 10 equal intervals. The marked point is 2 intervals from zero.
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. Name the point in decimal notation. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 10 equal intervals. The marked point is 5 intervals from zero.
Answer: 0.5
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. Name the point in decimal notation. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 10 equal intervals. The marked point is 7 intervals from zero.
Answer: 0.7
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.
h1. Explain why 0.45 lies between 0.4 and 0.5.
Answer: 45 hundredths lies between 40 and 50 hundredths.
Look for:
- Rename 0.4 and 0.5 as 40 and 50 hundredths.
- Identify 0.45 as 45 hundredths.
- Use 40 < 45 < 50 to justify its position between the endpoints.
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.
h2. Give three hundredth decimals between 0.6 and 0.7.
Answer: 0.61, 0.65 and 0.69 fit.
Look for:
- Provide three distinct hundredth decimals.
- Make each strictly greater than 0.60 and less than 0.70.
- Locate or compare each using the same hundredth-sized unit.
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. Why is a zoomed segment helpful?
Answer: Hundredth intervals can be spaced visibly while the unit remains specified.
Look for:
- Explain that enlarging the segment makes small intervals easier to see.
- Keep the endpoint values stated, such as 0.4 and 0.5.
- Retain the correct numerical value of each interval; a longer drawing does not change the unit.
Teacher response: Accepts uneven spacing on a fraction number line. Fold a paper strip into four equal lengths and transfer the fold positions to a 0–1 line. Compare every neighboring gap with the first gap before labeling fourths. 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. Repair a line that places 0.9 before 0.12.
Answer: 0.9 is 90 hundredths and must lie farther from zero than 12 hundredths.
Look for:
- Rename 0.9 as 0.90 or ninety hundredths.
- Compare ninety hundredths with twelve hundredths.
- Place 0.9 to the right of 0.12 on an increasing number line.
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.
e1. Compare 0.45 and 0.5. 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. How many hundredth steps equal one tenth?
Answer: Ten hundredth steps.
Look for:
- State ten hundredth steps.
- Connect 10/100 with 1/10.
- Explain that the ten steps cover the same distance as one tenth of the same unit.
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.
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/intervals-not-points/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/decimal-detective/. Use the number-line positions to challenge decimal digit-count rules.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.