Fraction Classroom · 2.0.0 · Teacher answers & responses

Beyond One Walk: teaching companion

Keep one fixed whole and one fixed step size while counting past one. 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 3 · 45 minutes. Extend unit fractions past one on a number line.

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. Make a floor number line from 0 to 2 with four equal steps per unit.

activity-2. Walk five fourth-length steps from zero and record the landing point.

activity-3. Repeat with thirds and halves; check that each unit stays the same length.

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: Make a floor number line from 0 to 2 with four equal steps per unit.

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

phase-3. 10 min · partner: Walk five fourth-length steps from zero and record the landing point.

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

phase-4. 7 min · discussion: Repeat with thirds and halves; check that each unit stays the same length.

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: Label the marked point beyond one. What stays the same when you pass one?

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. Starting at zero, take five steps of size 1/4. Where do you land?

Answer: 5/4, or 1 1/4.

Teacher response: If the learner writes 1/5 or restarts at 1/4 after reaching one. Label every landing 0/4 through 5/4. Leave 4/4 = 1 visible while taking the fifth identical step.

d2. A line marks 0, 1, and 2. If there are four equal intervals in each whole, how many quarter-steps reach 2?

Answer: Eight quarter-steps; 2 = 8/4.

Teacher response: If the learner calls the whole 0–2 line one unit and uses eighths. Bracket 0–1 as one whole and 1–2 as another equal whole; count four quarters in each bracket.

Check after reteaching

Complete this check independently after discussing the model.

r1. What fraction names six quarter-steps from zero? What does its denominator count?

Answer: 6/4 = 1 1/2; the denominator counts four quarter-steps in each one whole.

Exit ticket

Complete both original exit questions independently. Use a drawing or explanation when asked.

e1. Label the marked point beyond one. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 5 intervals from zero.

Answer: 5/4

Teacher response: Restarts the fraction count or changes the denominator after passing one. Place two equal-length unit strips end to end, both split into fourths. Label 0/4, 1/4, …, 8/4 continuously. Circle 4/4 at one and take one more unchanged quarter-step to 5/4.

e2. What stays the same when you pass one?

Answer: Each unit keeps the same length and the same equal step size.

Look for:

State that one whole keeps the same length on either side of 1.
State that the fractional step size stays fixed.
Explain that the count continues, for example from 4/4 to 5/4.

Focused reteach

Keep one fixed whole and one fixed step size while counting past one.

teach-1. Place two same-length quartered strips end to end and mark both whole boundaries.

teach-2. Count quarters from zero through 4/4, 5/4, and 8/4, keeping each jump equal.

teach-3. Locate 7/4 and explain why it is less than 8/4 but greater than 4/4.

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 labels the point beyond one as 5/4. e2 also meets its criterion: e2 states that the whole length and equal step size stay unchanged after one.

Teacher response: Continue to the linked next lesson. Compare distances by the number of shared-size fractional steps. Next original lesson: /lessons/same-denominator-race/.

next-2. e1 does not yet meet this criterion: e1 labels the point beyond one as 5/4.

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/beyond-one/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 states that the whole length and equal step size stay unchanged after one.

Teacher response: Ask the learner to compare the step immediately before and after one, then explain why the denominator remains four at 5/4. Short intervention: /interventions/beyond-one/.

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 fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 5 intervals from zero.

Answer: 5/4

c2. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 6 intervals from zero.

Answer: 6/4

c3. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 7 intervals from zero.

Answer: 7/4

c4. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 2 equal intervals. The marked point is 3 intervals from zero.

Answer: 3/2

c5. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 3 equal intervals. The marked point is 4 intervals from zero.

Answer: 4/3

c6. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 3 equal intervals. The marked point is 5 intervals from zero.

Answer: 5/3

c7. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 6 equal intervals. The marked point is 9 intervals from zero.

Answer: 9/6

c8. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 8 equal intervals. The marked point is 10 intervals from zero.

Answer: 10/8

s1. Write the fraction at the marked point. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 2 equal intervals. The marked point is 2 intervals from zero.

Answer: 2/2

Teacher response: Does not recognize that b copies of 1/b make one whole. Fill one outlined whole with b equal pieces and label the completed whole b/b. Exchange the complete collection for a card labeled 1; do the exchange again with a different partition.

s2. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 2 equal intervals. The marked point is 3 intervals from zero.

Answer: 3/2

s3. Write the fraction at the marked point. Number line: 0 to 1; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 4 intervals from zero.

Answer: 4/4

s4. Write the fraction at the marked point. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 5 intervals from zero.

Answer: 5/4

h1. Draw 5/4 using two same-sized strips.

Answer: Shade one full strip and one of four parts of a second strip.

Look for:

Use two strips representing the same whole size.
Split both into fourths and shade one full strip plus one quarter.
Label the combined five quarter-pieces 5/4 or 1 1/4.

h2. Explain why 7/4 is between one and two.

Answer: One is 4/4 and two is 8/4; seven fourths lies between them.

Look for:

Rename one as 4/4 and two as 8/4.
Locate seven quarter-steps after 4/4 and before 8/4.
Conclude 1 < 7/4 < 2 while keeping quarter-step size fixed.

h3. Give two fraction names for 1 on your line.

Answer: 2/2 and 4/4 are both at one.

Look for:

Give two different fractions with equal numerator and denominator, such as 2/2 and 4/4.
Explain that each includes all parts of one whole.
Place both fraction names at the point labeled 1.

h4. A partner labels the fifth quarter step 1/5. Correct it.

Answer: The step size is 1/4, and five steps make 5/4.

Look for:

Identify each step as 1/4 of the fixed unit.
Count five such steps from zero.
Correct the label to 5/4 and explain that five is the step count, not the part size.

e1. Label the marked point beyond one. Number line: 0 to 2; one whole is the distance from 0 to 1. Each whole has 4 equal intervals. The marked point is 5 intervals from zero.

Answer: 5/4

e2. What stays the same when you pass one?

Answer: Each unit keeps the same length and the same equal step size.

Look for:

State that one whole keeps the same length on either side of 1.
State that the fractional step size stays fixed.
Explain that the count continues, for example from 4/4 to 5/4.

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/beyond-one/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
Next original lesson: /lessons/same-denominator-race/. Compare distances by the number of shared-size fractional steps.
Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.