Extend one steady fraction step past 1 and connect improper fractions to whole-number landmarks.
Listen for the idea
A fraction step keeps its size after crossing a whole-number landmark.
Keep an unchanged unit interval across 1.
Count unit fractions to name points greater than 1.
Materials: Paper and ruler; Two equal-length paper strips.
A 12-minute route
2 minutes: Try the three diagnostic questions. Ask for a brief explanation.
3 minutes: Teach one worked example that addresses the difficulty.
5 minutes: Choose three practice tasks and discuss the reasoning.
2 minutes: Give one transfer task independently and decide the next step.
The second worked example, three unused practice tasks, and remaining transfer task are available for a follow-up session. Allow more time if the group needs to build the models.
Teaching move: Mark 0, 1, and 2 with equal spacing before adding fraction partitions. Repeat the same small step across the whole-number boundary without restarting the numerator.
Use the answer notes to interpret what you hear. Choose a response for each question to get a suggested teaching step. These choices describe this brief check and are not a mastery score.
1 Starting at zero, take five steps of size 1/4. Have you passed 1? Name your position.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Yes. Four steps reach 1 and the fifth reaches 5/4, one fourth past 1.
Listen for this difficulty: Assuming fractions cannot exceed one.
Try this response: Place four fourth-size pieces in the first unit and continue with one identical piece.
Look for:
Recognizes the point is beyond 1.
Names 5/4 using five fourths.
2 A line runs from 0 to 2 and is divided into eight equal intervals. Is each interval 1/8 of one unit?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: No. Each interval is 1/4 of one unit, because four intervals fit between 0 and 1.
Listen for this difficulty: Using the entire displayed line as the unit.
Try this response: Circle only the distance from 0 to 1 and count its intervals.
Look for:
Uses 0 to 1 as one unit.
Names each interval 1/4.
3 Fill the missing label when counting thirds: 1/3, 2/3, 3/3, __, 5/3.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 4/3.
Listen for this difficulty: Restarting at 1/3 after the first whole.
Try this response: Count four identical third-size pieces from the original zero.
Look for:
Writes 4/3.
Keeps counting from zero with denominator 3.
2 · Model together
Two worked examples
Use one now. Ask the learner to explain each step before revealing the next part of your drawing.
1 Locate 7/4 on a number line from 0 to 2.
Keep 0 to 1 and 1 to 2 the same length.
Split each whole unit into four equal intervals.
Count seven fourth-size steps from zero: four reach 1 and three more reach 7/4.
Check that eight fourths reach 2, so the point is one step before 2.
Result: 7/4 is three fourths after 1 and one fourth before 2.
Teaching notes
Listen for this difficulty: Partitioning the entire two-unit line into four parts.
Try this response: Mark 4/4 directly under 1 and 8/4 directly under 2.
Look for:
Uses four intervals per unit.
Locates the point correctly between 1 and 2.
2 Two same-sized strips are each one whole. Shade 5/3 altogether.
Divide each whole into three equal parts.
Count five parts of size 1/3 across both strips.
Three thirds fill the first whole; two thirds remain to shade.
Name the total 5/3, which is 1 and 2/3.
Result: Shade all three thirds of the first strip and two thirds of the second.
Teaching notes
Listen for this difficulty: Changing the denominator to six because two strips are present.
Try this response: Keep a label reading “one whole” above each separate strip.
Look for:
Shades five third-sized parts.
Explains 3/3 plus 2/3 makes 5/3.
3 · Guided practice
Choose three tasks
Mix a calculation or drawing with an explanation or error repair. Keep the other tasks for another short session.
1 Name six steps of size 1/3 from 0 as a fraction and as a whole number.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 6/3 = 2.
Listen for this difficulty: Stopping fraction counting at 3/3.
Try this response: Group every three thirds into one whole.
Look for:
Names 6/3.
Connects six thirds to 2.
2 Is 9/8 between 0 and 1, between 1 and 2, or beyond 2? Explain without drawing all the marks.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Between 1 and 2, because 8/8 is 1 and 16/8 is 2.
Listen for this difficulty: Using the denominator as a cap on the numerator.
Try this response: Compare the numerator to the number of eighths in one and two wholes.
Look for:
Chooses between 1 and 2.
Uses 8/8 and 16/8 as landmarks.
3 Write the next two fractions after 5/4 when counting by fourths.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 6/4 and 7/4.
Listen for this difficulty: Changing both numerator and denominator while counting.
Try this response: Say “six fourths, seven fourths” before writing symbols.
Look for:
Keeps denominator 4.
Increases the numerator by one each step.
4 Draw two equal whole strips. Show 4/3 and label each whole.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: One full strip and 1/3 of the second strip, with each strip labeled one whole.
Listen for this difficulty: Calling all six parts sixths of a single whole.
Try this response: Point to one original whole and count its thirds.
Look for:
Keeps equal-sized wholes.
Shows four third-sized parts.
5 A line has 0, 1, and 2 marked. There are two equal gaps in each unit. Name the point halfway between 1 and 2.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 3/2, also 1 and 1/2.
Listen for this difficulty: Naming it 1/2 by counting only from 1.
Try this response: Count three half-unit gaps from the original zero.
Look for:
Names the complete distance as 3/2.
Relates it to one whole and one half.
6 A student says 8/4 is 1 because eight pieces are shown. What must be checked, and what is 8/4?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Check how many fourths make the defined whole. Four fourths make 1, so eight fourths make 2.
Listen for this difficulty: Making all displayed pieces into a new whole.
Try this response: Return to the one-unit bracket and fit four fourths underneath it.
Look for:
Uses four fourths per whole.
Concludes 8/4 = 2.
4 · Independent transfer
Try a fresh situation
Choose one task without coaching. Check the explanation against both criteria in its teaching notes; use the other task for a later check.
1 A toy car travels 7 steps, each 1/3 meter long. Between which whole-meter marks does it stop? Name the exact distance.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: It stops between 2 and 3 meters at 7/3 meters, or 2 and 1/3 meters.
Listen for this difficulty: Restarting the distance count at each meter.
Try this response: Group six third-meter steps into two meters, then add the remaining step.
Look for:
Uses thirds of a meter as a fixed step.
Gives 7/3 meters between 2 and 3.
2 A drawing shows the line from 0 to 3 split into six equal gaps. Jo calls the fourth gap endpoint 4/6. Correct the label and explain.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Each gap is 1/2 of one unit. The fourth endpoint is 4/2 = 2, because two gaps make one unit.
Listen for this difficulty: Using the full displayed length instead of the 0-to-1 distance as one whole.
Try this response: Label 1 after two gaps and repeat that unit to reach 3.
Look for:
Identifies half-unit gaps.
Corrects the point to 4/2 or 2 with a unit explanation.
What to do next
If the explanation holds: Move on when the student keeps the unit fixed and explains a point beyond 1 using both a fraction and whole-number landmarks.
If the difficulty remains: Align two identical strips end to end, label each as one whole, and move one third-sized piece repeatedly across both.
Standards: 3.NF.A.2.b, 3.NF.A.3.c. Grade 3 alignment describes the mathematical work in this pack.
The tasks, worked examples, and teacher responses are original Fraction Classroom materials, version 2.0.0. The teaching approach draws on noticing the unit, moving between representations, and responding to student reasoning in the linked Illustrative Mathematics lesson. Its text and exercises are not reproduced here.