Keep the unit when crossing one

d1. Starting at zero, take five steps of size 1/4. Have you passed 1? Name your position.

Answer: Yes. Four steps reach 1 and the fifth reaches 5/4, one fourth past 1.

Look for:

• Recognizes the point is beyond 1.
• Names 5/4 using five fourths.

Teacher response: Likely difficulty: Assuming fractions cannot exceed one. Response: Place four fourth-size pieces in the first unit and continue with one identical piece.

d2. A line runs from 0 to 2 and is divided into eight equal intervals. Is each interval 1/8 of one unit?

Answer: No. Each interval is 1/4 of one unit, because four intervals fit between 0 and 1.

Look for:

• Uses 0 to 1 as one unit.
• Names each interval 1/4.

Teacher response: Likely difficulty: Using the entire displayed line as the unit. Response: Circle only the distance from 0 to 1 and count its intervals.

d3. Fill the missing label when counting thirds: 1/3, 2/3, 3/3, __, 5/3.

Answer: 4/3.

Look for:

• Writes 4/3.
• Keeps counting from zero with denominator 3.

Teacher response: Likely difficulty: Restarting at 1/3 after the first whole. Response: Count four identical third-size pieces from the original zero.

w1. Locate 7/4 on a number line from 0 to 2.

1. Keep 0 to 1 and 1 to 2 the same length.
2. Split each whole unit into four equal intervals.
3. Count seven fourth-size steps from zero: four reach 1 and three more reach 7/4.
4. Check that eight fourths reach 2, so the point is one step before 2.

Worked example: 7/4 is three fourths after 1 and one fourth before 2.

Look for:

• Uses four intervals per unit.
• Locates the point correctly between 1 and 2.

Teacher response: Likely difficulty: Partitioning the entire two-unit line into four parts. Response: Mark 4/4 directly under 1 and 8/4 directly under 2.

w2. Two same-sized strips are each one whole. Shade 5/3 altogether.

1. Divide each whole into three equal parts.
2. Count five parts of size 1/3 across both strips.
3. Three thirds fill the first whole; two thirds remain to shade.
4. Name the total 5/3, which is 1 and 2/3.

Worked example: Shade all three thirds of the first strip and two thirds of the second.

Look for:

• Shades five third-sized parts.
• Explains 3/3 plus 2/3 makes 5/3.

Teacher response: Likely difficulty: Changing the denominator to six because two strips are present. Response: Keep a label reading “one whole” above each separate strip.

p1. Name six steps of size 1/3 from 0 as a fraction and as a whole number.

Answer: 6/3 = 2.

Look for:

• Names 6/3.
• Connects six thirds to 2.

Teacher response: Likely difficulty: Stopping fraction counting at 3/3. Response: Group every three thirds into one whole.

p2. Is 9/8 between 0 and 1, between 1 and 2, or beyond 2? Explain without drawing all the marks.

Answer: Between 1 and 2, because 8/8 is 1 and 16/8 is 2.

Look for:

• Chooses between 1 and 2.
• Uses 8/8 and 16/8 as landmarks.

Teacher response: Likely difficulty: Using the denominator as a cap on the numerator. Response: Compare the numerator to the number of eighths in one and two wholes.

p3. Write the next two fractions after 5/4 when counting by fourths.

Answer: 6/4 and 7/4.

Look for:

• Keeps denominator 4.
• Increases the numerator by one each step.

Teacher response: Likely difficulty: Changing both numerator and denominator while counting. Response: Say “six fourths, seven fourths” before writing symbols.

p4. Draw two equal whole strips. Show 4/3 and label each whole.

Answer: One full strip and 1/3 of the second strip, with each strip labeled one whole.

Look for:

• Keeps equal-sized wholes.
• Shows four third-sized parts.

Teacher response: Likely difficulty: Calling all six parts sixths of a single whole. Response: Point to one original whole and count its thirds.

p5. A line has 0, 1, and 2 marked. There are two equal gaps in each unit. Name the point halfway between 1 and 2.

Answer: 3/2, also 1 and 1/2.

Look for:

• Names the complete distance as 3/2.
• Relates it to one whole and one half.

Teacher response: Likely difficulty: Naming it 1/2 by counting only from 1. Response: Count three half-unit gaps from the original zero.

p6. A student says 8/4 is 1 because eight pieces are shown. What must be checked, and what is 8/4?

Answer: Check how many fourths make the defined whole. Four fourths make 1, so eight fourths make 2.

Look for:

• Uses four fourths per whole.
• Concludes 8/4 = 2.

Teacher response: Likely difficulty: Making all displayed pieces into a new whole. Response: Return to the one-unit bracket and fit four fourths underneath it.

t1. A toy car travels 7 steps, each 1/3 meter long. Between which whole-meter marks does it stop? Name the exact distance.

Answer: It stops between 2 and 3 meters at 7/3 meters, or 2 and 1/3 meters.

Look for:

• Uses thirds of a meter as a fixed step.
• Gives 7/3 meters between 2 and 3.

Teacher response: Likely difficulty: Restarting the distance count at each meter. Response: Group six third-meter steps into two meters, then add the remaining step.

t2. 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.

Answer: Each gap is 1/2 of one unit. The fourth endpoint is 4/2 = 2, because two gaps make one unit.

Look for:

• Identifies half-unit gaps.
• Corrects the point to 4/2 or 2 with a unit explanation.

Teacher response: Likely difficulty: Using the full displayed length instead of the 0-to-1 distance as one whole. Response: Label 1 after two gaps and repeat that unit to reach 3.