Count spaces on a number line

d1. A number line from 0 to 1 has marks at 0, 1/4, 2/4, 3/4, and 1. How many equal intervals does it have?

Answer: 4 equal intervals, although it has 5 marks.

Look for:

• Counts 4 spaces.
• Distinguishes 5 marks from 4 intervals.

Teacher response: Likely difficulty: Counting endpoint marks as fractional parts. Response: Slide a finger across each gap, starting the first count after leaving zero.

d2. Starting at 0 on a line divided into sixths, a student counts the zero mark as “one” and labels the next mark 2/6. Repair the label.

Answer: The next mark is 1/6 because only one sixth-length interval has been traveled.

Look for:

• Labels the next mark 1/6.
• Explains zero is the starting position.

Teacher response: Likely difficulty: Including zero as a traveled step. Response: Ask what distance has been traveled before taking the first step.

d3. A line from 0 to 1 has three gaps of different lengths. Can its marks be labeled 0, 1/3, 2/3, 1 in order? Explain.

Answer: No. Thirds require three equal-length intervals.

Look for:

• Rejects the thirds labeling.
• Requires equal interval lengths.

Teacher response: Likely difficulty: Treating any three consecutive gaps as thirds. Response: Measure or copy the first gap to check whether it fits the other two.

w1. Draw a line from 0 to 1 and locate 3/4.

1. Mark and label 0 and 1 first.
2. Partition their distance into four equal intervals; this needs five marks including both ends.
3. Count three intervals from 0.
4. Label the endpoint of the third interval 3/4.

Worked example: 3/4 is the endpoint after three fourth-length steps from zero.

Look for:

• Uses four equal intervals in the unit.
• Locates the endpoint after three steps.

Teacher response: Likely difficulty: Placing 3/4 at the third mark when zero is counted as mark one. Response: Draw three travel arcs and read their endpoints: 1/4, 2/4, 3/4.

w2. A line from 0 to 1 has seven equally spaced marks including its endpoints. Name the point two intervals from 0.

1. Seven marks create six gaps.
2. The distance between neighboring marks is 1/6.
3. Travel two of those intervals from zero.
4. Name the endpoint 2/6; two sixths also make one third.

Worked example: 2/6, equivalent to 1/3.

Look for:

• Uses denominator 6.
• Counts two intervals rather than two marks.

Teacher response: Likely difficulty: Using seven marks to claim sevenths. Response: Underline the six gaps before writing a denominator.

p1. Draw a unit line with 3 equal intervals. How many marks are needed when both 0 and 1 are included?

Answer: 4 marks.

Look for:

• Draws three equal gaps.
• Counts four marks including endpoints.

Teacher response: Likely difficulty: Assuming the mark count equals the interval count. Response: Start with the zero mark and add one endpoint for each of three steps.

p2. On a unit line split into eighths, what fraction is reached after five equal steps from 0?

Answer: 5/8.

Look for:

• Names eighths as the step size.
• Labels 5/8.

Teacher response: Likely difficulty: Swapping the traveled steps and unit partition counts. Response: Name the step size first, then count five copies.

p3. A line has labels 0, A, B, C, 1, all equally spaced. Name B.

Answer: 2/4, or 1/2.

Look for:

• Identifies four equal intervals.
• Names B as 2/4 or 1/2.

Teacher response: Likely difficulty: Using five letters or labels as the denominator. Response: Count the four gaps and mark the two that reach B.

p4. A student draws marks at 0 cm, 2 cm, 5 cm, and 6 cm and labels them 0, 1/3, 2/3, 1. Repair the positions of the two interior marks.

Answer: Place the interior marks at 2 cm and 4 cm, making three equal 2 cm intervals.

Look for:

• Moves the second interior mark to 4 cm.
• Checks three equal 2 cm intervals.

Teacher response: Likely difficulty: Accepting unequal spacing when labels look correct. Response: Compare all three gap lengths to the first gap.

p5. Draw 2/3 on a line from 0 to 1. Add an arc over each interval you travel.

Answer: A unit split into three equal intervals with two arcs ending at 2/3.

Look for:

• Shows thirds with equal spacing.
• Uses two arcs from zero ending at 2/3.

Teacher response: Likely difficulty: Drawing two marks instead of traveling two intervals. Response: Ask the learner to describe the length of each arc.

p6. A point is three steps from zero on a unit line. Is that enough information to name its fraction? Explain.

Answer: No. The length of each step, or the number of equal steps in one whole unit, is also needed.

Look for:

• Recognizes missing step size.
• Explains that different unit partitions give different fractions.

Teacher response: Likely difficulty: Using the numerator without a defined unit fraction. Response: Compare three steps on a fourths line with three steps on an eighths line.

t1. A 12 cm line represents 0 to 1. Mark 5/6 and explain the difference between the number of marks and the number of intervals.

Answer: 5/6 is 10 cm from 0. Six 2 cm intervals require seven marks including the endpoints.

Look for:

• Locates 5/6 at 10 cm.
• Distinguishes six intervals from seven marks.

Teacher response: Likely difficulty: Using endpoint count to determine the denominator. Response: Build the line with six 2 cm steps, then count endpoints separately.

t2. A robot moves from 0 to 1 in three equal jumps. It stops after jump two. A friend calls the point 3/4 because there are four marks. Write a correction.

Answer: The robot stops at 2/3. Three equal jumps make the unit, and two jumps have been completed.

Look for:

• Corrects the location to 2/3.
• Explains numerator as traveled jumps and denominator as jumps per unit.

Teacher response: Likely difficulty: Mixing the count of all marks with the count of reached marks. Response: Have the student name what each number in 2/3 counts.