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?

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.

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.

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.

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.

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

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

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

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

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

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.

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.