Walk along a fraction number line

Full explanation

Learning goal: Construct and label number lines for unit fractions and fractions greater than one. This lesson targets Ohio 3.NF.2 and both subparts. Plan two 30-minute sessions. Use a ruler, blank strips, and pencils. Review equal parts and the meanings of numerator and denominator before beginning.

A number line locates numbers by their distance from zero. Begin by marking 0 and 1. The interval between them is one whole unit. To locate 1/4, divide that interval into four equal lengths. The endpoint of the first length starting at zero is 1/4. The next endpoint is 2/4, then 3/4, then 4/4 at 1.

Count spaces, not tick marks. Four equal spaces need five boundary marks, including both 0 and 1. Counting the marks as the denominator gives a wrong unit size. A useful check is to ask how many equal jumps of the proposed unit fraction land exactly on 1.

To locate a/b, make a equal jumps of length 1/b starting at 0. The letter b names the number of equal parts in one whole; a counts the jumps. This works even when a is greater than b. Five jumps of length 1/4 reach 5/4, which is one fourth beyond 1. Fractions are numbers; they are not confined to the inside of one shape or to the interval from 0 to 1.

Keep the unit length constant as the line continues. If 0 to 1 occupies eight centimeters, 1 to 2 must occupy eight centimeters too. Fourths on the second unit have the same length as fourths on the first. Uneven tick spacing can make a correct-looking label represent the wrong location.

Additional worked examples

Example 1: Locate 2/3. Draw a line and mark 0 and 1. Divide the interval into three equal spaces. Label the first endpoint after zero 1/3 and the second 2/3. Two jumps, each of length 1/3, reach 2/3. The third jump reaches 3/3, which is 1. Check both the size of each jump and the number of jumps.

Example 2: Locate 7/4. Draw a line from 0 to 2, with equal-length intervals from 0 to 1 and 1 to 2. Partition each unit into fourths. Count seven fourth-length jumps from 0: 1/4, 2/4, 3/4, 4/4, 5/4, 6/4, 7/4. Four jumps reach 1, and three more reach 7/4. The point is between 1 and 2, one fourth before 2. A label at the seventh tick when counting zero as tick one would be wrong because zero is the starting boundary, not a completed jump.

Example 3: Repair a scale error. A student splits the entire interval from 0 to 2 into four equal spaces and calls the first endpoint 1/4. Each space is actually 1/2 of one unit: four such spaces cover two wholes. To show fourths, the student needs four spaces between 0 and 1 and four more between 1 and 2, making eight spaces altogether.

Independent practice and scoring

Guided practice: Draw 0 to 1 and partition it into six equal lengths. Locate 1/6 and 5/6. Ask a partner to count the intervals without reading the labels. If the partner counts five or seven, revise the construction before labeling it again.

Independent construction: (1) Draw 3/4 on a number line. (2) Draw 4/3 on a line extending beyond 1. (3) Draw 0 to 2 in halves and label every endpoint. (4) Explain why six equal spaces from 0 to 1 require seven marks. These are paper-and-pencil performance tasks in addition to the selected-response practice test.

Solutions and rubric: (1) The third fourth-length endpoint from 0. (2) One third beyond 1. (3) 0, 1/2, 2/2 or 1, 3/2, 4/2 or 2, equally spaced. (4) Each space has two boundaries, but neighboring spaces share a boundary; zero supplies the initial mark. Award 2 points for each: 1 for accurate placement or explanation, 1 for a consistent scale and explicit unit reasoning. A right label on an uneven scale earns only partial credit.

Reteaching: Use folded strips laid along the line and touch each space while counting. Do not begin by memorizing tick labels. Extension: Show two differently sized drawings of 5/4 and explain why both can be correct when each drawing uses its own consistent unit scale. Exit task: explain where 1/3 comes from before locating 2/3.