Build fractions from equal parts and locate them on a line

Learning goal: Explain a fraction as a number of equal parts of a defined whole, including fractions beyond one. Begin by identifying the whole. In our model, one complete strip is one whole, including both colored and uncolored regions. The colored portion is a part of the whole, not a separate definition of the whole.

Partition the strip into four equal areas. Each area is one fourth, written 1/4. The denominator four tells how many equal parts make one whole. Select three of those parts: the selected area is 3/4 of the whole. The numerator three counts selected parts of size one fourth. All four parts together make 4/4, or one whole.

Equal size is essential. Cutting a shape into four pieces does not automatically make fourths. A strip with one large piece and three smaller pieces has four pieces, but the pieces are not all fourths. For area fractions, the pieces must have equal area; they need not always have identical shapes. Start with rectangular pieces that have the same size and shape so equality is clear, then compare other valid partitions.

A number line represents fractions as distances from zero. Define the distance from zero to one as one unit. Partition that distance into four equal intervals. Each interval has size 1/4, because four of them make the whole from 0 to 1. The interval that starts at 0 ends at the point 1/4, so that endpoint locates the number 1/4 on the line. The first endpoint after zero is 1/4, the second is 2/4, the third is 3/4 and the fourth is 4/4 = 1. Count intervals traveled, not the number of tick marks. Five marks, including zero and one, bound four intervals.

Fractions continue beyond one. Keep the same unit length from one to two and partition it into fourths too. Traveling five fourth-size intervals from zero reaches 5/4. This point is one fourth beyond one. A numerator larger than the denominator is a valid fraction: it counts more unit-fraction parts than fit in a single whole.

The strip model and number line tell related stories. Five selected quarter-pieces require one complete strip and one quarter of another equally sized strip. On the line, five quarter-intervals reach the same number, 5/4. Changing the size of the second strip would break the model because each quarter must refer to the same-sized whole.

Use different denominators within the third-grade sequence, including halves, thirds, fourths, sixths and eighths. For thirds, the zero-to-one unit has three equal intervals. Two such intervals locate 2/3. The position depends on equal distances, not on writing three labels wherever there is room. A careful drawing is mathematical evidence, so check the spacing.

Worked area example: Fold a rectangular paper into two equal halves, then fold each half in half. When opened, it has four equal regions. Shade one and label it 1/4; shade a total of three and label the shaded amount 3/4. The whole is still the complete original paper.

Worked line example: Draw a unit from zero to one twelve centimeters long for convenience. Mark every three centimeters, producing four equal intervals. The mark nine centimeters from zero is 3/4. Extend the same scale to fifteen centimeters from zero to locate 5/4. The centimeter choices are a drawing aid; the mathematical unit is the full zero-to-one distance.

Worked unit-fraction example: To locate 1/3, treat the interval from 0 to 1 as the whole and partition it into three equal parts. Each part has size 1/3. The part that starts at 0 ends at the first mark, and that endpoint is the number 1/3. Marking off two lengths of 1/3 from 0 reaches 2/3, and three reach 3/3 = 1.

Guided practice: Build a region model and a number line for the same fraction. Say the denominator's job, the numerator's job and the defined whole aloud. Compare an equal partition with an unequal one and explain which can name unit fractions directly.

Independent performance: Draw two different shapes partitioned into equal areas and label one region in each. Locate 2/3 and 7/4 on separate appropriately partitioned lines. Explain why 7/4 is beyond one. If a learner counts marks instead of intervals, trace the actual traveled distances from zero and count those equal units. Do not accept a numerically correct label on an unequal or contradictory drawing.