Display and interpret fractional measurements in graphs

Learning goal: Create and interpret line plots, bar graphs and picture graphs using fractional or decimal quantities. Use the key and axis units to solve problems, and distinguish a measurement value from the number of observations at that value.

A fictional ribbon investigation records eight lengths in feet: 1/2, 3/4, 3/4, 1, 1, 1, 1 1/4 and 1 1/2. A line plot places each length at its position on a number line and stacks one X for each observation. The tick marks increase by one fourth of a foot, so equal intervals represent equal differences in length.

The counts are one at 1/2, two at 3/4, three at 1, one at 1 1/4 and one at 1 1/2. The stack height tells how many ribbons have a given length. It does not tell the total feet in that stack. Three marks above one foot mean three one-foot ribbons, whose combined length is three feet.

Add the lengths to find the total: 1/2 + 2 × 3/4 + 3 × 1 + 1 1/4 + 1 1/2 = 7 3/4 feet. One way to check is to rename all lengths in quarters: 2 + 3 + 3 + 4 + 4 + 4 + 5 + 6 = 31 quarters, or 7 3/4. Counting only eight X marks would give the number of ribbons, not their total length.

The longest and shortest ribbons differ by 1 1/2 − 1/2 = 1 foot. Ribbons at least one foot long include the three at one foot and the two longer ribbons, making five. At least includes the boundary value; more than one foot would count only the two longer ribbons. Read comparison wording carefully before selecting marks.

A bar graph can display category totals instead of individual measurements. Suppose fictional bins hold 2.5, 1.75 and 3.25 pounds of paper. Use a pounds axis with equal quarter-pound increments beginning at zero, and label bins A, B and C. The paper in the three bins weighs 7.5 pounds in all. The paper in Bin C weighs 1.5 pounds more than the paper in Bin B. A bar's length represents the weight of the paper, not the weight of the empty bin and not the number of paper sheets.

A picture graph needs an explicit key. If one drawn square represents two cups of beads, three full symbols mean six cups. A half-symbol represents one cup under that key. A display of three full symbols for red beads and two full symbols plus a half for blue beads therefore shows six cups and five cups. The difference is one cup, not one whole symbol. Symbols must use consistent sizes and meanings.

Choose a graph according to the question. A line plot preserves how many measurements occur at each numerical value. A bar graph compares amounts across categories. A picture graph can make small category counts or amounts visible when the key is clear. The same data should not acquire different totals merely because its display changes.

A misleading graph can use uneven intervals, omit the key or change symbol sizes. Check the scale and unit before drawing a conclusion. If a line plot spaces 1/2 and 3/4 as far apart as 1 and 1 1/2, it gives unequal differences the same distance. Repair the axis rather than treating the distorted positions as evidence.

Worked calculation: The two three-quarter-foot ribbons together measure 1 1/2 feet. That equals the length of the single longest ribbon, although the number of objects differs. The graph distinguishes equal total length from equal object count.

Worked key interpretation: With a two-cup-per-square key, two and one half symbols represent five cups. To display seven cups, use three and one half symbols. Explain the fractional symbol using the key instead of rounding it to a full symbol.

Guided investigation: Measure safe classroom strips to the nearest suitable quarter or eighth inch with an actual ruler, record units and build a line plot. Use the real collected values, keeping them separate from this lesson's fictional ribbon data. State any rounding used in measurement.

Independent performance: Create a labeled line plot, bar graph and picture graph for specified or collected data. Solve a total, a difference and a threshold question, explaining the unit of each answer. Check mark counts, scales, symbol keys and arithmetic against the original record. Retain the measurement record so another reader can verify the display.