Read picture graphs, bar graphs and fraction line plots

Learning goal: Display data in picture graphs, bar graphs and line plots, then solve comparison and total problems using the scale and units. A graph organizes observations so patterns are easier to see. It does not change the observations. Before calculating, read the title, category labels, axis markings and key. A mark can stand for one observation or several, depending on the graph.

Use this made-up class data: three groups record the books they read in one week. Oak reads twelve, Pine eight and Elm six. In a picture graph with one circle representing two books, Oak needs six circles, Pine four and Elm three. The number of circles is not the number of books. Multiply each symbol count by two to recover the data. A different key would require a different number of symbols.

A bar graph can display the same counts. Use a zero baseline and equal intervals marked zero, two, four, six, eight, ten and twelve books. Oak's bar reaches twelve, Pine's eight and Elm's six. A bar twice as long represents twice as many books when both use the same scale and baseline. Do not give different categories different unit scales.

Use either graph to answer questions. The total is 12 + 8 + 6 = 26 books. Oak reads 12 − 8 = 4 more than Pine. Pine and Elm together read fourteen, which is two more than Oak. In the picture graph, Oak has two more circles than Pine, but the difference in books is four because each circle represents two.

A line plot is useful for repeated numerical measurements. Eight ribbon pieces in a made-up set have lengths 1/4, 1/2, 1/2, 3/4, 3/4, 3/4, 1 and 1 meter. Draw a number line with equal quarter-meter intervals. Place one X above the location for each ribbon. Stack repeated measurements vertically rather than moving them sideways to a different value. Three X marks above 3/4 mean three different ribbons of that length.

Count the marks to check the display: one at 1/4, two at 1/2, three at 3/4 and two at one meter give eight observations. Each X represents one ribbon, not one meter. The horizontal location states length; the vertical stack counts frequency. These two meanings should not be mixed when answering a question.

To find the combined ribbon length, account for every observation: 1/4 + 2 × (1/2) + 3 × (3/4) + 2 × 1. Rename halves and wholes as fourths to obtain (1 + 4 + 9 + 8)/4 = 22/4 = 5 2/4 meters. Simply adding the four labeled values would ignore repeated ribbons. Simply counting eight marks would report the number of ribbons, not their total length.

The longest ribbon is one meter and the shortest is one fourth meter, so their length difference is 3/4 meter. Three ribbons are at least three fourths but shorter than one meter; five ribbons are at least three fourths when the two one-meter ribbons are included. Read inclusive words such as at least carefully before counting.

Worked scale check: If a picture graph uses one square for four notebooks, five squares represent twenty notebooks. A difference of two squares means eight notebooks, not two. Write the key beside the calculation to show which unit was counted.

Worked line-plot calculation: The two half-meter ribbons together measure one meter. The three three-quarter-meter ribbons measure 9/4 = 2 1/4 meters. Their combined length is 3 1/4 meters. Verify that all five relevant X marks have been included once.

Guided practice: Rebuild the book data as a picture graph and a bar graph. Compare their answers to the same total and difference questions. Then plot the supplied ribbon measurements and check that the total number of marks equals the number of measurements.

Independent evidence: Collect a small set of real object lengths with a stated unit and measurement precision. Make a line plot and explain repeated values. Solve one frequency question, one length-difference question and one combined-length question. Separately make a scaled category graph, supply its key and explain how the key affects a comparison.