What makes a data picture clear and worth looking at

What to assume they know

• Drawing a bar graph, and choosing a scale that fits — drawing a bar graph and choosing its scale
• Reading a bar graph: what length is standing for — reading values off bars
• Subtracting two four-digit numbers
• Comparing several four-digit numbers and ranking them
• The idea that a picture can be turned through a quarter turn without changing what it records

What they should be able to do

• Explain why a table of numbers is a poor instrument for comparison, using the chapter's own seven-mountain example
• Convert a small table into a bar graph and state which questions became easier
• Distinguish a bar graph drawn with horizontal bars from one drawn with vertical bars, and name the second a column graph
• Decide, for a given quantity, whether vertical or horizontal bars suit it better, and give the reason
• Explain how the scale is used to make a data picture fit the space it has
• Rank the seven summits from the graph and state the difference between the extremes
• Argue that a clear picture is a mathematical requirement, not decoration, because the reader's answer is the point of drawing it at all

Where it usually goes wrong

• "A table is more accurate than a graph, so it is better." Both carry the same numbers here. The table is worse at the job the reader has, which is comparing. Accuracy and usefulness are different properties.
• "Rotating the graph changes the data." It does not change a single height. The seven values, the ordering and the scale step of 1000 m carry over unchanged to both pictures on p.102; only how far the marked axis is carried differs, to 9000 across and to 10000 up. Only the reader's eye is being helped.
• "'Column graph' is a different kind of graph." It is the name for a bar graph whose bars stand up. The book introduces the word and immediately explains it by analogy with the pillars of a building.
• "Horizontal bars are wrong." The chapter uses horizontal bars for traffic on p.87, for tigers on p.100 and for the mountains on p.102. The rule is about fit, not about legality.
• "Making it look good means adding things." Everything §4.5 does is rearrangement: same numbers, same widths, same scale steps. Nothing is added. What happens when things are added is the next topic.
• "The scale is only about arithmetic." The book also gives it a second job: making the picture fit the paper. Same decision, two reasons.
• "Everest is roughly four times the Australian summit, since 8848 is about four times 2228." That ratio is real, but the book asks for the difference, not the ratio, and an explanation that answers the wrong question models the wrong habit. Compute what was asked.

Questions to check understanding

Class 6 has no board paper; these are the shapes the book uses and that school and competency-based assessments repeat:

• Given a quantity, choose vertical or horizontal bars and justify the choice (the book's own two items, p.103)
• Convert a small table into a bar graph and state which comparison became easier
• Name the arrangement in which bars are vertical
• Read a ranking off a bar graph without computing anything
• Compute the difference between the largest and smallest bar
• Explain how the scale can be used to make a graph fit a given space
• Given two graphs of the same data drawn differently, say what is the same and what has changed

Examples worth working on the board

• The seven summits table (p.101). Continent against tallest mountain against height in metres: Asia / Everest / 8848; South America / Aconcagua / 6962; North America / Denali / 6194; Africa / Kilimanjaro / 5895; Europe / Elbrus / 5642; Antarctica / Vinson Massif / 4892; Australia / Koscuiszko / 2228. Note the spelling — see Notes.
• The two questions the book asks of that table (p.101). How much taller Everest is than the Australian summit; and whether Denali and Kilimanjaro are very different in height. The first difference is large and the second is small, and the point is that neither is visible in the table.
• The horizontal bar graph (p.102). The same seven values drawn as boxes running across, with a scale marked 0 to 9000 in steps of 1000, and each row labelled continent and mountain.
• The column graph (p.102). The identical data rotated a quarter turn, the upright line marked 0 to 10000 in steps of 1000, mountain names running along the bottom at an angle. Same numbers, same scale interval, different reading.
• The directional rule (p.103). Heights, being measured upward from the ground, suit vertical bars; lengths lying parallel to the ground — the book's example is distances between places on Earth — suit horizontal ones.
• The two applications (p.103). The height of the tallest person in each class of your school; and the longest river on each continent, which the book asks you to look up and then graph. Vertical for the first, horizontal for the second, by the rule just given.
• The scale as a fitting tool (p.101). The book names the scale explicitly as the way to make a data presentation sit in the space it has been given. This links §4.5 straight back to §4.4 and is worth showing as a link, not as a new idea.

Figures to have open

• The seven-summit table (p.101). Type, extracts cleanly, but redraw rather than reproducing the printed table.
• The horizontal bar graph of the seven summits (p.102). Drawn artwork; must come from a printed page.
• The column graph of the same seven summits (p.102). Drawn artwork. These two must be shown as a pair — the whole argument of the topic is the difference between them.
• A pillar-and-roof sketch for the word "column". Standard schematic; the book gives the analogy in words only.
• A pair of icons for the directional rule: something tall measured upward, something long measured across. Standard schematic.
• No infographic belongs here — the decorated mountain pictures on pp.104–105 are the next topic, and showing them now pre-empts its argument.

Where this sits in the book

• NCERT Class 6 Mathematics (Ganita Prakash), Chapter 4 "Data Handling and Presentation", §4.5 "Artistic and Aesthetic Considerations", pp.101–103.
• The seven-summit table and the two questions asked of it, p.101; the horizontal bar graph and the column graph, p.102; the directional rule and "Figure it Out", p.103.
• §4.4 "Drawing a Bar Graph", p.100 — the bar-graph rules this section builds on.
• Chapter Summary, p.106 — the bullet that gathers colour, accompanying pictures and bar direction under the heading of §4.5.
• Back pointer: Drawing a bar graph, and choosing a scale that fits. Forward: How a decorated graph can mislead an honest reader.