Pictographs, and why the scale has to be stated

What to assume they know

• Tally marks and frequency: organising data so it can be read — frequency, and reading a frequency table
• Multiplication of a small whole number by 5 and by 10
• Halves: half of 10 is 5, and recognising a half of a drawn symbol
• Comparing two counts, and finding the difference between them
• Multiples, and testing whether one number divides another exactly

What they should be able to do

• Read a pictograph correctly by first reading its key
• Compute a category's frequency as (number of symbols) × (value of one symbol)
• Interpret a half symbol as half the key's value, and justify the reading
• Explain why two pictographs of the same data can look completely different
• Choose a key for a given data set and defend the choice on grounds of space and of whether the values are multiples of it, or of half of it
• Identify data that a chosen key cannot represent exactly, and say why
• Draw a pictograph from a frequency table, with categories in rows or columns, a symbol per unit, and the key printed beside it
• State what must always be printed alongside a pictograph for it to mean anything

Where it usually goes wrong

• "Count the symbols and you have the number." Only when the key says one symbol is one thing. On p.80 five symbols mean fifty children. The habit to build is: read the key first, every time.
• "A half symbol means one more." (The book calls it a half picture, p.80.) It means half of whatever the key says. On Sangita's pictograph a half symbol is 5 students because her key is 10; on Jarina's it would be 2½, which is why she never draws one.
• "A longer row always means more." Only if both rows use the same key. Jarina's Class II row is seven symbols and Sangita's is three and a half, and they represent the same 35 students.
• "An empty row is a mistake in the drawing." Class V on p.81 has no symbols because nobody was absent. Zero is a legitimate frequency and must be drawn as nothing, not skipped.
• "A bigger key is always better because it saves space." It also settles what you can still draw exactly — its own multiples, and with a half symbol the values midway between them, and nothing else. That is exactly what the book's 33-and-27 question is testing.
• "You can just round 27 up to 30 and draw three symbols." Then the picture says something the data does not. If the key cannot express the value, change the key or say so — do not quietly adjust the data.
• "The key is a caption; the picture is the maths." Reverse it. Without the key the drawing carries no number at all.

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:

• Read a pictograph with a key greater than 1 and state a named category's value
• Find the total across all categories of a pictograph
• Find the difference between two categories, and test a stated comparison such as "D is half of E" or "this one is more than double that one"
• State how many symbols a given value needs under a given key
• Choose a suitable key for a listed data set and justify it
• Explain what changes in the picture when a single value increases by a stated amount (the book's own p.97 item: two more girls admitted to Class 2)
• Draw a pictograph from a frequency table, key included
• Say why a particular value cannot be drawn exactly under a given key

Examples worth working on the board

• Modes of travelling (p.79). Key: one symbol = 1 student. Drawn rows — private car 4 symbols, public bus 5, school bus 11, cycle 3, walking 7. The two questions put to it are printed overleaf at the top of p.80: which mode the most students use, and which the fewest. Neither asks for a total and none is printed. The row lengths are drawn, not typed, so this must come from the printed page.
• The sleep survey (p.80). Nand Kishor asked the children of a middle school in Berasia to say how frequently a night's sleep of nine hours or more had been reached. Key: one symbol = 10 children. Drawn — always 5 symbols, sometimes 2 whole symbols and a half, never 4 symbols. The book works all three readings itself on pp.80–81: 50, 25 and 40.
• Lakhanpal's absentees (p.81). Classes I to VIII with 3, 5, 4, 2, 0, 1, 5 and 7 students absent. Drawn as a pictograph at one symbol per student. Class V has no symbols at all — a row of length zero is a reading, not an omission.
• Students present (pp.81–82). The same eight classes with 30, 35, 20, 25, 30, 25, 30 and 20 present. At one symbol per student this needs 215 symbols; the book asks what difficulties that would cause before it offers a fix. (215 is an added sum, not printed.)
• Jarina's pictograph (p.82). Key: one symbol = 5 students. Rows drawn as 6, 7, 4, 5, 6, 5, 6 and 4 symbols for Classes I to VIII.
• Sangita's pictograph (p.82). Key: one symbol = 10 students. Rows drawn as 3, 3½, 2, 2½, 3, 2½, 3 and 2 symbols for Classes I to VIII. Three of those eight rows end in a half symbol, and they are exactly the three classes whose count is not a multiple of ten — Class II at 35, and Classes IV and VI at 25 each.
• The 33-or-27 question (p.83). The book asks what would go wrong if a class had 33 or 27 present. Neither is a multiple of 10 or of 5, and neither leftover (3 and 7) can be drawn as a clean part of a symbol.
• Books borrowed (p.84). Key: one symbol = 1 book. Drawn — Monday 5, Tuesday 4, Wednesday 2, Thursday none, Friday 5, Saturday 8. The book asks for the minimum day, the maximum day and the week's total.
• Magan Bhai's kites (pp.84–85). Chaman 250, Rani 300, Rukhsana 100, Jasmeet 450, Jetha Lal 250, Poonam Ben 700. The book fixes the key at one symbol = 100 kites, so three of the six — Chaman, Jasmeet and Jetha Lal — end in a half symbol and the other three do not. Rukhsana's claim that Poonam Ben bought more than double Rani's is a comparison to be tested, not asserted.
• Mudhol Hounds (§4.4, p.98). Six villages: 18, 36, 12, 48, 18 and 24 dogs. Here the key is not given — the student must choose one that draws every value with whole symbols. This is the cleanest example in the chapter of the key being derived from the data rather than handed down.

Figures to have open

• All the pictographs in this topic are drawn artwork and extract as nothing from the PDF text. Every one of them must be taken from a printed page: travel p.79, sleep p.80, absentees p.81, Jarina and Sangita p.82, books p.84, tractors p.96, girl students p.97.
• A key panel, enlarged: one symbol, an equals sign, a number. Standard schematic. This is the single most important image in the topic.
• A half symbol shown as a symbol cut down the middle, with the arithmetic beside it. Standard schematic.
• Redraw rather than reproduce: the symbol counts are the data and must be exact, but the book's smiling-face and open-book art is its own.
• No bar graph belongs here; the comparison with bar graphs is the next topic.

Where this sits in the book

• NCERT Class 6 Mathematics (Ganita Prakash), Chapter 4 "Data Handling and Presentation", §4.2 "Pictographs", pp.79–85. Within it the unnumbered bold sub-heading "Drawing a Pictograph" begins on p.81 — it carries no number, so cite it by name and page.
• The travel pictograph and the definition, pp.79–80; the worked sleep example and its solutions, pp.80–81; Lakhanpal, Jarina and Sangita, pp.81–82; the Math Talk on 33 and 27 and the summary bullets, p.83; "Figure it Out", pp.83–85.
• §4.4 "Drawing a Bar Graph", pp.96–98 — three further pictograph items (tractors p.96, girl students p.97, Mudhol Hounds p.98) sit inside §4.4's exercise block although they are pictograph questions.
• Chapter Summary, p.106 — pictographs, and the scale that must be specified.
• Back pointer: Tally marks and frequency: organising data so it can be read. Forward: Reading a bar graph: what length is standing for.