Drawing a bar graph, and choosing a scale that fits

What to assume they know

• Reading a bar graph: what length is standing for — reading a bar graph and what a stated scale means
• Tally marks and frequency: organising data so it can be read — building a frequency table from raw observations
• Division of a three- or four-digit number by 200, 100, 10, 5 and 4, exactly
• Finding the largest and smallest value in a list
• Halving, doubling, and comparing a number with a quarter of another
• Drawing a straight line, marking equal intervals along it

What they should be able to do

• Set up the two perpendicular reference lines and decide which carries the categories and which the frequencies
• Choose a scale from the smallest and largest frequencies in the data, and justify the choice
• Compute each bar's length as frequency ÷ scale and draw it to that length
• Draw bars of equal width with equal gaps between them, starting the markings at zero
• Redraw a given data set at a second scale and say what changed and what did not
• Recover a graph's scale from one bar whose length in units and value are both known
• Detect and correct a bar drawn at the wrong height, given the table it came from
• Answer comparison questions from a finished graph, including ratio comparisons such as "about half" and "less than a quarter"
• Build the frequency table first when handed unsorted observations, then graph it

Where it usually goes wrong

• "The scale is whatever the teacher says." It comes from the data. The book arrives at ten runs to the unit by looking at the smallest score, 0, and the largest, 100, and asking what would fit. Show the reasoning, not the result.
• "A bigger scale is a bigger graph." The opposite. One unit worth ₹200 turns a ₹3400 bar into 17 units; one unit worth ₹1 would need 3400.
• "Bar heights are drawn by eye." They are computed. The book prints a whole column of divisions on p.92 precisely so this is not left to judgement.
• "A frequency of zero means leave the category out." Match 6 keeps its slot and gets no height. Removing it would change the shape of the record.
• "Bars should touch, so the graph looks full." The chapter's own summary says the gaps are there to show the bars stand free and represent separate categories.
• "Start the numbering wherever the data starts." The book's summary states plainly that the markings must begin at zero. Starting at 15 makes a bar of 16 look tiny beside a bar of 28 in a way the numbers do not support.
• "To total a frequency table, add the left-hand column." This is exactly the trap the book sets with Bumrah's wickets. The left column lists the values; the total requires each value times its own frequency.
• "Every bar in a printed graph must be right." Two of the chapter's exercises hand the student a wrong graph on purpose. Checking a picture against its table is a skill the book is explicitly teaching.

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:

• Draw a bar graph from a given frequency table at a stated scale
• Draw a bar graph from a given table choosing your own scale, and justify it
• State the scale of a graph given one bar's length in units and its value
• Complete a partly drawn graph: add the axis numbering, correct a wrong bar, draw a missing bar
• Find the mistakes in a graph by checking it against its own table
• Build a frequency table from unsorted observations, then graph it
• Compute a total from a frequency table where the left column holds values rather than categories
• Answer ratio questions off a finished graph ("about half", "less than a quarter") by computing rather than estimating
• Add two bars, and compare two bars by difference

Examples worth working on the board

• The sweets graph (pp.89–90). Frequencies of 13 for gujiya, 9 for gulab jamun, 7 for rasgulla, 6 for jalebi and 3 for barfi. The book sets one unit length to one student, marks the upright line 0 to 14, and draws five bars. The smallest frequency is 3 and the largest 13 — that spread is why one-for-one is comfortable here.
• Smriti's runs (pp.90–91). Eight matches with 80, 50, 10, 100, 90, 0, 90 and 50 runs. The book reasons that stepping from 0 to 100 one run at a time would be tedious and sets one unit length to ten runs instead. Match 6 then has a bar of zero height.
• Imran's family's monthly spending (pp.91–92). House rent ₹3000, food ₹3400, education ₹800, electricity ₹400, transport ₹600, miscellaneous ₹1200. The book fixes one unit length at ₹200 and prints the division for every item: 3000 ÷ 200, 3400 ÷ 200, 800 ÷ 200, 400 ÷ 200, 600 ÷ 200, 1200 ÷ 200.
• The three questions on that graph (p.93). Which item takes the most and which the second most; whether electricity is about half of education; whether education is less than a quarter of food. The second and third are ratio questions, and the third is genuinely close.
• Samantha's tea-garden count (p.93). Mites 6, caterpillars 10, beetles 5, butterflies 3, grasshoppers 2 — a small data set to be graphed from scratch, with the scale left to the student.
• Pooja's ticket graph (pp.93–94). Tickets sold at Bhopal station over two hours: Vidisha 24, Jabalpur 20, Seoni 16, Indore 28, Sagar 16. Part of the drawn graph has been rubbed out — the upright line carries no numbers and Sagar has no bar at all. The book states that the Vidisha bar is 6 unit lengths and the Jabalpur bar 5, which is what makes the scale recoverable. As drawn, Seoni's bar stands at 4 unit lengths; Indore's stands between 4 and 5.
• Chinu's hour of traffic (pp.94–95). A printed grid of individual vehicles seen passing a house between 9 and 10 in the morning — bikes, cars, bicycles, auto rickshaws, scooters, buses and bullock carts, in no order. The task is to build the frequency table first.
• Bumrah's wickets (pp.95–96). Wickets taken 0 to 7 against matches 2, 4, 6, 8, 3, 5, 1, 1. The book has a character propose adding 0 + 1 + 2 + … + 7 to get the total wickets and asks whether that works. It does not: each wicket-count must be multiplied by its match-count first. The match column adds to 30, which is the check that the table is complete.
• The free-time survey (p.98). 120 students: playing 45, reading story books 30, watching TV 20, listening to music 10, painting 15. The book fixes the scale at one unit length to five students and asks which activity leads apart from playing.
• The saplings graph (p.99). A week of planting drawn as seven bars with the upright line marked in tens up to 70. Read at the printed gridlines the week's days come out as 50, 40, 30, 40, 50, 60 and 40 — see Notes on the Monday bar. The book asks for Wednesday and Thursday together, the week's total, and the greatest and least days.
• The tiger graph (pp.99–100). Table: 2006 about 1400 tigers, 2010 about 1700, 2014 about 2200, 2018 about 3000, 2022 about 3700. Beside it a bar graph the book says contains mistakes. Measured against its own printed scale only the 2022 bar matches the table; 2014 and 2018 appear interchanged, and 2006 and 2010 are both drawn short. This is the chapter's best single exercise and should be the explanation's closer.

Figures to have open

• The finished sweets bar graph (p.90), with the upright line marked 0 to 14. Drawn artwork; must come from a printed page. Redraw.
• The runs graph (p.91) with its zero-height bar. Drawn artwork.
• The division column on p.92 — this one is type and extracts, but it is the single most useful image in the topic and should be shown as a column, each row lining up with its bar.
• Imran's expenditure graph (p.92), marked in steps of ₹200 up to ₹3600. Drawn artwork.
• Pooja's part-erased ticket graph (p.94): five labelled slots, four drawn bars, an unnumbered upright line. Drawn artwork. This figure cannot be reconstructed from the text at all — the whole exercise is what the drawing shows and does not show.
• The saplings graph (p.99) and the tiger graph (p.100) with its table beside it. Both drawn artwork.
• A ruler motif for the recurring "height = frequency ÷ scale" step. Standard schematic.

Where this sits in the book

• NCERT Class 6 Mathematics (Ganita Prakash), Chapter 4 "Data Handling and Presentation", §4.4 "Drawing a Bar Graph", pp.89–100.
• The four-step procedure and the sweets graph, pp.89–90; the runs example, pp.90–91; the expenditure example with its division column, pp.91–92; the three reading questions, p.93; "Figure it Out" items 1–11, pp.93–100; the closing bar-graph rules and the Teacher's Note, p.100.
• §4.3 "Bar Graphs", pp.85–89 — the reading conventions this topic assumes.
• Chapter Summary, p.106.
• Back pointer: Reading a bar graph: what length is standing for. Forward: What makes a data picture clear and worth looking at.