Tally marks and frequency: organising data so it can be read

What to assume they know

• Turning an argument into a question you can settle with data — what data is, and why a list of replies is not yet an answer
• Counting in fives, and adding a run of small numbers
• Comparing whole numbers, and putting a short list in increasing order
• Reading a two-column table

What they should be able to do

• Record a running count using tally marks, bundling every fifth stroke
• Read a bundled tally group as a number without recounting the strokes
• Complete a frequency table from tally marks and state each frequency
• Define frequency as the count of occurrences of a value or category
• Explain what a frequency table can be used for and what it can no longer tell you, using the book's own sweets question
• Arrange a set of measures in ascending order and read off the largest, the smallest, and how many take a given value
• Answer "how many are greater than 4?" from ordered data and say why the ordering made it quick
• Choose between an ordered list and a frequency table for a given question, and justify the choice

Where it usually goes wrong

• "Tally marks are just a childish way of writing numbers." They exist for a situation numerals handle badly — a count that has to be updated while you are looking somewhere else. The crossed fifth stroke is a deliberate design feature, not decoration.
• **"A crossed bundle means five more."** The crossing stroke is the fifth. A bundle plus one loose stroke is six, not seven. This is the single most common slip when reading the printed sweets table.
• "Frequency means how often something happens in time." In this chapter it means the count of occurrences of a category or value. Nine students chose gulab jamun; nothing in that is about time.
• "Once you have the frequency table you have all the data." The book asks the killing question itself on p.77: the sweets table cannot say which child wanted which sweet. Summarising loses information — deliberately, and you should know which information you have chosen to lose.
• "Ordering the data changes it." Sorting the twenty-seven shoe sizes adds nothing and removes nothing. It only puts equal values next to each other, so the counting stops requiring you to scan the whole board each time.
• "'Larger than 4' includes 4." It does not. Fifteen students, not twenty-four, wear a size above 4 — a place where a careless read of ordered data goes wrong quietly.
• "There is one right way to organise data." The book's own Math Talk asks for other arrangements. Which organising is right depends on the question.

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:

• Complete a frequency table from given tally marks (the book's own p.76 item)
• Convert a list of raw observations into tally marks and a frequency table
• Read a bundled tally group and state the number it represents
• Given a frequency table, state which question it can answer and which it cannot, with a reason (the book's own p.77 item)
• From data in ascending order, state the largest, the smallest, the count of a named value, and the count above or below a stated value
• Explain why ascending order made a particular question quicker to answer
• Total a frequency column and check it against the number of observations

Examples worth working on the board

• The sweets tally (p.76). Five sweets, drawn tally marks, and a third column partly filled in. Jalebi shows one bundle of five plus one loose stroke and the printed frequency 6. Gulab jamun shows one bundle plus four loose strokes and the printed frequency 9. Gujiya shows two bundles and three loose strokes; barfi three loose strokes; rasgulla one bundle and two loose strokes. The last three frequency cells are blank for the student. The values the tallies encode are 13, 3 and 7.
• The class total. 6 + 9 + 13 + 3 + 7. The book does not print the sum; it is worth computing as a check that every child was recorded once.
• The frequency box (p.77). A boxed note on that page attaches the word frequency to the numbers in the last column, taking 6 and 9 as its two worked instances.
• The distribution question (p.77). The book asks whether the same table is enough to hand each sweet to the right child. It is not: the table records how many chose each sweet, not who chose what.
• The shoe sizes (p.77). Three printed rows of nine values: 4 5 3 4 3 4 5 5 4 / 5 5 4 5 6 4 3 5 6 / 4 6 4 5 7 5 6 4 5. Twenty-seven values in all. The book then prints them in ascending order.
• What the ordering yields. Size 3 occurs 3 times, size 4 occurs 9 times, size 5 occurs 10 times, size 6 occurs 4 times, size 7 occurs once. Largest 7, smallest 3. Ten students wear size 5; fifteen wear a size larger than 4.
• The die (§4.4, p.95). Roll a die thirty times, tally the outcomes, then read off which face came up least, most, and which tie. This is the chapter's invitation to build a frequency table from data nobody has pre-sorted.

Figures to have open

• The sweets frequency table with drawn tally marks and three blank cells (p.76). Redraw: the tallies and the two printed frequencies are the data and must be exact, but the layout is the book's. The tally strokes are artwork and do not extract from the PDF text — this figure must be drawn from the printed page.
• A single crossed bundle of five, enlarged, with the fifth stroke distinguished from the first four. Standard schematic.
• The twenty-seven shoe sizes as printed, three rows of nine (p.77), and the same values in ascending order. Both are printed on p.77; redraw.
• The blank tree table (p.78) if section 12 is shown. Standard schematic.
• No pictograph or bar graph belongs in this topic.

Where this sits in the book

• NCERT Class 6 Mathematics (Ganita Prakash), Chapter 4 "Data Handling and Presentation", §4.1 "Collecting and Organising Data" — the sweets tally table and its questions, p.76; the frequency box and the shoe-size data, p.77; the self-collection tasks, p.78; the Teacher's Note, p.79.
• §4.4 "Drawing a Bar Graph", p.95 — the die-rolling frequency-table task and the Bumrah wicket table, both of which are frequency tables built before any graph is drawn.
• Chapter Summary, p.105 — data organised in tabular form using tally marks, and frequencies as counts of occurrences.
• Back pointer: Turning an argument into a question you can settle with data. Forward: Pictographs, and why the scale has to be stated.