Line graphs, and what change over time looks like

What to assume they know

• Reading a bar graph and a clustered column graph, and reading a dot plot — all recalled, not taught, at the head of §5.2 (Part II p.116)
• Plotting a point against two axes, and reading a value off a marked axis with gridlines
• Subtracting to find a change, and comparing two changes
• Reading a scale whose labels step by more than one unit
• Twelve-hour and twenty-four-hour clock times, for the two astronomy items
• Rounding a reading to a sensible precision, and saying so

What they should be able to do

• State what a line graph adds to a column graph of the same data, and what it does not
• Decide whether a given data set may honestly be drawn as a line graph, and justify the decision by looking at the horizontal axis
• Read a value off a line graph to a stated precision, and read a change between two points
• Compare two changes by comparing the steepness of two segments
• Identify the axis labels, the units, the scale interval and the legend of an unfamiliar line graph before interpreting it
• Explain how a monthly maximum temperature is produced from many recorded readings, and what that leaves out
• Describe two series on one axis in a way that names both the shapes and the difference between them
• Draw a line graph from a two-row table, choosing an interval that fits the data
• Fill missing table values by reading them off a plotted line, and vice versa
• Read a pair of lines that are two measurements of one place rather than one measurement of two places

Where it usually goes wrong

• "A line graph is a bar graph with the tops joined." Visually yes, logically no. Joining the tops asserts that the quantity took the in-between values, and that is a claim about the world which the columns never made.
• "Any table can be drawn as a line graph." Only when moving along the horizontal axis means something. Draw six cities' rainfall as a line and the segments claim a city between Mumbai and Chennai. The chapter never does this.
• "A steeper line means a bigger quantity." It means a bigger change. Punjab's steepest segment is between January and February, where its values are the lowest on the page.
• "The graph tells me the temperature in the middle of March." It tells you March's figure and April's. The line between them is a straight-line guess drawn by whoever made the figure.
• "The monthly maximum is the usual temperature." It is the highest single reading from any station in the state that month. Kerala looking flat means its maxima are flat.
• "Two graphs that look different must show different data." Part II pp.116 and 117 are the same twelve pairs of numbers. Which form to choose is a question about the reader's question, not about the data.
• "If the numbers barely move, the line will look flat." Only if the axis says so. This chapter is careful — the births axis reaches down near half a million and the sleep figures on Part II p.126 are printed twice, once full-scale and once zoomed, precisely so the reader can see what zooming does. Point at the pair rather than warning in the abstract.
• "A source line under a figure is decoration." It is the only thing on the page that lets you find out how the numbers were made. Four figures in §5.2 carry one, and the provenance box exists to make the reader read them.

Questions to check understanding

• Draw a line graph from a two-row table, choosing and labelling a sensible interval
• Read a stated value, and a change between two times, off a line graph
• Say which of two segments represents the larger change, and justify it by steepness
• Fill a blank row of a table from a line already plotted, and plot a row that is only in the table
• Judge whether a data set may be drawn as a line graph and give the reason
• Name the axis units, the interval, the legend and the source of an unfamiliar figure before interpreting it
• Explain how a stated summary figure (a monthly maximum, a monthly average) was produced from raw readings
• Estimate an annual total from twelve monthly readings
• Read two curves for one place, such as sunrise and sunset, and compute the quantity between them

Examples worth working on the board

Values marked printed appear on the page. Values marked not in the book are added readings off the printed pages and the printed page, or arithmetic added here on them; the chapter prints no answers to any exercise item and Part II has no answer appendix.

• The temperature figure, twice (Part II p.116 as a clustered column graph and Part II p.117 as a line graph; printed). Title on both: Monthly Maximum Temperature. Vertical axis labelled in Celsius, 0 to 40 in tens; horizontal axis the twelve months; legend Kerala with a blue circle marker and Punjab with a red square marker. The prose says the data is for 2023. The page asks whether the two figures show the same information — they do, which is the pivot of the whole section.
• The twelve pairs of values (not in the book, read off Part II p.117 and cross-checked against the book's own stated readings). Kerala, January to December, in °C: about 30.5, 31.5, 32.5, 33, 32, 30.5, 29, 29, 29.5, 30, 31, 31. Punjab: about 19, 25.5, 29, 34, 37, 38, 34.5, 35, 35.5, 33, 28, 23. The book itself states five of these in its interpretation — Punjab's June high of 38, its July value just under 35, its December value of about 23 and its January low of about 19, and Kerala's April peak of about 33 and July low of about 29 — so the readings above are anchored, not free.
• The annotated version (Part II p.118, printed). The same line graph carries three callout labels: one naming the upward slope from January to June on the Punjab line, one naming the June peak of about 38, and one naming the July low of about 29 on the Kerala line. This figure is the model for section 4 and should be rebuilt beat by beat rather than shown finished.
• The two-step reading procedure (Part II pp.117–119, printed as bold steps with bullets under each). Step one asks what is given — how the figure is organised, what scale it uses, what the marks and colours mean. Step two asks what can be inferred, and wants summary statements. The same two steps are then reused verbatim on the space-launch figure in Interrogating a real dataset: traffic, and rainfall.
• The provenance box (Part II p.117, printed as a tinted box with an owl). Its point is that knowing how the data was collected fixes the scope of what may be claimed, exposes bias and gaps, and settles how much confidence a conclusion deserves.
• How a monthly maximum is made (Part II p.117, printed). A state has several weather stations recording locally; the month's figure is the largest of all those readings. Not in the book: so a single hot afternoon at one station sets the value for the whole state and month, and nothing in the figure reports the typical day. That is the cleanest available example of the provenance box's point.
• The marker-shape note (Part II p.117, printed as a green side box). Different marker shapes keep the series distinguishable in greyscale printing and for readers who cannot easily separate the colours. Worth one beat: it is an accessibility decision made visible in a maths textbook.
• The two summary statements the page reaches (Part II p.118, printed): Punjab's monthly maxima climb to a June high and fall away again, while Kerala's stay nearly level all year, so Punjab is the more variable of the two — colder in winter and hotter in summer. Alongside sits a cartoon whose speaker has one foot in each of two buckets and declares that on average all is well: the joke names exactly what a single summary number throws away.
• The follow-up questions (Part II pp.118–119, printed, unanswered): why the two trends differ and what governs a region's temperature; whether other states show other kinds of trend; which states resemble Punjab and what they have in common; what the monthly minimum temperatures would look like. A Math Talk flag then asks what questions occur to the reader.
• The fifty-two-bar argument (Part II p.120, printed in a boxed aside about the space-launch data). Thirteen years by four series drawn as clusters would be fifty-two columns, crowded and hard to read; joined-up points show the trend and let a reader follow one series across the years. Not in the book: the box is an argument about the reader, not about the data — the same numbers are in both pictures.
• Item 1 (Part II p.122, printed). Average customers visiting a shop, Monday to Sunday: 16, 19, 10, 14, 20, 22, 35. Average customers actually buying something: 10, 8, 7, 11, 12, 16, 26. The task is to draw the line graph. Not in the book: the two lines on one axis make the gap the subject — 6, 11, 3, 3, 8, 6, 9 across the week — and the fraction who buy swings from about two in five on Tuesday to about three in four on Thursday and Sunday. An interval of two customers to a gridline fits both rows on one axis.
• Item 2, the births figure (Part II p.123, printed). Vertical axis Live births, labelled 0.5M, 1M, 1.5M and 2M; horizontal axis Time, labelled at six-month intervals from Jul 2017 to Jan 2020; the source line credits Nambiar and others, forthcoming, on seasonal variation in Indian births. Not in the book, read off the printed page: thirty-six monthly points running from about April 2017 to about March 2020, oscillating between about 1.4 million and about 1.95 million, with every year peaking around August to October and bottoming in April or May. The point standing on the Jul 2017 gridline reads about 1.7 million. The three January values read about 1.58, 1.68 and 1.71 million, so January rises slightly year on year. Adding the twelve months of 2019 gives about 20.7 million births, so "about 20 million" is the right answer to the last part and the method — twelve readings, added — is what the item is really asking for.
• Item 9, the salt table (Part II p.129, printed): retail price of iodised salt in rupees each January from 2016 to 2025 for Andaman and Nicobar Islands, Assam, Gujarat, Mizoram, Uttar Pradesh and West Bengal. The item allows rounding to whole rupees, asks for a line graph of any three states, a comparison of the variation in Gujarat against Uttar Pradesh, and which state rose most across the decade. Not in the book: the ten-year rises are about ₹5 for the islands, ₹6.35 for Assam, ₹2.70 for Gujarat, ₹9.80 for Mizoram, ₹8.66 for Uttar Pradesh and ₹14.52 for West Bengal — so West Bengal on the absolute question, and West Bengal again if you ask which multiplied most, at about two and a half times. Gujarat is the flattest series, sitting at 14.75 for three years running. Assam's price doubled between 2016 and 2017 (₹6 to ₹12) and then only wobbled — it holds at 12 through 2018, 2019 and 2020, rises to 15 in 2021, then runs 14, 12.02, 13.72, 12.35. "Doubled once and then wobbled" is the exact shape; do not say it stopped moving. It is still the most interesting shape on the page, and it is invisible in any single number.
• Item 14, sunrise and sunset (Part II pp.131–132, printed as two panels). Left panel: Kibithu and Ghuar Moti. Right panel: Srinagar and Kanyakumari. Both panels carry a time axis labelled 04:00, 08:00, 12:00, 16:00 and 20:00 against the twelve months, and each place contributes two curves. The item asks which curves are sunrise and which sunset, where the sun rises earliest in January, what the day length there is, and which place has the longest day of the year. Not in the book readings: the lower band is sunrise. Kibithu rises at about 06:00 in January and about 04:00 in June, and sets at about 16:30 in January; Ghuar Moti rises at about 07:40 in January. Srinagar swings from about 07:35 and 17:35 in January to about 05:15 and 19:45 in June; Kanyakumari holds near 06:30 and 18:25 all year. So the earliest January sunrise is Kibithu's, its January day runs about ten and a half hours, and Srinagar has both the longest day of the year and the largest swing, while Kanyakumari's day barely changes.
• Item 15, the moon (Part II p.132, printed). Moonrise as yellow diamonds and moonset as blue squares, plotted for each day 1 to 31 of one month against a full twenty-four-hour axis. The item asks on which dates the new moon and the full moon fell. Not in the book: each series climbs by roughly fifty minutes a day and therefore runs off the top of the axis and restarts at the bottom — moonset does this between about day 5 and day 7, moonrise between about day 20 and day 21. Full moon is where moonrise sits near 18:00 and moonset near 06:00, which the graph puts near day 14; new moon is where moonrise sits near 06:00 and moonset near 18:00, near day 29. The fifteen-day gap between them is the check that the reading is right.

Figures to have open

• The temperature data in both forms (Part II pp.116–117). Both are needed in section 1 and the argument fails with either one missing. Redraw from the twelve pairs above rather than reproducing the printed art, and keep the circle and square markers.
• The annotated temperature graph (Part II p.118), rebuilt one callout at a time.
• A segment-under-magnification figure: two measured points, the straight line between them, and two or three other curves that would fit the same two points. Not in the chapter, and it is the figure that makes section 2 honest.
• A legitimate-axis test figure: the same values drawn against months and against unordered categories. Standard schematic.
• The customers table and its graph (Part II p.122), built.
• The births figure (Part II p.123), redrawn from the readings above with its source line kept.
• One panel of the sunrise-and-sunset figure (Part II pp.131–132) with the daylight band shaded between the two curves — the shading is the explanation's addition and is what makes day length visible.
• The moon figure (Part II p.132), redrawn, with the wrap at midnight marked.
• No photograph is needed.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 5, "Tales by Dots and Lines", §5.2 "Visualising and Interpreting Data", Part II pp.116–119. §5.2 opens at the foot of Part II p.116 with a one-paragraph recall of the graph types already known; the unnumbered bold subheading "Line Graphs" follows immediately, with a further bold label "Temperature" under it. The clustered column graph is on Part II p.116, the line graph and the provenance box on Part II p.117, the annotated graph and the interpretation on Part II p.118, and the last follow-up question and its Math Talk flag at the top of Part II p.119.
• The boxed argument for preferring a line graph is on Part II p.120, inside the next subsection; it is quoted here because it is the chapter's only explicit statement of why the form was chosen.
• SUMMARY, Part II p.133, third bullet: line graphs show change over time. That single line is the chapter's summary of this topic.
• Exercise items owned by this topic: Part II p.122 item 1; Part II p.123 item 2, the births figure; Part II p.129 item 9; Part II pp.131–132 item 14; Part II p.132 item 15.
• Sibling topics: the two interrogation datasets, and the two valid-inference items on Part II p.130, are Interrogating a real dataset: traffic, and rainfall; the rainfall-days item on Part II pp.122–123 is also filed there. Infographics and activity strips are Infographics and activity strips: what a visual shows and what it hides; the sleep curves, which are the chapter's one smooth-looking line graph, are Telling a story with data, and letting it raise the next question.