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How to Perform in the new G3 SEC Examinations | Learner’s Guide Vol 0099 | Mathematics: Graph-Scale Workshop — Read a Non-Zero Axis Before You Judge the Change

G3 SEC Mathematics · Learner’s Guide Vol 0099. Learn to read the scale, calculate the actual change and evaluate graph-based claims.

A graph can tell the truth numerically and still leave readers with the wrong impression. Imagine two bars showing test scores of 78 and 82. When the vertical axis begins at zero, the bars look similar. When the axis starts at 75, the difference appears enormous. Neither picture changes the recorded scores. The scale changes how the difference looks.

This G3 SEC Mathematics Learner’s Guide Vol 0099 teaches you to read graphs before believing their visual impact. You will check axis titles, units, starting points, tick spacing, intervals, gradients, totals and the purpose of a graph. You will also learn to calculate the actual change and explain why a display can be misleading without calling every non-zero axis dishonest. Many statistical graphs use thoughtful non-zero limits for legitimate reasons. The task is to interpret the display correctly.

The route builds on Volume 0091 on journeys and time graphs, which linked slopes to distance and speed. It also extends Volume 0079 on weighted comparisons, where numbers needed fair denominators. Here, the problem is the link between numerical data and the picture used to represent them.

Chapter index

1. The ten-second graph check

Before reading a value from a graph, look at the horizontal axis and vertical axis. What quantity appears on each? What are the units? Where does each scale begin? What does each tick represent? Is the interval spacing uniform? Is a break in the axis indicated? If you cannot answer those questions, wait before making a comparison.

Suppose a vertical bar graph shows the weekly number of library visits. The axis labels are 90, 100, 110 and 120, not 0, 10, 20 and 30. One bar reaches 100, another reaches 115. Numerically the increase is 15 visits, or 15/100 × 100%=15%. Visually the second bar may look more than twice as tall above the 90 baseline. That exaggerated ratio of displayed heights is not the ratio of the underlying totals.

This is a useful distinction: absolute difference, percentage difference and visual height ratio are three different things. Here, absolute difference is 15 visits. Relative increase is 15%. The visible portion of the first bar rises 10 units above the truncated baseline, while the second rises 25; its displayed excess height is 2.5 times as large. A viewer who calls the second week’s attendance “two and a half times the first” has mistaken the drawing’s clipped lengths for the data.

A clear answer to “Why might this bar graph be misleading?” says, “The vertical axis starts at 90 instead of zero, which exaggerates the apparent difference between the two bars. The underlying values are 100 and 115, only 15% apart.” It names a feature, explains its effect and supplies a correct comparison. Calling the graph “fake” would be too broad when the values may have been plotted correctly.

Not every graph must start at zero. A line graph of hourly room temperature may use 20 to 30 °C to reveal a small but scientifically important change. The graph becomes misleading if the reader assumes the truncated display tells them the relative size of entire quantities. Interpret the purpose and measurement before judging the choice of axis.

2. The current K310 examination context

The official 2027 G3 SEC Mathematics syllabus K310 includes reading and interpreting tables, bar charts, pictograms, line graphs, pie charts, histograms, cumulative-frequency diagrams, box-and-whisker plots and other statistical representations. Under data handling, it explicitly includes explaining why a statistical diagram leads to misinterpretation of data. Our workshop therefore develops a named syllabus skill, but these examples are not official specimen questions.

K310 Paper 1 lasts 2 hours 15 minutes and contains about 26 compulsory short-answer questions worth 90 marks. Paper 2 also lasts 2 hours 15 minutes, has 9 to 10 compulsory questions worth 90 marks and concludes with a real-world application question. The two papers each contribute 50% to the assessment. The syllabus also makes clear that essential working matters and that non-exact numerical answers generally should be given to three significant figures unless a question states another level of accuracy.

Do not confuse statistical graph questions with coordinate geometry or journey graphs simply because all involve axes. A bar graph’s category position is not a physical time interval. A distance–time graph can represent a changing quantity where gradient matters. A histogram uses rectangles with a frequency interpretation tied to class intervals. A cumulative-frequency graph supports percentile reading. You must recognise the representation before choosing the method.

The Mathematics Hub owns the larger subject route; the G3 SEC learner-guide index keeps the wider examination plan connected. This article owns the narrower skill of graph scale and visual inference, not the entire Statistics and Probability strand.

For learners coming from Primary School, think back to simple picture graphs. A picture representing ten objects cannot suddenly be treated as one object when you compare categories. Secondary graph literacy makes the same demand with more subtle scale choices and representations.

3. Why bars and lines need different reading habits

In a bar graph, the size of each bar is often compared visually. A zero baseline is normally important when bars represent amounts whose lengths should be proportional to those amounts. If two values are 40 and 80, a zero-based bar twice as tall for 80 communicates a genuine factor of two. With a baseline at 35, the visible bar lengths are 5 and 45, so the second appears nine times as tall even though the values differ by a factor of two.

In a line graph, the purpose may be to show variation from one time to another rather than compare complete bar lengths. A non-zero vertical axis may be appropriate when a small fluctuation is the focus. Yet the reader still has to notice the starting point, scale and time interval. A line that looks steep on a tall, narrow graph may look gentle on a wide, short version using the same data.

Steepness on a drawing is affected by the relative physical sizes of the horizontal and vertical axes. A numerical gradient, in contrast, depends on changes in the measured variables and their units. If a line rises 12 metres over 3 seconds, its physical rate is 4 m/s in the relevant model. Making the paper graph wider does not change the numerical rate.

A misleading graph question might involve a bar chart with truncated values, an uneven tick interval, a selected date range or a graph title that promises more than its data show. Do not diagnose every picture with the same slogan “start at zero”. Identify which representation is being used and how its form affects the inference.

Suppose a line graph of a substance’s temperature spans 28 to 32 °C. Such a narrow axis can be helpful for measuring a 1 °C difference. The graph should be clearly labelled, and conclusions should describe a one-degree change rather than saying the sample’s temperature “doubled” because the visible line height doubled. Temperature in degrees Celsius is also not a ratio scale suitable for claims about doubling the absolute physical temperature.

The broader lesson is to read exact values and units rather than trust an angle, colour or apparent length by itself.

4. Worked bar case: two school clubs

A school club recorded 120 members last year and 132 this year. A publicity poster shows two bars with a vertical axis beginning at 115 and ending at 135. The visible heights are 5 units for last year and 17 for this year. A casual reader might see the new bar as more than three times the old.

Calculate properly. The increase is 132−120=12 members. The percentage increase is 12/120 × 100%=10%. The membership is 1.10 times the previous total, not 3.4 times. The graph’s cropped lengths have a ratio of 17/5=3.4, which describes only how much each bar extends above the chosen baseline.

If you are asked to make a fairer display, propose a bar graph starting at zero with equal intervals. That preserves proportional bar heights for amounts. You can also keep the precise values printed above the bars. The revised graph will probably make the increase look modest, which accurately reflects a 10% change while still showing improvement.

Suppose someone argues that the original graph is automatically invalid. The counts may be perfectly accurate. The concern is that the visual framing could exaggerate the perceived magnitude. A careful response criticises the potential interpretation, not necessarily the data collection. Data quality and data display quality are separate questions.

Now imagine membership increased from 120 to 132 but the school as a whole grew from 600 to 750 students. Club membership as a share of the school changed from 120/600=20% to 132/750=17.6%. The club gained 12 members in absolute terms, yet its share of the whole school decreased by 2.4 percentage points. No graph can tell the complete story without identifying which comparison matters.

This extension connects graph literacy to changing denominators, the skill in Volume 0087 on successive transfers. A chart may be clear about absolute totals while remaining insufficient for proportional conclusions. Always match your claim to the values that were actually graphed.

5. An uneven axis can change the story

Another problem occurs when tick marks appear evenly spaced on the page but represent unequal numerical increments. Imagine a vertical axis labelled 0, 10, 20, 50 and 100 at equal physical distances. If the graph does not clearly identify a changed scale, it may distort the relation between plotted heights and values. The same gap on paper does not consistently represent the same numerical difference.

To check a graph, subtract adjacent tick values. In a uniform linear axis the numeric gaps between equally spaced ticks should be constant. If 0 to 10 and 10 to 20 are represented by equal gaps, but 20 to 50 receives the same gap, the scale has changed. Unless the representation explicitly and appropriately uses a non-linear scale, a reader should be cautious.

Not all non-linear scales are invalid. Scientific graphs sometimes use logarithmic axes, where equal physical spacing represents equal multiplicative factors rather than equal additions. But such a graph requires clear labelling and appropriate interpretation. The existence of a log scale does not mean the data are wrong; it means linear differences cannot be read off as though every step adds a fixed number.

For ordinary K310 comparison questions, you should be ready to identify both linear and non-linear presentation issues where relevant, without importing advanced logarithmic plotting techniques that the task has not taught. A safe answer focuses on the stated scale: “The axis intervals are unequal despite equal spacing, so the visual distances do not represent equal changes in value.”

A numerical table is sometimes the simplest remedy. If graph values are 5, 15, 25 and 60, record them explicitly and compare with arithmetic. Differences are 10, 10 and 35. A distorted axis cannot change those differences once you have read the actual values correctly.

It is worth asking whether the graph uses a break symbol. A clearly marked break warns readers that the axis has skipped a range. This can improve communication in a suitable context, but the viewer must not compare full bar heights as though the omitted section were displayed. The important thing is disclosure and correct inference.

6. Percentage change versus percentage points

Suppose the share of students using a library rises from 40% to 50%. The increase is 10 percentage points. Relative to the original 40%, it is an increase of (50−40)/40 × 100%=25%. Both statements can be correct. They answer different questions.

A graph might label its vertical axis “percentage of students” and plot 40 and 50. If a headline says “Usage rose by 10%”, it may be ambiguous or wrong depending on what is meant. The change in percentage points is 10; the relative percentage increase is 25%. Use the term the problem requests and make the basis explicit.

Take another example: a pass rate increases from 80% to 88%. That is an increase of 8 percentage points and a relative increase of 10%. A bar chart drawn from 75% to 90% can make the difference look enormous, even though the percentage-point change is 8. The quality of the conclusion depends on knowing the underlying denominator and context.

A pass rate also depends on the group assessed. Eighty successes out of 100 candidates and 88 successes out of 100 produce the stated percentages. But 80 successes out of 100 compared with 88 successes out of 110 gives rates of 80% and 80%, despite an increased number of successes. If class sizes change, calculate actual rates from the counts before comparing.

In a practical answer, separate the exact measure and the broader interpretation. “The pass rate rose from 80% to 88%, an increase of 8 percentage points” is clear. “Performance improved by 10% relative to the original pass rate” can also be mathematically accurate, but only if it fits the question and is properly explained. Do not switch between them halfway through the response.

This is why units and labels matter even for percentages. A percentage is always a fraction of something. When the “something” changes, the visual trend alone may not capture the meaning. Volume 0085 on reference audits deepens this habit across subjects.

7. Line graphs: a steep picture is not always a fast process

Consider a distance–time graph where an object travels 24 km in 2 hours. The average speed over that interval is 24/2=12 km/h. If the line is drawn across a short horizontal axis and tall vertical axis, it looks steep. If the same data use a wide horizontal axis and short vertical axis, it looks gentle. The physical slope of the ink changed; the speed did not.

For a line segment, calculate gradient as change in vertical quantity divided by change in horizontal quantity. Label the units. If the graph measures cumulative distance in kilometres against time in hours, the gradient has units of km/h. If it measures temperature in degrees Celsius against time in minutes, the gradient has units of °C/min. The formula’s shape is similar, but the result describes a different phenomenon.

A line graph can also be misleading when data points are joined as though the relationship between observations were known. Suppose a sensor records values once an hour. A straight line connecting two points is a visual interpolation, not proof that the quantity changed uniformly between readings. In a mathematical question that explicitly assumes linear change, you can use the slope. In a data-evaluation question, you should recognise that intermediate behaviour may not have been measured.

Pay attention to non-zero time origins. If a graph starts at 08:15 rather than 00:00, its first plotted point is not the beginning of the day. Do not calculate total elapsed time using the horizontal distance on the page alone. Use the axis labels and clock arithmetic to find the actual duration.

Sometimes the graph displays only the final ten minutes of an hour-long journey. A steep increase in that window may be important, but cannot by itself describe the entire hour’s average speed. The interval being graphed is part of the claim. Volume 0091 provides longer journey cases with stops and changing speed.

A good graph explanation names both numerical change and relevant interval. “The measured distance increased by 12 km between 09:00 and 10:00, so the average rate over that hour was 12 km/h.” It does not say “the graph looks steep, so speed was very high.”

8. Cumulative-frequency graphs and axis meaning

A cumulative-frequency graph records how many observations lie at or below specified boundaries, subject to the representation used. Its vertical axis is a cumulative count, not the frequency within each class. A reader who mistakes the two will obtain the wrong interpretation even if every tick mark is evenly spaced.

Suppose a class has 40 students. A cumulative-frequency curve shows that 28 scored at or below a given mark. That tells us 28 of the 40 observations lie in the cumulative group up to that boundary. To find how many lie between two boundaries, subtract the cumulative totals at those boundaries. For example, if 12 are at or below 30 and 28 are at or below 50, then 16 lie in the interval above 30 and up to 50, under the stated convention.

A graph whose vertical axis begins at 10 rather than zero might make changes in cumulative totals look larger visually. But the meaning remains a count accumulated from the beginning of the distribution. Read the actual values and subtract correctly. You should not claim a negative frequency merely because the line is displayed from a truncated baseline.

Quartiles divide an ordered dataset into four parts, and a cumulative-frequency diagram can help locate approximate quartile positions. For 40 observations, you may use positions around one quarter and three quarters of the cumulative total, subject to the task’s taught method and conventions. Do not confuse the 25th percentile with a value of 25 on the marks axis. Percentile rank and measured mark are different quantities.

The interquartile range is the difference between upper and lower quartile values. A graph with a dramatic-looking steep section does not automatically have a large interquartile range. You must locate the relevant cumulative positions, read the corresponding horizontal-axis values and subtract them. The scale’s physical appearance is not a substitute for reading the data.

This connects to the official K310 Statistics and Probability strand, which includes cumulative-frequency diagrams, quartiles and percentiles, measures of spread and comparisons. The Mathematics Hub provides the broader route; this chapter focuses on preventing the display from changing the meaning of the counts.

9. A complete worked comparison with three representations

A neighbourhood activity centre records 100 visits on Monday, 110 on Tuesday and 105 on Wednesday. In a table, the values are straightforward. From Monday to Tuesday, the increase is 10 visits, or 10%. From Tuesday to Wednesday, the decrease is 5 visits, or 5/110 ×100%≈4.55%. The two percentage changes use different starting values.

Now imagine a bar chart starting at 95 and ending at 115. The displayed bar segments have lengths 5, 15 and 10 above the baseline. Tuesday’s segment looks three times Monday’s, despite Tuesday having only 10% more visits. The table exposes the correct comparison.

Next imagine a line graph with days on the horizontal axis and visits on the vertical. Joining the three points makes a peak on Tuesday. This accurately communicates the up-then-down pattern across the recorded days, provided the scale is properly labelled. But it says nothing about what happened within each day, because only daily totals are supplied.

Finally imagine a headline: “Attendance is soaring.” Three days of values, 100, 110 and 105, do not necessarily justify a broad long-term claim. Tuesday has more visits than Monday, but Wednesday partly reverses the rise. A cautious description might say “Visits rose on Tuesday and eased on Wednesday; all three totals were between 100 and 110.” This is informative without hiding the variation.

The mathematics and interpretation are complementary. Correct arithmetic protects you from visual exaggeration. Correct reading of scope protects you from generalising a short sequence into a permanent trend. Neither requires distrust of graphs as a category.

When you work under time pressure, write the table values beside the plotted points or bars if allowed. That simple step makes the numerical comparison visible and reduces the temptation to judge by visual height alone.

10. Questions you should ask before trusting a public chart

First, what does the chart count? A bar labelled “participants” may count distinct people, visits, registrations or completed activities. Those are different. If the chart lacks a definition, a precise interpretation may be impossible. The title and notes are part of the source.

Second, does the graph show totals or rates? A total may increase because the population increased. A rate may remain stable. If a clinic reports more appointments, the reason could involve more patients rather than higher appointment rates per person. You need the relevant denominator for a fair claim.

Third, how long is the observation period? Two points cannot establish a stable long-term trend, even though they establish a difference between those points. If the chart omits earlier months, an apparent rise could be part of a larger decline. Ask whether the stated claim matches the window shown.

Fourth, is the scale linear and clearly marked? Uneven or undisclosed increments can distort visual differences. A legitimate alternative scale should be described and interpreted according to its own rules, not quietly treated as ordinary linear spacing.

Fifth, are the categories comparable? If one school reports scores out of 100 and another reports scores out of 80, raw values cannot be compared directly as percentages. Convert appropriately or explain the limitation. A chart can be meticulously drawn while its categories are fundamentally mismatched.

Sixth, are uncertain measurements presented as exact? Rounded values, estimates and samples should not be treated as perfectly precise population totals. A graph’s smooth line can make uncertain points look more authoritative than the source justifies.

Finally, does the conclusion outrun the display? “The number rose” may be correct. “The policy caused the rise” is a causal claim requiring a different sort of evidence. Volume 0093 on sequence versus cause helps separate those tasks.

11. Original examination-style practice

A. A bar chart shows sales of 84 and 91 items, with the vertical axis starting at 80. Calculate the actual increase and percentage increase. Explain why the second bar might appear much taller.

B. A club has 200 members one year and 220 the next. A chart starts its count axis at 190. State the percentage change and one risk of comparing the visible bar lengths.

C. The share of pupils joining a programme rises from 35% to 42%. Calculate the percentage-point increase and the relative percentage increase.

D. A line graph of temperature shows 27 °C at 12:00 and 30 °C at 12:30. Find the average recorded rate of change in °C/min over that interval. Does the graph prove the temperature rose at a constant rate between the points?

E. The vertical axis is marked 0, 10, 20, 50 and 100 at equal distances. Explain what is unusual and how a reader could be misled if the graph is presented as having a linear scale.

F. A cumulative-frequency table shows 9 observations at or below 10, and 25 at or below 20. How many observations are above 10 and at or below 20?

G. A graphic claims a school doubled its reading rate because the visible height of a bar above a 70% baseline doubled. What two values would you need to assess the actual change?

H. A bus covers 30 km in 45 minutes. A distance–time line is redrawn on a wider chart. Does its average speed change? Calculate it in km/h.

I. A poster shows three daily visitor counts: 100, 110 and 105. Write one accurate trend statement and one exaggerated conclusion that the three values do not establish.

J. Two schools report 90 passes out of 100 candidates and 99 passes out of 120. Which school has the higher pass rate, and by how many percentage points?

K. A graph title says “More students are learning better” but its only data are app log-ins. Explain what the graph directly measures and name one outcome it does not establish.

L. A chart uses a clearly labelled logarithmic axis. Should you automatically call it inaccurate? Explain the kind of caution that is needed when interpreting equal visual gaps.

Do the arithmetic before interpreting the display. When asked to explain a misleading graph, identify a particular graphical feature and connect it to a possible false conclusion. A vague answer such as “it looks wrong” is not enough.

12. Worked answers, with the source of each error

A. The increase is 91−84=7 items. The relative increase is 7/84 ×100%=8.33% to three significant figures. Visible segments above 80 have lengths 4 and 11 units, so the second looks 2.75 times as tall even though the actual totals are close. The truncated baseline exaggerates the apparent ratio.

B. Percentage increase is 20/200 ×100%=10%. The visible lengths above 190 are 10 and 30, a ratio of three. That is not the membership ratio; the actual totals are in a ratio of 1.1. Starting at zero would give proportional bar lengths for the two amounts.

C. The programme share increases by 42−35=7 percentage points. Its relative percentage increase is 7/35 ×100%=20%. Both can be correct if properly labelled.

D. Change is 3 °C in 30 minutes, giving an average of 3/30=0.1 °C/min. Two endpoint readings alone do not establish a constant rate between them. A straight joining line may be a display choice unless a model specifies linear behaviour.

E. Equal physical gaps represent unequal numerical increments. On an ordinary linear axis this distorts how changes appear. A clearly explained non-linear axis may be legitimate, but an unlabeled change in scale misleads readers who assume equal spacing means equal additions.

F. The number within the interval is 25−9=16 observations. The earlier cumulative total must be subtracted from the later one. Do not call 25 the number inside the interval.

G. You need the actual percentage values represented by both bars and the axis origin. With those, calculate the numerical difference or relative increase. A ratio of visible clipped heights is not the ratio of the percentages.

H. Average speed is 30/(45/60)=40 km/h. Redrawing the graph wider changes the apparent angle of the line, not the ratio of actual distance and time.

I. Accurate: “Visits rose from 100 to 110, then fell to 105.” Unsupported: “Visits are rising continuously and will keep rising.” The third value contradicts continuous rise; a three-day series cannot establish a future trend.

J. First school pass rate is 90/100=90%. Second is 99/120=82.5%. The first is higher by 7.5 percentage points. More passes in absolute number at the second school do not imply a higher pass rate.

K. The graph measures app log-ins, not directly learning achievement. It cannot alone establish improvement in comprehension, examination scores or durable understanding. Additional relevant assessments and evidence would be needed.

L. No. A logarithmic axis may be valid for its purpose if clearly labelled. Equal gaps indicate multiplicative rather than equal additive steps. Read the values and scale type, and avoid applying ordinary linear interpretations to the picture.

Each error has a diagnostic name: truncated baseline, uneven intervals, wrong denominator, confused percentage points, misread cumulative totals, visual-angle inference or unsupported causal claim. Name the one you made, then revisit that skill.

13. Practical revision: build a graph that can survive scrutiny

A useful practice exercise is to take a small table and display it in two ways. For bar charts showing positive amounts, make one with a zero baseline and another with a clearly marked truncated baseline. Ask which comparison each picture encourages. Then write the actual differences beside the graphs so visual impression and arithmetic remain connected.

For a line graph, plot the same two or three points with different page proportions. Calculate numerical slopes from coordinates and units. Explain why the graph looks more or less dramatic even though the calculated rate is unchanged. This is a good test of whether you understand that slope on the paper and slope in the measured variables are not always the same visual thing.

Work from tables before drawing more complicated statistical diagrams. A table records the numerical values plainly; a graph displays their pattern. A capable student can move between the two without changing the meaning. Volume 0069 on source-to-answer translation develops this related transfer skill across subjects.

Once you can read values accurately, practise interpreting public charts. Newspaper, transport and school examples often provide context, but use them critically. Check the definition of the measure and whether the chart supports the headline. Do not assume every visual choice has a deceptive intention; you can criticise the effect without asserting motive.

Under exam conditions, show the calculation that establishes the correct comparison. If the question asks why a chart misleads, a small accurate calculation strengthens your explanation. Use exact values where possible and follow the stated rounding rule if approximation is needed.

Keep the workload sustainable. Five careful graph interpretations with corrections are more useful than fifty charts judged by appearance. The aim is to make one reliable habit automatic.

14. A seven-day route from reading to evaluating

Day 1: Read axis labels, units, origins and tick spacing in ten simple charts. Do not interpret trends until all four are identified.

Day 2: Compare zero-based and truncated bar displays of the same data. Calculate absolute and relative changes, then explain the visual difference in one sentence.

Day 3: Work on percentage change versus percentage points. Use examples with different denominators so that “10%” and “10 percentage points” are not casually swapped.

Day 4: Calculate gradients on line graphs, including one time axis with minutes and one with hours. Describe what the gradient represents in context, not merely the arithmetic result.

Day 5: Read cumulative-frequency values and subtract to find interval frequencies. Practise identifying quartile positions without confusing cumulative count with measurement value.

Day 6: Critique a headline and chart as a pair. Say what the data directly establish, what presentation choices emphasise and which extra claim would need more evidence.

Day 7: Attempt Questions A–L independently. Group errors by graph type, scale, denominator or interpretation. Revisit the weakest group three days later with unfamiliar data.

Parents can ask, “Where does the axis begin?” and “What difference do the actual numbers show?” Those are approachable questions that reveal whether a learner has been persuaded by the picture or has read the mathematics. As confidence develops, ask whether the chart is measuring the thing the headline claims.

15. Take the graph back to the data

A sound graph is a representation of measured values, not a replacement for them. It helps readers see patterns, but it should not tempt us to forget units, interval widths, population sizes and the origin of the scale. The best learner can see both the pattern and the numerical basis for it.

For further work, move from this narrow workshop to the Mathematics Hub and the G3 SEC learner-guide route. If interpreting a rate is the difficulty, revisit Volume 0091 on journeys. If deciding what a chart supports is the problem, Volume 0093 on sequence versus cause is a useful companion.

Official examination references: SEAB 2027 G3 syllabus list and Mathematics K310 syllabus. The graphs described here are original teaching examples, not official SEC examination diagrams.