The mean as the point where the distances balance

What to assume they know

• Computing a mean as total ÷ count, and a median as the middle value of sorted data — both are recalled, not taught, at the head of this chapter
• Reading a dot plot: one dot per value, stacked when a value repeats — carried in from earlier classes; this chapter uses dot plots from its first page without re-explaining them
• Subtracting to find the distance between two points on a number line, with half-units
• Adding a short list of decimals, and dividing a whole number by 3 or 4 to get a decimal answer
• That a number line's tick labels may step by 2 while the drawn grid lets a dot sit on the odd values between them

What they should be able to do

• Compute the mean of a small collection from its dot plot and mark it on the same axis
• Show, for a given collection, that the total of the distances below the mean equals the total above it
• State why that equality must hold, using the fact that the surpluses and shortfalls measured from the mean cancel
• Produce a collection whose mean is nowhere near the midpoint of its smallest and largest values, and explain why the midpoint of the extremes is a different statistic
• Test a proposed centre other than the mean and show that one side's total grows while the other's shrinks, so no second balancing point exists
• Decide, without finishing the division, whether a stated number can be the mean of a plotted collection
• Connect the fair-share reading of the mean to the balance reading, and say why they are one statement

Where it usually goes wrong

• "The mean is halfway between the smallest and the largest." True for two values, which is exactly why the chapter opens there, and false as soon as there are three. Put {2, 4, 9}: halfway is 5.5, the mean is 5. The midpoint of the extremes ignores everything in between; the mean cannot.
• "The mean has to be one of the values." {2, 6, 10, 12} has mean 7.5 and no value anywhere near it. The mean is a position on the line, not a member of the collection.
• "Balance means the same number of values on each side." That is the median. In {10, 10, 11, 17} three values sit below 12 and one above, and the balance still holds — because one value 5 away pays for three values that are 2, 2 and 1 away. Counting values and measuring distances are different tests, and the chapter is about the second.
• "Distances left and right — so the mean is where the arrows are longest." It is the totals that match, not the individual lengths — and the four printed diagrams on Part II p.104 make the point better than a single count would. Measured against each collection's mean: {6, 7, 8} gives 1 against 1, equal; {2, 4, 9} gives 3 and 1 against 4, longest on the right; {10, 10, 11, 17} gives 2, 2, 1 against 5, longest on the right; but {2, 6, 10, 12} gives 5.5 and 1.5 against 2.5 and 4.5, so the longest single arrow is on the left. Two right, one equal, one left. Use the fourth diagram deliberately: it breaks the pattern a student is on the point of generalising from the first three.
• "If I move the centre a little, the two sides stay nearly balanced, so any nearby point works too." The printed pair of failed diagrams is the argument against this: move up by half a unit with three values below and one above and the left total gains 1.5 while the right loses 0.5. The imbalance appears immediately and grows steadily, which is why the balancing point is unique.
• "The mean tells me what the collection looks like." It tells you where the collection balances and nothing else.

Questions to check understanding

• Given a dot plot, compute the mean and mark it on the axis
• Given a collection and its mean, list the distances on each side and show the totals agree
• Given a proposed centre, show it is not the mean by comparing the two totals
• Say whether the mean and the midpoint of the extremes coincide for a stated collection, and if not, which is larger
• Judge whether a stated value can be the mean of a plotted collection, giving a reason that does not require the division
• Construct a collection whose mean is a stated value not in the collection
• Explain in a sentence or two why a collection has only one balancing point

Examples worth working on the board

Values marked printed appear on the page. Values marked not in the book are worked out here on the chapter's data and must be presented as added here, not the book's — this chapter prints no answer key and Part II has no answer appendix.

• The two-number case (Part II p.103, printed). 3 and 7 give 5; 8 and 9 give 8.5. A dot plot on an axis labelled 0, 2, 4, 6, 8 carries the two dots and the label Mean = 5, with a vertical rule drawn at 5. The page's own remark is that for two numbers the mean lands exactly halfway between them.
• The eight collections set for the reader (Part II pp.103–104, printed as dot plots with no means given). All eight are drawn on ruled axes whose labels step by 2 while the grid allows dots on the values between. Read off the printed pages and confirmed on the printed page:
  • Part II p.103: {6, 7, 8} · {3, 6, 9} · {2, 4, 9} · {4, 11, 15}
  • Part II p.104: {11, 13, 17, 19} · {5, 6, 15, 16} · {10, 10, 11, 17} · {3, 5, 10, 12} Not in the book means, in the same order: 7 · 6 · 5 · 10 · 15 · 10.5 · 12 · 7.5. Note: only one of the eight, {10, 10, 11, 17}, is quoted in the running prose, and its mean of 12 is the one mean the page states.
• The four balance diagrams (Part II p.104, printed, values and distance labels both). Each is a dot plot with a vertical rule at the mean, arrows measuring each dot's distance to that rule, and the two totals written underneath:
  • {6, 7, 8}, Mean = 7, LHS = 1, RHS = 1
  • {2, 4, 9}, Mean = 5, LHS = 3 + 1 = 4, RHS = 4
  • {10, 10, 11, 17}, Mean = 12, LHS = 2 + 2 + 1 = 5, RHS = 5
  • {2, 6, 10, 12}, Mean = 7.5, LHS = 5.5 + 1.5 = 7, RHS = 2.5 + 4.5 = 7 The fourth is the one to watch: 7.5 is nowhere near the midpoint of 2 and 12, which is also 7 — see the note in §Notes about this figure not matching the eighth collection above.
• The midpoint-of-extremes contrast (not in the book, from printed data). For {2, 4, 9} the extremes average to 5.5 but the mean is 5. For {10, 10, 11, 17} the extremes average to 13.5 and the mean is 12. For {4, 11, 15} the extremes average to 9.5 and the mean is 10 — the midpoint can sit on either side, which is the cleanest way to show it is a different statistic rather than a worse one.
• The two rejected centres (Part II p.105, printed). Both are drawn on {10, 10, 11, 17}. Trying a centre above the mean: LHS = 2.5 + 2.5 + 1.5 = 6.5 against RHS = 4.5. Trying one below it: LHS = 1 + 1 = 2 against RHS = 6. Not in the book: the candidates those distance labels correspond to are 12.5 and 11 respectively, and the totals differ by 2 in one direction and 4 in the other — the imbalance grows with the distance moved, at a rate equal to the number of values on each side.
• A collection to test the claim on (Part II p.128, item 7, printed as a cross-marked dot plot). Counts read off the printed page, one column per value from 14 to 23: 2, 2, 3, 5, 4, 4, 3, 1, 0, 1 — twenty-five throws in all, and the column above 22 is empty. The question asks whether 17 is the average. Not in the book: the total is 443 and the mean is 17.72, so it is not; and the balance test settles it faster than the division does, because measured from 17 the values above outweigh the values below.
• Three album dot plots (Part II p.114, item 4, printed on axes running 0 to 7.5 in half-minute steps; the question asks which one has a mean of 5.57 minutes). Read off the printed page:
  • A: 5, 5, 5.25, 5.5, 5.75, 6, 6.5 — seven songs
  • B: 0.5, 0.75, 1.5, 1.5, 2, 3.75, 4.25, 5 — eight songs
  • C: three songs at 3.5, three at 4, then 4.25 and 4.5 — eight songs Not in the book: A totals 39 minutes over 7 songs, which is 5.571…, so A is the album; B averages about 2.4 and C about 3.9, and both can be rejected without arithmetic because every one of their values sits below 5.57.

Figures to have open

• A dot plot with measured distance arrows and running LHS/RHS totals. This is the chapter's own device (Part II pp.104–105) and the whole topic lives on it. Redraw rather than reproduce; the essential features are the vertical rule at the mean, one arrow per dot, and the two sums written beneath their own sides.
• The eight collections as a set of small labelled dot plots. Standard schematic; the axes must keep the printed tick steps of 2 with dots allowed on the values between, because a student re-reading the page has to recognise them.
• A balance-beam figure with the values as equal weights hung at their positions and the fulcrum at the mean. Not in the chapter, and worth adding once in section 5 — it makes the distance-times-one-weight idea visible without a word of algebra.
• The 25-throw dot plot from Part II p.128, item 7. Note it is drawn with crosses rather than filled dots, the only place in the chapter that switches marker.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 5, "Tales by Dots and Lines", §5.1 "The Balancing Act", Part II pp.103–105. The two-number opening and the first four collections are on Part II p.103; the second four collections, the four balance diagrams and the question about a second centre are on Part II p.104; the two rejected-centre diagrams and the conclusion that the centre is unique are at the top of Part II p.105.
• Three Math Talk flags sit in this stretch (Part II p.104, twice, and Part II p.105) and one Try This (Part II p.107); the first two mark the "explain how the mean is the centre" prompts that section 4 is built from.
• SUMMARY, Part II p.133, first bullet: states the distance-balance property and points back to the fair-share reading from the previous class. The SUMMARY box occupies that page alone.
• Exercise items owned by this topic: Part II p.114 item 4 (three album dot plots) and Part II p.128 item 7 (the 25-throw dot plot). Both are in Figure it Out sets whose other items belong to sibling topics.
• Forward pointer: what happens to this balance when a value is added or every value is shifted is Which added values move the mean, and in which direction, Part II pp.105–108.