Which added values move the mean, and in which direction

What to assume they know

• The mean as the balancing point of the distances, and that the surpluses above it pay for the shortfalls below it — The mean as the point where the distances balance
• Mean = total ÷ count, and the same equation read as total = mean × count
• Reading and adding values off a dot plot, including stacked dots and half-units
• Writing a general collection as x₁, x₂, … xₙ, and knowing that n stands for how many there are
• Combining like terms and splitting a fraction over addition, e.g. turning (S + 3n)/n into S/n + 3
• The distributive property, in the form (a + b + c) × 5 = 5a + 5b + 5c — What happens to a product when you nudge one factor

What they should be able to do

• Predict, given the mean of a collection and one new value, whether the mean will rise, fall or stay put, before computing anything
• Compute the new mean after an insertion and check it against that prediction
• Explain why an inserted value equal to the mean leaves the mean unchanged, and say what happens when such a value is removed
• Construct two values whose insertion leaves the mean unchanged, and explain the condition they have to satisfy
• Construct three values that leave the mean unchanged with two of them below the mean, and again with two above it
• State and justify the effect on the mean of adding a fixed number to every value, and of multiplying every value by a fixed number
• Reproduce the chapter's algebraic argument for the additive case, and adapt it to subtraction
• Use the shift rule to correct an average that was computed from measurements carrying a constant error, without re-measuring anything
• Decide when the information given is not enough to say what happens to an average

Where it usually goes wrong

• "More values means a bigger average." This is printed as a tempting option in Part II p.128 item 6 for exactly that reason. Two more values can raise the average, lower it or leave it alone; what decides is where they sit relative to the current mean, not how many there are.
• "The mean moves to the new value." It moves towards it and stops short. Inserting 17 into a collection averaging 11 moved the mean to 12.2, not to 17, because the newcomer's surplus is shared among all five values.
• "Adding a value below the mean might still raise it." Never. Its gap is negative, so the total gains less than one average share and the mean must fall. This is the "always true" answer to Part II p.115 item 6(ii).
• "Adding 10 to every one of 11 values adds 110 to the mean." It adds 110 to the total, and the total is then shared by 11 again. This is the single most common wrong turn in the section, and the algebra on Part II p.107 exists to close it: 3n/n is 3, not 3n.
• "If the average is wrong I have to measure everything again." The shoes item is built to defeat this. A constant error in every measurement is a constant error in the mean, so it can be removed at the end with one subtraction.
• "Two values that keep the mean must both equal the mean." They only have to have gaps that cancel. There are infinitely many pairs.
• "Doubling the values doubles the mean, so halving them halves the spread and leaves the mean alone." Scaling does both: it multiplies the mean and stretches the distances by the same factor. The relative position of the mean inside the data is what survives, which is what the chapter's heading is pointing at.
• "Every claim about an average can be settled from an average." Item 8 is the counter-case in the other direction: the mean is settled without the group size, and the median is not settled at all. Knowing when the information runs out is part of the skill.

Questions to check understanding

• Given a collection's mean and count, and one new value, state the direction of change and then compute the new mean
• Given a mean before and after an insertion, recover the inserted value
• Supply two values whose insertion leaves a stated mean unchanged, and explain the condition
• Supply three values with two of them below the mean that leave it unchanged
• State the new mean when every value is increased by a fixed number, decreased by one, doubled, or halved
• Reproduce the algebraic argument for adding a fixed number, and write the matching one for subtracting
• Correct an average that was computed from measurements carrying a known constant error
• Decide whether the given information settles what happens to an average, and say what else would be needed
• Judge an "always true / sometimes true / never true" statement about the mean and justify the verdict

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 stated inputs; this chapter prints no answers and Part II has no answer appendix, so nothing derived may be presented as the book's.

• The insertion figure (Part II p.105, printed as three stacked dot plots on an axis labelled 4 to 24). The first plot is labelled Mean = 11, the second shows one green dot dropped in with the words new value inserted and the mean rule still at 11, the third is labelled Mean = 12.2. Read off the printed page: the original four values are 5, 5, 10 and 24, and the inserted value is 17. Not in the book, and worth showing as the check: the original total is 44, which over 4 values is 11; adding 17 makes 61 over 5, which is 12.2. The inserted value's gap from 11 is +6, and 6 ÷ 5 = 1.2, which is exactly how far the mean moved.
• The Math Talk stretch (Part II p.105, printed, answers not given): three questions and a fair-share prompt, not four questions — what taking an existing value away does; whether the mean then rises, falls or holds still; and what a newcomer already sitting at the mean does, going in or coming out. The fourth item is an instruction rather than a question — it directs the reader back to last year's fair-share reading of the average and asks them to account for the result that way — and it sits outside the Math Talk bracket. Added answers: removal reverses insertion — take away a value above the mean and the mean falls; a newcomer at the mean itself changes nothing in either direction, because its gap is zero and there is nothing to redistribute.
• The experiment data for "Unchanging Mean!" (Part II p.106, printed as a dot plot labelled Mean = 9, axis 2 to 15). Read off the printed page, seventeen values: 2.5, 5, 6.5, 7, 7.5, 8, 8, 8, 8, 9, 10, 10.5, 11, 12, 12, 13, 15. The four dots at 8 are drawn as a stack of four and the two at 12 as a stack of two. Verified: the total is 153 and 153 ÷ 17 = 9 exactly, which is the check that the reading is right.
• The printed mean-preserving triple (Part II p.106, printed as a second copy of the same dot plot with three green dots added and three distances marked 2, 2 and 4). The added values are 7, 7 and 13. Not in the book: two gaps of −2 and one of +4 cancel, so the new total 180 over 20 values is still 9. The page then sets the mirror task — two values above and one below.
• The shift data (printed, and it runs across two pages — cite them separately): 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5, and the same eleven values each raised by 10: 18, 13, 20, 23, 14, 16, 17, 17, 18, 18, 15. Part II p.106 carries the eleven values, the plot labelled Mean = 7.18 on an axis running 0 to 12, and beneath it a second plot that is unlabelled, on an axis running 10 to 22 — those two are the "two dot plots" the printed Hint points at. Part II p.107 carries the label Mean = 17.18, on a plot with the 10-to-22 axis, under the sentence about the relative position of the mean; and the plot labelled Mean = 16.18 on an axis running 9 to 21, which is what the collection becomes when every value drops by 1. (Part II p.107 also repeats a Mean = 7.18 plot in red at the foot of the page, for the doubling beat — do not confuse it with p.106's.) Not in the book: the total is 79 over 11 values, so the mean is 7.1818…, printed rounded to 7.18 — a good place to say out loud that the displayed mean is rounded and the exact one is 79/11.
• The chapter's additive argument (Part II p.107, printed in full). With n values averaging a, adding 3 to every value gives a new total of (x₁ + x₂ + … + xₙ) + 3n; dividing by n splits into a + 3n/n, which is a + 3. The page then asks the reader to run the same argument for subtracting 2, and to explain the result through fair shares as well.
• The doubling and multiplying case (Part II pp.107–108, printed). The same eleven values doubled are shown on an axis running 0 to 26 with the label Mean = 14.36, against the original Mean = 7.18. The argument is then written for a factor of 5: each value becomes 5xᵢ, the distributive property pulls the 5 out of the total, and the mean becomes 5a.
• Item 1 (Part II p.113, printed task, no answers). Find the mean of: the first 50 natural numbers; the first 50 odd numbers; the first 50 multiples of 4. Not in the book: 25.5, then 50, then 102. The observation the item is fishing for is that the second and third are the first computed from the first — the odd numbers are each twice a natural number less one, so their mean is 2 × 25.5 − 1, and the multiples of 4 are each four times a natural number, so their mean is 4 × 25.5. This item is where the two rules of this topic pay off together.
• Item 2 (Part II p.113, printed as a dot plot with a vertical rule at 9, axis 0 to 16). One dot is missing and the mean has to come out at 9. Read off the printed page, the ten dots shown are 4, 7, 8, 8 and five dots stacked at 9, plus one at 11. Not in the book: those ten total 83; eleven values averaging 9 need a total of 99; so the missing dot goes at 16, the right-hand end of the printed axis.
• Item 3 (Part II p.114, printed). Heights of 24 students average 150.2 cm, measured with shoes that add 1 cm each, and the options offered are 174.2, 126.2, 150.2, 149.2, 151.2, none of these, and insufficient information. Not in the book: every value carries the same +1, so the mean carries it too and the corrected average is 149.2 cm — option (d). Nobody needs re-measuring, which is the point of part (i).
• Item 6, parts (i) and (ii) (Part II p.128, printed). Two students join a class whose average height was 150.2 cm; the offered claims are that the average must rise because there are two more values, that it stays the same, that the two new heights must be measured, and that everybody must be measured again. Not in the book: only the third is right, and the fourth is unnecessary because the old total is recoverable as 24 × 150.2 = 3604.8 cm. With the two heights given as 149 cm and 152 cm, the gaps are −1.2 and +1.8, so the new total is 3905.8 over 26 values and the average rises to about 150.22 cm — a rise of roughly 0.02 cm, which is worth showing precisely because "it increases" and "it barely moves" are both true. Part (iii) of the same item is about the median and belongs to Whether adding a value raises or lowers the median.
• Item 8 (Part II pp.128–129, printed). Last month a group's mean weight was 65.3 kg and the median 67 kg; this month one person is 2 kg lighter and two are 1 kg heavier. Not in the book: the total is unchanged, so the mean is still 65.3 kg whatever the group size — the size is never given and never needed. The median half of the question goes to Whether adding a value raises or lowers the median.
• Item 5 (Part II p.127, printed as three statements to test with algebra). Not in the book: the average of two even numbers is a whole number but need not be even — 2 and 4 give 3. The average of two multiples of 5 need not even be a whole number — 5 and 10 give 7.5. The average of five multiples of 5 is always a whole number, because the total is a multiple of 5, but it need not be a multiple of 5 — 5, 5, 5, 5 and 10 give 6, while 5, 10, 15, 20 and 25 give 15. Answer the item in its own frame. Part II p.127 item 5 asks the reader to check whether each statement is true, justifying with algebra if needed; it is not an always/sometimes/never item — that frame belongs to Part II p.115 item 6, and the two must not be run together. Statements (ii) and (iii) are printed with "any", so they are universal claims, and (i) reads as a general claim too. Universally quantified, all three are false, and the three counterexamples above are exactly what refutes them. The genuinely interesting sub-result — that the average of five multiples of 5 is always a whole number — is a fact about integrality, not a defence of statement (iii), and should be presented as the bonus it is.

Figures to have open

• The three-panel insertion figure (Part II p.105). Redraw with the same structure: identical dot plot three times, the newcomer in a second colour, the mean rule shifting between panel one and panel three. This carries sections 2 to 4 on its own.
• A gap-and-share diagram: the new value's gap drawn as a bar, then that bar cut into as many equal pieces as there are values now, one piece handed to each. Not in the chapter; it is the picture that makes ÷(n+1) obvious rather than algebraic.
• The mean-9 dot plot with the three added dots (Part II p.106), including the printed distance labels 2, 2 and 4, because the cancellation is visible in the arrows before it is visible in the arithmetic.
• The sliding dot plot for section 8: one figure, three positions, one axis. Standard schematic.
• The chapter's two algebraic derivations set as displayed lines (Part II pp.107–108). These should be re-typeset, not photographed.
• No photograph or external dataset is needed 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.105–108. Within that stretch the chapter prints two unnumbered bold subheadings, which can be named but not cited by number: "Unchanging Mean!" (opening at the foot of Part II p.105 and running through Part II p.106) and "Relatively Unchanged!" (Part II pp.106–108). The insertion figure and the Math Talk stretch — three questions plus a fair-share prompt — are on Part II p.105; the eleven shift values and the Mean = 7.18 plot with its unlabelled shifted twin are on Part II p.106, while the Mean = 17.18 and Mean = 16.18 plots are on Part II p.107; the additive algebra and the Try This flag are on Part II p.107; the multiplication argument closes on Part II p.108 immediately above "Tinkering with Median".
• SUMMARY, Part II p.133, second bullet: states the direction-of-change result for inserted values and adds that the median behaves similarly.
• Exercise items owned by this topic: Part II p.113 items 1 and 2; Part II p.114 item 3; Part II p.115 item 6(ii); Part II p.127 item 5; Part II p.128 item 6 parts (i) and (ii); Part II pp.128–129 item 8, mean half only.
• Backward pointer: the distributive property as used on Part II p.108 is Part I printed Chapter 6, and in this unit What happens to a product when you nudge one factor.
• Forward pointer: the same set of questions asked about the median is Whether adding a value raises or lowers the median, Part II pp.108–109.