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Chapter 1 · A Square and A Cube

Why the first n odd numbers add up to n²

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What to assume they know

  • What makes a number a perfect square — square numbers, perfect squares, the notation n²
  • A table of squares for the natural numbers, at least as far as 12²
  • Subtracting across three and four digits, and recognising when a subtraction has gone below zero
  • Odd numbers, and that consecutive odd numbers differ by 2
  • The Class 6 dot-pattern work the chapter explicitly calls back to

What they should be able to do

  • Compute the differences between consecutive squares and identify them as the odd numbers in order
  • Explain, from the L-shaped border, why the gap from (n − 1)² to n² is 2n − 1
  • State that the first n odd numbers total n², and justify it rather than assert it
  • Write down the nth odd number as 2n − 1 and use it to find a square from its predecessor
  • Test a given number for squareness by subtracting 1, 3, 5, … and reading the outcome
  • Read the root of a perfect square off the count of subtractions performed
  • Say what a wordless argument still has to establish before it counts as a proof
  • Reason about how many whole numbers lie strictly between two consecutive squares

Where it usually goes wrong

  • "The odd-number pattern is a coincidence someone noticed." The L-shaped border makes it unavoidable. An explanation that shows the list without the border has taught the pattern and not the reason.
  • **"Adding the nth odd number means adding 2n + 1."** Off by one. The nth odd number is 2n − 1; the quantity 2n + 1 is what carries you from n² to (n + 1)². Both appear in this topic and students routinely swap them.
  • "A picture is not a proof." The chapter says otherwise, and it is right — but only when the picture shows the general step. One instance drawn is an illustration. Show a border being added at an unlabelled size.
  • "If the subtraction chain does not land on zero, keep subtracting." Once a subtraction takes you below zero the question is settled and the answer is no.
  • "The number of subtractions is the number you started with." It is the square root. Nine subtractions from 81 mean 81 = 9², not 81 = 81.
  • "Between 16² and 17² there are 17 numbers." Students reach for n or n + 1. The gap itself is 2n + 1 and the count strictly between is one less than that — a different quantity, and worth separating.
  • "The gap between consecutive squares is constant." It grows, by exactly 2 each time, which is the same statement as "the differences are the odd numbers".

Questions to check understanding

  • Continue the difference pattern for the next few squares and name what the differences are
  • Given k², find (k + 1)² by adding the right odd number, showing the step
  • Decide by successive subtraction whether a given number is a perfect square, and give its root from the count
  • Count the numbers falling strictly between two given consecutive squares, with reasoning
  • Choose the correct option for (k + 1)² given k², from a list like the chapter's item 3
  • Fill blanks in a square identity pattern and state the rule you used
  • Explain in words why the first n odd numbers total n² — the reasoning form the board now favours over the computation

Examples worth working on the board

  • The four differences (Part I p.5): 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9. The results 3, 5, 7, 9 are set in red on the printed page.
  • The running sums (Part I p.5), also with the totals in red: 1 = 1; 1 + 3 = 4; 1 + 3 + 5 = 9; 1 + 3 + 5 + 7 = 16; 1 + 3 + 5 + 7 + 9 = 25; 1 + 3 + 5 + 7 + 9 + 11 = 36.
  • The nested-L dot array (Part I p.5, right of the running sums). A 6-by-6 array of solid dots with red brackets drawn inside it, each bracket enclosing one more L than the last, so the six sums line up with the six nested shells. Verified on the printed page — the array carries no printed numbers.
  • The five-panel growth figure (Part I p.6). Dot arrays captioned, left to right, 1 + 3, then 1 + 3 + (3 + 2), then 1 + 3 + 5, then 1 + 3 + 5 + (5 + 2), then 1 + 3 + 5 + 7. Arrows run from the first panel to the second and from the third to the fourth. The arrays are 2-by-2, 3-by-3, 3-by-3, 4-by-4 and 4-by-4 respectively, with the newly added L outlined in red. Verified on the printed page. The (3 + 2) and (5 + 2) captions are the whole argument in miniature: the next odd number is the previous one plus two, because you gain one cell on the row and one on the column.
  • The chapter's note on wordless argument (Part I p.6), set in a blue panel with an owl illustration: reasoning in mathematics can sometimes be presented without words, and such a picture can stand as a complete argument on its own. Paraphrased; do not read the printed sentence aloud.
  • The squareness test on 25 (Part I p.6): 25 − 1 = 24, 24 − 3 = 21, 21 − 5 = 16, 16 − 7 = 9, 9 − 9 = 0. Five subtractions land exactly on zero, so 25 is what the first five odd numbers come to, and 25 = 5².
  • 36² from 35² (Part I p.6). Given 35² = 1225, so 1225 is the total of the first 35 odd numbers. The 36th odd number is needed. The chapter states the 1st, 2nd, 3rd and 6th odd numbers as 1, 3, 5 and 11, gives the general form 2n − 1, then 71 for n = 36, and 1225 + 71 = 1296 = 36². All printed.
  • The test failing on 38 (Part I p.6): 38 − 1 = 37, 37 − 3 = 34, 34 − 5 = 29, 29 − 7 = 22, 22 − 9 = 13, 13 − 11 = 2, 2 − 13 = −11. The chain shoots past zero, so 38 is not a square.
  • The blocks-of-a-hundred table (Part I p.7). Ten shaded cells, labelled 1–100, 101–200, 201–300, 301–400, 401–500, 501–600, 601–700, 701–800, 801–900, 901–1000, each with a blank rule under it for the count of squares in that block. Two questions sit above it: how many whole numbers lie between two consecutive squares, and what the largest square below 1000 is. Inputs only.
  • Figure it Out item 3 (Part I p.10). Given 125² = 15625, what is 126²? The five printed options are 15625 + 126, 15625 + 26², 15625 + 253, 15625 + 251, 15625 + 51². This item is the 2n + 1 rule under examination.
  • Figure it Out item 7 (Part I p.10): the count of numbers falling between 16² and 17², and between 99² and 100². The route is the same border count — the gap from n² to (n + 1)² is 2n + 1, so the whole numbers strictly between them number one fewer. Give the route; leave the two values to the student.
  • Figure it Out item 8 (Part I pp.10–11), an unrelated square identity that still rewards the same eye for structure: 1² + 2² + 2² = 3²; 2² + 3² + 6² = 7²; 3² + 4² + 12² = 13²; then 4² + 5² + 20² = (_)²; and 9² + 10² + (_)² = (_)². Across the three printed rows the third base is the product of the first two and the right-hand base is that product plus one — use that observation as the way in, not the filled blanks.
  • Figure it Out item 5 at Part I p.17, parts (iii) and (iv): 67² − 66² and 43² − 42², to be compared with the cube versions in the same item. Direct applications of the border count.

Figures to have open

  • A square growing by one L-shaped border, drawn at a size the student cannot count, so the argument reads as general. This is the explanation's central figure and the chapter's own five-panel version (Part I p.6) is the model; redraw rather than reproduce.
  • The 6-by-6 nested-L dot array set beside the running sums (Part I p.5), with each shell lighting up as its sum is read.
  • The chapter's ten-block hundreds table (Part I p.7). Standard schematic.
  • A number line marked with consecutive squares so the widening gaps are visible at a glance. An added figure; it carries sections 1 and the item 7 hook.

Where this sits in the book

The book

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