Write down two consecutive numbers.

Square each of them and find the difference.

Do the same for other pairs of consecutive numbers.

What do you notice?

## A Mathematics Lesson Starter Of The Day

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Topics: Starter | Number

• Mr Frost, John Summers High School
•
• The difference of the squares of two consequetive numbers will always equal the sum of those two numbers.

sum of the numbers:
a + (a-1) = 2a - 1

Difference of the square of the numbers
a2 - (a-1)2 = a2 - (a2 - 2a + 1)
= 2a - 1
• Mr Frost, John Summers High School, Flintshire
•
• or

Difference in squares
a2 - (a + 1)2 = a2 - (a2 + 2a + 1)
= 2a + 1

Sum of the numbers

a + (a + 1) = 2a + 1
• David Longman, Bedfordshire Middle School
•
• As an extension of this idea

a² - b² = (a + b) x (a - b) wherever a is greater than b
• Steve Eastop, Margate, Kent
•
• The difference between the results of squaring each consecutive number and then subtracting the lesser result from the greater result always results in an ODD INTEGER (i.e. a positive or negative whole number indivisible by two). In other words, the result to such a calculation will always be a member of the set {...-5,-3,-1,1,3,5,7,9,11,....}.
In general, algebraically, let the two consecutive numbers be: (N-1) and (N) respectively.(whereby N is the larger of the two).
Then (N)^2 - (N-1)^2 = (N^2) - ((N-1)(N-1)) (expanding and simplifying)
= N^2 - (N^2 - N - N + 1)
= N^2 - N^2 + N + N + 1
= (2N +1).
Hence, whatever integral value of N you assign, 2N will always be even and thus (2N + 1) will be odd as already stated above!

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## Answers

 22 - 12 = 3 32 - 22 = 5 42 - 32 = 7 52 - 42 = 9 62 - 52 = 11 72 - 62 = 13 82 - 72 = 15 92 - 82 = 17 102 - 92 = 19 112 - 102 = 21 122 - 112 = 23 132 - 122 = 25 142 - 132 = 27 152 - 142 = 29 162 - 152 = 31 172 - 162 = 33 182 - 172 = 35 192 - 182 = 37 202 - 192 = 39 212 - 202 = 41 222 - 212 = 43 232 - 222 = 45 242 - 232 = 47 252 - 242 = 49 262 - 252 = 51 272 - 262 = 53 282 - 272 = 55 292 - 282 = 57 302 - 292 = 59 312 - 302 = 61 322 - 312 = 63 332 - 322 = 65 342 - 332 = 67 352 - 342 = 69 362 - 352 = 71 372 - 362 = 73 382 - 372 = 75 392 - 382 = 77 402 - 392 = 79 412 - 402 = 81 422 - 412 = 83 432 - 422 = 85 442 - 432 = 87 452 - 442 = 89 462 - 452 = 91 472 - 462 = 93 482 - 472 = 95 492 - 482 = 97 502 - 492 = 99 512 - 502 = 101 522 - 512 = 103

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## How Not To Be Wrong

The maths we learn in school can seem like an abstract set of rules, laid down by the ancients and not to be questioned. In fact, Jordan Ellenberg shows us, maths touches on everything we do, and a little mathematical knowledge reveals the hidden structures that lie beneath the world's messy and chaotic surface. In How Not to be Wrong, Ellenberg explores the mathematician's method of analyzing life, from the everyday to the cosmic, showing us which numbers to defend, which ones to ignore, and when to change the equation entirely. Along the way, he explains calculus in a single page, describes Gödel's theorem using only one-syllable words, and reveals how early you actually need to get to the airport.

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