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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Previous Day | This starter is for 10 October | Next Day

 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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