Consecutive Squares

A Maths Starter of The Day

Write down two consecutive numbers.

Square each of them and find the difference.

Do the same for other pairs of numbers. What do you notice?

 

 

 

 

 

Answers

The first thing you may notice is that all of your answers are odd numbers.

Did you also notice that your answers are always the sum of the two consecutive numbers?

22 - 12 = 332 - 22 = 542 - 32 = 7
52 - 42 = 962 - 52 = 1172 - 62 = 13
82 - 72 = 1592 - 82 = 17102 - 92 = 19
112 - 102 = 21122 - 112 = 23132 - 122 = 25
142 - 132 = 27152 - 142 = 29162 - 152 = 31
172 - 162 = 33182 - 172 = 35192 - 182 = 37
202 - 192 = 39212 - 202 = 41222 - 212 = 43
232 - 222 = 45242 - 232 = 47252 - 242 = 49
262 - 252 = 51272 - 262 = 53282 - 272 = 55
292 - 282 = 57302 - 292 = 59312 - 302 = 61
322 - 312 = 63332 - 322 = 65342 - 332 = 67
352 - 342 = 69362 - 352 = 71372 - 362 = 73
382 - 372 = 75392 - 382 = 77402 - 392 = 79
412 - 402 = 81422 - 412 = 83432 - 422 = 85
442 - 432 = 87452 - 442 = 89462 - 452 = 91
472 - 462 = 93482 - 472 = 95492 - 482 = 97
502 - 492 = 99512 - 502 = 101522 - 512 = 103

 

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Keywords: Starter | Number Patterns

  • Mr Frost, John Summers High School
  • Tuesday, October 03, 2006
  • "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
  • Wednesday, October 04, 2006
  • "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
  • Saturday, October 04, 2008
  • "As an extension of this idea

    a² - b² = (a + b) x (a - b) wherever a is greater than b"

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