Consecutive Squares

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?

Two Squares

A Mathematics Lesson Starter Of The Day


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 = (N2) - ((N-1)(N-1)) (expanding and simplifying) = N2 - (N2 - N - N + 1) = N2 - N2 + 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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Laptops In Lessons

Teacher, do your students have access to computers?
Do they have iPads or Laptops in Lessons?

Whether your students each have a TabletPC, a Surface or a Mac, this activity lends itself to eLearning (Engaged Learning).

Laptops In Lessons

Here a concise URL for a version of this page without the comments.

Transum.org/go/?Start=October10

Here is the URL which will take them to another activity involving square numbers.

Transum.org/go/?to=sqpg

Student Activity


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