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 such as tablets, iPads or Laptops?  This page was really designed for projection on a whiteboard but if you really want the students to have access to it here is a concise URL for a version of this page without the comments:

However it would be better to assign one of the student interactive activities below.

Laptops In Lessons

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

Student Activity

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.


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