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!

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
Click here to enter your comments.

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



Sign in to your Transum subscription account to see the answers

Your access to the majority of the Transum resources continues to be free but you can help support the continued growth of the website by doing your Amazon shopping using the links on this page. Below is an Amazon search box and some items I have chosen and recommend to get you started. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.


Online Maths Shop

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.

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.


©1997-2021 WWW.TRANSUM.ORG