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

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.

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

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 link. As an Amazon Associate I earn a small amount from qualifying purchases which helps pay for the upkeep of this website.

Educational Technology on Amazon

## GCSE Revision and Practice

Whatever exam board you use for GCSE Mathematics, this book by David Rayner remains an all-round winner. With this latest edition presented in full colour and completely updated for the new GCSE(9-1) specifications, this uniquely effective text continues to increase your chance of obtaining a good grade.

This book is targeted at the Higher tier GCSE, and provides a wealth of practice with careful progression, alongside substantial revision support for the new-style grading and exam questions. With all the new topics included, and a dedicated section on using and applying mathematics, this unique resource can be used either as a course book over two or three years or as a revision text in the run-up to exams. more... #ad

## Maths T-Shirts

 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: Transum.org/go/?Start=October10 However it would be better to assign one of the student interactive activities below.

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

Transum.org/go/?to=sqpg

## Curriculum Reference

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

For Students:

For All: