Practice Finding Squares Using (a + d)(a - d) + d²

Introduction

This worksheet helps learners master a fast and elegant mental maths trick for finding squares of numbers up to 200 using the identity a² = (a + d)(a − d) + d². It is designed for speed practice, clarity, and strong number sense.

The Core Squaring Identity

This trick is based on the algebraic identity:

a² = (a + d)(a − d) + d²

Here, d represents the difference between the number a and its nearest tens value.

By adjusting the number up and down by the same amount, the squaring process becomes simpler and more structured.

How This Squaring Method Works

If you are new to this technique, it is strongly recommended to begin with the interactive blog post Finding Squares Using (a + d)(a - d) + d² that explains this trick.

The blog includes visual flow charts that show each step clearly, making the logic easy to grasp before attempting timed practice.

To apply this method mentally:

1. Identify the nearest tens number to a
2. Compute d, the difference between a and that tens value
3. Multiply (a + d) and (a − d)
4. Add d² to the result

This approach reduces squaring to a manageable multiplication followed by a small adjustment.

Worked Example

Example: Finding 28²

• Let a = 28, d = 8
• (a + d)(a − d) + d²
• (28 + 8)(28 − 8) + 8²
• (36)(20) + 64
• 720 + 64 = 784

Left-to-Right Mental Calculation Flow

This worksheet follows a left-to-right mental calculation style, preferred by many mathemagicians and mental-math experts.

Learners begin with the larger multiplication(a + d)(a − d), focusing on higher place values first, and then refine the result by adding d².

This flow feels natural, intuitive, and efficient for in-mind calculations.

Practice Scope and Structure

This worksheet provides extensive practice for squaring numbers up to 200.

A full practice session includes 150 problems, allowing learners to build speed, confidence, and pattern recognition through repetition.

Skills Developed Through This Worksheet

• Fast mental squaring up to 200
• Strong understanding of number adjustment techniques
• Left-to-right arithmetic fluency
• Confidence in speed maths and mental calculation

Online, Interactive, and Speed-Focused

This is an online, interactive worksheet where learners enter the final square directly into the input box.

Instant feedback helps correct mistakes quickly and reinforces accurate mental processing. No printing or downloading is required.