# Picturing Square Numbers

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

## Problem

*Picturing Square Numbers printable worksheet*

The diagram shows that 1 + 3 + 5 + 7 = 16.

**What is the sum of the first $30$ odd numbers?**

What is the sum of the first $60$ odd numbers?

Can you describe a method for working these out quickly?

Can you make $3249$ by adding odd numbers in this way?

What is the value of:

$1 + 3 + ... + 149 + 151 + 153$?

$83 + 81+ ... + 5 + 3 + 1$?

$51 + 53 + 55 + ... + 149 + 151 + 153$?

$2 + 4 + ... + 150 + 152 + 154$?

$2 + 6+ ... + 298+ 302+ 306$?

Explain how you worked these out.

## Getting Started

How could you describe the squares that are formed?

What is the relationship between the square numbers and the first and last numbers in their respective sequence of consecutive odd numbers?

## Teachers' Resources

Why do this problem?

This problem develops the links between visualisation, verbal description and algebraic representation.

### Possible approach

*This printable worksheet may be useful: **Picturing Square Numbers**.*

Kick off by talking about odd numbers:

What do the first $10$ odd numbers add up to?

What do the first $5$ odd numbers add up to?

What is the $50$th odd number anyway? The $100$th?

$125$ is an odd number. Which is it?

Show students this image or the interactivity

Ask for comments on the arrangement of dots.

"How can this help us explain the relationship between square numbers and the sum of odd numbers?"

"How many more dots will I need to add to make the next square? And the next? And the next?"

"How many more dots will I need to go from the $100$th square to the $101$th?"

Set students off to work in pairs on the questions set in the main body of the problem: Picturing Square Numbers

### Key questions

What is the $5$th, $10$th, $455$th odd number?

What is the sum of the first $10$, $20$, $50$, ... $n$ odd numbers?

### Possible support

This task could be used as a context for working hard on odd numbers and their structure, practising doubling numbers and mental addition. Tasks could include adding sets of odd numbers, imagining the last layer on the $30$th square, the $57$th square, working out which square would have $43$ as its last layer.

To prepare students for looking closely at other sequence pattern diagrams, the interactivity could support discussion between students - how they imagine the next diagram will look, whether different students see it differently.

### Possible extension

A suitable extension task is provided in this worksheet .

For another problem that uses a similar idea go to Picturing Triangle Numbers

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