# Picturing Square Numbers

*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.

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?

Well done to all of you who sent in correct solutions to this problem. A lot of you spotted the connection with square numbers.

Hannah from Millom School in Cumbria sent in a nicely articulated solution:

The diagram shows that the sum of the first $4$ odd numbers is $16$ because there are $4$ rows and $4$ columns of counters. For the sum of the first $20$ odd numbers there are $20$ rows and $20$ columns. So if I do $30 \times 30$ (or $30$ squared) I get an answer of $900$. For the sum of the first $60$ odd numbers there are $60$ rows and $60$ columns. So if I do $60 \times 60$ (or $60$ squared) I get an answer of $3600$.

If you want the sum of the first $n$ odd numbers the answer would be $n$ squared.

I worked out that $153$ is the $77$th odd number. I did this by adding one (to get $154$) and then divided the answer by $2$.

The sum of the first $77$ odd numbers is $77\times77$ which is $5929$.

To find $51 + 53+ 55+\ldots+ 149 + 151 + 153$ I used the answer from the previous question which was $5929$.

As we were starting at $51$ this time and not $1$, I needed to find the sum of all the odd numbers from $1$ up to $49$. I found that $49$ is the $25$th odd number (by adding $1$ to $49$ and then dividing the answer by $2$) So the sum of the odd numbers from $1$ to $49$ is $25$ squared which is $625$.

Finally I took $625$ away from $5929$ to give an answer of $5304$.

David decided to use algebra to explain his thinking:

The sum of the first $30$ odd numbers $= 30^2 = 900$.

The sum of the first $60$ odd numbers $= 60^2 = 3600$

Quick Method: The sum of the first $n$ odd numbers $= n^2$

What is the sum of $1 + 3 + \ldots + 149 + 151 + 153$?

The formula for odd numbers is $2n-1$

We have: $2n-1 = 153$

$2n = 154$

$n = 77$

So $153$ is the $77$th odd number. The sum of the first $77$ odd numbers $= 77^2 = 5929$. Therefore, the sum of $1 + 3 + \ldots+ 149 + 151 + 153 = 5929$

What is the value of $51 + 53 + 55 + \ldots+ 149 + 151 + 153$?

The answer is the sum of ($1 + 3 + \ldots + 149 + 151 + 153$ - which is already worked out) minus the sum of ($1 + 3 + \ldots +49$)

$49$ is the $25$th odd number (as $2n-1 = 49 \Rightarrow 2n = 50$, so $n = 25$)

Therefore the value of $51 + 53 + 55 + ... + 149 + 151 + 153 = 77^2 - 25^2 = 5304$

Ian from Colton Primary School and Hannah from Thorner C of E Primary School also completed particularly nice solutions, but we don't have space to show them here .

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