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Up and Down Staircases

Stage: 2 Challenge Level: Challenge Level:1

Many of you worked on this problem and sent us solutions.

Angelica and Beth from St Louise Primary Primary School in Scotland sent in the following account of what they did:

For the Up and Down Staircase Problem, we made a table with the number of steps up and down at the top and the number of cubes at the bottom.
Then we drew out pictures with the cubes making 1 step up and down, then 2 steps up and down, and so on to 7 steps up and down, and filled in the table.
As we made the table, we could see that the number of cubes needed was the number of steps squared. So, say you needed to know how many cubes you need to make twelve stairs up and down. You would find twelve by twelve. You could also see, say, 144 cubes and know that you could make twelve stairs up and down (c = s x s) when c = number of cubes and s = stairs up and down.

Rowan Class in Saxmundham Primary School sent in the following and the picture of their work.

The children of Rowan class worked on Up and Down Staircases by building staircases from blocks.
Molly and Willow were the first to spot a pattern in the numbers of blocks in the different staircases.
Here is their work:
1 step staircase - 1 block (1 x 1 = 1)
2 step staircase - 4 blocks (2 x 2 = 4)
3 step staircase - 9 blocks (3 x 3 = 9)
4 step staircase - 16 blocks (4 x 4 = 16)
To work out the number of blocks in a staircase you must multiply the number of steps by the number of steps. You can work out bigger numbers with a calculator.
27 step staircase - 729 blocks (27 x 27 = 729)

Amy and Izzy said -
The pattern is ...
all you do is time (multiply) the steps you are doing e.g. if you were doing a 5 step staircase you 5 x 5 = 25

Just as I was gathering solutions together I had many come in from Lindfield East Public School in Australia. So thank you Joanna and Paris, Mark and Matthew, Megan, Chanhee, Isaac and Saarthak, Sean, Angus, Alexy and Iliana, Miles and Joey, Luke and Nick.
Here, as a sample, are the two solutions from the last four pupils:

Firstly we worked out the first question by drawing out all the blocks on grid paper, the answer was 25 blocks.
The second question was: explain how to work out any number of blocks by only looking at the number of steps. The formula we used was X up multiplied by X down equals the total amount of blocks. We worked out this formula by finding all the similarities between the number of steps and
the number of blocks.
They were: Half of the blocks equaled the number of steps up and the number of steps down seperately but this did not work with the 5 and 1 block ones. The only other one was the number of blocks up times the number of blocks down equals the number of blocks in total. This was right with all of them so we made a formula for it and submitted it.


The amount of blocks on the centre column is the amount of steps you can go up and down on.
E.g. If you want three steps up and down, the centre column must be three blocks high. Then the next to columns beside the middle column must be 1 minus the middle column (2 on each column). Then the next two columns will be 1 minus those columns (1 on each column). It will keep on going until the columns on the ends of the shape is one block high. Then just add them up.
The problem is to find how many blocks you need to make 5 steps.
The centre column will be 5 blocks high.
The next two columns will be 4 blocks high on each column.
The next two columns will be 3 blocks high on each column.
The next two columns will be 2 blocks high on each column.
Then finally the last columns will be 1 block high on each column.
The formula is the centre column squared = the amount of  blocks e.g. 5$^2$ = 25

The following solution came in was from a teacher, Kalley, who wrote about the work that her pupil Hannah, who loves our activities, did on this particular challenge:

Here is how we did it ...
We built the stages revealed on the website and then continued to investigate the next two stages and found
For 3 up and 3 down you need 9 blocks
For 4 up and 4 down you need 16 blocks

We noticed that on the bottom row there was a pattern where you added two extra blocks on this row going in an odd number pattern: 1, 3, 5, 7... Which we then predicted for the next stage of 5 up and 5 down which was 9 as we thought!

We then looked at our results and noticed a pattern in the number of steps needed and the blocks needed to build it was square numbers! So we predicted what 6 up 6 down would be ... 36 ... and we were right when we built it!
Again we checked with 7... and we were correct!
I then challenged Hannah to see what 15 up and 15 down would be ... 225 blocks! But unfortunately we were running out of blocks to check, so midway we reverted to 11 up and 11 down ... 121 blocks... which we checked and were correct with our prediction.
We then went further and thought we would choose a random number ... 88 up and 88 down and using our secret of squared numbers we predicted you would need 7744 blocks!

Here are some pictures of our working out...

Well done all of you for your thoughts, pictures and accounts of what you did.