You have found several different ways of answering this problem which is great.
Fiona at Tattingstone School and Douglas in year 5 at Burgoyne Middle School both decided to draw the squares. However, their approaches weren't the same. This is what Fiona says:
The first thing I did was to get some squared paper to draw the figure on. I found out I couldn't start the middle square as one square because I couldn't draw the other squares accurately so I did the middle square as a four and that gave me points for the rest of the figure.
large outside traingles numbered 1-4 starting at top left , then top right, bottom left, bottom right To find the total area of the numbered triangles:
I noticed that 1 and 4 were two halves of a square and also 2 and 3 are two halves of a square.
Now I just have to count every square in this top rectangle as one unit:
My answer is 32 square units.
 diagram with rectangle top four rows and eight little squares length highlighted
Here is what Douglas did:

1) I drew the squares starting with the inside square that measured 1cm by 1cm.
2) I measured the sides and calculated the area of each of the outside triangles:
Height 4cm width 4cm
Area of 1 triangle =8cm ²
Area of 4 triangles =32cm²

So they agree the area of the outside triangles is 32 square units in total.
Caroline, also from Tattingstone, tried another way of answering the problem. She explains:
one square
 The area of the first square is 1cm 2
square with smaller square above set inside it
 The area of the new triangles is the same as the dotted triangles, which means that the area of the new square is 2cm 2
third sqare with two smaller squares set inside
 The area of the new triangles is the same as the dotted triangles, which is twice the amount of the second square so the area of the new square is 4cm 2
The area of the fourth square is two times the area of the third so it equals 2 x 4 = 8cm 2
The area of the fifth square is twice the area of the fourth so it equals 2 x 8 = 16cm 2
The area of the next square is 2 times the area of the fifth so it equals 2 x 16 = 32cm 2
When we add the 7th square we just add up the four outside triangles, which are the same as the dotted inside triangles, so the total of the new triangles is 32cm 2
Joshua who is in Year 5 at Maldon Court Preparatory School used a very similar method and so did Tom Neill. Thank you to you all.