We had a good number of solutions submitted for this task.
Steph from York House School in CanadaӬ wrote:
We figured this out by putting the 1 to 10's on the very left side and the top side. And then we put together the puzzle pieces.

Matilda from St Philip's Primary School Cambridge”¨ sent in this explanation:
First of all, I did the pieces with the 1 times table on it. After that, I completed the 10 times. Then, using the left over pieces, I filled the empty space and put them into an appropriate position to finish the jigsaw.

Connor from The Pike School in America wrote:
ӬI did this like a puzzle. I looked for all the numbers going up by 1, 2, 3, 4, 5, 6 and so on. Most of the time the pieces matched. That strategy worked for me.

Here are two solutions submitted from The American School In Japan:ӬӬ
Firstly, Hironari's: 
Well, what I did to find out the places for pieces of numbers was that, well, I knew how the multiplication squares worked like so I started by putting the pieces that had the numbers go up by one.
After that, I put in the pieces that went up by twos, threes and on and on. I didn't just do that way left to right, I did up and down too!
I learned that when you do these kind of multiplication puzzles, you can't just put the closest numbers to each other! It is multiplication so the smallest numbers times small numbers doesn't just equal small numbers like addition!

And then Mia's:Ӭ
First you had to solve 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 and the puzzle pieces fit exactly in to the square then I did 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 which is the two times two pattern and realised there are patterns. Finding the pattern was easy but are there any other solutions?

Here are a few submissions from Westridge SchoolӬ in America:

Chloe and Addison said:
We found a solution for this problem by entering the first ones row. Then we looked at the other numbers connected to the same block as the ones row. When you start to put more blocks together, you start to see patterns like; 
1, 2, 3, 4, 5...  2, 4, 6, 8, 10...   3, 6, 9, 12, 15...   and so on.

Maddie wrote:ӬӬ
To start, I put together pieces with the 1 and the 10 times tables.
To put together the pieces, I first started with the ones that have parts of the edges, the 1 and 10 times tables. Then I put in a piece that had part of the 2 and 3 times tables, and after that I put in one with a big long part of the 5 times table. Next, I put in a piece with the 4 and 6 times tables, and lastly, one with the 8 times table. The last one was easy because it just fit right in. 

Ruby and Frankie said:ӬӬ
For this problem, we figured out the jigsaw puzzle by knowing our multiplication tables. But, the problem required more then just knowing our times tables. We had to know where the puzzle pieces fit best with the other pieces. This is how we figured it out!

Emerson said:Ӭ
I solved this problem by first looking at the numbers at the right corner because all of the numbers there are in consecutive order. Then I looked at the numbers that crossed into the same row and I multiplied them. I then moved the pieces accordingly to the answers and slowly put the pieces together and created the multiplication table.

Aditya from St Columbas CollegeӬ gave this explanation:
I first tried to find 1, which gave me 2-6 vertically too. Then I tried to find 2 horizontally, which also gave me 3-5 vertically.
Having 5, I found 6-10 and fit it in horizontally.
Then I found 4, 6 and 8 to go into the vertical 2x table.
I had 3 and 6, so I had to find a piece with 9, 12 and 15 vertically together and one with 18 and 21. That completed 4x table also so I moved onto the 5x table.
Having 35 in place meant I had to slot in 40, 45 and 50.
I had a gap between 6 and 18 so I put in piece with 12 at the top. The second number in the 8x table is 16, so I slot in the piece with 16 on the top. Then I put in the next one in according to shape.
So to cut a long story short, I worked systematically through vertical times tables and this allowed me to know my next move instead of putting pieces in random places like you would when doing a puzzle.

Thank you for your submissions and particularly to those who tried to describe how they went about it.