Emerald and Brooke's solutions are in an attached word document with diagrams.
Here is the text:
Emerald
First I tried putting any numbers into the circles. After a few goes I made this triangle where the sides add up to 11, 12 and 13.
Because 11, 12, 13 was so close, I swapped the 3 for the 2 to make 13 one smaller and 11 one higher. Then all the sides total 12.
Brooke
I tried to add up to nine lots of different ways only using numbers 1 to 6. There were only 3 different ways 6 + 1 + 2 = 9, 1 +5 + 3 = 9 and 3 + 4 + 2 = 9.
I found that I couldn’t have 5 and 2 on the same side because you would need another 2 to make it add up to 9.
I put the three sets of numbers into the triangle to see if they match up to make nine.
I found that I needed to hide the numbers 4, 5 and 6 on a middle, not a corner because they were only used once in the ways of making 9.
For the 10 triangle the ways to make 10 are 5 + 3 + 2 = 10, 5 + 1 + 4 = 10, 6 + 1 + 3 = 10.
You can’t have a 4, 2 or 6 on a corner because they are only in one way of making 10.
Emerald
To make the 11 triangle I started using Brooke’s method. I made 11 by adding 6 + 4 + 1 and 5 + 2 + 4. I put those sides on a triangle with the 4 at the top because it was in both sentences.
The only number missing was 3, so that had to go in the empty circle. That made the bottom side total 14. I swapped the 5 and the 2 to make the bottom side less. All the sides equal 11.
Brooke
I guessed that we could make other triangles with the numbers 1 to 6.
I tried to make triangle totals of 3, 5, 8, 13 and 15, but none of them could be done. For the 8 triangle there were not two numbers to go with 6 that would make 8. For the 13 and 15 triangles there were not two numbers to go with 1 to make 13, 15 or higher numbers.