Why do this problem?
This problem offers great opportunities for mental arithmetic and estimation. It can also be used as an opening to discussing the order of operations.
Display the numbers $1$ - $9$ on the board and ask students to add them up. (They might do this in any order, perhaps noticing that pairs from either end add to $10$.) As they explain their working, record it in order on the board, for example:
$1+ 9 +2 + 8 + 3 + 7 + 4 + 6 + 5 = 45$
Ask if they can suggest a way to make the answer bigger, but still only using the numbers $1$ - $9$. Again, record the calculations on the board in the order that the children say them. This is likely to involve a multiplication sign. Ask if they can make it even bigger. Again, record the calculations.
Then offer the problem. Allow some time for students to work, possibly in pairs, and provide calculators for them to use to check their arithmetic if necessary. Provide a central wall space for students to record their solutions. This would make an ideal 'simmering' activity that could go on for a week or more. (See the extension questions below.)
How close can you get to 100 with just adding?
What operation might you use to make the result bigger?
Which sorts of calculations make the most difference to the total?
Which numbers less than $100$ is it possible to make?
What other questions can you suggest?
The numbers could be written on separate pices of paper, together with several $+$, $-$, $\times$ and $\div$ signs. Being able to rearrange the numbers can sometimes help to see patterns or number bonds that help with calculations. These
printed digit and operation cards could also be used. And whilst the
problem offers great opportunities for mental arithmetic and estimation, pupils who are less confident at these could use a calculator. This would help to support their estimation skills and include them in a whole class activity.
An additional challenge would be for the students to decide on their own target number and see if they can make it using $1$ - $9$.