Why do this problem?

This problem offers students an opportunity to add positive and negative numbers while challenging them to work systematically.

Possible approach

Introduce the scenario as it appears in the problem. Negative weights could be seen as strong helium balloons lifting up the pan of the weighing scales.

Students could suggest a few examples for the class to work through and clarify the procedure. Ask about the largest total and the smallest total, giving time for students to explain to their partners why $20$ and $^-60$ are correct.
Set the class working on the main task:
Can you make all the numbers in between? Can you show us how?
Is there always a unique way of producing a total, or can different combinations produce the same total?

There are different strategies for tackling this problem so if you observed that students approached the task in different ways, ask (carefully selected) students to describe how they did it. This could lead to a discussion of the merits of the different approaches.

Key questions

Can you make all the numbers in between? Can you show us how?
Is there always a unique way of producing a total, or can different combinations produce the same total?

Possible extension

Possible support

Organise students into small teams with a big central sheet of paper. Teams can write the numbers from $20$ to $^-60$, and then students can fill in any sum that they find. Ask students to pass their suggested sum to a team mate for checking before it is written on the team sheet. If appropriate, the teams could be racing each other to get as many totals as possible, but organise a penalty system for errors (thus really encouraging the checking process!).

Teachers may like to take a look at the article Adding and Subtracting Negative Numbers