This is a variation on the problem

Consecutive
Numbers . Teachers may need to choose between the two problems
- doing both may involve too much overlap.

### Why do this problem?

This problem could replace a "standard practice exercise" for
adding and subtracting negative numbers. It provides opportunities
for a lot of calculation, in a context of experimenting,
conjecturing, testing conjectures, etc.

### Possible approach

Ask for four consecutive negative numbers, write them on the
board, then place $+$ and/or $-$ signs in the three gaps between
them. "Assuming we are only allowed to add and subtract, how else
could the gaps be filled?" Make a note of the suggestions until
students are confident that all possibilities have been listed.
"How do we know we have listed them all?"

In pairs, students calculate the answers to the list of 'sums'
from the board. Students compare notes with neighbours to resolve
disagreements. Students then work, either individually or in pairs,
on new sets of four consecutive negative numbers, repeating the
process above.

To direct attention to more than just routine calculation, collate
sets of some students' results on the board, and ask the group for
general descriptive comments - encouraging conjectures and
explanations. Students can then go back to working in pairs to test
the validity of what they have heard suggested.

Encourage students to move from "there's always a zero" to the
reason why this is true - isolate and examine the cases where it is
zero.

At the end of the lesson a plenary discussion can offer students a
chance to present their findings, explanations and proofs.

### Key questions

How do you know you have considered all the possible
calculations?

Do the answers seem random, or can any/all be predicted?

How do you KNOW that what you say will ALWAYS work?

### Possible extension

What happens if we allow a $+$ or a $-$ sign before the first
number?

What happens if it doesn't have to be FOUR consecutive
negatives?

### Possible support

Students could use

Consecutive
Numbers for a similar problem which offers opportunities to
experiment/conjecture/justify but doesn't require negative
numbers.

Students could start by considering the solutions when they
add and/or subtract three consecutive negative numbers.

Teachers may like to take a look at the article

Adding
and Subtracting Negative Numbers