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If the odd numbers on two dice are made negative, which of the totals cannot be achieved?

# Consecutive Negative Numbers

##### Age 11 to 14 Challenge Level:

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

This printable worksheet may be useful: Consecutive Negative Numbers.

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