Draw the number line. Every question in this topic becomes counting along it, and counting is something a child could already do at eight.
Maths carries 25% of the marks. Almost all of it arrives as worded problems rather than bare calculations, so the reading is part of the maths.
Draw the line
A horizontal line with zero in the middle, negatives to the left, positives to the right. Adding moves right. Subtracting moves left. That is the entire model.
What is minus 5 plus 8?
Start at minus 5. Move 8 places right. You pass zero after 5 moves, with 3 left to go, so you land on 3.
Children who try to do this with rules alone get into trouble. Children who draw the line almost never do.
| Move | Direction on the line | Example |
|---|---|---|
| Adding | Right, towards the positives | minus 5 plus 8 lands on 3 |
| Subtracting | Left, towards the negatives | 2 minus 6 lands on minus 4 |
| Subtracting a negative | Right, because two minuses make a plus | minus 2 minus minus 6 is 4 |
| Finding a difference | Count the gap, breaking at zero | minus 3 to 4 is 3 then 4, so 7 |
Difference questions
The commonest form at this level, usually dressed as temperature.
The temperature rises from minus 3 to 4 degrees. By how much has it risen?
Count the gap on the line: minus 3 up to 0 is 3, then 0 up to 4 is 4. Total: 7 degrees.
Break the journey at zero, every time. It stops children answering 1, which is what you get by subtracting carelessly.
Why subtracting a negative adds
This is the one that genuinely confuses, so here is the version that sticks.
What is minus 2 minus minus 6?
Subtracting means finding the difference. On the line, minus 2 and minus 6 are 4 apart, and minus 2 is the higher one, so the answer is 4.
Phrased as a rule: two minus signs together become a plus. So minus 2 minus minus 6 is the same as minus 2 plus 6, which is 4.
Teach the number line version first and the rule second. A child who only has the rule applies it in the wrong places.
Where they actually appear
- Temperature. The overwhelming majority of questions.
- Height above and below sea level.
- Money owed, in a simple bank balance.
- Floors of a building, with basements counted as negatives.
- Sequences that run backwards past zero, which is where this meets our guide to sequences.
Almost all of it is worded, so the first job remains converting the sentences into numbers on a line before any arithmetic starts.
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The line turns it into counting
This is a topic where children fail on rules they half remember rather than on anything conceptually hard. Two minuses make a plus, except they apply it to a multiplication they have not met yet, or to a question where it does not belong.
Draw the line instead. Zero in the middle, negatives to the left. Adding goes right, subtracting goes left. Now the whole topic is counting along a line, which a child could do years ago.
The one refinement I would add is breaking the journey at zero for difference questions. From minus three to four is three and then four, which is seven. Done in one leap it is exactly the sort of question a child gets wrong at speed, and speed is what causes most of the mistakes in this paper.
Mushtaque Ahmed, Founder of GrammarPath and Bright Mentors
Frequently asked questions
How do you teach negative numbers for the 11+?
With a number line. Zero in the middle, negatives left, positives right; adding moves right and subtracting moves left. Minus 5 plus 8 means starting at minus 5 and moving 8 places right, landing on 3.
How do you find the difference between a negative and a positive?
Break the journey at zero. From minus 3 to 4 is 3 up to zero plus 4 beyond it, which is 7. Splitting at zero prevents the careless subtraction that produces 1.
Why does subtracting a negative add?
Because subtracting means finding the difference, and on a number line minus 2 and minus 6 are 4 apart. As a rule, two minus signs together become a plus, so minus 2 minus minus 6 is the same as minus 2 plus 6, which is 4.
Where do negative numbers appear in the 11+?
Mostly temperature, along with height above and below sea level, money owed, floors of a building with basements, and sequences that run backwards past zero. Nearly all of it arrives as worded problems.
Should my child learn the rules or use a number line?
The number line first, the rules second. A child who has only memorised rules tends to apply them in the wrong places, whereas a child who can picture the line can always fall back on counting.