11 plus Negative Numbers questions

What a Negative Numbers question looks like

Negative Numbers is one of the number topics in 11 plus maths, and every question in it turns on the same idea: a value below zero, and what happens when you count past zero in one direction or the other. Its nearest neighbour, Place Value & Ordering, compares, rounds and orders whole numbers above zero, while this is the topic where the minus sign changes what “bigger” and “smaller” mean. It is one of the few eleven plus number topics that asks a child to think in a new direction rather than simply to calculate faster.

The settings are mostly temperatures, with bank balances, debts and depths below sea level making up most of the rest. Some questions come with a figure to read: a thermometer, a marked number line, a bar chart of daily low temperatures, a table of city temperatures or a sorting diagram of two overlapping groups. A few use bare numbers with no story at all.

Across all of those settings the questions do five jobs. They add or subtract across zero, they order and compare values below zero, they find the gap between a cold value and a warm one, they read a point off a scale, and in the hardest few they land on a half degree or share a change out evenly over several days. Multiplying and dividing two negative numbers does not come up: that is later school work.

Example one

The temperature at midnight was −8°C. By noon it had risen by 13°C. What was the temperature at noon?

Start at −8 and remember that “risen” means counting upwards. The useful move is to break the journey at zero. Getting from −8 up to 0 uses 8 of the 13 degrees, which leaves 5 degrees still to go, and those 5 take you above zero. The temperature at noon was 5°C.

The tempting wrong answer is 21°C. That is what a child gets by reading the two numbers and ignoring the minus sign: 8 + 13 = 21. The other one to watch is 13°C, which is the size of the rise rather than the temperature it finished at.

Example two

The temperature on Monday was −3°C. On Tuesday it was 4°C. By how many degrees did the temperature rise?

This one asks for a gap, and the same move works. Count in two hops, with zero as the stopping point. From −3 up to 0 is 3 degrees. From 0 up to 4 is another 4 degrees. Add the hops: 3 + 4 = 7. The temperature rose by 7°C.

The wrong answer built into questions of this shape is 1°C, which comes from doing 4 − 3 and treating −3 as though it were 3. A child who has drawn the line rarely falls for it, because one degree does not look like the distance between a frosty Monday and a mild Tuesday.

Example three

The third shape gives its numbers in a table and asks a comparison question. The table lists the temperature in four cities, and the question is which city is coldest.

Lisbon is above zero, so it can go straight away. That leaves three cities below zero, and the coldest is the one that sits furthest below. Helsinki at −13°C is further down than Oslo at −6°C, which is further down than Berlin at −1°C. Helsinki is the answer.

Berlin is the trap for a child who covers up the minus signs and picks the smallest number they can see, because 1 is smaller than 6 and smaller than 13. Oslo catches the child who spots one large-looking negative and stops comparing.

How to answer it

1. Decide the direction before calculating anything. Rising, warming, earning and swimming up all count upwards. Falling, dropping, cooling, spending, owing and diving down all count the other way.
2. Picture a number line with zero in the middle, or sketch one in the margin, and mark the starting value on it. Most questions on this topic become obvious once the line is there.
3. Break the journey at zero. Count how many steps it takes to reach zero, take those off the total, then carry on with whatever is left on the far side.
4. For “how much warmer”, “how much deeper” or “what is the difference”, count the two hops and add them. An answer of that kind is a size, so it never carries a minus sign.
5. Read the question again before choosing. It matters whether you were asked for the value at the end or for the size of the change, because both numbers are usually sitting there in the options.

Common mistakes

• Dropping the minus sign. −8 + 13 quietly becomes 8 + 13 = 21. It is the most common wrong answer in the whole topic, and it is a reading slip rather than an arithmetic one.
• Treating a bigger-looking negative as a bigger number. −9 is colder than −1, and −11 is lower than −7. Asked for the lowest value, children often pick the one nearest zero.
• Counting the wrong way on a direction word. If it is −6°C and the temperature drops by 2 degrees, the answer is −8°C, not −4°C. “Drops” means further from zero, not back towards it.
• Stopping halfway through a two-step story. A diver at −20 m who swims up 7 m and then dives down 4 m finishes at −17 m, not at the −13 m she passed through in the middle.
• Answering the other question. Giving the size of the rise when the final temperature was asked for, or the final temperature when the rise was asked for, costs the same mark as getting the sum wrong.

How to practise 11 plus Negative Numbers questions

The most useful thing at home costs nothing: draw a number line from −20 to 20 on a strip of paper and leave it on the table. Ask your child to put a finger on the starting value and walk it, rather than doing the sum in their head, until the direction no longer needs thinking about. It helps to say the direction word out loud too, because “dropped”, “owes” and “dived” are what each question really hinges on, and a child who has settled which way to move has usually finished the hard part. Ten focused minutes beats an hour of mixed practice, which is the pattern we set out in how much 11 plus practice a child needs each day. 11+ Daily has a Negative Numbers project alongside the rest of the number work, on laptop, tablet or phone, and the other maths topics are listed on our 11 plus maths practice page. Every question is original to 11+ Daily, nothing is a past paper and nothing is licensed from another publisher, and the bank is re-checked in rounds.

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