Add and subtract integers

Integer

\(\displaystyle -4\)

Which integer is the opposite of negative four?

Read addition as a signed change

The first integer gives the starting position. The integer being added gives the change

Adding a positive integer moves right. Adding a negative integer moves left

In the second expression, the plus sign is the addition operation. The minus sign belongs to the number being added

Brackets keep the negative number together and make these two roles clear. They do not change its value

Use the sign to choose the direction

Start at negative two and add positive three

Move three units right: from negative two to negative one, then zero, then one

Now use the same starting integer but add negative three

Move three units left: from negative two to negative three, then negative four, then negative five

The magnitude of the change is three units in both cases, but its sign gives the direction

Add a positive integer and cross zero

Evaluate the expression

Start at -4 on the number line

The second integer tells you the signed change from the starting value

Adding 7 means moving 7 units right

First move 4 units right to reach zero

There are 3 units left to move in the same direction

The final position is 3

Check by undoing the change: add the opposite integer to the result

Keep moving right through zero

Add a negative integer

Evaluate the expression

Start at 3 on the number line

The second integer tells you the signed change from the starting value

Adding -8 means moving 8 units left

First move 3 units left to reach zero

There are 5 units left to move in the same direction

The final position is -5

Check by undoing the change: add the opposite integer to the result

Move left for a negative change

Add two integers with the same sign

Two positive integers combine rightward distances from zero

Two negative integers combine leftward distances from zero

Starting three units left and moving five more units left finishes eight units left of zero

Adding two negatives does not produce a positive result. Keep track of the operation instead of using a rule about two minus signs

Combine two leftward distances

Evaluate the expression

Start at -6 on the number line

The second integer tells you the signed change from the starting value

Adding -4 means moving 4 units left

Count 4 equal unit steps left from -6

The final position is -10

Check by undoing the change: add the opposite integer to the result

Add two negative integers

Add two positive integers

Opposite changes cancel, while zero changes nothing

Opposite integers have equal distances from zero in opposite directions. Their sum is zero

Adding zero leaves the starting integer unchanged

Starting at zero and adding an integer reaches that integer

Recognise an exact cancellation

Add zero to a negative integer

Evaluate the expression

Start at -7 on the number line

The second integer tells you the signed change from the starting value

A change of zero does not move the point

The final position is -7

Check by undoing the change: add the opposite integer to the result

Start at zero

Changing the order of addition keeps the sum

For two integers, swapping the order of addition does not change the sum

Starting at three and moving five units left reaches negative two. Starting at negative five and moving three units right reaches the same result

Use this as a check for addition, but do not assume it also works for subtraction

Subtracting a positive moves left

Subtracting a positive integer reverses the change made by adding it

Subtracting three means moving three units left from the starting value. The same rule works when the starting value is negative

Subtracting a positive integer is therefore the same as adding its negative opposite

Subtract a positive integer and cross zero

Evaluate the expression

Start at 2 on the number line

Subtracting a number has the same effect as adding its opposite

The opposite of 7 is -7

Adding -7 means moving 7 units left

First move 2 units left to reach zero

There are 5 units left to move in the same direction

The final position is -5

Check by adding the subtracted integer back to the result

Start negative and move farther left

Subtracting a negative adds its opposite

Subtraction undoes addition. Adding negative three moves left three units, so subtracting negative three reverses that change and moves right three units

The first integer stays unchanged. Replace the subtraction operation with addition, then replace the second integer with its opposite

The same rule works whether the final result is negative, zero or positive

Do not turn the first integer positive just because a negative is being subtracted

Start at negative two again. Compare subtracting positive three with subtracting negative three. The labelled ticks are two units apart

Which statement matches these two settings?

Subtract a negative and stay below zero

Evaluate the expression

Start at -8 on the number line

Subtracting a number has the same effect as adding its opposite

The opposite of -3 is 3

Adding 3 means moving 3 units right

Count 3 equal unit steps right from -8

The final position is -5

Check by adding the subtracted integer back to the result

Subtract a negative and cross zero

Which rewrite preserves the expression?

Distinguish subtracting zero from subtracting from zero

Subtracting zero leaves the starting integer unchanged

Subtracting an integer from zero gives its opposite

Subtracting an integer from itself gives zero, including when that integer is negative

Subtract zero

Subtract a negative integer from itself

Subtract a negative integer from zero

The order matters in subtraction

Swapping the integers in a subtraction generally changes its result

These two results are opposites, not equal. Keep the starting value and the value being subtracted in their stated order

Undo the operation to check

To check an addition, subtract the added integer from the result

The check returns to the starting integer, negative six

To check a subtraction, add the subtracted integer back to the result

Keep the sign on the integer that you add back

Addition

\(\displaystyle -7+4=-3\)

Which calculation checks the addition by undoing the added change?

Find a temperature after a rise

The temperature starts at the first reading and rises by the stated number of degrees Celsius. Find the final temperature

The starting temperature is negative five degrees Celsius

A rise of eight degrees is a positive change

Five degrees of the rise reach zero, leaving three more degrees of increase

The final temperature is three degrees Celsius

A rise does not always end above zero

Find the signed change between two readings

The signed change is the final reading minus the initial reading. A positive change is a rise and a negative change is a fall

A temperature changes from negative five degrees Celsius to positive two degrees Celsius

The positive change is a rise of seven degrees Celsius

Reversing the readings reverses the sign of the change

From positive two to negative five is a fall of seven degrees Celsius, recorded as a signed change of negative seven degrees Celsius

Keep the direction of the temperature change

Keep position, operation and sign separate

For addition, start at the first integer. Move right for a positive added integer and left for a negative one

Opposite integers add to zero. Adding or subtracting zero leaves the value unchanged

For subtraction, keep the first integer and add the opposite of the second

An addition can be checked with subtraction, and a subtraction with addition

In a context, distinguish the final reading from the signed change. The signed change is final minus initial