Subtracting Integers
Common Core Standard: Understand subtraction of rational numbers as adding the additive inverse, p - q = p+ (-q). Show
that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this
principle in real-world contexts.
Common Core Standard: Apply properties of operations as strategies to add and subtract rational numbers.
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Zero pairs – two integers who have the same absolute value, so that when added the sum is zero
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Example 3 and -3 are zero pairs
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3 + -3 = 0
Hot and Cold Cubes?
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This is how students are first introduced to positive and negative integers
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Positive integers are hot cubes and negative integers are cold cubes
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This is done to give a concrete example and help to make the integers make more sense
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Example: A chef has a pot with 5 hot cubes in it. Then she takes out 7 cold cubes.
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The number sentence for this is 5 – (-7)
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Since, there aren’t 7 cold cubes originally in the pot; the chef has to add in 7 cold cubes. BUT, this
will change the temperature immediately. So, the chef must add in 7 hot cubes and 7 cold cubes,
because this will create zero pairs. Therefore, it will not change the temperature of the pot.
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Now, the chef can remove the 7 cold cubes.
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This leaves the 5 original hot cubes and 7 more hot cubes.
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So 5 – (-7) = 12
When is the answer positive or negative not using hot and cold cubes?
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To determine if a difference is positive or negative, 1st determine which number in the subtraction problem
has the larger absolute value
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The number’s symbol with the larger absolute value is the symbol of the absolute value
Example Not using Hot and Cold Cubes:
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-6 – 5
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The distance between these two numbers (-6 and 5) is 11
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This is distance can be seen on number lines
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This means that the difference is a number that has an absolute value of 11
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So it difference is either 11 or -11
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Which number has the larger absolute value in the problem?
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|−6| = 6 and |5| =5, so -6 has the larger absolute value
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So, the answer is -11, because the answer must be a negative
Real World Example:
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Jill has $15. She owes someone $7. Determine how much money she has left after she pays this
back.
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Naturally, the answer is already known $8
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Since $7 is owed, it can be written as -$7
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However, $15 - -$7 is not $8
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Instead think about it as how much money is left, if you combine both the $15 she currently has
and the -$7 she owes
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This means $15 + -$7 = $8, which is the same as $15 - $7
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Know that subtraction is the same as adding an opposite
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Example: -4 – 8
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This is the same as -4 + -8
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Solution Path 1: Since, there are already 4 cold cubes and 8 cold cubes are added in, this means that the
temperature will only get colder, so add the integer -4 + -8 = -12
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Solution Path 2: On a number line, begin at -4, then add -8 or towards the negatives, on the number line, 8
times. The end point will be -12.
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Solution Path 3: The absolute values of -4 and -8 are 4 and 8. The sum of these is 12. Since both are
negative, the sum is -12.
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It does not matter which solution path is used; nor does it matter if none of these solution paths are used.
Example: -8 - (-1)
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There is -1 within -8, so take it away, this leaves it with -7
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-8 – (-1) = -7