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Subtract Rational Numbers as Adding the Inverse

Students understand that subtracting a rational number is the same as adding its additive inverse: p − q = p + (−q). They apply this to subtract positive and negative fractions, decimals, and integers in real-world and mathematical contexts.

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Subtract Rational Numbers as Adding the Inverse

Grade 7 · The Number System · question 1 of 3

0 of 20

20 rungs. A star at the top.

3 − (−4) =

How it is taught

0:00 of 2:56

Subtracting a number is adding its additive inverse.

The additive inverse of a number is the number that sums with it to zero.

Read it: Adding positive and negative numbers

−5 + 7 starts at −5 on the number line and hops 7 to the right. The hop passes 0 and lands on 2, so −5 + 7 = 2.

  1. Turn any subtraction into adding the opposite: 3 − 5 is 3 + (−5).
  2. Pair any opposites first: a number and its opposite make 0.
  3. Add what is left on the line: hop right for a positive number, left for a negative.

When the two numbers have the same sign, the hop keeps going the same way: add the distances and keep the sign.

When the signs are opposite, the hop turns back: take the shorter distance from the longer, and the longer one's sign wins. Here 7 − 5 = 2.

Subtracting a number is the same as adding its opposite, and a number plus its opposite is 0.

Same signs: add the distances and keep the sign. Opposite signs: subtract the distances and keep the sign of the longer one.

Adding the distances and keeping the first sign: −5 + 7 = −12.

The hop of 7 goes right, back past 0: 7 − 5 = 2.

−12 − 11 = ___ — −23. Subtracting 11 adds its opposite: −12 + (−11) = −23.

3 − 7 = 3 + (−7) = −4

It's three degrees outside, and then the temperature drops seven degrees. What's the new temperature?

We can write that as three minus seven.

Here's the big idea: subtracting a number is the same as adding its opposite.

The opposite of seven — its additive inverse — is negative seven.

So three minus seven becomes three plus negative seven.

Let's see this on a number line.

Start at three.

Adding negative seven means jumping seven steps to the left.

It was 3°. Then it dropped 7°.What's the new temperature?3 − 7 = ?p − q = p + (−q)subtract → add the opposite+7 → −7additive inverse3 + (−7) = ?−7703+ (−7)−43 − 7 = −4ANSWER−4degrees
Grade 7 · Subtracting is the same as adding the opposite

We land on negative four.

So three minus seven equals negative four. Subtracting really is just adding the inverse!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics
Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal
1 star — one skill mastered, every rung of its ladder climbed
Climbing now
Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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