Add with pieces of two types

Try 2 tokens +1 and 5 tokens −1. Before clicking “Cancel pairs”, predict the balance.

Em telas pequenas, deslize o gráfico na horizontal para ler os detalhes.

Zero-valued pairs can be removed without changing the sum.

An account revealed little by little

  1. Write 2 + (−5).
  2. Form two pairs: each pair has a +1 and a −1 chip.
  3. Three tokens remain −1.
  4. Therefore, 2 + (−5) = −3.
Question to check Example
Did I confuse addition with the multiplication rule? −2 + (−3) = −5, while (−2) × (−3) = 6
Did I transform subtraction into the correct opposite? 2 − (−5) = 2 + 5
Is the result in the predicted position? 2 + (−5) ends to the left of zero

Compare, find the opposite and measure the distance

On the line, the number that is furthest to the right is the largest: −5 < −2 < 0 < 3. The numbers −4 and 4 are opposites: they are the same distance from zero, on different sides. Zero is its own opposite.

The module indicates the distance to zero: |−4| = |4| = 4 e |0| = 0. Distance is not negative; therefore, module and value of the number are different ideas.

Adding is applying a variation

Start at −3 and apply +5: you walk five units to the right and reach 2. Start at 2 and apply −4: you walk four units to the left and arrive at −2.

Em telas pequenas, deslize o gráfico na horizontal para ler os detalhes.

The dot sign indicates the position; the sign of variation indicates the direction of movement.

Subtracting is adding the opposite

To calculate a − b, we add a to the opposite of b:

  • 2 − 5 = 2 + (−5) = −3.
  • 2 − (−5) = 2 + 5 = 7.
  • −3 − (−3) = −3 + 3 = 0.

In the figure, try starting at 2 and applying a variation of −5. Then apply +5.

Multiply and divide

When multiplying and dividing two non-zero numbers, equal signs produce a positive result and different signs produce a negative result:

Operation Result
3 × (−4) −12
(−3) × (−4) 12
(−12) ÷ 3 −4
(−12) ÷ (−3) 4

One way to investigate the product of negatives is to observe a pattern: 3 × (−4) = −12, 2 × (−4) = −8, 1 × (−4) = −4, 0 × (−4) = 0. When decreasing the first factor by one unit, the result increases 4. Continuing, (−1) × (−4) = 4 and (−2) × (−4) = 8.

Attention: division by zero is not defined. Furthermore, an integer division can come out of ℤ: 5 ÷ 2 = 2,5. Addition, subtraction and multiplication of integers always produce integers.

Challenge of expressions

Calculate −4 + 7 − 5 and (−3) × 4 + 8. Record the multiplications first, if any.

Check out the strategies

−4 + 7 − 5 = 3 − 5 = −2. In the second expression, (−3) × 4 + 8 = −12 + 8 = −4.

Continue with the sign rule and the powers.