Math Detours · Grade 7 · Sharpen · 1

Why minus times minus is plus

Not a rule to swallow. Follow the pattern down a times table past zero, or cancel a debt, and the sign has no choice.

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Tonight's idea

You have seen the pattern: \(2 \times (-2) = -4\), \(1 \times (-2) = -2\), \(0 \times (-2) = 0\). Each answer is 2 more, so \((-1) \times (-2)\) has to be \(2\). (That was Below zero.) Here are two more reasons.

Debts. A debt of $5 is \(-5\). Three debts make \(3 \times (-5) = -15\). Now a friend cancels three of your debts. Taking away three lots of \(-5\) is \((-3) \times (-5)\). You are $15 better off, so the answer is \(+15\).

73 82 7 × 8 is left 3 × 2 right strip3 × 10 = 30 bottom strip 10 × 2 = 20
Cut off both strips and the orange corner goes twice. Put it back once.

A rectangle. Work out \(7 \times 8\) as \((10 - 3) \times (10 - 2)\). Start with the 10 by 10 square: 100. Cut off the right strip, 30, and the bottom strip, 20. The corner has now gone twice, so put it back once: \(100 - 30 - 20 + 6 = 56\). Correct. That \(+6\) is \((-3) \times (-2)\). If it were \(-6\) you would get \(7 \times 8 = 44\).

Same signs multiply to a positive. Different signs multiply to a negative.

Watch one: \(19 \times 18\) in your head

Write it as \((20 - 1) \times (20 - 2)\). There are four pieces.

The square: \(20 \times 20 = 400\). The two strips: \(20 \times (-2) = -40\) and \((-1) \times 20 = -20\). The corner: \((-1) \times (-2) = +2\).

Total: \(400 - 40 - 20 + 2 = 342\).

Check another way: \(19 \times 18\) is \(19 \times 20\) take away \(19 \times 2\), which is \(380 - 38 = 342\). The corner had to be \(+2\). With \(-2\) the total would be 338, and that is wrong.

Your turn

  1. Work out: (a) \((-5) \times 3\); (b) \((-4) \times (-6)\); (c) \(7 \times (-8)\); (d) \((-9) \times (-9)\).
  2. A leaking tank loses 5 litres every hour. Write the change each hour as \(-5\). (a) Write a multiplication for the change over the next 4 hours, and work it out. (b) Compared with now, how much water was in the tank 3 hours ago? Write it as \((-3) \times (-5)\) and explain the sign in words.
  3. Use the four pieces, as in the example. (a) \(9 \times 8\) as \((10 - 1) \times (10 - 2)\). (b) \(98 \times 97\) as \((100 - 2) \times (100 - 3)\).
  4. Kim says \((-3) \times (-2)\) should be \(-6\). Everyone agrees that \((-3) \times 0 = 0\), and that \(0\) is the same as \(2 + (-2)\). So \((-3) \times 2\) and \((-3) \times (-2)\) must add up to 0. Show that Kim's answer breaks this, and find the only answer that works.

Stretch: work backwards

(a) Two numbers multiply to make \(12\) and add to make \(-7\). Find them. (b) Two numbers multiply to make \(-12\) and add to make \(1\). Find them. (c) Can a number multiplied by itself ever be negative? Say why.

Answers

  1. (a) \(-15\). (b) \(24\). (c) \(-56\). (d) \(81\).
  2. (a) \(4 \times (-5) = -20\): 20 litres less. (b) \((-3) \times (-5) = 15\): 15 litres more than now. Going back in time undoes a loss, and undoing a loss is a gain.
  3. (a) \(100 - 20 - 10 + 2 = 72\). (b) \(10{,}000 - 300 - 200 + 6 = 9{,}506\).
  4. \((-3) \times 2 = -6\). With Kim's answer the two parts add to \(-6 + (-6) = -12\), not 0. To reach 0 the second part must be \(+6\). So \((-3) \times (-2) = 6\), and nothing else will do.

Stretch. (a) The product is positive and the sum negative, so both are negative: \(-3\) and \(-4\). (b) The product is negative, so the signs differ: \(4\) and \(-3\). (c) No. A positive times itself is positive, a negative times itself is positive, and \(0 \times 0 = 0\).

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