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Ratios and Data 6-8 / Week 08 / Tuesday
2/6
Week 08 · Multiplying and Dividing Signed Numbers

Tuesday

The sign rules
// Repeated hops and the sign rules
⏱ about 20 min

Tuesday: The Sign Rules

Rocket writes (-2) × (-3) on the whiteboard and stares at it. "A negative hop, a negative number of times. That makes no sense."

Raven writes a column instead: 3 × (-2), 2 × (-2), 1 × (-2), 0 × (-2). "Fill these in. What do you notice?"

Rocket writes -6, -4, -2, 0. "Each answer goes up by 2."

"Keep the pattern going," Nova says, her light sliding down the column. "What comes after 0 × (-2)?"

"(-1) × (-2)," Rocket says slowly. "And the pattern says... 2. Then (-2) × (-2) is 4."

He sits back. "A negative times a negative is positive. The pattern made me say it."

"Patterns are proof you can see," Raven says. "Now we can score every game both ways."

The pattern that settles it

Multiplication has rules that never break. One is that the pattern of products keeps going in even steps.

Look at the column. 3 × (-2) = -6 and 2 × (-2) = -4. Then 1 × (-2) = -2 and 0 × (-2) = 0.

Each step down adds 2. The next step must be (-1) × (-2) = 2, and then (-2) × (-2) = 4.

So a negative times a negative is positive. In particular, (-1) × (-1) = 1.

MultiplicationProductStep
3 × (-2)-6
2 × (-2)-4+2
1 × (-2)-2+2
0 × (-2)0+2
(-1) × (-2)2+2
(-2) × (-2)4+2

Here is a second reason, using the distributive property. Start with (-1) × (1 + (-1)).

Inside the parentheses, 1 + (-1) = 0, so the whole thing is (-1) × 0 = 0.

Distribute instead: (-1) × 1 + (-1) × (-1) = -1 + (-1) × (-1). This must also be 0.

The only number that makes -1 + ? = 0 true is 1. So (-1) × (-1) = 1.

The sign rules

  • Same signs give a positive answer: 4 × 3 = 12 and (-4) × (-3) = 12.
  • Different signs give a negative answer: 4 × (-3) = -12 and (-4) × 3 = -12.
  • Division follows the same two rules: -12 ÷ (-3) = 4 and -12 ÷ 3 = -4.
  • A negative fraction can wear its sign in three places: -(12/3) = (-12)/3 = 12/(-3). All three are -4.
  • You can never divide by 0. Every other quotient of two whole numbers is a rational number.
  1. Multiply or divide the sizes, ignoring the signs.
  2. Count the negative numbers. An even count gives a positive answer. An odd count gives a negative answer.
  3. Write the answer with its sign, and check it against the pattern or a hop.
SAME SIGNS, DIFFERENT SIGNS
  • Read the question.
  • Tap your answer.
(-4) × (-3) = ?
(-5) × 6 = ?
-24 ÷ (-6) = ?
18 ÷ (-3) = ?
WHICH SIGN?
  • Read the question.
  • Tap your answer.
Is the product of a negative number and a negative number positive or negative?
Is the product of a positive number and a negative number positive or negative?
Is the quotient of a negative number and a negative number positive or negative?
Is the quotient of a negative number and a positive number positive or negative?
THREE AT A TIME
  • Read the question.
  • Tap your answer.
(-2) × (-3) × (-1) = ?
(-1) × (-1) × (-1) × (-1) = ?
-7 ÷ 2 = ?
(-9) × 0.5 = ?
Number sentenceTrue or false?
(-2) × (-3) = -6?
(-1) × (-1) = 1?
-12 ÷ (-3) = 4?
-12 ÷ 3 = 4?
WHY THIS EXERCISEEach false row is a common slip. Naming them now stops them at Field Day.
Try it
Copy the pattern table into your Numbers Notebook, but use -5 instead of -2.
Start at 3 × (-5) and keep stepping down to (-3) × (-5). Does each step add 5?
Draw the pattern as dots on a number line from -6 to 4. Label each dot with its multiplication.

That is the hardest idea of the week, and you have it. Tomorrow you score the paper ball toss in the Numbers Lab.

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