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Remember, one way to think of a multiplication problem like “$2(3)$” is as “$2$ copies of $3$.” When you have a multiplication problem where only the second number is negative, like $2(-3)$, this is a good way to think about it. (That is, as finding “$2$ copies of $-3$.”)

Solve these multiplication problems, as pictured on the grid to the left. |

What if the first number in a multiplication problem is negative? You can see what to do in that case by following a pattern.

Solve these multiplication problems. If you’re not sure what to do, use the picture. |

Notice that each row of the table has one less copy of $3$ than the previous row. That is, each row has either $3$ less green squares or $3$ more pink squares than the row above it.

Look at the table from negTimesPos. If you take the answer in the first row ($6$) and subtract the answer in the second row ($3$), you get $3$.

What do you get if you take the answer in the second row and subtract the answer in the third row? |

What do you get if you take the answer in the fourth row and subtract the answer in the fifth row? |

What do you think you would get if you subtracted the answer in any row from the answer in the row above it? |

Solve these multiplication problems. |

Look at the table above. What is $-2(5)$? |

What is $2(5)$? |

Now, what is the opposite of $2(5)$? |

So following this pattern gives us a rule for multiplying any positive number by any negative number, in either order:

To multiply a positive number by a negative number, multiply the two numbers as though they were both positive, and then take the opposite.

Solve the following multiplication problems. |

What if you want to multiply two negative numbers?

Solve these multiplication problems. Notice that each problem has one fewer copy of $-2$ than the one before it. Use the picture if you need help. |

Look at the table above. What is $-3(-2)$? |

What is $3(2)$? |

Following these patterns tells you everything you need to know about multiplying with negative numbers:

- To multiply a positive and a negative number, multiply them as if they were both positive, and then take the opposite.
- To multiply two negative numbers, multiply them as if they were both positive.

Solve these multiplication problems. |

When you’re dividing a negative number by a positive number, it’s easiest to think of
division in terms of *splitting up*. That is, when you see something like “$${-12}/4$$”
or “$-12$ divided by $4$,” think of it as “$-12$ split into $4$ groups.”

What number is represented by each of the narrow pink rectangles that divides up the wide pink rectangle on the grid? (Remember that each pink square represents $-1$.) |

When you’re dividing a negative number by another negative number, it’s easiest to think of
division in terms of *counting out*. That is, when you see something like
“$${-18}/{-3}$$” or “$-18$ divided by $-3$,” think of it as “the number of
‘$-3$’s in $-18$.”

How many copies of each narrow pink rectangle are there in the wide pink rectangle on the grid? |

To divide a positive number by a negative number, you can again use the “pattern” method to figure out what you should do.

Solve these division problems which follow a pattern. Notice that each problem has one fewer copy of $-2$ than the one before it. |

Look at the table above. What is $$6/{-2}$$? |

Now, what is the opposite of $$6/2$$? |

As you can see, the rules for division with negative numbers are exactly the same as the rules for multiplication:

- If one number is positive and one is negative, divide them as if they were both positive, and then take the opposite to get your final answer.
- If both numbers are negative, divide them as if they were both positive.

Solve these division problems. |