1.01a - Multiplying and dividing integers

Signs of products

If a game removes a penalty of six points four times, does your score fall or rise? We need to distinguish four negative changes from reversing four negative changes.

You already know that $4\times(-6)$ means four lots of negative six:

Multiplication can be written in either order, so $(-6)\times4=-24$ too. To understand a negative multiplier, keep the pattern consistent. As the multiplier falls by one, the product increases by six:

This explains the positive result for two negative factors. It also keeps multiplication consistent with addition: multiplying $(-4)+4=0$ by $-6$ must give zero, so the two products must be opposites.

Find the size using positive multiplication, then attach the sign justified by the factors.

Now change only the sign of the second factor. The size stays the same; the result reverses.

The opening game question now has an answer. Four penalties of six points give a change of $-24$; removing those four penalties gives a change of $24$.

Use both the signs and the size in this new product.

Written multiplication

For a larger product, separate the sign decision from the written arithmetic.

To calculate $243\times16$, split $16$ into $10+6$. Every one of the $243$ groups contains both portions, so the partial products must be added. This is why a tens row represents ten times the digit, rather than the digit alone.

Change only the tens digit of the multiplier, from $16$ to $26$. The ones contribution stays $1458$.

Writing the partial products in columns is equally valid. Line up units with units, tens with tens and hundreds with hundreds.

Use the same method with a new three-digit factor.

Signs of quotients

Division asks which number, multiplied by the divisor, gives the original number.

For example, $(-24)\div6$ asks “What times six is negative twenty-four?” The answer is $-4$, since $(-4)\times6=-24$. Similarly, $(-24)\div(-6)=4$, since $4\times(-6)=-24$. The multiplication rule therefore explains the division rule.

Keep the order: $a\div b$ asks what must multiply $b$ to give $a$.

Change only the sign of the divisor. A positive quotient is now needed to make a negative product.

Division order is not interchangeable in the way multiplication order is.

Find the new quotient without changing the order.

Written division and zero

Written division works through place values, carrying any remainder to the next smaller place.

In $1728\div6$, the initial one thousand cannot form a whole thousand in each of six groups. Combine it with the seven hundreds to make seventeen hundreds. Dividing those gives two hundreds per group and five hundreds left. Those five hundreds become fifty tens, which join the two tens to make fifty-two tens.

Change only the divisor's sign. The place-value division does not change.

A zero inside an answer still has a place. For instance, $1224\div6=204$: after dividing twelve hundreds, the two tens are too few for one ten per group, so write zero tens and carry them into the twenty-four ones.

Zero as the number being divided is allowed: $0\div(-6)=0$, because $0\times(-6)=0$. Zero as the divisor is different. There is no answer to $6\div0$, because no number multiplied by zero gives six. For $0\div0$, every number passes the multiplication check, so there is no unique quotient. Division by zero is undefined in both cases.

Show the division method and keep the zero in its correct role.

Quick Recap

• For two nonzero factors or division inputs, equal signs give a positive result and different signs give a negative result.
• Use positive-number written arithmetic for the size, then justify the sign.
• Partial products and division remainders must retain their place values.
• Zero divided by a nonzero number is zero; division by zero is undefined.

Removing four penalties of six points raises the score by $24$: $(-6)\times(-4)=24$.