Guided Practice

When a whole number is multiplied by a negative number, you can think of it as repeated subtraction. For example:

Another way to write 3 × –4 is 0 – 4 – 4 – 4 = –12.

Try out some problems like this on the right.

–5 × 3 =

–8 × 5 =

–1 × 9 =

–7 × 7 =

When you multiply positive and negative whole numbers, you can think of the product as so many groups of negatives. For example, 2 × –5 can be represented as 2 groups of –5.

The model shows 2 × –5, which is equal to –10.

The product 2 × –5 is also equal to 2 × –1 × 5. And, using the Commutative Property, this can be rewritten as –1 × 2 × 5 or –2 × 5.

Thus, a × –b = –a × b.

You can multiply a positive and negative whole number by using a number line.

For example, –9 × 3 = –27.

You can also add to show this product:

–9 + –9 + –9 = –27.

The product of two numbers, a and b, with the same sign is positive: a × b = –a × –b = ab.

4 × 7 = 28

–4 × –7 = 28

The product of two numbers, a and b, with different signs is negative: –a × b = a × –b = –ab.

–3 × 6 = –18

3 × –6 = –18

You can think of 3 × –4 as groups

of .

You can write 3 × –4 using repeated addition as

+ + .

Using the Zero Property, you can see that

–3 × (–4 + 4) = .

The Distributive Property tells us that

–3 × (–4 + 4) = × –4 + –3 × .

So, –3 × –4 + = 0. This means that

–3 × –4 must be equal to 12.

The model shows 5 groups of –1. Write the product represented by the model.

× =

The beginning and end of the video show two equal products. Complete each true equation.

3 × –4 = 4 ×

a × –b = a × × b

Write the product represented by the model.

× =

Write the product represented by the model.

× =

Write the product represented by the model.

× =

Determine the product. Draw on the blank number line to help you.

–8 × 8 =

Determine the product. Draw on the blank number line to help you.

5 × –4 =

Determine the product. Draw on the blank number line to help you.

6 × –8 =

Determine the product. Draw on the blank number line to help you.

–4 × 2 =

Determine the product. Draw on the blank number line to help you.

5 × –5 =

Determine each product.

–3 × –3 =

–3 ×

3 =Determine each product.

–4 × 9 =

–9 × –4 =

Determine the product.

–10 × –0.5 =

Determine the product.

1/3 × –9 =

Determine the product. Write the product as a decimal.

–0.6 × –0.1 =

The distance between ANY two numbers on a number line is the absolute value of their difference. For example:

|–2 – 4| = |–6| = 6

Remember, the absolute value | | makes any expression positive.

So, the distance between 2 and –4 on the number line is 6.

Determine the distance between the numbers.

The distance between –5 and 5 is .

Determine the distance between the numbers. Write your answer as a fraction with a slash (/) in lowest terms.

The distance between 3/4 and –0.1 is .

Determine the distance between the numbers. Write your answer as a decimal.

The distance between –0.45 and –0.99 is .

Determine the distance between the numbers.

The distance between 6 and –24 is .

Determine the distance between the numbers. Write your answer as a fraction with a slash (/) in lowest terms.

The distance between –9/10 and –1/2 is .

The operations of multiplication and division are inverse operations. So, if you know how to multiply with two negative numbers, you also know how to divide with them.

–9 × –4 = 36, so

36 ÷ –4 = –9

36 ÷ –9 = –4

You can use the same rules that you used to multiply two numbers. If two numbers have the same sign, then their quotient is positive. If two numbers have different signs, then their quotient is negative.

–36 ÷ 9 = –4

–36 ÷ –9 = 4

–3 × –7 = 21, so

21 ÷ –3 = , and 21 ÷ –7 = .

Enter each quotient.

–20 ÷ 4 =

–20 ÷ –2 =

Enter each quotient.

–28 ÷ 7 =

–28 ÷ –4 =

Enter each quotient.

56 ÷ –7 =

–56 ÷ –4 =

Enter each quotient.

63 ÷ 7 =

–63 ÷ 21 =

Enter each quotient.

–81 ÷ –9 =

81 ÷ –3 =

Enter each quotient.

–6 ÷ –2 =

–6 ÷ –3 =

Enter each quotient.

–3/4 ÷ 1/4 =

1/2 ÷ –1/2 =

Enter each quotient.

–2/3 ÷ 4/3 =

2/5 ÷ –1/2 =

Enter each quotient.

–3/8 ÷ 4/5 =

3/4 ÷ –5/6 =

clear

For every $5 you gain, Bo gains $3. Represent this as 5→ 3↑ on the graphing tool on the left. (Click on the arrows to rotate them. Click on the coordinate plane to show the result.)

Your gains are measured on the -axis. Bo's gains are measured on the -axis.

If you gain $15, then Bo gains $.

The amount that Bo gains is the amount you gain times .

For every –$5 you gain (a loss of $5), Bo gains

–$3 (loses $3). Press clear. Then represent this situation as 5← 3↓ on the graphing tool.

If you "gain" –$15, then Bo "gains" $.

The amount that Bo "gains" is the amount you "gain" times .

The points you plotted for the gaining and losing situations all lie on the same line. The slope of this line is the ratio of the change in y to the change in x.

The slope of the line is / .

For every –$6 you gain (a loss of $6), Bo gains +$8. Represent this situation on the graphing tool as 6← 8↑, measuring your "gains" on the x-axis and Bo's gains on the y-axis.

If you gain –$12, Bo gains $.

The amount Bo gains is the amount you gain times .

The slope of the line is / .

A submarine changes its depth by –6 meters every 3 seconds. Represent this situation on the graphing tool as 3→ 6↓, measuring time in seconds on the x-axis and elevation in meters on the y-axis.

The elevation of the sub in meters is the time in seconds times .

The slope of the line is / .

For every 2 shirts you sell, you make $2. Represent this situation on the graphing tool as 2→ 2↑, measuring shirts sold on the x-axis and dollars made on the y-axis.

The money made is the number of shirts sold times .

The slope of the line is / .

For every $5 a store takes off of a price, the number of customers who buy it goes up by 4. Represent this situation on the graphing tool as 5← 4↑, measuring dollars off on the x-axis and increase in customers on the y-axis.

The increase in customers is the change in the price times .

The slope of the line is / .

For every 7 miles a car drives, it spends 1 quart of fuel. Represent this situation on the graphing tool as 7→ 1↓, measuring miles on the x-axis and fuel on the y-axis.

The change in fuel used is the number of miles driven times .

The slope of the line is / .

For every 5 days the senator did not spend campaigning, she lost 8 votes. Represent this situation on the graphing tool as 5← 8↓, measuring days on the x-axis and change in votes on the y-axis.

The change in votes is the change in campaign days times .

The slope of the line is / .