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Pre-Algebra View details

6. Multiply and Divide Integers

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Chapter 1

6.

Multiply and Divide Integers

Understanding integers, especially in the context of multiplication and division, is essential for various real-world applications. For instance, negative numbers often appear in financial calculations, such as debt or temperature measurements like below-zero weather conditions. Multiplication and division of integers are not just academic exercises; they are tools that can help in planning budgets, analyzing data, and even in coding algorithms for computer programs. The lesson explains these concepts in a straightforward manner, making it easier to grasp the logic behind the operations and their practical uses.

Student Learning Objectives:
Multiply and divide integers with the same sign Multiply and divide integers with different signs

	12 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Multiply and Divide Integers

Slide of 12

If an ice cream cone costs $3, how many ice cream cones could be bought with $12? How much money is needed to buy ten milkshakes that cost $4 each? The answers to these questions can be found by using division and multiplication, respectively. This lesson will explore how these operations are useful for finding the answers to many similar daily situations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

What is Multiplication?

Multiplying a positive integer a by a positive integer b is the same as adding b to itself a times.

Showing that a multiplication is a repeated sum

Consider the following questions!

What happens if b is negative?
What happens if both a and b are negative?

Discussion

Multiplying Integer Numbers

Multiplication is the same as repeated addition. However, when multiplying integers, the signs of the factors determine whether the product is positive or negative.

Method

Multiplying Integers With the Same Sign

The product of two integers is always positive if and only if the factors have the same sign. This means that when the factors are both positive or both negative, ignore the signs and multiply them as if they were whole numbers. Consider the following multiplication of integers. -3*(-4) There are two steps to follow when the factors have the same sign.

1

Verify the Signs of the Factors

If both factors are positive, keep the numbers as they are. On the other hand, if the factors are both negative, ignore the signs of the factors to reduce the multiplication to a multiplication of two positive integers. In this example, the signs of the integers will be removed because both are negative. -3*(-4) ⇕ 3*4 The expression is now a multiplication of two whole numbers.

2

Find the Product Using a Number Line

The first factor indicates how many intervals will be needed to find the product. The second factor indicates the size of each interval. The intervals are drawn on a number line starting from zero. In this case, 3* 4 indicates that three equal intervals of length 4 are needed.

Multiplication of 3x4 on a number line

The endpoint represents the product of the given multiplication of integers with the same sign. 3*4=12 ⇔ -3*(-4)=12 When integers with the same sign are multiplied, the result is always positive.

Discussion

Multiplying Integers With Different Sings

The product of two integers is always negative if and only if one factor is negative and the other is positive. Change the negative factor to its opposite and perform the multiplication as if both were whole numbers. Next, change the result to its opposite to get the final product. Consider the following multiplication of integers. -7*2 Follow these three steps to find the product of two integers with different signs.

1

Check the Signs of the Factors

Verify that one factor is positive and the other is negative. Change the negative factor to its opposite. The result is a multiplication of two positive integers. For the given example, the first factor is negative 7, so change it to its opposite 7. -7*2 ⇒ 7*2

2

Find the Product of the Resulting Multiplication

Next, find the product of the multiplication of two whole numbers. In this case, 7*2 means that seven equal intervals of length 2 are needed. Use a number line to find this product.

Multiplication of 7x2 on a number line.

The endpoint is the product of the resulting multiplication. This means that 7*2=14.

3

Change the Product to Its Opposite

Change the product from the previous step to its opposite to get the product of the initial multiplication of integers with different signs. For the given example, the product of whole numbers is 14. Its opposite is -14. 7*2= 14 ⇒ -7*2= -14 In summary, multiply two integers with different signs as if they were whole numbers and change the result to its opposite to get the final product.

Example

The Piano Girl

LaShay is going to perform at her school's piano recital. She wants to calculate how long it would take her to play her part.

She has a total of 8 songs to play and each song is an average of 3 minutes long. About how long will it take her to play all the songs?

Hint

To find the total time, multiply the number of songs by the average length of a song.

Solution

LaShay is performing 8 songs with an average time of 3 minutes per one song. We multiply the number of songs by the average length of one song to find out how much time she needs. Total Time (min) = 8* 3 This is a multiplication of two positive integers. We can find it using a number line. The factors are 8 and 3, so we need 8 intervals of length 3.

The endpoint is 24, so LaShay will play for 24 minutes. Total Time (min)= 8* 3 = 24

Example

Auntie's Pizza Shop

In the afternoons, LaShay often helps her aunt with her pizza shop. LaShay's aunt pays her $100 for each day she helps her. However, every day LaShay is late, she loses $30.

a

LaShay helped her aunt for 15 days last month, but she was late 6 days. How much money did LaShay lose last month? Give the answer as a negative integer.

b

How much money did LaShay earn at the pizza shop last week?

Hint

a

Multiply the number of days she was late by the amount of money she lost each day for being late. Use a number line to find the product.

b

Multiply the days worked by the amount paid per day. Add the amount from Part A to this product.

Solution

a

A loss means something negative. A loss of $30 can be represented by - 30. Since LaShay was late 6 out of the 15 days, we multiply 6 by -30 to find her total loss.

6* ( -30) We are multiplying a positive integer by a negative integer, so the product is negative. Let's ignore the signs for now and multiply the numbers as if they were whole numbers. Visualize this as 6 equal intervals of 30 on a number line.

The product of 6 and 30 is 180. Finally, we change 180 to its opposite, - 180, because we know that the product should be negative. 6 * ( - 30) = - 180 LaShay lost $180 last month.

b

Let's start by finding how much LaShay could earn if she was on time every day. Her daily pay is $100 and she worked for 15 days, so multiply these two numbers. Since we are multiplying two positive integers, the result will be positive.

She could earn $1500. In Part A, we found that her loss for being late was - $180. Her pay is the sum of these integers. 1500+(-180) We can simplify this expression by rewriting the addition of a negative as the subtraction of a positive. 1500 - 180 Finally, we calculate the difference. 1500-180=1320 LaShay earned $1320 at the pizza shop.

Discussion

Dividing Integer Numbers

The division of integer numbers is similar to the division of whole numbers. However, the signs of the dividend and the divisor determine the sign of the quotient.

Method

Dividing Integers With the Same Sign

The quotient of two nonzero integers a and b is positive if a and b have the same sign. This will be illustrated using the following division of integers. -15÷(-3) Follow these two steps when dividing two integers with the same sign.

1

Verify the Signs of the Dividend and the Divisor

If the dividend and the divisor are positive, keep the numbers as they are. If the dividend and the divisor are negative, ignore their sings to reduce the division to a division of two positive integers. In the given example, the signs of the integers will be removed because both are negative. -15÷(-3) ⇔ 15÷ 3 The expression is now a division of two whole numbers.

2

Find the Quotient Using a Number Line

The quotient can be found by moving to the right starting from zero on a number line. Use intervals the size of the divisor until the dividend is reached. The number of jumps needed to reach the dividend equals the quotient. In this case, move to the right of zero in intervals of 3 since the divisor is 3.

Division of 15/3 on a number line

It took 5 jumps of 3 units to reach 15. This means that the quotient of the initial division is 5. 15÷ 3=5 ⇔ -15÷(-3)=5 The quotient of two numbers with the same sign is always positive.

Discussion

Dividing Integers With Different Sings

The quotient of two nonzero integers is always negative if and only if one integer is negative and the other is positive. Change the negative number to its opposite and perform the division as if they were whole numbers. Next, change the result to its opposite to get the quotient of the initial division. This process will be illustrated with the following division. -12÷4 There are three steps to follow to find the quotient when dividing two numbers with different signs.

1

Check the Signs of the Integers

Identify which of the numbers involved in the division is negative. Next, change it to its opposite to get a division of two positive integers. In the given example, the dividend -12 is negative. Its opposite is 12. -12÷4 ⇒ 12÷4

2

Find the Quotient of the Resulting Division

Find the quotient of the division of two whole numbers. For this example, 12÷4 means how many jumps of 4 are needed to reach 12 on a number line, starting from zero.

Division of 12/4 on a number line.

It took three jumps of 4 units to reach 12. This means that 12÷4=3.

3

Change the Quotient to Its Opposite

Change the quotient found in the previous step to its opposite to get the quotient of the initial division of integers with different signs. For the given example, the quotient is 3. Its opposite is -3, so the quotient of - 12 ÷ 4 is - 3. 12÷4= 3 ⇒ -12÷4= -3 In summary, divide two integers with different signs as if they were whole numbers and change the result to its opposite to get the final quotient.

Example

Waiting for the Ferris Wheel

Jordan and her friends are waiting to ride the Ferris wheel at a carnival.

They are at the end of a line of 36 people waiting for the ride. Each car can hold 4 people at a time. How many cars will pass before Jordan and her friends can ride?

Hint

Divide the number of people in line by the number of people that can ride in a car.

Solution

There are 36 people before Jordan and her friends and each car can fit 4 people. Let's divide 36 by 4 to find how many cars will pass before they can ride the Ferris wheel. 36 ÷ 4 We are dividing two positive integers. On the number line, we move from 0 to 36 in steps of 4.

Jordan and her friends need to wait for 9 cars.

Example

The Skiing Competition

Heichi took place in a skiing competition.

He made 6 mistakes and lost 96 points. If each mistake was worth the same number of points, how much was each mistake worth? Give the answer as a negative integer.

Hint

Divide the number of points lost by the number of mistakes made.

Solution

Heichi lost 96 points for 6 mistakes, each worth the same number of points. Since a loss can be represented by a negative number, we write the total loss as -96. To find the points per mistake, we divide -96 by 6. -96÷ 6 We are dividing a negative integer by a positive integer, so the result is negative. First, let's ignore the signs and divide the numbers as if they were whole numbers using long division.

The quotient of 96 and 6 is 16. Finally, we change 16 to its opposite, - 16, because we know that the quotient should be negative. - 96 ÷ 6 = - 16 Heichi lost 16 points for each mistake.

Pop Quiz

Multiplying and Dividing Random Integers

Use the rules for multiplying and dividing integers to find the result of the given question. Remember, the quotient or product is always positive if the two integers have the same sign. If not, it is always negative.

Random question generator of division and multiplication of integers

Closure

Facts About Negative Numbers

Negative numbers have been used for over 2000 years. Brahmagupta wrote some of the first rules for working with them in the 7th century, but people did not use negative numbers to solve equations until the 16th century.

Historical facts about negative numbers on a number line.

Negative numbers were controversial at first. Some mathematicians did not like the idea of quantities less than 0, while others thought negative numbers were imaginary or even evil.

Despite the early controversies, negative numbers are very useful in math, science, and technology. They help us measure things like temperature and debt and describe relationships between numbers.

Multiply and Divide Integers

Exercise 2.1

If Dominika reads 17 pages of a book every day, how many pages will she have read after 11 days?

We want to calculate the number of pages Dominika will have read after 11 days. We are told that she reads 17 pages each day. We can find this number by multiplying the number of pages read per day by the number of days. Number of Pages Read = 17 * 11 This is a multiplication of two positive integers, so the product will also be positive. Let's use digit by digit multiplication to find this result.

The product is 187. Dominika will have read 187 pages after 11 days if she reads 17 pages each day.

Magdalena burns 650 calories by running for one hour. She ran for six hours this week. How many calories did she burn by running this week?

We want to find how many calories Magdalena burned by running this week. The total calories burned is given by the product of the calories burned per hour and the number of hours run. Calories Burned ⇕ Calories Burned per Hour*Hours Run We are told that she ran for a total of 6 hours this week while burning 650 calories per hour. Let's plug this information into our expression. Calories Burned = 650 * 6 This situation represents a multiplication of two positive integers. The product will also be positive. Let's find it by using digit by digit multiplication.

The product is 3900. This means that Magdalena burned 3900 calories this week by running.

Emily ran 816 meters in 24 minutes. What was her average running speed in meters per minute?

We want to calculate Emily's average speed. We can do this by dividing the distance she ran by the time it took her to run the distance. Speed = Distance÷Time In this case, Emily ran 816 meters in 24 minutes. Let's plug these values into the expression we wrote previously. Speed = 816 ÷ 24 We have a division of two positive integers. This means that the quotient of this division will also be positive. We can use long division to find this quotient.

The quotient is 34. This means that on average, Emily ran 34 meters per minute.

Kriz is a rock climber learning to rappel down the face of a granite rock formation. On an outing with a climbing guide, Kriz descends the rock face in five equal descents.

The total height of the rock formation is 240 feet. What integer represents Kriz's change in altitude in feet each time they descend?

We are asked to calculate the integer that represents Kriz's change in altitude in feet each time they descend. We can find this integer by dividing the total distance descended by the number of rappels down. Change in Altitude = Distance÷ Number of Rappels Note that the elevation of the boulder is 240 feet. This is the total distance Kriz descended. However, because they went down, we will represent this distance as negative 240. In addition, the number of rappels down is 5. Change in Altitude = - 240 ÷ 5 We wrote a division of two integers with different signs, so the quotient must be negative. We can begin by finding the quotient ignoring the signs of the numbers and putting the negative sign in the result at the end. Let's use long division to find this number.

The quotient of this division is 48. Remember to change it to its opposite -48 to get the result of the initial division. This means that the integer representing Kriz's change in altitude in feet each time they descend is -48.

Multiply and Divide Integers

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