Divide Integers

DIVIDE INTEGERS

INTRODUCTION

The objective for this lesson on Divide Integers is, the student will explore division with integers and apply this understanding to solve problems in mathematical and real world situations.

The skills students should have in order to help them in this lesson include, division of whole numbers, use of a number line and addition, subtraction and multiplication of integers.

We will have three essential questions that will be guiding our lesson. Number one, explain how to divide integers that have the same signs. Number two, explain how to divide integers that have different signs. And number three, how does division of integers compare to multiplication of integers? Justify your thinking.

Begin by completing the warm-up on basic division to prepare for this lesson on dividing integers.

SOLVE PROBLEM – INTRODUCTION

The SOLVE problem for this lesson is, the temperature in the northern part of Wyoming is very cold in the winter. One night the temperature had a decrease of thirty degrees in five hours. If the temperature fell at a steady rate throughout the five-hour period, what integer could be used to represent the average temperature loss every hour?

In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, what integer could be used to represent the average temperature loss every hour?

Now that we have identified the question, we need to put this question in our own words in the form of a statement. This problem is asking me to find the integer that represents the average temperature loss every hour.

During this lesson we will learn how to divide integers in order to complete this SOLVE problem at the end of the lesson.

DIVIDE INTEGERS – DISCOVERY ACTIVITY

Now we’re going to divide integers with a discovery activity.

What is the value of the yellow tile? Positive one

What is the value of the red tile? Negative one

Take a look at the wording for the division of the integer problems. Examples one and two use the following wording: If you divvy up blank into blank equal groups, what will be in each group? Answer: number and color of counters.

POSITIVE DIVIDED BY POSITIVE

Let’s look at integer division at the concrete level dividing a positive by a positive.

What is the wording for Problem one? If you divvy up positive six into three equal groups, what will be in each group?

How can you model the first value of positive six? Six yellow unit tiles

Let’s model how to divide the tiles into three equal groups by moving one tile at a time into three separate groups until all six tiles are divided.

How many yellow tiles are in each group? There are two.

Is our answer positive or negative? Our answer is positive. Explain your answer. There are positive two tiles in each group. The tiles are yellow.

POSITIVE DIVIDED BY POSITIVE

Let’s look at integer division at the concrete level with a negative value divided by a positive value.

What is the wording for Problem two? If you divvy up negative six into three equal groups, what will be in each group?

How can you model the first value of negative six? With six red unit tiles.

Let’s model how to divide the tiles into three equal groups by moving one tile at a time into three separate groups until all six tiles are divided.

How many red tiles are in each group? Two

Is our answer positive or negative? Negative. Explain your answer. There are negative two tiles in each group. The tiles are red.

NEGATVIE DIVIDED BY NEGATIVE

Integer at the concrete level with a negative divided by a negative.

Is it possible to have a negative number of groups? No Because we cannot have a negative number of groups, when we divide a negative by a negative, we use the following wording: Splitting up blank items into groups of blank, how may groups can your make?

Discuss the wording for Problem three. Splitting up negative six items into group of negative three, how many groups can you make? Let’s represent the first value with six red tiles. Then, we can move each individual group of negative three.

How many groups are there? Two

There are two groups, so the answer is two.

DIVIDE INTEGERS – PICTORIAL AND VERBAL

Use the letter Y to represent the yellow tiles. How many Y’s do we start with? Six – let’s draw the Y’s.

What is the wording for the problem? If you divvy up positive six into three equal groups, what will be in each group? What is our solution? Two yellow tiles which is equal to positive two.

What is the quotient of positive six divided by positive three? Positive two

What is the process in Problem one? In Problem one, we divvy up positive six into three equal groups.

In Problem one, when you divided the two positive values, what was the quotient? It was a positive two.

What conclusion can you draw about dividing two positive integers? When you divide two positive integers, the quotient will be positive.

Let’s look at our second example. Negative six divided by three. Use the letter R to represent the red tiles. How many R’s do we start with? Six – Let’s draw the R’s.

What is the wording for the problem? If you divvy up negative six into three equal groups, what will be in each group?

What is our solution? Two red tiles, which equal to negative two.

What is the quotient of negative six divided by positive three? Negative two

What is the process in Problem two? In Problem two, we divvy up negative six into three equal groups.

In Problem two, when you divided the negative and positive values, what was the quotient? It was a negative two. What conclusion can you drawn about dividing a negative and a positive integer? When you divide a negative and a positive integer, the quotient will be negative.

Negative six divided by negative three. Use the letter R to represent the red tiles. How many R’s do we start with? Six. Let’s draw the R’s.

What is the wording for the problem? Splitting up negative six items into groups of negative three, how many groups can you make? What is our solution? Two groups or positive two.

What is the quotient of negative six divided by negative three? Positive two

What is the process in Problem three? In Problem three, we divvy up negative six into groups of negative three.

In Problem three, when you divided a negative value by a negative value, what was the quotient? It was a positive two.

What conclusions can you draw about dividing a negative integer by a negative integer? When you divide a negative integer by a negative integer, the quotient will be a positive integer.

Positive six divided by negative three. Can we show a model of this problem? No

Explain your thinking. You cannot divide into negative groups, and you cannot find the number of negatives in a group of positives.

In Problem two, when you divided a positive value by a negative value, what was the quotient? It is a negative two.

What conclusion can you draw about dividing a positive value by a negative value? When you divide a positive value by a negative value, the quotient is negative.

THE RELATIONSHIP BETWEEN DIVISION AND MULTIPLICATION OF INTEGERS

Let’s look at our conclusions:

When you divide two integers that have like signs the answer is positive.

When you divide two integers that have different signs, the answer is negative.

Let’s look at Problem one, six divided by three.

What is Problem one asking you to find? If you divvy up positive six into three equal groups, what is in each group? What is the quotient of Problem one? Two

What is the opposite operation of division? Multiplication What is the multiplication problem that corresponds to the division fact for Problem one? Three times two equals six.

What is the multiplication problem asking us to find? Three groups of two items, which is six items.

How are the division and multiplication problems similar? The signs of the corresponding parts of the two equations are the same.

How are the division and multiplication problems different? One is multiplication, one is division.

What is Problem two asking you to find? If you divvy up negative six into three equal groups, what is in each group?

What is the quotient of Problem Two? Negative two

What is the opposite operation of division? Multiplication

What is the multiplication problem that corresponds to the division fact for Problem two? Three times negative two equals six.

What is the multiplication problem asking us to find? Three groups of negative two items, which is negative six items.

How are the division and multiplication problems different? One is multiplication, one is division

What is Problem three asking you to find? Splitting up negative six into groups of negative three, how many groups can you make?

What is the quotient of Problem three?

What is the opposite operation of division? Multiplication

What is the multiplication problem that corresponds to the division fact for Problem three? Two times negative three is equal to negative six.

What is the multiplication problem asking us to find? Two groups of negative three items, which is negative six items.

How are the division and multiplication problems similar? The signs of the corresponding parts of the two equations are the same. How are the division and multiplication problems different? One is multiplication, one is division

Is there a model for Problem four? No

Based on Problems one through three, what is the corresponding multiplication problem? Negative two times negative two is equal to positive six.

How are the division and multiplication problems different? One is multiplication, one is division

What is the conclusion you can make about the integer rules for multiplication and division? The same rules apply for division and multiplication of integers.

DIVIDING INTEGERS PROPERTIES AND REAL WORLD CONTEXT

Identify the two integers in the first row. Eighteen and six

Explain what the negative sign outside of the parentheses mean. It means that after determining the quotient, it will be multiplied by a negative, which will make the quotient a negative three.

Looking at the second column, what is different about this problem than the problem in the first column? The negative sign is in the numerator or the dividend.

Do you think this will affect the quotient? Explain your thinking. No, because the rules of integers tell us that if the signs are different, then the quotient will be negative value.

Looking at the third column, what is different about this problem than the problem in the second column? The negative sign is in the denominator or the divisor.

Will that impact the quotient? Explain your thinking. No, because the rules of integers tell us that in division if the signs are different, the answer will be negative.

What conclusions can we make about the three representations? All three expressions have the same quotient and so they are equivalent.

Complete the table for the next three rows.

Conclusion is: when dividing two rational numbers, if the quotient is negative, then the expression can be written with the negative value in the numerator or the denominator.

Write the quotients for Problems one and two. Forty five divided by fifteen is equal to three.

Negative thirty six divided by twelve is equal to negative three.

Discuss scenarios for the first equation. Here is a sample answer: There are forty five students going on the field trip. Each school van will fit fifteen students.

What scenarios did you create for Problems one and two?

Let’s look at this real world problem. The seventh grade paid one thousand two hundred dollars for t-shirts to sell for a fundraiser. If they bought three hundred t-shirts, what integer represents the amount paid per shirt?

Write an equation to represent the scenario. We have negative one thousand two hundred divided by three hundred is equal to negative four. The amount paid out is represented by a negative integer, the number of t-shirts by a positive integer, the cost per shirt paid out again represented by a negative integer.

Alicia owes her mom twenty dollars. If she pays her back four dollars per week, how many weeks will it take her to pay back the twenty dollars?

Write an equation to represent the scenario. Negative twenty which represents the amount she owes her mom divided by negative four, which represent the amount she can pay per week is equal to five, which represents the number of weeks.

INTEGER RULES

Let’s look at the integer rules for division.

Two integers with the same signs, and two integers with different signs.

Take a look back at the problems we completed during the lesson.

Completing the sentences in the box, let’s identify the rules for integer division.

When we have two integers with the same signs: we divide and the answer is positive.

When we have two integers with different signs: we divide and the answer is negative.

DIVIDING INTEGERS FOLDABLE

We have already completed the additions, subtraction and multiplication sections.

Inside, on the right side of the fourth flap, write the rules for dividing integers. On the left side, write examples for dividing integers. SOLVE PROBLEM – COMPLETION

We are now going to go back to the SOLVE problem from the beginning of the lesson. The question was, the temperature in the northern part of Wyoming is very cold in the winter. One night the temperature had a decrease of thirty degrees in five hours. If the temperature fell at a steady rate throughout the five-hour period, what integer could be used to represent the average temperature loss every hour?

We completed the S Step by Studying the Problem. We underlined the question and we completed this statement. The problem is asking me to find the integer that represents the average temperature loss every hour.

In the O Step, we Organize the Facts. We first begin by identifying the facts. We go back to our original problem and we place a vertical line or a strike mark at the end of each fact. The temperature in the northern part of Wyoming is very cold in the winter./ One night the temperature had a decrease of thirty degrees/ in five hours./ If the temperature fell at a steady rate throughout the five hour period,/ what integer could be used to represent the average temperature loss every hour?

After we identify the facts we eliminate the unnecessary facts. In this problem we can eliminate the first fact, the temperature in the northern part of Wyoming is very cold in the winter. Because we do not need that information in order to determine the integer to represent the average temperature loss per hour.

Then we list our necessary facts. We have a decrease of thirty degrees, which is represented by negative thirty. Our time period is five hours.

STOP! What are we trying to find? An integer that can be use to represent the average temperature loss per hour. What does this mean? We need to divide the total loss by the number of hours.

What integer can we use to represent the total change in temperature? Negative thirty

What integer can we use to represent the number of hours? Five

What does this mean? We will be dividing a negative value by a positive value.

How do we divide integers? Explain your thinking. Divide the values. If the signs of the dividend and the divisor are the same, the sign of the quotient is positive. If the signs of the dividend and the divisor are different, the sign of the quotient is negative.

Let’s move to our L Step, Line Up a Plan. Write in words what your plan of action will be. Divide the total decrease in temperature by the number of hours.

Choose an operation or operations. Division Verify Your Plan with Action. First estimate your answer. The estimate we have is a loss of about five degrees per hour.

Carry out your plan. Negative thirty divided by five is equal to negative six degrees.

E, Examine Your Results.

Does your answer make sense? Compare your answer to the question. Yes, because we are looking for the loss of temperature per hour.

Is your answer reasonable? Compare your answer to the estimate. Yes, it is close to our estimate of about five degrees per hour.

Is your answer accurate? Check your work. Yes.

Write your answer in a complete sentence. The integer that represents the loss of degrees per hour is negative six.

CLOSURE

Let’s go back and discuss the essential question from this lesson. Our first question was, explain how to divide integers that have the same signs. We divide, and our answer will be positive.

Number two was, explain how to divide integers that have different signs. We divide, and our answer will be negative.

Number three, how does the division of integers compare to multiplication of integers? Justify your thinking. When the signs are the same, the answer is positive. When the signs are different the answer is negative. These rules apply to both multiplication and division of integers.