Comparing, Ordering, and Absolute Value of Rational Numbers

COMPARING, ORDERING, AND ABSOLUTE VALUE OF RATIONAL NUMBERS

INTRODUCTION

The objective for this lesson on Comparing, Ordering, and Absolute Value of Rational Numbers is, the student will understand, describe, order, and compare positive and negative rational numbers as well as numbers with absolute value.

The skills students should have in order to help them in this lesson include, opposites and comparing positive rational numbers.

We will have three essential questions that will be guiding our lesson. Number One, when looking at points plotted on a number line, how can you tell which is the larger number? Explain your thinking. Number Two, describe what you are finding when identifying the absolute value of a number. And Number Three, when calculating the absolute value of a number, what will be the final sign on the value? Explain.

Begin by completing the warm-up for this lesson on comparing positive rational numbers, to prepare for the lesson on comparing, ordering, and absolute value of rational numbers.

SOLVE PROBLEM – INTRODUCTION – POSITIVE AND NEGATIVE INTEGERS

The SOLVE problem for the first part of this lesson is on positive and negative integers.

Lanita signs into her online banking account to view her statement and sees two new transactions. The first transaction listed is a negative forty five dollars from Tuesday. The second transaction is posted as a positive three thousand five hundred dollars on Friday. What could each of these transactions represent? Be as specific as possible in your description.

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 could each of these transactions represent? Be as specific as possible in your description.

After 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 specific transactions that occurred as related to the two numbers given.

During this first part of the lesson we will be reviewing positive and negative integers in order to complete this SOLVE problem at the end of the first section of the lesson.

REVIEW OF PLOTTING POSITIVE AD NEGATIVE INTEGERS

What do you notice about the numbers that are listed? They are both positive and negative integers.

What does the negative sign tell us when plotting on the number line? The negative sign tells us that we are moving to the left of zero to plot the point.

What does the positive sign tell us when plotting on a number line? The positive sign tells us that we are moving to the right of zero to plot the point.

As we move farther to the right of zero, what happens to the numbers? They become larger.

As we move farther to the left of zero, what happens to the numbers? They become smaller.

What is the first number in our list? Negative seven

How do we plot this number? The negative tells us the point is to the left of zero, and the seven tells us to move seven units.

What is the next number in our list? One

How do we plot this number? The number is positive, which tell us the point is to the right of zero, and the one tells us to move one unit.

Let’s plot the rest of the points. Positive four, negative four, positive six, positive nine, and negative two.

SOLVE PROBLEM – COMPLETION

Now let’s go back and complete the SOLVE problem from the beginning of the lesson. The question was, Lanita signs into her online banking account to view her statement and sees two new transactions. The first transaction listed is negative forty five dollars from Tuesday. The second transaction is posted as positive three thousand five hundred dollars on Friday. What could each of these transactions represent? Be as specific as possible in your description.

In the S Step, we Study the Problem. Underline the question and complete this statement. This problem is asking me to find the specific transactions that occurred as related to the two numbers given.

In the O Step, we Organize the Facts. First we identify the facts. We go back to our original question and read each sentence and make a vertical line or strike mark after each fact. Lanita signs into her online banking account to view her statement and sees two new transactions. / The first transaction listed is negative forty-five dollars from Tuesday. / The second transaction is posted as positive three thousand five hundred dollars on Friday. /

The next step in O is to eliminate the unnecessary facts. We strike through any fact that does not contain information that will help us determine what each of the transactions represent.

Then we list the necessary facts. The first transaction is negative forty five dollars. The second transaction is a positive three thousand five hundred dollars.

In the L Step, we Line Up a Plan. Write in words what your plan of action will be. Identify and describe the meaning of the values based on the signs and their potential meanings in the context of the problem.

Then we choose an operation or operations. In this particular problem we do not have a specific operation.

V, Verify Your Plan with Action. First estimate your answer. One will be a deposit and the other a type of withdrawal.

Carry out your plan. The first transaction was a negative forty five dollars. With the sign being negative, it must represent a withdrawal or deduction that was made to pay a bill. Therefore, we can describe this transaction as either an ATM withdrawal in the amount of forty five dollars or that Lanita wrote a check to pay a bill, maybe a bill for her mobile phone. The second transactions was a positive three thousand five hundred dollars. The sign for this transaction is positive, and it is a rather large sum, so we might assume that it represents the deposit of a paycheck into her account for her earnings.

E, Examine Your Results.

Does your answer make sense? Compare your answer to the question. Yes, we found the specific transactions that occurred based on the numbers given.

Is your answer reasonable? Compare your answer to the estimate. Yes, the first is a withdrawal, and the second is a deposit.

Is your answer accurate? Check your work. Yes.

Write your answer in a complete sentence. The negative forty five dollars represents an ATM withdrawal or a mobile phone bill payment, while the positive three thousand five hundred dollars represents the deposit of a paycheck.

Remember these are the examples that are given of a description of a withdrawal or a positive deposit. Your answers may vary as long as it is a negative amount is represented or a positive amount is represented in your example, the answer is correct.

SOLVE PROBLEM – INTRODUCTION – COMPARING, ORDERING, AND ABSOLUTE VALUE OF RATIONAL NUMBERS

This is the introduction to the SOLVE problem on this section of the lesson entitled, comparing, ordering, and absolute value of rational numbers.

The SOLVE problem for this part of the lesson is, Rhonda and James were discussing their recent purchases. Rhonda bought a new laptop computer and her balance is currently at negative four hundred thirty two dollars and sixteen cents, while James purchased a television and has an account balance of negative five hundred sixty six dollars and five cents. Who owes more money on the purchase made? Use a number line to justify your answer.

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, who owes more money on the purchase made? Use a number line to justify your answer.

Now that we have identify 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 person who owes more money.

During this lesson we will learn how to compare, order and use absolute value of rational numbers, in order to complete the SOLVE problem at the end of the lesson.

MODELS OF INTEGER SETS

The following students received numbers to participate in the class raffle. Use the number line to plot the students’ numbers. Label each point with the letter that corresponds to each student.

What is the number that Student A received? Four

How can we plot this number on the number line? The number is positive, so we move to the right of zero four spaces.

Plot the point and label it “A.”

What is the number that Student D received? Negative three

How can we plot this number on the number line? The number is negative, so we move to the left of zero three spaces.

Plot the point and label it “D.”

Plot the rest of the points in the first and second rows. Student B is at negative one, Student C positive eight, Student E zero, and Student F negative six.

The high temperature today is twenty-five degrees Fahrenheit, while the low could drop to ten degrees Fahrenheit. Last year, temperatures were much colder with a high of eight degrees Fahrenheit and a low of two degrees Fahrenheit below zero. Plot all of the temperatures on the thermometer provided.

What is different about the number line in this scenario compared to the number line in Scenario One?

What happens to our thermometer, or number line, as the numbers rise above zero? The numbers increase in value.

What happens to our thermometer, or number line, as the numbers descend below zero? The numbers decrease in value.

Let’s plot the four temperatures from this year and last year. Positive twenty five, positive ten, positive eight, and negative two.

Ethan started a new online t-shirt business. At the end of each day, he records the profit or loss for that day. He earns a profit if he sells more than the cost of the materials, but will see a loss if materials are more expensive than the money he earns that day. Look at the table below for the past seven days. Plot the points representing the profit or loss for the past week and label each day on the number line.

There’s the table with each day and the sales noted

What do the positive values on this number line represent in this scenario? A profit for Ethan’s company.

What do the negative values on this number line represent in this scenario? A loss for Ethan’s company.

How can we plot the value for Sunday? It is positive, so it will be at one hundred twenty to the right of zero. Plot the point.

Is the value for Sunday loss or profit? It is a profit because it is positive.

Let’s plot the values for Monday Through Saturday. Monday was a profit of eighty dollars. Tuesday was a profit of one hundred dollars. Wednesday was a negative seventy or a loss of seventy dollars. Thursday was a negative thirty or loss of thirty dollars. Friday was a positive two hundred or a profit of two hundred dollars. And Saturday was a positive two hundred fifty or a profit of two hundred fifty dollars.

CONCRETE – COMPARING RATIONAL NUMBERS

As a review, what do we know about numbers as we move farther to the left of zero? The numbers decrease in value or get smaller as we move to the left in the negative direction.

What do we know about numbers as they move farther to the right of zero? The numbers increase in value or get larger as we move to the right in the positive direction.

For each of the comparisons in the table, use a greater than symbol (>) or less than symbol (<) to create a true inequality statement. Use the number line to assist you.

What is an inequality statement? A statement where two numbers being compared are not equal. One is larger than the other.

Place an algebra tile on the number line where the point of four should be plotted. Place an algebra tile on the number line where the point of six should be plotted.

How can we tell which number is larger? The number farther to the right will be the larger number.

Which number if farther to the right? Six

Which inequality symbol, greater than or less than, would we want to use to complete the statement? Less than

How do you know that we should use the less than symbol? When we read inequality statements, we read them from left to right. Therefore, we would want to use less than because the statement would read “four is less than six.”

Place an algebra tile on negative seven. Place an algebra tile on negative two.

What do you notice about these values? They are both negative, and negative two is farther to the right than negative seven.

Which number is larger? Negative two, because it is farther to the right than negative seven.

Which inequality symbol should we use to complete the statement? Less than

How do you know that we should use the less than symbol? When we read inequality statements, we read them from left to right. Therefore, we would want to use less than because the statement would read “negative seven is less than negative two.”

Complete the rest of the table. Five is greater than negative three. Two is greater than negative four. Zero is greater than negative five. And one is greater than negative one.

COMPARING RATIONAL NUMBERS IN REAL-WORLD CONTEXT

For the next activity, you may refer back to the number lines that we created as well as the description of the full scenario.

If the student with the greatest number gets to choose the first prize, who will be the first to choose a prize? Explain you answer. Student C because eight is the number the farthest to the right, meaning it is the largest value.

If students continue to choose in order from the greatest number to the least, who would get to choose directly before Student B? Justify your answer. Student E because zero is to the right of negative one on the number line.