dividing rational numbers practice and problem solving d

DIVIDING RATIONAL NUMBERS WORD PROBLEMS

Problem 1 :

A diver needs to descend to a depth of 100 feet below sea level. She wants to do it in 5 equal descents. How far should she travel in each descent ?

To find how far she should travel in each descent, we have to divide 100 by 5. 

Take the reciprocal of the divisor 5.

5 ----> reciprocal ----> 1/5

Step 2 : 

Multiply 100 by 1/5

(100) x (1/5)  

Step 3 :  Simplify

(20) x (1/1)

Step 4 :  Multiply 

(20) x (1/1)  =  20    

So, she should travel 20 feet in each descent.

Problem 2 : 

A teacher wants to give pizza to her students and each student can eat 1/4 of a pizza. If there are 36 students in her class, how many pizzas does she need ?

Given :  One student can eat 1/4 of a pizza.

To find the number of pizzas required for 36 students, we have to multiply 36 by 1/4. 

36 ⋅ 1/4  =  9

So, the teacher needs 9 pizzas.

Problem 3 : 

Mason made 3/4 of a pound of trail mix. If he puts 3/8 of a pound into each bag, how many bags can Mason fill?

To find the number of bags, we have to divide 3/4 by 3/8. 

(3/4)  ÷ (3/8)  =  (3/4) ⋅ (8/3)

(3/4)  ÷ (3/8)  =  24 / 12

(3/4)  ÷ (3/8)  =  2

So, Mason fill can fill the trail mix in 2 bags. 

Problem 4 : 

Raymond bought 5 rolls of paper towels. He got 99 ⅘  meters of paper towels in all. How many meters of paper towels were on each roll ?

To find no. of meters of paper towels were on each roll, we have to divide 99 4/5 by 5. 

5 ----> reciprocal ----> 1/5

Multiply 99 4/5 by 1/5

(99 4/5) x (1/5)  =  (499/5) x (1/5)    

(99 4/5) x (1/5)  =  499/25

(99 4/5) x (1/5)  =  19 ²⁴⁄₂₅     

So, 19 ²⁴⁄₂₅  meters of paper towels were on each roll.

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Hexingo - Operations with rational numbers Online practice for grades 7-8

Practice adding, subtracting, and multiplying a mixture of fractions and decimals (some are negative) in this fun game! To win, you need to get four correct answers in a row on the hexagon-tiled board.

You can choose to include or not include three different operations: addition, subtraction, and multiplication. The game will give you math problems that always include both a fraction and a decimal, such as 1/2 + 0.2 or −0.5 − 1/5. There are also two difficulty levels to choose from.

• → Curriculum
• → 7th Grade
• → Unit 1: Rational Numbers

Dividing Rational Numbers Lesson Plan

Get the lesson materials.

Dividing Rational Numbers Fractions Decimals Guided Notes Sketch & Doodles

Ever wondered how to teach dividing rational numbers, including fractions, integers, and decimals, in an engaging way to your middle school students?

In this lesson plan, students will learn about dividing rational numbers and their real-life applications. Through artistic and interactive guided notes, check for understanding questions, a color by code activity, and a maze worksheet, students will gain a comprehensive understanding of dividing rational numbers.

The lesson culminates with a real-life example that explores how dividing rational numbers can be applied to splitting a bill at a restaurant.

• Standards : CCSS 7.NS.A.2 , CCSS 7.NS.A.2.a , CCSS 7.NS.A.2.c
• Topics : Integers & Rational Numbers , Fractions , Decimals
• Grade : 7th Grade
• Type : Lesson Plans

Learning Objectives

After this lesson, students will be able to:

Divide rational numbers, including fractions, integers, and decimals

Solve division problems involving positive and negative rational numbers

Apply division of rational numbers to real-life situations

Prerequisites

Before this lesson, students should be familiar with:

Basic operations with rational numbers (adding, subtracting, and multiplying)

Basic understanding of fractions and decimals

Knowledge of how to determine the greatest common factor (GCF) and least common multiple (LCM) of numbers

Colored pencils or markers

Dividing Rational Numbers Fractions Decimals Guided Notes

Key Vocabulary

Rational numbers

Introduction

As a hook, ask students why dividing rational numbers, including fractions, integers, and decimals, is important in real life. Refer to the real-life math application on the last page of the guided notes as well as the FAQs below for ideas.

Use the guided notes to introduce the concept of dividing rational numbers. Walk through the key points of the topic, including the steps and techniques involved in dividing rational numbers. Refer to the FAQ below for a walk through on this, as well as ideas on how to respond to common student questions.

Check for Understanding : Have students walk through the "You Try!" section of the guided notes. Call on students to talk through their answers, potentially on the whiteboard or projector. Based on student responses, reteach concepts that students need extra help with.

Based on student responses, reteach concepts that students need extra help with. If your class has a wide range of proficiency levels, you can pull out students for reteaching, and have more advanced students begin work on the practice exercises.

Have students practice dividing rational numbers including fractions, integers, and decimals using the color by code activity included in the resource. Walk around the classroom to answer any student questions and provide assistance as needed.

Fast finishers can work on the maze activity for extra practice. You can assign these activities as homework for the remainder of the class.

Real-Life Application

Bring the class back together, and introduce the concept of rational number division applied to splitting a bill with friends. Refer to the FAQ for more real life applications that you can use for the discussion!

Additional Self-Checking Digital Practice

If you’re looking for digital practice for dividing rational numbers, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

Here's an activity to try:

Multiplying & Dividing Rational Numbers Digital Pixel Art

Additional Print Practice

A fun, no-prep way to practice dividing rational numbers is Doodle Math — they’re a fresh take on color by number or color by code. It includes multiple levels of practice, perfect for a review day or sub plan.

Multiplying & Dividing Rational Numbers | Doodle Math: Twist on Color by Number

What is dividing rational numbers? Open

Dividing rational numbers involves dividing numbers that can be expressed as fractions or decimals. It is the process of finding how many times one number can be evenly divided by another number.

How do you divide fractions? Open

To divide fractions, you multiply the first fraction by the reciprocal (flipped) form of the second fraction. This can be done by multiplying the numerators together and the denominators together. Simplify the resulting fraction if possible.

How do you divide decimals? Open

Dividing decimals is similar to dividing whole numbers. Use long division to divide the decimal dividend by the decimal divisor. Place the decimal point in the quotient directly above the decimal point in the dividend.

Can you divide positive and negative rational numbers? Open

Yes, you can divide positive and negative rational numbers. The rules for dividing positive and negative numbers are the same as for multiplying them. The result of the division will have a positive quotient if both numbers have the same sign, and a negative quotient if the numbers have different signs.

What is the difference between dividing fractions and dividing decimals? Open

The main difference is in the representation of the numbers. Dividing fractions involves dividing numbers expressed as fractions, while dividing decimals involves dividing numbers expressed as decimal numbers. The processes and calculations are similar, but the final answers may be in different forms.

How can dividing rational numbers be applied in real life? Open

Dividing rational numbers is commonly used in real-life situations such as dividing the bill for a pizza among friends, calculating the cost per unit of a product, or determining the average speed of a moving object. It helps in solving problems that involve sharing, distributing, or comparing quantities.

Are there any tips or tricks for dividing rational numbers? Open

One tip for dividing rational numbers is to always simplify the fraction before dividing. This makes the calculation easier and reduces the chances of errors. Additionally, keeping track of the signs (+/-) and placing the decimal point correctly when dividing decimals will help in obtaining accurate results.

What are some common mistakes to avoid when dividing rational numbers? Open

Common mistakes to avoid when dividing rational numbers include forgetting to simplify the fraction, reversing the order of the fractions when finding the reciprocal, misplacing the decimal point when dividing decimals, and forgetting to consider the signs of the numbers being divided.

Are there any resources available to practice dividing rational numbers? Open

Yes, there are various resources available for practicing dividing rational numbers. This lesson plan includes guided notes, practice worksheets, color by code activities, and a real-life math application.

Want more ideas and freebies?

Get my free resource library with digital & print activities—plus tips over email.

Worksheet on Word Problems on Rational Numbers

Practice the questions given in the worksheet on word problems on rational numbers. The questions are related to various types of word problems on four fundamental operations on rational numbers.

1.  From a rope 11 m long, two pieces of lengths 13/5 m and 33/10 m are cut off. What is the length of the remaining rope?  2.  A drum full of rice weighs 241/6 kg. If the empty drum weighs 55/4 kg, find the weight of rice in the drum.  3.  A basket contains three types of fruits weighing 58/3 kg in all. If 73/9 kg of these be apples, 19/6 kg be oranges and the rest pears. What is the weight of the pears in the basket?  4.  On one day a rickshaw puller earned $80. Out of his earnings he spent $68/5 on tea and snacks, $51/2 on food and $22/5 on repairs of the rickshaw. How much did he save on that day? 

5. Find the cost of 17/5 meters of cloth at $147/4 per meter. 6. A car is moving at an average speed of 202/5 km/hr. How much distance will it cover in 15/2 hours? 7. Find the area of a rectangular park which is 183/5 m long and 50/3 m broad. 8. Find the area of a square plot of land whose each side measures 17/2 meters. 9. One liter of petrol costs $187/4. What is the cost of 35 liters of petrol? 10. An airplane covers 1020 km in an hour. How much distance will it cover in 25/6 hours? 11. The cost of 7/2 meters of cloth is $231/4. What is the cost of one meter of cloth? 12. A cord of length 143/2 m has been cut into 26 pieces of equal length. What is the length of each piece? 13. The area of a room is 261/4 m \(^{2}\) . If its breadth is 87/16 meters, what is its length? 14. The product of two rational numbers is 48/5. If one of the rational number is 66/7, find the other rational number. 15. Rita had $300. She spent 1/3 of her money on notebooks and 1/4 of the remainder on stationery items. How much money is left with her? 16. Adrian earns $16000 per month. He spends 1/4 of his income on food; 3/10 of the remainder on house rent and 5/21 of the remainder on the education of children. How much money is still left with him?

Answers for the worksheet on word problems on rational numbers are given below to check the exact answers of the above rational problems.

2. 317/12 kg

3. 145/18 kg

5. $2499/20

7. 610 m \(^{2}\)

8. 289/4 m \(^{2}\)

10. 4250 km

● Rational Numbers - Worksheets

Worksheet on Rational Numbers

Worksheet on Equivalent Rational Numbers

Worksheet on Lowest form of a Rational Number

Worksheet on Standard form of a Rational Number

Worksheet on Equality of Rational Numbers

Worksheet on Comparison of Rational Numbers

Worksheet on Representation of Rational Number on a Number Line

Worksheet on Adding Rational Numbers

Worksheet on Properties of Addition of Rational Numbers

Worksheet on Subtracting Rational Numbers

Worksheet on Addition and Subtraction of Rational Number

Worksheet on Rational Expressions Involving Sum and Difference

Worksheet on Multiplication of Rational Number

Worksheet on Properties of Multiplication of Rational Numbers

Worksheet on Division of Rational Numbers

Worksheet on Properties of Division of Rational Numbers

Worksheet on Finding Rational Numbers between Two Rational Numbers

Worksheet on Operations on Rational Expressions

Objective Questions on Rational Numbers

Math Homework Sheets 8th Grade Math Practice From Worksheet on Word Problems on Rational Numbers to HOME PAGE

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● Rational Numbers

Introduction of Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Rational Numbers in Descending Order

Representation of Rational Numbers on the Number Line

Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Addition of Rational Numbers

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Same Denominator

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Rational Expressions Involving Addition and Subtraction

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Product of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Reciprocal of a Rational  Number

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

Rational Numbers between Two Rational Numbers

To Find Rational Numbers

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7.1 Multiply and Divide Rational Expressions

Learning objectives.

By the end of this section, you will be able to:

• Determine the values for which a rational expression is undefined
• Simplify rational expressions
• Multiply rational expressions
• Divide rational expressions
• Multiply and divide rational functions

Be Prepared 7.1

Before you get started, take this readiness quiz.

Simplify: 90 y 15 y 2 . 90 y 15 y 2 . If you missed this problem, review Example 5.13 .

Be Prepared 7.2

Multiply: 14 15 · 6 35 . 14 15 · 6 35 . If you missed this problem, review Example 1.25 .

Be Prepared 7.3

Divide: 12 10 ÷ 8 25 . 12 10 ÷ 8 25 . If you missed this problem, review Example 1.26 .

We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression .

Rational Expression

A rational expression is an expression of the form p q , p q , where p and q are polynomials and q ≠ 0 . q ≠ 0 .

Here are some examples of rational expressions:

Notice that the first rational expression listed above, − 24 56 − 24 56 , is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.

We will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.

Determine the Values for Which a Rational Expression is Undefined

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

Determine the values for which a rational expression is undefined.

• Step 1. Set the denominator equal to zero.
• Step 2. Solve the equation.

Example 7.1

Determine the value for which each rational expression is undefined:

ⓐ 8 a 2 b 3 c 8 a 2 b 3 c ⓑ 4 b − 3 2 b + 5 4 b − 3 2 b + 5 ⓒ x + 4 x 2 + 5 x + 6 . x + 4 x 2 + 5 x + 6 .

The expression will be undefined when the denominator is zero.

Determine the value for which each rational expression is undefined.

ⓐ 3 y 2 8 x 3 y 2 8 x ⓑ 8 n − 5 3 n + 1 8 n − 5 3 n + 1 ⓒ a + 10 a 2 + 4 a + 3 a + 10 a 2 + 4 a + 3

ⓐ 4 p 5 q 4 p 5 q ⓑ y − 1 3 y + 2 y − 1 3 y + 2 ⓒ m − 5 m 2 + m − 6 m − 5 m 2 + m − 6

Simplify Rational Expressions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.

Simplified Rational Expression

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

For example,

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.

Equivalent Fractions Property

If a , b , and c are numbers where b ≠ 0 , c ≠ 0 , b ≠ 0 , c ≠ 0 ,

Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see b ≠ 0 , c ≠ 0 b ≠ 0 , c ≠ 0 clearly stated.

To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

Removing the x ’s from x + 5 x x + 5 x would be like cancelling the 2’s in the fraction 2 + 5 2 ! 2 + 5 2 !

Example 7.2

How to simplify a rational expression.

Simplify: x 2 + 5 x + 6 x 2 + 8 x + 12 x 2 + 5 x + 6 x 2 + 8 x + 12 .

Simplify: x 2 − x − 2 x 2 − 3 x + 2 . x 2 − x − 2 x 2 − 3 x + 2 .

Simplify: x 2 − 3 x − 10 x 2 + x − 2 . x 2 − 3 x − 10 x 2 + x − 2 .

We now summarize the steps you should follow to simplify rational expressions.

Simplify a rational expression.

• Step 1. Factor the numerator and denominator completely.
• Step 2. Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors.

We’ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples.

Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.

Example 7.3

Simplify: 3 a 2 − 12 a b + 12 b 2 6 a 2 − 24 b 2 3 a 2 − 12 a b + 12 b 2 6 a 2 − 24 b 2 .

Simplify: 2 x 2 − 12 x y + 18 y 2 3 x 2 − 27 y 2 2 x 2 − 12 x y + 18 y 2 3 x 2 − 27 y 2 .

Simplify: 5 x 2 − 30 x y + 25 y 2 2 x 2 − 50 y 2 5 x 2 − 30 x y + 25 y 2 2 x 2 − 50 y 2 .

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is − a − a and − a = −1 · a . − a = −1 · a .

The numerical fraction, say 7 −7 7 −7 simplifies to −1 −1 . We also recognize that the numerator and denominator are opposites.

The fraction a − a a − a , whose numerator and denominator are opposites also simplifies to −1 −1 .

This tells us that b − a b − a is the opposite of a − b . a − b .

In general, we could write the opposite of a − b a − b as b − a . b − a . So the rational expression a − b b − a a − b b − a simplifies to −1 . −1 .

Opposites in a Rational Expression

The opposite of a − b a − b is b − a . b − a .

An expression and its opposite divide to −1 . −1 .

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat a + b a + b and b + a b + a as opposites. Recall that in addition, order doesn’t matter so a + b = b + a a + b = b + a . So if a ≠ − b a ≠ − b , then a + b b + a = 1 . a + b b + a = 1 .

Example 7.4

Simplify: x 2 − 4 x − 32 64 − x 2 . x 2 − 4 x − 32 64 − x 2 .

Simplify: x 2 − 4 x − 5 25 − x 2 . x 2 − 4 x − 5 25 − x 2 .

Simplify: x 2 + x − 2 1 − x 2 . x 2 + x − 2 1 − x 2 .

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Multiplication of Rational Expressions

If p , q , r , and s are polynomials where q ≠ 0 , s ≠ 0 , q ≠ 0 , s ≠ 0 , then

To multiply rational expressions, multiply the numerators and multiply the denominators.

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, x ≠ 0 , x ≠ 0 , x ≠ 3 , x ≠ 3 , and x ≠ 4 . x ≠ 4 .

Example 7.5

How to multiply rational expressions.

Simplify: 2 x x 2 − 7 x + 12 · x 2 − 9 6 x 2 . 2 x x 2 − 7 x + 12 · x 2 − 9 6 x 2 .

Simplify: 5 x x 2 + 5 x + 6 · x 2 − 4 10 x . 5 x x 2 + 5 x + 6 · x 2 − 4 10 x .

Try It 7.10

Simplify: 9 x 2 x 2 + 11 x + 30 · x 2 − 36 3 x 2 . 9 x 2 x 2 + 11 x + 30 · x 2 − 36 3 x 2 .

Multiply rational expressions.

• Step 1. Factor each numerator and denominator completely.
• Step 2. Multiply the numerators and denominators.
• Step 3. Simplify by dividing out common factors.

Example 7.6

Multiply: 3 a 2 − 8 a − 3 a 2 − 25 · a 2 + 10 a + 25 3 a 2 − 14 a − 5 . 3 a 2 − 8 a − 3 a 2 − 25 · a 2 + 10 a + 25 3 a 2 − 14 a − 5 .

Try It 7.11

Simplify: 2 x 2 + 5 x − 12 x 2 − 16 · x 2 − 8 x + 16 2 x 2 − 13 x + 15 . 2 x 2 + 5 x − 12 x 2 − 16 · x 2 − 8 x + 16 2 x 2 − 13 x + 15 .

Try It 7.12

Simplify: 4 b 2 + 7 b − 2 1 − b 2 · b 2 − 2 b + 1 4 b 2 + 15 b − 4 . 4 b 2 + 7 b − 2 1 − b 2 · b 2 − 2 b + 1 4 b 2 + 15 b − 4 .

Divide Rational Expressions

Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.

Division of Rational Expressions

If p , q , r, and s are polynomials where q ≠ 0 , r ≠ 0 , s ≠ 0 , q ≠ 0 , r ≠ 0 , s ≠ 0 , then

To divide rational expressions, multiply the first fraction by the reciprocal of the second.

Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.

Example 7.7

How to divide rational expressions.

Divide: p 3 + q 3 2 p 2 + 2 p q + 2 q 2 ÷ p 2 − q 2 6 . p 3 + q 3 2 p 2 + 2 p q + 2 q 2 ÷ p 2 − q 2 6 .

Try It 7.13

Simplify: x 3 + 8 3 x 2 − 6 x + 12 ÷ x 2 − 4 6 . x 3 + 8 3 x 2 − 6 x + 12 ÷ x 2 − 4 6 .

Try It 7.14

Simplify: 2 z 2 z 2 − 1 ÷ z 3 − z 2 + z z 3 + 1 . 2 z 2 z 2 − 1 ÷ z 3 − z 2 + z z 3 + 1 .

Divide rational expressions.

• Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
• Step 2. Factor the numerators and denominators completely.
• Step 3. Multiply the numerators and denominators together.
• Step 4. Simplify by dividing out common factors.

Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

Example 7.8

Divide: 6 x 2 − 7 x + 2 4 x − 8 2 x 2 − 7 x + 3 x 2 − 5 x + 6 . 6 x 2 − 7 x + 2 4 x − 8 2 x 2 − 7 x + 3 x 2 − 5 x + 6 .

Try It 7.15

Simplify: 3 x 2 + 7 x + 2 4 x + 24 3 x 2 − 14 x − 5 x 2 + x − 30 . 3 x 2 + 7 x + 2 4 x + 24 3 x 2 − 14 x − 5 x 2 + x − 30 .

Try It 7.16

Simplify: y 2 − 36 2 y 2 + 11 y − 6 2 y 2 − 2 y − 60 8 y − 4 . y 2 − 36 2 y 2 + 11 y − 6 2 y 2 − 2 y − 60 8 y − 4 .

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.

Example 7.9

Perform the indicated operations: 3 x − 6 4 x − 4 · x 2 + 2 x − 3 x 2 − 3 x − 10 ÷ 2 x + 12 8 x + 16 . 3 x − 6 4 x − 4 · x 2 + 2 x − 3 x 2 − 3 x − 10 ÷ 2 x + 12 8 x + 16 .

Try It 7.17

Perform the indicated operations: 4 m + 4 3 m − 15 · m 2 − 3 m − 10 m 2 − 4 m − 32 ÷ 12 m − 36 6 m − 48 . 4 m + 4 3 m − 15 · m 2 − 3 m − 10 m 2 − 4 m − 32 ÷ 12 m − 36 6 m − 48 .

Try It 7.18

Perform the indicated operations: 2 n 2 + 10 n n − 1 ÷ n 2 + 10 n + 24 n 2 + 8 n − 9 · n + 4 8 n 2 + 12 n . 2 n 2 + 10 n n − 1 ÷ n 2 + 10 n + 24 n 2 + 8 n − 9 · n + 4 8 n 2 + 12 n .

Multiply and Divide Rational Functions

We started this section stating that a rational expression is an expression of the form p q , p q , where p and q are polynomials and q ≠ 0 . q ≠ 0 . Similarly, we define a rational function as a function of the form R ( x ) = p ( x ) q ( x ) R ( x ) = p ( x ) q ( x ) where p ( x ) p ( x ) and q ( x ) q ( x ) are polynomial functions and q ( x ) q ( x ) is not zero.

Rational Function

A rational function is a function of the form

where p ( x ) p ( x ) and q ( x ) q ( x ) are polynomial functions and q ( x ) q ( x ) is not zero.

The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make q ( x ) = 0 . q ( x ) = 0 .

Determine the domain of a rational function.

• Step 3. The domain is all real numbers excluding the values found in Step 2.

Example 7.10

Find the domain of R ( x ) = 2 x 2 − 14 x 4 x 2 − 16 x − 48 . R ( x ) = 2 x 2 − 14 x 4 x 2 − 16 x − 48 .

The domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain.

Try It 7.19

Find the domain of R ( x ) = 2 x 2 − 10 x 4 x 2 − 16 x − 20 . R ( x ) = 2 x 2 − 10 x 4 x 2 − 16 x − 20 .

Try It 7.20

Find the domain of R ( x ) = 4 x 2 − 16 x 8 x 2 − 16 x − 64 . R ( x ) = 4 x 2 − 16 x 8 x 2 − 16 x − 64 .

To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.

Example 7.11

Find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) = 2 x − 6 x 2 − 8 x + 15 f ( x ) = 2 x − 6 x 2 − 8 x + 15 and g ( x ) = x 2 − 25 2 x + 10 . g ( x ) = x 2 − 25 2 x + 10 .

Try It 7.21

Find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) = 3 x − 21 x 2 − 9 x + 14 f ( x ) = 3 x − 21 x 2 − 9 x + 14 and g ( x ) = 2 x 2 − 8 3 x + 6 . g ( x ) = 2 x 2 − 8 3 x + 6 .

Try It 7.22

Find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) = x 2 − x 3 x 2 + 27 x − 30 f ( x ) = x 2 − x 3 x 2 + 27 x − 30 and g ( x ) = x 2 − 100 x 2 − 10 x . g ( x ) = x 2 − 100 x 2 − 10 x .

To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.

Example 7.12

Find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) = 3 x 2 x 2 − 4 x f ( x ) = 3 x 2 x 2 − 4 x and g ( x ) = 9 x 2 − 45 x x 2 − 7 x + 10 . g ( x ) = 9 x 2 − 45 x x 2 − 7 x + 10 .

Try It 7.23

Find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) = 2 x 2 x 2 − 8 x f ( x ) = 2 x 2 x 2 − 8 x and g ( x ) = 8 x 2 + 24 x x 2 + x − 6 . g ( x ) = 8 x 2 + 24 x x 2 + x − 6 .

Try It 7.24

Find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) = 15 x 2 3 x 2 + 33 x f ( x ) = 15 x 2 3 x 2 + 33 x and g ( x ) = 5 x − 5 x 2 + 9 x − 22 . g ( x ) = 5 x − 5 x 2 + 9 x − 22 .

Section 7.1 Exercises

Practice makes perfect.

In the following exercises, determine the values for which the rational expression is undefined.

ⓐ 2 x 2 z 2 x 2 z , ⓑ 4 p − 1 6 p − 5 4 p − 1 6 p − 5 , ⓒ n − 3 n 2 + 2 n − 8 n − 3 n 2 + 2 n − 8

ⓐ 10 m 11 n 10 m 11 n , ⓑ 6 y + 13 4 y − 9 6 y + 13 4 y − 9 , ⓒ b − 8 b 2 − 36 b − 8 b 2 − 36

ⓐ 4 x 2 y 3 y 4 x 2 y 3 y , ⓑ 3 x − 2 2 x + 1 3 x − 2 2 x + 1 , ⓒ u − 1 u 2 − 3 u − 28 u − 1 u 2 − 3 u − 28

ⓐ 5 p q 2 9 q 5 p q 2 9 q , ⓑ 7 a − 4 3 a + 5 7 a − 4 3 a + 5 , ⓒ 1 x 2 − 4 1 x 2 − 4

In the following exercises, simplify each rational expression.

− 44 55 − 44 55

56 63 56 63

8 m 3 n 12 m n 2 8 m 3 n 12 m n 2

36 v 3 w 2 27 v w 3 36 v 3 w 2 27 v w 3

8 n − 96 3 n − 36 8 n − 96 3 n − 36

12 p − 240 5 p − 100 12 p − 240 5 p − 100

x 2 + 4 x − 5 x 2 − 2 x + 1 x 2 + 4 x − 5 x 2 − 2 x + 1

y 2 + 3 y − 4 y 2 − 6 y + 5 y 2 + 3 y − 4 y 2 − 6 y + 5

a 2 − 4 a 2 + 6 a − 16 a 2 − 4 a 2 + 6 a − 16

y 2 − 2 y − 3 y 2 − 9 y 2 − 2 y − 3 y 2 − 9

p 3 + 3 p 2 + 4 p + 12 p 2 + p − 6 p 3 + 3 p 2 + 4 p + 12 p 2 + p − 6

x 3 − 2 x 2 − 25 x + 50 x 2 − 25 x 3 − 2 x 2 − 25 x + 50 x 2 − 25

8 b 2 − 32 b 2 b 2 − 6 b − 80 8 b 2 − 32 b 2 b 2 − 6 b − 80

−5 c 2 − 10 c −10 c 2 + 30 c + 100 −5 c 2 − 10 c −10 c 2 + 30 c + 100

3 m 2 + 30 m n + 75 n 2 4 m 2 − 100 n 2 3 m 2 + 30 m n + 75 n 2 4 m 2 − 100 n 2

5 r 2 + 30 r s − 35 s 2 r 2 − 49 s 2 5 r 2 + 30 r s − 35 s 2 r 2 − 49 s 2

a − 5 5 − a a − 5 5 − a

5 − d d − 5 5 − d d − 5

20 − 5 y y 2 − 16 20 − 5 y y 2 − 16

4 v − 32 64 − v 2 4 v − 32 64 − v 2

w 3 + 216 w 2 − 36 w 3 + 216 w 2 − 36

v 3 + 125 v 2 − 25 v 3 + 125 v 2 − 25

z 2 − 9 z + 20 16 − z 2 z 2 − 9 z + 20 16 − z 2

a 2 − 5 a − 36 81 − a 2 a 2 − 5 a − 36 81 − a 2

In the following exercises, multiply the rational expressions.

12 16 · 4 10 12 16 · 4 10

32 5 · 16 24 32 5 · 16 24

5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2 5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2

12 a 3 b b 2 · 2 a b 2 9 b 3 12 a 3 b b 2 · 2 a b 2 9 b 3

5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p 5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p

3 q 2 q 2 + q − 6 · q 2 − 9 9 q 3 q 2 q 2 + q − 6 · q 2 − 9 9 q

2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y 2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y

z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2 z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2

28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49 28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49

72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36 72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36

3 c 2 − 16 c + 5 c 2 − 25 · c 2 + 10 c + 25 3 c 2 − 14 c − 5 3 c 2 − 16 c + 5 c 2 − 25 · c 2 + 10 c + 25 3 c 2 − 14 c − 5

2 d 2 + d − 3 d 2 − 16 · d 2 − 8 d + 16 2 d 2 − 9 d − 18 2 d 2 + d − 3 d 2 − 16 · d 2 − 8 d + 16 2 d 2 − 9 d − 18

6 m 2 − 13 m + 2 9 − m 2 · m 2 − 6 m + 9 6 m 2 + 23 m − 4 6 m 2 − 13 m + 2 9 − m 2 · m 2 − 6 m + 9 6 m 2 + 23 m − 4

2 n 2 − 3 n − 14 25 − n 2 · n 2 − 10 n + 25 2 n 2 − 13 n + 21 2 n 2 − 3 n − 14 25 − n 2 · n 2 − 10 n + 25 2 n 2 − 13 n + 21

In the following exercises, divide the rational expressions.

v − 5 11 − v ÷ v 2 − 25 v − 11 v − 5 11 − v ÷ v 2 − 25 v − 11

10 + w w − 8 ÷ 100 − w 2 8 − w 10 + w w − 8 ÷ 100 − w 2 8 − w

3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64 3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64

r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45 r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45

p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12 p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12

v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4 v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4

x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 ) x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 )

2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z ) 2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z )

2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16 2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16

3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15 3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15

12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5 12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5

4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4 4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4

For the following exercises, perform the indicated operations.

10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 ÷ 5 m 2 + 10 m 2 m − 10 10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 ÷ 5 m 2 + 10 m 2 m − 10

4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 ÷ 108 n 2 − 24 n n + 6 4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 ÷ 108 n 2 − 24 n n + 6

12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 · p − 7 9 p 3 − 9 p 2 12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 · p − 7 9 p 3 − 9 p 2

6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 · 4 q 2 + 12 q 12 q + 6 6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 · 4 q 2 + 12 q 12 q + 6

In the following exercises, find the domain of each function.

R ( x ) = x 3 − 2 x 2 − 25 x + 50 x 2 − 25 R ( x ) = x 3 − 2 x 2 − 25 x + 50 x 2 − 25

R ( x ) = x 3 + 3 x 2 − 4 x − 12 x 2 − 4 R ( x ) = x 3 + 3 x 2 − 4 x − 12 x 2 − 4

R ( x ) = 3 x 2 + 15 x 6 x 2 + 6 x − 36 R ( x ) = 3 x 2 + 15 x 6 x 2 + 6 x − 36

R ( x ) = 8 x 2 − 32 x 2 x 2 − 6 x − 80 R ( x ) = 8 x 2 − 32 x 2 x 2 − 6 x − 80

For the following exercises, find R ( x ) = f ( x ) · g ( x ) R ( x ) = f ( x ) · g ( x ) where f ( x ) f ( x ) and g ( x ) g ( x ) are given.

f ( x ) = 6 x 2 − 12 x x 2 + 7 x − 18 f ( x ) = 6 x 2 − 12 x x 2 + 7 x − 18 g ( x ) = x 2 − 81 3 x 2 − 27 x g ( x ) = x 2 − 81 3 x 2 − 27 x

f ( x ) = x 2 − 2 x x 2 + 6 x − 16 f ( x ) = x 2 − 2 x x 2 + 6 x − 16 g ( x ) = x 2 − 64 x 2 − 8 x g ( x ) = x 2 − 64 x 2 − 8 x

f ( x ) = 4 x x 2 − 3 x − 10 f ( x ) = 4 x x 2 − 3 x − 10 g ( x ) = x 2 − 25 8 x 2 g ( x ) = x 2 − 25 8 x 2

f ( x ) = 2 x 2 + 8 x x 2 − 9 x + 20 f ( x ) = 2 x 2 + 8 x x 2 − 9 x + 20 g ( x ) = x − 5 x 2 g ( x ) = x − 5 x 2

For the following exercises, find R ( x ) = f ( x ) g ( x ) R ( x ) = f ( x ) g ( x ) where f ( x ) f ( x ) and g ( x ) g ( x ) are given.

f ( x ) = 27 x 2 3 x − 21 f ( x ) = 27 x 2 3 x − 21 g ( x ) = 3 x 2 + 18 x x 2 + 13 x + 42 g ( x ) = 3 x 2 + 18 x x 2 + 13 x + 42

f ( x ) = 24 x 2 2 x − 8 f ( x ) = 24 x 2 2 x − 8 g ( x ) = 4 x 3 + 28 x 2 x 2 + 11 x + 28 g ( x ) = 4 x 3 + 28 x 2 x 2 + 11 x + 28

f ( x ) = 16 x 2 4 x + 36 f ( x ) = 16 x 2 4 x + 36 g ( x ) = 4 x 2 − 24 x x 2 + 4 x − 45 g ( x ) = 4 x 2 − 24 x x 2 + 4 x − 45

f ( x ) = 24 x 2 2 x − 4 f ( x ) = 24 x 2 2 x − 4 g ( x ) = 12 x 2 + 36 x x 2 − 11 x + 18 g ( x ) = 12 x 2 + 36 x x 2 − 11 x + 18

Writing Exercises

Explain how you find the values of x for which the rational expression x 2 − x − 20 x 2 − 4 x 2 − x − 20 x 2 − 4 is undefined.

Explain all the steps you take to simplify the rational expression p 2 + 4 p − 21 9 − p 2 . p 2 + 4 p − 21 9 − p 2 .

ⓐ Multiply 7 4 · 9 10 7 4 · 9 10 and explain all your steps. ⓑ Multiply n n − 3 · 9 n + 3 n n − 3 · 9 n + 3 and explain all your steps. ⓒ Evaluate your answer to part ⓑ when n = 7 n = 7 . Did you get the same answer you got in part ⓐ ? Why or why not?

ⓐ Divide 24 5 ÷ 6 24 5 ÷ 6 and explain all your steps. ⓑ Divide x 2 − 1 x ÷ ( x + 1 ) x 2 − 1 x ÷ ( x + 1 ) and explain all your steps. ⓒ Evaluate your answer to part ⓑ when x = 5 . x = 5 . Did you get the same answer you got in part ⓐ ? Why or why not?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction

• Authors: Lynn Marecek, Andrea Honeycutt Mathis
• Publisher/website: OpenStax
• Book title: Intermediate Algebra 2e
• Publication date: May 6, 2020
• Location: Houston, Texas
• Book URL: https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction
• Section URL: https://openstax.org/books/intermediate-algebra-2e/pages/7-1-multiply-and-divide-rational-expressions

© Jul 24, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

• Inspiration

Using Rational Numbers

How to add, subtract, multiply and divide rational numbers

A rational number is a number that can be written as a simple fraction (i.e. as a ratio ).

In general ...

So a rational number looks like this:

But q cannot be zero, as that is dividing by zero .

How to Add, Subtract, Multiply and Divide

When the rational number is something simple like 3 , or 0.001 , then just use mental arithmetic, or a calculator!

But what about when it is in p q form?

Here we will see those operations in a more general Algebra style.

You might also like to read Fractions in Algebra .

Let us start with multiplication, as that is the easiest.

Multiplication

To multiply two rational numbers multiply the tops and bottoms separately , like this:

Here is an example:

To divide two rational numbers, first flip the second number over (make it a reciprocal) and then do a multiply like above:

Addition and Subtraction

We will cover Addition and Subtraction in one go, as they are the same method.

Before we add or subtract, the rational numbers should have the same bottom number (called a Common Denominator ).

The easiest way to do this is to

Multiply both parts of each number by the bottom part of the other

Like this (note that the dot · means multiply):

Here is an example of addition:

And an example of subtraction (the middle step is skipped to make it quicker):

Simplest Form

Sometimes we have a rational number like this:

But that is not as simple as it can be!

We can divide both top and bottom by 5 to get:

Now it is in "simplest form", which is how most people want it!

Be Careful With "Mixed Fractions"

We may be tempted to write an Improper Fraction (a fraction that is "top-heavy", i.e. where the top number is bigger then the bottom number) as a Mixed Fraction :

For example 7 / 4 = 1 3 / 4 , shown here:

But for mathematics the "Improper" form (such as 7 / 4 ) is actually better .

Because Mixed fractions (such as 1 3 / 4 ) can be confusing when we write them down in a formula, as it can look like a multiplication :

So prefer using Improper Fractions when doing mathematics.

7.1.2B Problem Solving with Rational Numbers

Standard 7.1.2.

Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest.

Use proportional reasoning to solve problems involving ratios in various contexts.

For example : A recipe calls for milk, flour and sugar in a ratio of 4:6:3 (this is how recipes are often given in large institutions, such as hospitals). How much flour and milk would be needed with 1 cup of sugar?

Standard 7.1.2 Essential Understandings

In this standard, students will develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication and division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying and dividing with negative numbers.

The focus of instruction at the 7th grade level is on being able to comfortably translate between decimal and fractional forms of a number for both positive and negative values. Students should be able to compare numbers and manipulate the values to derive other forms of the numbers to make comparing less inhibiting and more accessible. Students will also use ratios and proportional reasoning to solve problems in various contexts. Students will be able to use information given to help find missing values. Their knowledge of equivalent fractions and scaling will enable them to use ratios and solve proportions.

All Standard Benchmarks

7.1.2.1 Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents. 7.1.2.2 Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense. 7.1.2.3 Understand that calculators and other computing technologies often truncate or round numbers. 7.1.2.4 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. 7.1.2.5 Use proportional reasoning to solve problems involving ratios in various contexts. 7.1.2.6 Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value.

7.1.2 Group B - Problem Solving with Rational Numbers

7.1.2.4 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. 7.1.2.5 Use proportional reasoning to solve problems involving ratios in various contexts. For example, a recipe calls for milk, flour and sugar in a ratio of 4:6:3. (This is how recipes are often given in large institutions, such as hospitals.) How much flour and milk would be needed with 1 cup of sugar?

What students should know and be able to do [at a mastery level] related to these benchmarks:

• Understand that simple interest does not use interest earned as new principal, but compound interest does;
• Model addition and subtraction of integers with physical materials and the number line;
• Perform calculations with and without the use of a calculator;
• Understand the rules for calculating with positive and negatives;
• Understand calculating with positive exponents;
• Scale values up and down;
• See the relationship as a proportional relationship;
• Use ratios accurately;
• Compare ratios;
• Be able to differentiate mathematical characteristics of proportional thinking from nonproportional contexts;
• Know the mathematical characteristics of proportional situations.

Work from previous grades that supports this new learning includes:

• Use and read output on a calculator;
• Know how to change percents to decimals and decimals to percents;
• Know how to calculate a percent of a number, such as 25% of 1000;
• Use and find percents;
• Use and find fractions and equivalent values;
• Multiply and divide;
• Input into a calculator, using correct keystrokes;
• Perform mental math;
• Understand equivalent fractions;
• Scale up and down;
• Use ratios;
• Know multiplication facts to 12's;
• Be proficient at problem solving ;
• Know how to work backwards;
• Understand and use the terms including numerator, denominator, greatest common factor and least common multiple;
• Know how to simplify fractions.
• Make use of estimation strategies.

NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems:

• Work flexibly with fractions, decimals, and percents to solve problems;
• Understand and use ratios and proportions to represent quantitative relationships.

Understand meanings of operations and how they relate to one another:

• Understand the meaning and effects of arithmetic operations with fractions decimals, and integers;
• Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

Compute fluently and make reasonable estimates:

• Select appropriate methods and tools for computing for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situations and apply the selected methods;
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

Common Core State Standards (CCSS)

7.NS ( The Number System ) Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

• 7.NS. 1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram;
• 7.NS.1.b. Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts;
• 7.NS.1.c . Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts;
• 7.NS.1.d. Apply properties of operations as strategies to add and subtract rational numbers;
• 7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers;
• 7.NS.2a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts;
• 7.NS.2.c. Apply properties of operations as strategies to multiply and divide rational numbers;
• 7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.

6.NS (The Number System) Apply and extend previous understandings of numbers to the system of rational numbers.

• 6.NS.7c. Understand ordering and absolute value of rational numbers. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.
• 6.EE (Expressions and Equations) Apply and extend previous understandings of arithmetic to algebraic expressions.

6.EE.1. Write and evaluate numerical expressions involving whole-number exponents.

7.EE (Expressions and Equations) Use properties of operations to generate equivalent expressions.

• 7.EE.3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations as strategies to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
• 7.RP (Ratios and Proportional Relationships) Analyze proportional relationships and use them to solve real-world and mathematical problems.

7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Misconceptions

Student misconceptions and common errors.

• Students may not realize that finding the percent of a number always involves division.
• Students incorrectly enter the percent into the calculator instead of the decimal value, and thus get a value that does not make sense. Many students do not know how to use the % key on a calculator.
• Students get confused with the difference between simple and compound interest.
• Students may not realize that using a proportion is the only way to solve these problems.
• Students want to use addition to get equivalent values, not multiplication.
• Students forget how to multiply with fractional values.
• If using a fractional scale factor (cutting a recipe in $\frac{1}{2}$), students may just add $\frac{1}{2}$ to the values they have, instead of dividing all the values by 2.

In the Classroom

In this vignette, students will mix juice.

Mixing Juice

Teacher: How many of you have made orange juice from a can before?

Student 1: I make it all the time.

Student 2: My mom makes ours.

Others raise their hands to indicate they have either made it or helped make it.

Teacher: When you made juice or helped, what did you have to do?

Student 3: Make sure we didn't add too much water or it wouldn't taste very good.

Teacher: Anything else?

Student 2: Make sure we add enough water or it would be too sour and strong tasting.

Teacher: Today we are going to look at some different recipes for different juice mixes. Each recipe is different from the others, so we are going to use our math skills and see if we can decide which juice would be the most orangey, and which would be the least orangey.

Student 1: Won't it depend on the brand of juice we use, too?

Teacher: You are right; that could be a factor, and that could be something we explore at a different time, but for today, we are going to assume that all the juices are the same brand, so that won't be a factor.

Teacher: OK, here is your information. Cam and Scott are in charge of making juice for the 200 students at camp. They mix water and frozen concentrate to make the juice. They were given the following recipes from different campers:

Teacher: OK, your task is to figure out which one is the most orangey. Good luck!

As the teacher circulates around, she hears some of the following conversations.

Student 1 : I think Mix V will be the most orangey because it has 5 cups of concentrate, and no other juice has that much, so it has to be the most orangey.

Student 2 : I think Mix S and Mix T will be the same because they each have only one more cup of concentrate than juice, so they should taste the same, shouldn't they?

Teacher: Well, how about if we use some concepts we've talk about in class before. What could we do that might help us?

Student 3: Write them as fractions?

Student 2: Find out what percent of each is concentrate?

Student 1: We could write ratios for each recipe.

Teacher: OK. It sounds like we have many ideas. There are multiple ways to do this problem. Just make sure you understand what you are doing, and if you get an answer, make sure you know what it represents in terms of the problem.

Student 1: Well for Mix S, $\frac{2}{3}$ of the juice is concentrate.

Teacher: Let's look at that again. While fractions offer one way of working on this problem, what you said is not accurate. Remember what you know about fractions. If you say that $\frac{2}{3}$ of the juice is concentrate, then which number in the fraction tells you how many total cups are in the juice?

Student 2: Well, the denominator is the total, so the total would be 3 cups. In Mix S, though, there are 3 cups of concentrate AND 2 cups of water, so there's really 5 cups of ingredients in the juice.

Teacher: So is $\frac{2}{3}$ of Mix S concentrate?

Student 1: No, I guess it isn't. But shouldn't we be able to use the fraction $\frac{2}{3}$?

Teacher: Let's talk about that. If we use $\frac{2}{3}$, what does the 2 represent?

Student 3: The 2 cups of water.

Teacher: And what does the 3 represent?

Student 4: The 3 cups of concentrate.

Teacher: So if we wanted to know what fraction is concentrate, what fraction would we use?

Student 2:  $\frac{2}{5}$, because 5 is the total amount of cups in the juice and 2 is the number of cups of concentrate.

Teacher: OK. Keep working and I'll come back to you. Again, we are trying to figure out which one is the most orangey.

The teacher continues to circulate around the room, listening to conversations that different groups are having.

Teacher: Who has an idea here? Which mix does your group think is the most orangey?

Student 2: We think it is Mix T. We set up ratios of concentrate to water, and compared the decimal values. So this is what we got:

Teacher: What do those decimal values represent?

Student 2: The ratio of concentrate to water.

Teacher: Did anyone do this another way?

Student 3: We found out what percent of the juice was concentrate. So here is what we got. We agree that Mix T is the most orangey.

Teacher: So, we can see from these two strategies that Mix T is the most orangey. Which one is the least orangey?

Student 1: Mix U. If you look at the percent that is concentrate, we can see that Mix U has the least percent of concentrate, so it is the least orangey.

Teacher: Let's move on to the next part of the problem. Assume each camper gets $\frac{1}{2}$ cup of juice. For each mix, you need to now figure out how many batches are needed to be made to serve all of the campers. Remember, there are 200 campers.

Student 4: I think we need to first figure out how many cups are in each batch of juice.

Teacher: Why would you need to know that?

Student 4: So we can scale the recipe up to make enough to feed the campers.

Teacher: Good thought process. Go ahead and give it a try and let me know how it goes.

Student 3: Can't we just multiply 200 by $\frac{1}{2}$ to get 100? We need 100 cups of juice made.

Teacher: OK. You are right that you need 100 cups, but that is 100 cups of juice. That doesn't say how many cups of each of the 2 ingredients you need.

Student 3: Oh. That's right. We need to do that. Hmm. Maybe putting the information into tables could help us.

Teacher: Nice. Any other way this could have been figured out?

Student 4: We used proportional reasoning to solve it. We knew that we needed 100 cups total. We also knew how many batches we needed. So, we knew we needed 100 cups total, and there were 5 total cups in the batch made. Since we needed two cups of concentrate, then we can scale that up. Here's the proportion we set up:

         $\frac{2}{5}$ = $\frac{x}{100}$

Student 4: So, we can solve and get 40 for x, just like the other group did. So, we need 40 cups of concentrate and 60 cups of water. We did that same process for the rest of them, too, like our table shows. By doing it our way, we won't have as much left over, because they rounded the number of batches, and we set up proportions so we have a closer number of cups that would be needed. We still have Mix T using the most concentrate in proportion to the amount of water used.

Teacher: OK. Nice work! To finish the lesson, here is your final task: Which of the following will taste most orangey: 2 cups of concentrate and 3 cups of water; 4 cups of concentrate and 6 cups of water; or 10 cups of concentrate and 15 cups of water?

The students use various strategies that were discussed in class.

Teacher: Anyone have an answer?

Student 2 : I think they are all the same orangey-ness?

Teacher: How could that be if they all have different numbers of concentrate and water?

Student 1: Because when we compared all of the ratios, they were equivalent: 2 to 3 is equivalent to 4 to 6, is equivalent to 10 to 15. If we write them all as fractions we can see that even better.

Teacher: Can you come up to the board and show us what you are doing?

Student 1: Sure. Here is what I did:

$\frac{2}{3}$ = $\frac{4}{6}$ = $\frac{10}{15}$

They all simplify to being the same concentrate, $\frac{2}{3}$, so, they are all proportional to one another.

Teacher: Nice job today. We will review this tomorrow.

Teacher Notes

• To find percent increase or percent decrease, have the students subtract the two values being compared. Then, have them take that answer and divide it by the beginning value to get the ratio of change to starting value. Multiply this value by 100 to get the percent change. For example, the population changed from 1000 to 1150. What was the percent increase? The students would subtract 1150 - 1000 to get 150. They then need to divide 150 by 1000 to get 0.15. Multiply this value by 100 (change a decimal to a percent), and the percent increase is 0.15 × 100 which equals 15%. This  website provides a calculator that will do this.
• The scale on a map suggests that 1 centimeter represents an actual distance of 5 kilometers. The map distance between two towns is 8 centimeters. What is the actual distance? In this situation, a table can help highlight this relationship.

Drawing pictures may help students see that the rate is a scalar concept.

• To help students learn the difference between compound and simple interest, work on simple interest first. In simple interest, the amount of interest earned is proportional to the number of months invested. For example, a deposit of $500 in an account earns 1% simple interest each month. After 1 month, $5 of interest would be earned, because $500×1% = $5. ($500x0.01 = $5).  After two months, there will be $10 of interest earned. Interest is only earned on the original deposit with simple interest. With compound interest, interest is not only calculated on the original deposit, but also the interest that has previously been earned. So, one month, the interest earned would be $500x1% = $5. The balance after that first month is $505. After two months, the interest is earned on the total balance of $505. As shown in the table, the amount added each month is not constant, therefore, compound interest earned is not an example of a proportional relationship, whereas simple interest earned is showing a proportional relationship.

• Proportion problems This website will explain proportions, provide examples, and provides sample problems.
• Ratios and proportions This site includes ratios, comparing ratios, and proportions.
• Ratio and proportion factsheets
• Ratios and proportions in everyday life This site addresses ratios and proportions and how knowledge of these mathematical concepts is used in everyday life. It includes lesson plans, animation, online and printable worksheets, online exercises, games, quizzes, and a link to eThemes Resource on Math: Equivalent Ratios. 
• Real-world proportions This site includes problems with solving proportions (algorithms) and real-world applications.
• Math in daily life: Cooking by numbers This webpage that helps make a connection between proportions and the real-life situation of cooking.
• Burns, M., and Sheffield, S. (2004). Jim and the Beanstalk. In Math and Literature (p. 60) . Sausalito, CA: Math Solutions Publications.
• Burns, M., and Sheffield, S. (2004). How Big is a Foot. In Math and Literature (p. 47). Sausalito, CA: Math Solutions Publications.

• interest: fee paid on loans or earned on invested money, based on the principal amount and the interest rate.
• simple interest: interest paid only on the original principal, not on the interest accrued.
• compound interest: interest computed on accumulated interest as well as on the principal.
• proportion: an equation which states that two ratios are equal; a relationship between two ratios. Example: $\frac{\text{hours spent on homework}}{\text{hours spent in school}}=\frac{2}{7}$

Note that this does not necessarily imply that "hours spent on homework" = 2 or that "hours spent in school" = 7. During a week, 10 hours may have been spent on homework while 35 hours were spent in school. The proportion is still true because $\frac{10}{35}=\frac{2}{7}$.

• proportional reasoning: a mathematical way of thinking in which students recognize proportional versus non-proportional situations and can use multiple approaches, not just the cross-products approach, for solving problems about proportional situations.

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

• What conclusions can be drawn about the student's understanding of applying relationships to solve problems in various contexts?
• Can students scale values up or down and understand why they are doing it?
• Can students tell if a relationship is proportional by looking at the verbal description of it, or do they need to mathematically figure it out?
• Are students able to transfer prior knowledge about equivalent fractions to the concept of proportionality?
• Do students understand why their procedures work?
• What connections have been made as students explored the mathematical characteristics of proportional situations?
• What models would help students in understanding the concepts addressed in the lesson?
• What aspects of proportional relationships are students still struggling with?

Cramer, K. & Post, T. (1993, February). Making connections: A case for proportionality. In Arithmetic Teacher, 60(6), 342-346.

• Massachusetts Comprehensive Assessment System Spring 2010 Test Items http://www.doe.mass.edu/mcas/2010/release/g7math.pdf
• Absolute value http :// www . purplemath . com / modules / absolute . htm
• Adding and subtracting negative numbers http :// www . purplemath . com / modules / negative 2. htm
• Lappan, G., Fey, J., Fitzgerald, W., Friel, S., Philips, E. (2009). Accentuate the Negative, CMP2. Pearson Prentice Hall.
• Lappan, G., Fey, J., Fitzgerald, W., Friel, S., Philips, E. (2009). Comparing and Scaling, CMP2. Pearson Prentice Hall.
• Rational Number Project: Proportional reasoning: the effect of two context variables, rate type, and problem setting http :// www . cehd . umn . edu / rationalnumberproject /89_6. html
• Dacey, L.S., and Gartland, K. (2009). Math for All: Differentiating Instruction. Sausalito, CA: Math Solutions.

Answer: a DOK: Level 3 Source: Minnesota Grade 7 Mathematics MCA - III Item Sampler Item, 2011, Benchmark 7.1.2.4

Answer: c DOK: Level 3 Source: Massachusetts Comprehensive Assessment System Release of Spring 2009 Test Items

Assume you borrow $900 at 7% annual compound interest for four years.

• How much money do you owe at the end of the four years? Show or explain your work.
• What is the total interest you will have to pay? Show or explain your work.

Answers: Part A: $900 + $252 = $1179.72; Part B: $279.72 DOK: Level 2 Source: Test Prep: Modified from MCA III Test Preparation Grade 7, Houghton Mifflin Harcourt Publishing Company, Attn: Contracts, Copyrights, and Licensing, 9400 South Park Center Loop, Orlando, FL 32819.

Tim is mixing 1 L of juice concentrate with 5L of water to make juice for his 10 guests. After he pours the mix into 10 different cups, he realizes that the juice is not sweet enough, so he adds 0.1 L of syrup into each of the cups. What is the final amount of juice in each cup?             A. 0.5 L             B. 0.7 L             C. 1.7 L             D. 2.0 L   Answer: b DOK: Level 2 Source: Test Prep: MCA III Test Preparation Grade 7, Houghton Mifflin Harcourt Publishing Company, Attn: Contracts, Copyrights, and Licensing, 9400 South Park Center Loop, Orlando, FL 32819.

Answer: b DOK: Level 2 Source: Minnesota Grade 7 Mathematics MCA - III Item Sampler Item, 2011, Benchmark 7.1.2.5

Answer: b DOK: Level 2 Source: Minnesota Grade 7 Mathematics Modified MCA - III Item Sampler Item, 2011, Benchmark 7.1.2.5

Answer: d DOK: Level 2 Source: Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

Answer: b DOK: Level 2 Source: Massachusetts Comprehensive Assessment System Release of Spring 2010 Test Items

Differentiation

• Provide students with multiplication tables.
• Students may need to be given a place value chart to help them in rounding to the correct place (hundreds vs. hundredths).
• Use pictures and or tables to help the students see the pattern; always label the values they are trying to scale, such as 10 in = 3 ft, 3 in = ________ft. By keeping the labels with the numbers, fewer errors in relationships will be made.
• Students may see the word "ratio" as "radios." Assist them by clarifying the meaning and pronunciation of both words.
• Use a graphic organizer displaying operations with integer rules .
• Introduce √ of a number to plot on a number line.
• Do multi-step conversion-type problems to show proportional relationship.
• Explain the concept of compound interest using exponential growth and exponential equations.
• Create a gameboard activity. Have students work in groups of four to create gameboards marked with mathematical questions they must answer to be able to move ahead on the boards. They should use at least three addition, three subtraction, three multiplication and three division operations. They should also use positive numbers, negative numbers, decimals, and fractions. Students will fill in operation symbols and numbers on the boards. When they are done, the class can play the different games.

Parents/Admin

Administrative/peer classroom observation.

Parent Resources

• Math games, problems and puzzles

Related Frameworks

7.1.2a applying & making sense of rational numbers.

• 7.1.2.1 Arithmetic Procedures
• 7.1.2.2 Explain Arithmetic Procedures
• 7.1.2.3 Calculators & Rational Numbers
• 7.1.2.6 Absolute Value
• 7.1.2.4 Solve Problems with Rational Numbers Including Positive Integer Exponents
• 7.1.2.5 Proportional Reasoning

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Adding and subtracting rational numbers to solve problems, lesson plan.

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• Grade Levels 7th Grade
• Related Academic Standards CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. CC.2.1.7.E.1 Apply and extend previous understandings of operations with fractions to operations with rational numbers.
• Assessment Anchors M07.A-N.1 Apply and extend previous understandings of operations to add, subtract, multiply, and divide rational numbers. M07.B-E.2 Solve real-world and mathematical problems using numerical and algebraic expressions, equations, and inequalities.
• Eligible Content M07.A-N.1.1.1 Apply properties of operations to add and subtract rational numbers, including real-world contexts. M07.B-E.2.1.1 Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50 an hour (or 1.1 × $25 = $27.50).
• Competencies

Students will compute and solve problems using rational numbers. They will:

• add and subtract rational numbers.
• solve real-world problems by adding and subtracting rational numbers.

Essential Questions

• How is mathematics used to quantify, compare, represent, and model numbers? 
• How are relationships represented mathematically?
• How can expressions, equations and inequalities, be used to quantify, solve, and/or analyze mathematical situations?
• What makes a tool and/or strategy appropriate for a given task?
• How can recognizing repetition or regularity assist in solving problems more efficiently?
• Rational Number: A number expressible in the form a / b, where a and b are integers, and b ≠ 0.
• Repeating Decimal: The decimal form of a rational number in which the decimal digits repeat in an infinite pattern.

60–90 minutes

Prerequisite Skills

• Lesson 2 Exit Ticket ( M-7-5-2_Exit Ticket and KEY.docx )
• Lesson 2 Small-Group Practice worksheet ( M-7-5-2_Small Group Practice and KEY.docx )
• Lesson 2 Expansion Worksheet ( M-7-5-2_Expansion and KEY.docx )
• Lesson 2 Computations Worksheet ( M-7-5-2_Computations and KEY.docx )
• Lesson 2 Word-Problem Examples ( M-7-5-2_Word Problem Examples and KEY.docx )

Related Unit and Lesson Plans

• Computing and Problem Solving with Rational Numbers
• Adding and Subtracting Rational Numbers on a Number Line
• Multiplying and Dividing Rational Numbers to Solve Problems

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

• http://ca.ixl.com/math/grade-7/add-and-subtract-rational-numbers

IXL’s Grade 7 Add and Subtract Rational Numbers will give students additional practice with addition and subtraction of rational numbers.

• http://ca.ixl.com/math/grade-8/add-and-subtract-rational-numbers-word-problems

IXL’s Grade 8 Add and Subtract Rational Numbers: Word Problems will give students additional practice with solving word problems that involve rational numbers.

Formative Assessment

• The modeling activity can be used to assess students’ prior knowledge and understanding regarding addition of rational numbers with unlike denominators.
• Activity 1 can be used to assess each student’s ability to create a word problem involving the addition and/or subtraction of rational numbers while also understanding the solution process.
• Use the exit ticket to quickly evaluate student mastery.

Suggested Instructional Supports

Instructional Procedures

As students come into class, have them evaluate the following expressions using a number line.

• 0.75 + 2.95      (3.7)

Walk around the classroom as students are working through the example problems. Briefly discuss the answers and make sure students are comfortable modeling addition and subtraction of rational numbers on a number line before moving on.

“In Lesson 1 of this unit, we learned how to model addition and subtraction of rational numbers on a number line. Today, we are going to focus on performing these computations without the use of a number line. We will then use these skills to solve some real-life problems.”

Computations: Adding and Subtracting Rational Numbers

Before presenting some real-world problems, give students the opportunity to practice adding and subtracting rational numbers without the help of a number line. If necessary, go over the following examples together as a class.

the other is not. Often, when computing with fractions, it is best to write all numbers in fraction form.”

common denominator. The lowest common denominator in this case would be 5.”

numerators as indicated. The denominator will stay as is.”

it is, but we may want to rewrite the fraction as a mixed number to get a better idea of the value.”

Example 2:      − 4.64 + 9.85

• −4.64 + 9.85     “Think about the number line. Based on the signs of each addend, do

you suspect our final answer here will be positive or negative?” (Positive, the absolute value of 9.85 is larger than the absolute value of −4.64.)

• Think:            9.85 – 4.64    
•                             

Distribute the Lesson 2 Computations Worksheet ( M-7-5-2_Computations and KEY.docx ). Instruct students to complete the worksheet individually. Walk around the room as students work to be sure they are on task and performing the computations accurately. Following the worksheet, provide time for students to discuss any problems they encountered, questions they have, or revelations they discovered. First, ask students to describe the computation process used to find each sum or difference. Then confirm their understanding by restating the correct process.

Problem Solving with Rational Numbers

Now it is time for students to apply their understanding of computation to solving real-world problems. Discuss the following examples together as a class.

• “The amount by which his savings have decreased is equal to the difference of 1018.20 and 920.45, written as 1018.20 – 920.45 or 97.75. Thus, Steven’s savings decreased by $97.75.”

Distribute Lesson 2 Word-Problem Examples ( M-7-5-2_Word Problem Examples and KEY.docx ). Have students discuss the solution process for each example problem in a manner similar to the process demonstrated above. Confirm the correct ideas students express. Then say: “Look through the problems you just received. Think of how the example word problems can be solved. Do you need to add or subtract the rational numbers? How will you go about doing this for fractions with unlike denominators, or for mixed numbers?”

Activity 1: Write-Pair-Share

Ask the whole class to think of some real-world contexts that involve the addition or subtraction of rational numbers. Students should make a list of at least five real-world contexts and provide one word problem. Ask students to share their ideas with a partner. Give students about 5 minutes to share contexts and word problems. During this time, each partner may ask questions of the other partner. Then, the whole class can reconvene. One member from each partner group will share the list of real-world contexts and word problems with the class. The teacher may wish to post the real-world contexts and word problems in a file on the class Web page or use them as a classroom display. These student examples would then serve as a reference tool.

Have students complete Lesson 2 Exit Ticket ( M-7-5-2_Exit Ticket and KEY.docx ) at the close of the lesson to evaluate students’ level of understanding.

Use the suggestions in the Routine section to review lesson concepts throughout the school year. Use the small-group suggestions for any students who might benefit from additional instruction. Use the Expansion section to challenge students who are ready to move beyond the requirements of the standard.

• Routine: Throughout the school year, encourage students to be on the lookout for real-world situations that involve the addition or subtraction of rational numbers. Students can present the problems to the teacher, who will facilitate class participation in solving the rational number problem.
• Small Groups: Students who need additional practice can be pulled into small groups to work on the Lesson 2 Small-Group Practice worksheet ( M-7-5-3_Small Group Practice and KEY.docx ). Students can work on the matching together or work individually and compare answers when done.
• Expansion: Students who are prepared for a greater challenge can be given the Lesson 2 Expansion Worksheet ( M-7-5-2_Expansion and KEY.docx ). The worksheet includes more difficult numeric expressions involving rational numbers.

Related Instructional Videos

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