CorrectionKey=B DO NOT EDIT--Changes must be made through ... · When students solve one-step equations, they first look at the equation to identify the operation involved. Then they

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Interactive Whiteboard Interactive example available online

ADDITIONAL EXAMPLE 1Use inverse operations to solve each equation.

A x - 5.6 = -1.7 x = 3.9

B y + 3 __ 4 = 6 y = 5 1 __ 4

C 0.7n = -3.5 n = -5

D c ___ 2.5 = -4.2 c = -10.5

EngageESSENTIAL QUESTION

How do you use one-step equations with rational coefficients to solve problems? Sample answer: You write an equation for the situation and solve the equation by using inverse operations.

Motivate the LessonAsk: What kinds of real-world quantities can negative numbers represent? What kinds of quantities can fractions or decimals represent?

ExploreConnect to Daily LifeHave students suggest a story or problem that might fit the equation -8.5 + x = -2 using a real-world context, such as owing money, degrees below zero, or feet below sea level.

ExplainEXAMPLE 1

Talk About ItCheck for Understanding

Ask: Which operation is shown in each part of the example? What is the inverse operation for each part? Part A: addition; subtraction; Part B: subtraction; addition;

Part C; multiplication; division; Part D: division; multiplication.

Questioning Strategies Mathematical Practices • Does it matter which side of the equals sign the variable is on when you solve the equation? no

• How do you check that your solution is correct? Substitute your answer for the variable in the original equation. If both sides remain equal, then it is the solution.

YOUR TURNEngage with the Whiteboard

Have a student volunteer circle the symbol that represents the operation in each exercise and then identify the inverse operation needed to solve the equation. Be sure to note that the lack of a symbol in exercise 3 means multiplication.

Avoid Common ErrorsSome students may focus on determining which inverse operation is needed to solve the equation, but only perform the inverse operation on the side of the equation that contains the variable. Remind students to perform the same operation on both sides of the equation.

One-Step Equations with Rational Coefficients6.2

L E S S O N

Florida StandardsThe student is expected to:

Expressions and Equations—7.EE.2.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

Mathematical Practices

MP.7.1 Using Structure

179 Lesson 6.2

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–12.2 m

–4.55 m

0 m

Writing and Solving One-Step Addition and Subtraction Equations Negative numbers often appear in real-world situations. For example, elevations below sea level are represented by negative numbers. When you increase your elevation, you are moving in a positive direction. When you decrease your elevation, you are moving in a negative direction.

A scuba diver is exploring at an elevation of −12.2 meters. As the diver rises to the surface, she plans to stop and rest briefly at a reef that has an elevation of −4.55 meters. Find the vertical distance that the diver will travel.

Write an equation. Let x represent the vertical distance between her initial elevation and the elevation of the reef.

−12.2 + x = −4.55

Solve the equation using an inverse operation.

−12.2 + x = −4.55+12.2 +12.2 x = 7.65

The diver will travel a vertical distance of 7.65 meters.

Reflect 4. Make a Prediction Explain how you know whether the diver is

moving in a positive or a negative direction before you solve the equation.

EXAMPLE 2

STEP 1

STEP 2

5. An airplane descends 1.5 miles to an elevation of 5.25 miles. Find the elevation of the plane before its descent.

YOUR TURN

7.EE.2.4

Add 12.2 to both sides.

The initial elevation is less than the elevation of the reef,

so the diver is moving from a lower elevation to a higher

elevation, that is, a positive direction.

x – 1.5 = 5.25; x = 6.75; the initial elevation of the

plane is 6.75 miles.

Unit 3180

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7_MFLESE056701_U3M06L2.indd 180 4/8/14 7:32 PM

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–7.5 = –1.5n

ESSENTIAL QUESTION

L E S S O N

6.2One-Step Equations with Rational Coefficients

One-Step EquationsYou have written and solved one-step equations involving whole numbers. Now you will learn to work with equations containing negative numbers.

Use inverse operations to solve each equation.

x + 3.2 = − 8.5

30 = −0.5a


EXAMPLEXAMPLE 1

Ax + 3.2 = - 8.5 - 3.2 - 3.2 x = - 11.7

B - 2 _ 3 + y = 8

- 2 _ 3 + y = 8

+ 2 _ 3 + 2 _ 3

y = 8 2 _ 3

C 30 = −0.5a -0.5 -0.5 -60 = a

D - q ___ 3.5 = 9.2

- q ___ 3.5 (-3.5) = 9.2 (-3.5)

q = -32.2

How do you use one-step equations with rational coefficients to solve problems?


YOUR TURNYOUR TURNYOUR TURN

1. 4.9 + z = −9 2. r – 17.1 = −4.8

3.

- 3c = 36

7.EE.2.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations … to solve problems by reasoning about the quantities.

7.EE.2.4

Subtract 3.2 from both sides.

Add to both sides. 2 _ 3

Divide both sides by −0.5.

Multiply both sides by −3.5.

z = −13.9 r = 12.3 c = −12

179Lesson 6.2

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PROFESSIONAL DEVELOPMENT

Math BackgroundA solution of an equation is the value of the variable that makes the equation true. Operations that undo each other, such as addition and subtraction, and multiplication and division, are called inverse operations. Inverse operations are used to solve equations. To solve an equation, the operation performed on the variable must be undone, leaving the variable alone, or isolated, on one side of the equation. The value on the other side of the equals sign should be the solution of the equation, if the solution process was performed properly.

Integrate Mathematical Practices MP.7.1

This lesson provides an opportunity to address this Mathematical Practice standard. It calls for students to look for and make use of structure. When students solve one-step equations, they first look at the equation to identify the operation involved. Then they identify and use the inverse operation to solve the equation.

One-Step Equations with Rational Coefficients 180




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ADDITIONAL EXAMPLE 2Amy has some money in her checking account. If she writes a check for $42.50, her checking account will be overdrawn by $23.75. Find how much money is in Amy’s checking account now. x - 42.50 = -23.75; Amy's checking account contains $18.75.

ADDITIONAL EXAMPLE 3Tyler received a notice from his bank that the balance in his checking account was –$12, not $0 as he expected. He realized that he forgot to record the time when he used his debit card to buy bus tokens for $1.50 each. How many bus tokens did Tyler buy? -1.50t = -12; Tyler bought 8 bus tokens.

EXAMPLE 2Focus on Modeling Mathematical PracticesHave students draw a diagram that illustrates the problem or explain how to use the given illustration.

Questioning Strategies Mathematical Practices • Why is x added to -12.2 in the equation in Step 1? Sample answer: It indicates that the scuba diver is rising in a positive direction.

• Explain why the equation -12.2 + x = -4.55 can be rewritten as x - 12.2 = -4.55. By the Commutative Property of Addition, the addends can be switched, and subtracting is the same as adding the opposite.

YOUR TURNConnect Vocabulary ELL

Make sure students understand that the word descends means that the airplane is going down, indicating subtraction.

EXAMPLE 3Talk About ItCheck for Understanding

Ask: What is the product when you multiply a fraction by its reciprocal? The product is 1.

Questioning Strategies Mathematical Practices • Explain why - 3 __ 4 is multiplied by - 4 __ 3 instead of being multiplied by - 3 __ 4 . You need to multiply by the reciprocal of the coefficient of x to get a coefficient of 1. The product of - 3 __ 4 and - 3 __ 4 is not 1.

• How many hours will it take for the temperature to decrease by 5 °F? 6 2 __ 3 hours

YOUR TURNFocus on ModelingMake sure students understand how to express the relationship for each situation. Remind them to use unit analysis to help them write the correct equation.

ElaborateTalk About ItSummarize the Lesson

Ask: Which operations are inverses of each other? Why must you perform the same operation on both sides of the equation when isolating the variable? Sample

answer: Addition and subtraction are inverse operations that undo each other. Multiplication and division are inverse operations that undo each other. In order for the rewritten equation to have the same solution as the original, you must keep the equation in balance by performing exactly the same operation on both sides.

GUIDED PRACTICEEngage with the Whiteboard

Have a student cross out the unnecessary information in the table in Exercise 1.

Avoid Common ErrorsExercise 2 Remind students to multiply by negative two thirds, not two thirds, to isolate x.

181 Lesson 6.2


Guided Practice

The table shows the average temperature in Barrow, Alaska, for three months during one year.

1. How many degrees warmer is the average temperature in November than in January? (Examples1 and 2)

Write an equation. Let x represent

.

x + = , or x - =

Solve the equation. Show your work.

The average temperature in November

is warmer.

2. Suppose that during one period of extreme cold, the average daily temperature decreased 1 1 _ 2 °F each day. How many days did it take for the temperature to decrease by 9 °F? (Examples 1 and 3)

Write an equation. Let x represent

.

x =

Solve the equation. Show your work.

It took days for the temperature to decrease by 9 °F.

Use inverse operations to solve each equation. (Example 1)

3. -2x = 34 4. y – 3.5 = −2.1 5.

2 _ 3 z = -6

6. How does writing an equation help you solve a problem?

STEP 1

STEP 2

STEP 1

STEP 2

ESSENTIAL QUESTION CHECK-IN???

Month Average Temperature (°F)

January -13.4

June 34.0

November -1.7the number

the number of days

(-13.4) 13.4 -1.7-1.7

11.7 °F

-1 1 _ 2 -9

Sample answer: It helps me describe the problem

6

x = -17 y = 1.4 z = -9

x - 13.4 = -1.7

+ 13.4 +13.4

x = 11.7

( - 2 _ 3 ) ( - 3 _ 2 x ) = ( - 2 _ 3 ) (-9)

x = 18 __ 3

x = 6

it takes the average temperature to decrease by 9 °F

of degrees warmer the average temperature is in Nov. than in Jan.

precisely and solve it using inverse operations.

Unit 3182

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Writing and Solving One-Step Multiplication and Division ProblemsTemperatures can be both positive and negative, and they can increase or decrease during a given period of time. A decrease in temperature is represented by a negative number. An increase in temperature is represented by a positive number.

Between the hours of 10 P.M. and 6 A.M., the temperature decreases an average of of a degree per hour. How many minutes will it take for the temperature to decrease by 5 °F?

Write an equation. Let x represent the number of hours it takes for the temperature to decrease by 5 °F.

Solve the equation using an inverse operation.

Convert the number of hours to minutes.

It takes 400 minutes for the temperature to decrease by 5 °F.

EXAMPLEXAMPLE 3

3 _ 4

STEP 1

- 3 _ 4 x = -5

STEP 2

- 3 _ 4 x = -5

- 4 _ 3 ( - 3 _ 4 x ) = - 4 _ 3 (-5)

x = 20 __ 3

STEP 3

20 __ 3 hours × 60 minutes ________ 1 hour

= 400 minutes

6. The value of a share of stock decreases in value at a rate of $1.20 per hour during the first 3.5 hours of trading. Write and solve an equation to find the decrease in the value of the share of stock during that time.

7. After a power failure, the temperature in a freezer increased at an average rate of 2.5 °F per hour. The total increase was 7.5 °F. Write and solve an equation to find the number of hours until the power was restored.

YOUR TURN

Why is multiplying by - the inverse of multiplying by - ?

4 _ 3 3 _ 4

Math TalkMathematical Practices

Why is multip

Math Talk

7.EE.2.4

Multiply both sides by - . 4 _ 3

x ___ 3.5 = -1.2; x = -4.2; $4.20

2.5x = 7.5; x = 3; 3 hours

Multiplying by - is the same as dividing by - .

4 _ 3 3 _ 4

181Lesson 6.2

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ManipulativesHave students use an actual balance scale to emphasize the concept of keeping both sides equal, or balanced. Write the equation x + 5 = 13 on the board. Have students place 5 tiles on the left side and 13 on the right side. Have students stack tiles on the side with 5 tiles until the scale balances. When they place 8 more tiles on the scale, the scale should balance. So, the solution to x + 5 = 13 is x = 8.

Visual CuesSuggest that students use colored pencils to circle the number that must be moved (or operated on) in order to isolate the variable. Then students can use a different color to write the step of performing the inverse operation on both sides of the equation.

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP

DIFFERENTIATE INSTRUCTION




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Lesson Quiz available online

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6.2 LESSON QUIZ


1. a - 5 __ 6 = -3 2 __ 3

2. k + 7.2 = 3.4

3. - 3 __ 5 d = 15

4. w ____ -1.3 = -6.2

5. The height of the water in an above ground pool is 3 feet. The pool needs to be drained. As the water drains, the height of the water changes at a rate of - 1 __ 2 inch per minute. Write and solve an equation to find how many minutes it will take to drain the pool.

6. The melting point of the chemical bromine is -7.2 °C. The boiling point of bromine is 58.8 °C. Write and solve an equation to find how much greater the boiling point of bromine is than the melting point.

EvaluateGUIDED AND INDEPENDENT PRACTICE

Concepts & Skills Practice

Example 1One-Step Equations

Exercises 1–5

Example 2Writing and Solving One-Step Addition and Subtraction Equations

Exercises 1, 6–9, 12–15

Example 3Writing and Solving One-Step Multiplication and Division Equations

Exercises 2, 10, 11, 16, 18

Additional ResourcesDifferentiated Instruction includes: • Leveled Practice Worksheets

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

7 2 Skills/Concepts MP.4.1 Modeling










17 3 Strategic Thinking MP.7.1 Using Structure


19 3 Strategic Thinking MP.3.1 Logic

20 3 Strategic Thinking MP.7.1 Using Structure

21 3 Strategic Thinking MP.4.1 Modeling

7.EE.2.4

7.EE.2.4

Answers1. a = -2 5 __ 6

2. k = -3.8

3. d = -25

4. w = 8.06

5. - 1 __ 2 n = -36; 72 minutes

6. -7.2 + x = 58.8; 66 °C

183 Lesson 6.2


Work Area

16. A diver begins at sea level and descends vertically at a rate of 2 1 _ 2 feet per second. How long does the diver take to reach

-15.6 feet?

17. Analyze Relationships In Exercise 16, what is the relationship between the rate at which the diver descends, the elevation he reaches, and the time it takes to reach that elevation?

18. Check for Reasonableness Jane withdrew money from her savings account in each of 5 months. The average amount she withdrew per month was $45.50. How much did she withdraw in all during the 5 months? Show that your answer is reasonable.

19. Justify Reasoning Consider the two problems below. Which values in the problems are represented by negative numbers? Explain why.

(1) A diver below sea level ascends 25 feet to a reef at -35.5 feet. What was the elevation of the diver before she ascended to the reef?

(2) A plane descends 1.5 miles to an elevation of 3.75 miles. What was the elevation of the plane before its descent?

20. Analyze Relationships How is solving -4x = -4.8 different from solving - 1 _ 4 x = -4.8? How are the solutions related?

21. Communicate Mathematical Ideas Flynn opens a savings account. In one 3-month period, he makes deposits of $75.50 and $55.25. He makes withdrawals of $25.15 and $18.65. His balance at the end of the 3-month period is $210.85. Explain how you can find his initial deposit amount.

FOCUS ON HIGHER ORDER THINKING

6.24 sec

Sample answer: the elevation is the product of the rate and the time.

$227.50; sample answer: $45.50 ≈ $50, and $50 × 5 = $250,

which is close to $227.50.

(1) The elevations of the diver and the reef; both are

below sea level. (2) The change in the plane's elevation;

the plane is moving from a higher to a lower elevation.

In the first case, you divide both sides by -4. In the

second, you multiply both sides by -4. The second

solution (19.2) is 16 times the first (1.2).

Add the deposits and the withdrawals. Let x represent

the amount of the initial deposit. Write and solve the

equation x + deposits - withdrawals = $210.85.

Unit 3184

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Name Class Date

Independent Practice6.2

The table shows the elevation in feet at the peaks of several mountains. Use the table for 7–9.

Mountain Elevation (feet)

Mt. McKinley 20,321.5

K2 28,251.31

Tupungato 22,309.71

Dom 14,911.42

7. Mt. Everest is 8,707.37 feet higher than Mt. McKinley. What is the elevation of Mt. Everest?

8. Liam descended from the summit of K2 to an elevation of 23,201.06 feet. How many feet did Liam descend? What was his change in elevation?

9. K2 is 11,194.21 feet higher than Mt. Kenya. Write and solve an equation to find the elevation of Mt. Kenya.

10. A hot air balloon begins its descent at a rate of 22 1 _ 2 feet per minute. How long will it take for the balloon's elevation to change by -315 feet?

11. During another part of its flight, the balloon in Exercise 10 had a change in elevation of -901 feet. What was its rate of descent?

The table shows the average temperatures in several states from January through March. Use the table for 12–14.

State Average Temperature (°C)

Florida 18.1

Minnesota -2.5

Montana -0.7

Texas 12.5

12. Write and solve an equation to find how much warmer Montana's average 3-month temperature is than Minnesota's.

13. How much warmer is Florida's average 3-month temperature than Montana's?

14. How would the average temperature in Texas have to change to match the average temperature in Florida?

15. A football team has a net yardage of -26 1 _ 3 yards on a series of plays. The team needs a net yardage of 10 yards to get a first down. How many yards do they have to get on their next play to get a first down?

7.EE.2.4

29,028.87 ft

26 1 _ 2 feet per minute

-2.5 + x = -0.7; x = 1.8; 1.8 °C warmer

18.8 °C warmer

It would have to increase by 5.6 °C.

36 1 _ 3 yards

5050.25 ft; -5050.25 ft

28,251.31 - x = 11,194.21;

x = 17,057.1; 17,057.1 feet

14 minutes

183Lesson 6.2

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Activity available online my.hrw.comEXTEND THE MATH PRE-AP

Activity Solve the equations in order. Use the value of the variables you find in each of the following equations.

1. Solve for a: 3a = -4.2

2. Solve for b: a + b = -7.8

3. Solve for c: 1 2 __ 3 c = a-b

4. Solve for d: a + b _ c = -2d

1. a = -1.4

2. b = -6.4

3. c = 3

4. d = 1.3



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CorrectionKey=B DO NOT EDIT--Changes must be made through ... · When students solve one-step equations, they first look at the equation to identify the operation involved. Then they