Alg2 Lesson 2.3: Inverting Operations

Lesson 2.3: Inverting

Operations

Obj: I can use inverse operations to solve equations.

EQ: How do I use logarithmic operations to solve
exponential equations?

Roles:
Facilitator
Scribe
Resourcer
Includer

Lesson Goals:
● Creative Thinking
● Talk through controversies and conflict
● Recognize and reduce ambiguity
● Encourage thinking based on formulas and prior info
● Help explain ideas to each other
● Own your ideas and work
● Record ideas in your journal
● Answer Questions on Slides
● Follow your team roles

Facilitator

• Make sure that all peers are staying on task.

• Give advice or suggestions to resolve the problem.

• Be sure everyone is able to explain.

Scribe

• Make sure peers organize their results on their own papers.

• Remind peers to use color, arrows, and other math tools to
communicate your mathematics, reasons, and connections.

• Be ready to join the teacher for a huddle.

Resourcer

• Make sure peers are getting the materials needed.

• Make sure that all materials are put away neatly.

• Make sure that peers are logged in to the needed site.

• Help troubleshoot any technology difficulties that may arise.

Includer

• Make sure that all peers are talking about their work.

• Helps keep peers’ voice volume low.

• Encourages everyone to ask questions.

• Communicates conflicts or questions to the teacher.

● Check off tasks & skills on calendar.

● Select skills to work on.

● Work on Deltamath.

Remember to work on the following too…

Poll

Which best describes what you'll do during this lesson?

• Includer

• Facilitator

• Scribe

• Resourcer

• I'm working on this ahead of time at home

Part 1: Basic Inverting

Operations

Inverses are Opposites
that undo each other.

Write the opposite of each scenario.

Scenario

Inverse

Turn on the TV.

Place a book in your backpack.

Type the word “math” into your device.

Add 3.

Multiply by 5.

Subtract 2.

Divide by -4.

Square a number.

Put a sock on, and then put a shoe on.

Add 3 to a number, and then multiply by 5.

Square a number, and then subtract 2.

One way of thinking about an equation is as a statement or a question. For each of the following equations, one partner should write a sentence that translates the mathematical equation into words. You do not need to solve the equations. The other partner should then write the inverse actions out in words. Be ready to share your statements and inverse actions with the class.

Eq into Words

Equation

Inverse Actions of Words

A number multiplied by 3 and
added to 4 equals 19.

3x + 4 = 19

Subtract 4 from 19, and then divide by 3.

x/5 - 1 = -7

x² + 1 = 50

(x + 4)² - 5 = 20

5(x - 3)² + 2 = 11

2^x = 8

Word Cloud

Write two things that are opposites of one another here.

Part 2: Introducing a New

Inverting Operation

We examined the equation 2^x = 8 and realized we
do not know any inverse operations to use to solve for x. Work with a partner to complete the following problems to determine an estimate for the value of x.

Write these & answer them in your journal…

1.

Create a possible scenario that can be modeled by 2^x = 8. Describe in words a context that could lead to this mathematical equation.

2.

Determine a low estimate and a high estimate for the value of x. Remember to consider input and output values.

3.

Discuss whether you think the value of x is closer to your low estimate or closer to your high estimate and explain why.

4.

Write 2^x = 8 as a system of two equations in two variables. Then graph this system of equations using a graphing utility.

5.

Look for the intersection point of the two equations you graphed. What is it? How does this compare with the estimated value of x you settled on for problem 3?

Random Question of the Day Time

https://wheelofnames.com/4ke-epz We’ll spin the wheel as a class and spend a minute or so
discussing our answers.

Part 3: Evaluating

Logarithmic Expressions

Handout 2.3.A: Exploring Logarithmic Expressions

For each of these logarithmic expressions, complete the following:

(a) Write an exponential equation that the logarithmic expression can be used to solve.

(b) Determine the value of each logarithmic expression. Show how you determined your answer.

(c) If the value is not rational, explain how you determined an approximate value.

Drag and Drop

log₁₀(100) = ​

Placeholder

log₅(5) = ​

Placeholder

log₆(1/36) = ​

Placeholder

log₂(1) = ​

Placeholder

Drag these tiles and drop them in the correct blank above

2

1

-2

0

1/2

6

5

Drag and Drop

log₂(13) = ​

Placeholder

log₇(35) = ​

Placeholder

log₄(1/32) = ​

Placeholder

log₂(-4) = ​

Placeholder

Drag these tiles and drop them in the correct blank above

0

Between 3 and 4

Between -3 and -2

Undefined

Between 1 and 2

-2

5

8

Poll

Which expression(s) on Handout 2.3.A were the most challenging for you?

• 1

• 2&4

• 3&7

• 5&6

• 8

Drag and Drop

Problem 8 on Handout 2.3.A is undefined since it is ​

Placeholder

to raise 2 to any real number power to yield ​a negative value.

Given log_b(c) where b > 0, what value(s) of c make this expression undefined? ​

Placeholder

Given log_b(b) where b > 0, what is always true? The expression is ​

Placeholder

.

Given log_b(1) where b > 0, what is always true? The expression is ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

not possible

possible

c ≤ 0

c < 0

1

0

undefined

Handout 2.3.B: Using Logarithms to Solve Exponential Equations

For each of the following equations, express the solution as a logarithmic expression. Then determine its exact or
approximate value to three decimal places.

Dropdown

2^x = 32; x = ​

Select answer

18^x = 18; x = ​

Select answer

10^x = 1/100; x = ​

Select answer

3^x = 1; x = ​

Select answer

Dropdown

2^x = 1/76; x = ​

Select answer

3^x = 80; x = ​

Select answer

5^x = 120; x = ​

Select answer

6^x = -36; x = ​

Select answer

Multiple Choice

What is the relationship between 4^x = 20 and x = log_4(20)?

• They are inverses of each other, they are equivalent.

• They are inverses of each other, they are not the same.

Common Log Vs Natural Log

Log(x) is the same as Log_10(x). This is known as the common logarithm.

Ln(x) is the same as Log_e(x). This is known as the natural logarithm.

e is Euler’s number, an irrational number commonly used in continuous
compounding in scenarios such as population growth.

Key Takeaways

● We now know there is an inverting operation, called taking

a logarithm, that is used as an inverse operation to undo an exponential expression.

● Evaluating a logarithmic expression is equivalent to

solving its related exponential equation.

Poll

How do you feel about today's lesson?

• I can explain it!

• I feel confident about it!

• I need more practice.

• I need to go back and do the slides over..