Embedded Files
LESSON OVERVIEW
Learning Goals
At the end of the lesson, students will be able to...
Define the natural logarithm from the exponential function
Use natural logarithms in algebraic and practical contexts
Solve equations and natural logarithmic expressions
Content Descriptors
E1.3: The exponential function and natural logarithms
Work with natural logarithms in a variety of practical and abstract contexts
define the natural logarithm ln(x) = log_e(x) from the exponential function f(x) = e^x (ACMMM159)
recognise and use the inverse relationship of the functions y=e^x and y = ln(x) (ACMMM160)
use the natural logarithm and the relationships e^(ln(x)) = x where x > 0, and ln(e^x) = x for all real x in both algebraic and practical contexts
use the logarithmic laws to simplify and evaluate natural logarithmic expressions and solve equations
Working Mathematically
Recognises and uses concepts taught before to relate to new information and make understandings and connections to new content (Understanding)
Uses understanding and practice in a variety of scenarios to evaluate, explain, and reach conclusions about content
Prior Knowledge
An understanding of logarithms as a function and logarithm laws
Features of the logarithm function learned in earlier lessons and it relationships with the exponential function
Potential Misconceptions/Student Challenges
Misconceptions with logarithm laws, treating log as a variable whereas the other is by possibly participants thinking log(x)/log(y) = log(x/y). (Liang and Wood, 2005)
Students struggle with the topic of logarithms in general. (Hewson, 2013).
LIT
Students understanding and defintion of a logarithm scale and Richter scale should be expanded
NUM
Body activity 1 should reinforce and expand students skills using logarithmic laws to simplify and evaluate natural logarithmic expressions and solve equations
ICT
Website - Desmos
LESSON STRUCTURE
Introduction (10 minutes) - Revision Activity
Body Activity 1 (20 minutes) - Practice Solving Equations
Body Activity 2 (20 minutes) - Richter scale investigation
Conclusion (10 minutes) - Exit slip (AFL)
LESSON ACTIVITIES
Introduction Activity: Revision of natural logarithm features
Duration: 10 minutes
Resources
Desmos: https://www.desmos.com/calculator
Activity Description
The purpose of this activity is to activate students' prior knowledge which will be necessary for this lesson. Students will be given a task to review the following concepts: the inverse relationship between logarithmic and exponential functions: y=log_a x and y=a^x, the graphical relationship between y=log_e x and y=e^x, the equivalence of y=e^x in exponential form and log_e y=x in logarithmic form. Students will use graphing software Desmos to look at this relationship and sketch the graphs of y = e^x, y = log_ex and y = x on the same set of axes to reflect the exponential curve in the line y = x, and will also compare the graph of the natural logarithm with logarithms in other bases.
Teacher Action
Teacher will instruct students to use Desmos to explore the relationship between logarithmic and exponential functions
Teacher will monitor students as they use Desmos and make sure all students understand the relationships and what they are investigating and making sure all students recall what they have already learned
Student Action
Students will use Desmos for investigation and understanding
Body Activity 1: Using logarithm laws to simplify and evaluate natural logarithmic expressions and solve equations
Duration: 20 minutes
Resource:
Math In Focus year 11 Advanced textbook
Questions from exercise 8.04 Logarithms
or
MathsOnline video and questions if appropriate
Activity Description
The purpose of this activity is to give students practice using logarithm laws and evaluating natural logarithmic expressions and solving equations. It gives students the time to also reinforce these laws. Students will be able to complete questions from textbook to have practice with this concept and teacher will show students how to use calculator and how to solve some of these questions on board. If there is an appropriate MathsOnline task this can be used to use video to demonstrate concept to students and give them questions to practice. The results of students on MathsOnline are immediately available to teacher to assess which students are doing well and which are struggling.
Teacher Action
Teacher will provide students with questions to complete from textbook or provide MathsOnline
Teacher will monitor students and help students who are struggling and make sure students are on track
Student Action
Students will complete questions from textbook or MathsOnline
Differentiated Version
Hard
Students doing well can access more challenging questions, either via the textbook or the MathsOnline will present more difficult questions for them
Easy
Students who are struggling are able to keep attempting MathsOnline questions which will progress only at the level they are understanding
Body Activity 1 Worksheets
Worksheet 5.1
(Grove, 2018)
Body Activity 2: Richter Scale Investigation
Duration: 20 minutes
Resources:
Activity Description
The purpose of this activity is to introduce students to how logarithms are linked to earthquakes and the Richter scale and provide them with the opportunity to work with logarithms in this context. Students will understand how natural logarithms are linked and used in the Richter scale for determining the magnitude of the earthquake and will also practice solving expressions and equations that determine the magnitude of earthquakes using logarithms.
Teacher Action
Teacher will provide students with link to website for them to read through section on "Logarithms and Earthquakes."
Teacher will then provide students with more practice questions with data on earthquakes for students to use the logarithm formula to find either intensity or magnitude of earthquakes
Teacher will monitor students and provide assistance where needed
Student Action
Students will take their time to understand how logarithms are linked to Richter scale and understand the formula
Students will practice using formula with a variety of different data
Body Activity 2 Sample from website
Sample from website, more questions and data using formula below provided
Conclusion Activity: Exit Cards (AFL)
Duration: 10 minutes
Activity Description
The purpose of this activity is to assess what students have learned in the lesson. It gives an opportunity for students to reinforce what they have learned in the lesson and think about anything they may not have understood quite well. What concepts did they retain/understand? What did they want further clarification on? The teacher can also make findings on concepts that students may not have understood as well and from this information, and an activity can be incorporated into the next lesson if there is something needing to be addressed.
Teacher Action
The teacher hands out Exit Cards and allows the students some time to reflect on the work for the day and what they have learned
Teacher will instruct students to write down their answers to the Exit Cards.
Students will be reminded this should include all activities done throughout the lesson
Student Action
Students reflect on their work for the lesson and fill out the Exit Cards
Students hand these to the teacher as they leave the class
Conclusion Activity Worksheets
Worksheet 5.2
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