Lesson 1: Introducing logarithms

Learning Goals

At the end of the lesson, students will be able to...

• Recall and sketch graphs involving logarithmic functions.

• Solve problems where they are required to recognise and apply the inverse relationship between logarithms and exponentials.

ICT

• Website (Interactive) - GeoGebra

• Website (Informative + questions) - Khan Academy

LESSON STRUCTURE

1. Introduction (10 minutes) - Diagnostic activity

2. Body Activity 1 (15 minutes) - Logs as indices

3. Body Activity 2 (15 minutes) - Inverse relationships of logs and exponentials & sketching

4. Conclusion (10 minutes) - Frayer model (AFL)

Duration: 10 minutes

Resource: Index laws from Australian Mathematical Science Institute (AMSI) (2011)

Activity Description

This diagnostic activity aims to gauge the prior learning and understanding of students on index laws from Stage 4, and logarithms/exponentials. Further, this activity is intended as a refresher and warm-up that calcifies the foundational knowledge and skills required for logs/exponentials.

Teacher Action

• The teacher starts the lesson by showing the [incomplete] index laws to the class and hands out the activity card between pairs/groups of threes (if odd).

• When students have completed, go through with the class on the following:

  • Fractional indices

  • Negative indices (incl. fractional powers)

  • Writing logs as indices

Student Action

• Students participate in a teacher posed questions to the class.

• Students work on the filling in the gaps for the index laws in pairs

Differentiated version

• More support needed?

  • The equations can be cut so students can match them up instead.

• Extension:

  • Students can work on the activity individually.

Duration: 15 minutes

Resources:

1. KhanAcademy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/a/intro-to-logarithms?modal=1

2. Worksheet 1.1 - Exercise 8.05 (Grove, 2018)

Activity Description

The purpose of the activity is to introduce students to the concept of logarithms as indices, connecting the content refreshed from the introduction activity to this new concept. The Khan Academy activity is compiled of an information section, questions to check student's knowledge and also an explanation, so that students can see how the answer can be achieved and can immediately learn from their own mistakes then and there without having to wait for a teacher to check so that they can address these potential issues as they occur.

Teacher Action

• The teacher

  • introduces the concept of logarithms, connecting this to student's knowledge of incides through the example of: 2^4 = 16 <=> log_2(16) = 4

  • dissects the features of the expression with the logarithm, identifying the base, exponent and power in both representations of the above example.

  • starts students off on the activity

  • actively observe student's progress and address any questions or provides clarification of answers provided by Khan Academy.

Student Action

• Students

  • write down notes from the initial explanation of logs

  • works through the activity by

    1. reading the initial information on the site and taking down some notes

    2. answering questions to test their understanding

    3. copying their answers into their books alongside annotations of correct solutions if applicable so that if reviewed later on they can see how it should have been solved and learn from these mistakes

Differentiated Version

• More support needed?

  • If students are struggling with the online nature of Khan Academy and/or work better with a more structured format of learning, they can instead attempt some textbook exercises from Worksheet 1.1 below questions 1-3.

• Extension:

  • If students have time, they complete a small independent inquiry about where logarithms are in the world around them and/or how they are used in everyday scenarios.

Worksheet 1.1 - Easier version

Duration: 15 minutes

Resource:

1. Geogebra - https://www.geogebra.org/m/wqmpavbx

2. Worksheet 1.2 - Exercise 8.06 (Grove, 2018)

Activity Description

The purpose of this activity is to build on students' familiarity with the index laws and highlight the inverse relationship between functions of the form y = a^x and x = log_a(y) through graphical representation. This activity also allows students to have their own go at exploring log graphs and developing their graphing skills.

Teacher Action

• The teacher

  • displays an image of a logarithmic graph on the board and explains and annotates features of the graph. These features include the presence of an asymptote and the nature of the changing slope of the curve.

  • explains the graph with an everyday example by asking students:

    • "Remember when we use graphs to represent a journey that you may go on? Well, how may we do the same with a graph looking like this? Can someone suggest a story to go with this graph?"

    • Possible student answer: someone has been released down a water slide and their speed is initially super fast and begins to slow down as the slide flattens out at the bottom. The graph would start at the point where the person starts down the slide.

  • works through an example of a question involving logarithmic graphs (see Worksheet 1.2 examples)

    • depending on how well the class grasps the concepts, there are a few examples that the teacher can work through

  • tasks students to try solving a few questions involving logarithmic graphs and/or sketching these graphs (see Worksheet 1.2)

• After this part of the activity, the teacher

  • identifies that there is another graph called an exponential graph that students will explore in a later lesson and that this graph is inversely related to the logarithmic graph

  • gives a demonstration using the Geogebra resource. This resource demonstrates this inverse relationship.

  • provides students with the link so that they can toggle with this interactive at their own pace.

Student Action

• Students

  • take notes when applicable

  • actively listen and participate in student-teacher discussion

  • participates in the questions from Worksheet 1.2

  • comment on and observe the teacher demonstration of the inverse relationship Geogebra interactive

  • toggle with this interactive at their own pace if time or after class

Differentiated Version

• More examples can be supplied for the students of logarithmic graphs before continuing on to answering questions involving these

• Students can complete as few or as many of the questions from Worksheet 1.2 as they can. Students up for a challenge will work to complete the latter half of the exercise.

Worksheet 1.2

Duration: 10 minutes

Activity Description

The purpose of this activity is to check on how do students interpret and communicate concepts and laws. Students' Frayer models can reveal potential misconceptions and provide a general consensus as to where students are at overall. Such data is helpful in curating the next lesson if there is something needing to be addressed.

Teacher Action

• The teacher hands out the Frayer model (p. 2 of Worksheet 1.2 below) shared between 2-3 students (ideally in pairs), informing students that they will be working together in reining their prior knowledges and what they could remember during the lesson in the models.

Student Action

• Students collaboratively reflect on their work for the lesson and fill out the Frayer model in their respective groups.

• Students hand these to the teacher as they leave the class

Worksheet 1.2 - Frayer model (p. 2)