Lesson 3: Logarithmic laws and applications (Pt. 2)

Learning Goals

At the end of the lesson, we will be able to:

• Explain the usefulness of logarithmic scales to represent data in a simpler manner.

• Recognise situations in which logarithmic laws are relevant.

• Apply logarithmic rules to solve questions involving logarithms in a variety of contexts.

ICT

• Students may use ICT to complete further research in the Challenge Yourself section of Worksheet 3.1

LESSON STRUCTURE

1. Introduction (10 minutes) - What is a log scale? (LIT)

2. Body Activity 1 (15 minutes) - Everyday log scales

3. Body Activity 2 (20 minutes) - Practice makes perfect (NUM)

4. Conclusion (5 minutes) - Exit tickets (AFL)

Duration: 10 minutes

Resource

Example 3.a (see below)

Activity Description

This activity aims to link the concept of a logarithmic graph with the new concept of logarithmic scales. This activity relates to students' prior knowledge of what a linear scale is to compare and contrast it to the existence and function of a logarithmic scale. This activity is also a literacy activity as it involves breaking down, establishing and defining key mathematical terminology as well as looking at a standard exemplar.

Teacher Action

• The teacher

  • draws up a table on the whiteboard with two columns, linear scale vs logarithmic scale.

  • asks students what they think a linear scale is and write down some characteristics of a linear scale. Include a visual example.

    • see example 3.a for example responses

  • asks students what they think a logarithmic scale is

    • not as many responses or correct answers are expected here since this is a new concept

  • explains what a logarithmic scale is and writes up some features as well as drawing an example

    • see example 3.a below for example responses

    • emphasis on the concept of magnitudes

Student Action

• Students

• actively participate in class discussion, answering questions posed by the teacher.

• write down the comparison table in their books for future reference.

Example 3.a - Sample Whiteboard Notes

Duration: 15 minutes

Resource:

Worksheet 3.1 (see below)

Activity Description

The purpose of this activity is to provide students with real-world examples and applications of the logarithmic scale and requires students to interact with these scales to explain how they are used to represent certain data. The focus of this activity is on the logarithmic pH scale where students will be interpreting the scale and using it to answer provided questions individually. This activity builds on this lesson's introduction, tasking students to solve questions involving logarithmic scales.

Teacher Action

• The teacher

  • introduces the concept of logarithmic scales being present in everyday contexts that students may have previously been unaware of.

  • tasks the class with completing Worksheet 3.1

    • while this is occurring, the teacher walks around to ensure students are on track or if they need any help

Student Action

• Students complete Worksheet 3.1 individually.

Differentiated Version

• Easier version

  • Students can work with the person next to them if they are struggling to answer the questions.

  • Alternatively, the teacher can ask students to try and visualise the problems by drawing out actual number line scales and getting students to think about how the problem can be represented on the scale.

• Harder version

  • Worksheet 3.1 includes an extension activity for students who either finish their work early or for students who are wanting some more challenging questions

  • After this, students can explain their findings from this part of the worksheet to the teacher.

Worksheet 3.1

Worksheet 3.1 - Solutions

Duration: 20 minutes

Resource:

Maths In Focus Mathematics Advanced Year 11 - Exercise 8.07 (Grove, 2018).

(See Worksheet 3.2 below)

Activity Description

The purpose of this activity is to build student fluency through the application of logarithmic principles developed throughout the lesson which in turn assists the development of their numeracy skills. Therefore, students will be answering questions individually that are abstract and some that are grounded in real-life applications.

Teacher Action

• The teacher

  • sets the context of the activity with a scaffolded approach using Worksheet 3.2

    • Start by first completing the first example (example '20' on the worksheet) step by step using method 1.

    • Follow this up by asking whether students think there is another way to solve this example.

    • Solve the same problem again using method 2.

    • Pose the second example (example '21' on the worksheet) to the class and this time instead of having it solved only by the teacher, ask students to tell them what they should do next to solve the problem and why. Asking the why is important to gauge their level of conceptual understanding of the content.

  • After working through both examples, hand out Worksheet 3.2 and ask students to have a go independently at solving the questions in the exercise.

Student Action

• Students will

  • actively listen and participate as the teacher works through examples at the start of the activity.

  • work through the questions on Worksheet 3.2

• If students feel comfortable with their ability to answer certain questions in the worksheet, they can do every second part of that question so that they can allocate more time on solving the other questions that may require more time and thinking for them to solve.

• If students finish, they should mark their work and use this feedback as a personal indicator of areas that need further practice. Otherwise, they should complete the worksheet for homework and mark it once finished.

Differentiated Version

• Easier version

  • examples of how to solve (a) parts of questions OR first steps to solving them provided

    • if students are struggling, the teacher can go through an example of the first part of the question or write the first step/s to guide the student's thoughts as to how to approach and begin to solve the question.

• Harder version

  • If students finish the worksheet

    • they should try and graph the key function in questions 5 to 10 from Worksheet 3.2

Worksheet 3.2

Worksheet 3.2 - Solutions

Duration: 5 min

Resource:

Exit Tickets - Worksheet 3.3

Activity Description

The aim of this activity is to provide students with an opportunity to reflect upon their learning during the lesson as they write responses to the exit tickets. Additionally, when students hand in their exit tickets to the teacher, this enables the teacher to assess each student's learning and development from that lesson and use these responses to address common issues towards the start of the next lesson (lesson 4).

There are four parts to the exit ticket which are as follows:

a) Something old

(What is one thing you knew prior to this lesson or one thing you learnt previously that helped you in this lesson?)

In this section, the teacher is looking to see whether students are able to identify the connections to the previous two lessons on logarithms for this lesson and/or if they are able to identify the fact that this lesson is dependent on other concepts that they have learnt which is important in order to develop a stronger conceptual understanding of a concept.

b) Something new

(What is one thing new that you have learnt this lesson?)

In this section, the teacher is checking whether students were able to recall one of the key concepts/big ideas from the lesson. Therefore, the teacher can tell whether they need to re-emphasise or be more clear when explaining concepts.

c) Something wonderous

(What is one part of the lesson that you feel you need some further explanation about or would like to explore more?)

In this section, the teacher is checking if students are able to critically reflect on the content and their own learning interactions with the content. This is mostly just a tool to assist students to become more critical learners.

d) Something to review

(What is one thing in the lesson that you feel you need to work on?)

In this section, the teacher is specifically looking for potential student misconceptions or misunderstandings about parts of the lesson. This is important so that the teacher can address these as early as possible and the tickets also enable the teacher to also determine whether this is a common issue throughout the class or something that the teacher can work on individually with a student to assist with their learning of that concept.

Teacher Action

• The teacher

  • asks students to start reflecting on what they have learnt that lesson as the teacher gives each student an exit ticket (see Worksheet 3.3).

  • collects the exit tickets at the end of the lesson to review and analyse where students are in their learning and whether there are any key areas of concern to address in the next lesson.

Student Action

• Students

  • work independently to complete the exit ticket (see Worksheet 3.3).

  • return their exit tickets to the teacher on their way out of the classroom.

Worksheet 3.3