Lesson 2: Logarithmic laws and applications (Pt. 1)

Learning Goals

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

• Identify and apply logarithmic laws to solve problems

• Correctly apply the change of base law to logarithmic expressions

ICT

• Website (Questions) - KhanAcademy quiz "Evaluate logarithms" (& advanced Qs)

• Website (Questions) - Khan Academy page "Logarithm change of base rule intro"

• Website (Questions) - KhanAcademy quiz "Use the logarithm change of base rule"

• Website (Questions) - KhanAcademy quiz "Evaluate logarithms: change of base rule"

LESSON STRUCTURE

1. Introduction (10 minutes) - Warm up

2. Body Activity 1 (20 minutes) - Proving log laws

3. Body Activity 2 (15 minutes) - Proving change of base

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

Duration: 10 minutes

Resource:

1. Questions from KhanAcademy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/e/logarithms_1

2. Advanced questions from KhanAcademy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/e/logarithms_1.5

Activity Description

The purpose of this activity is to warm up the students by getting them to use their prior knowledge, especially from lesson 1.

Teacher Action

• The teacher

  • Suggest students turn to the people around them so that they form groups of 2-3

  • Hands out some butchers' paper for each group to use for the question, but also inform students of KhanAcademy's annotation tools (where students can scribble over the quiz page*)

  • Write the main log laws from the syllabus (NESA, 2017) on the board where students can look back at them throughout the lesson

  • Display the first question on the board for the class as an exemplar

    • NB: Notify students that we will be using log laws (in which we will prove later in the lesson!)

  • Helps connect the hints to the actual question for groups that may need help starting off or stuck on a step in their working out

Student Action

• Students work in groups of 3 to solve the questions on KhanAcademy

• Students need to discuss their approach with each other as they write down their working on the butcher’s paper

  • *If students are using their tablets, KhanAcademy's annotation features will be useful

Differentiation

• Even though the hints provided by KhanAcademy should be used sparingly, it must be emphasised that there is no shame in using them. However, it is encouraged for students to take notes of the hints they used so they can look back at them later!

  • Not all students may feel comfortable in asking their teachers directly for help, so the hints provided can support them.

• KhanAcademy offers advanced questions which students can work on alone/in groups.

Non-tech alternatives

• If students do not have sufficient access to computers, print outs of the KhanAcademy questions and hints (stored in ziplock bags) for student groups will help.

  • A "focus group-style" activity where ziplock bags (of assorted questions and hints) are on tables and student groups move around the classroom as they complete the questions will also help.

Duration: 20 minutes

Resource:

Worksheet 2.1 for proof examples and log laws (see below)

Activity Description

The purpose of this activity is to introduce and prove the different log laws. Students will participate in a jigsaw activity and peer teach the proofs for different log laws after these laws have been established by the teacher.

Teacher Action

• The teacher

  • identifies that there are six different log laws that are used to simplify problems involving the logarithms that students have explored so far.

  • writes these different log laws (see Worksheet 2.1 below) on the board and assigns students to a different log law so that the class is roughly divided into six even groups

  • gives each group a designated log law and associated proof and 5 minutes to discuss and write down their allocated proof

  • splits up the initial groups into new groups so that each new group has at least one person with each different log law

  • gives the new groups the remaining time to peer teach/share their proofs with the rest of the group and take notes of these proofs in their books

Student Action

• Students

  • write down the log laws and associated proofs as they participate in the jigsaw activity

Additional Notes

• Alternative proofs are also provided by KhanAcademy

• This activity also aims to allow teachers to observe how students construct, communicate and justify their mathematical reasoning to their peers in how they reach the conclusion of the proofs.

• Given the proofs are already provided (i.e. students know the 'end goal' of each law), students are more likely to participate in the activity cf. being imposed with a task where they themselves have to start the proofs from scratch. However, it is important to note that each class demographic is different, so the option of allowing students to derive proofs themselves can be ideal for some.

Worksheet 2.1

Duration: 15 minutes

Resources:

1. Khan Academy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:change-of-base/a/logarithm-change-of-base-rule-intro

2. Quiz "Use the logarithm change of base rule" from KhanAcademy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:change-of-base/e/evaluate-logarithms-using-the-change-of-base-rule

3. Quiz "Evaluate logarithms: change of base rule" from KhanAcademy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:change-of-base/e/rewrite-logarithmic-expressions-using-the-change-of-base-rule

Activity Description

The purpose of this activity is to prove, identify and use the change of base law.

Teacher Action

• The teacher

  • introduces the change of base law to students, writing it up clearly on the board and annotating the different components of the law

  • provides students with some time to exercise this new law by reading through and completing the Khan Academy questions (Resource 1)

  • actively walks around the classroom to view student progress and ask students to explain their thinking at random intervals

Student Action

• Students

  • write down the change of base law and contribute to general class discussion and exploration of the new law

  • work through the Khan Academy questions to exercise their knowledge, ask questions to the teacher if stuck, or discuss possible approaches with the person next to them

Differentiated Version

• Students have the option to explore more into the idea of changing bases with the quizzes provided by Khan Academy (Resource 2: Use the logarithm change of base rule & Resource 3: Evaluate logarithms: change of base rule).

• Additionally, students can spend some time on an independent inquiry researching the practical relevance of changing the base of logarithms.

Duration: 5 minutes

Resources:

1. Worksheet 2.2: Solutions on p. 1; student cut-outs on pp. 2-3

2. KhanAcademy - https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:change-of-base/a/logarithm-properties-review

Activity Description

The purpose of this activity is for students to consolidate the log laws which will be fundamental for their learning in the following lessons and journey in applying logs in novel and dynamic contexts (e.g. the alternative assessment). Instead of merely rote-learning the laws, students are given the chance to collectively (or otherwise) practice recalling the laws in a dynamic activity where they are also required to configure the Tarsia puzzle.

Teacher Action

• The teacher hands out the Tarsia puzzles (see Worksheet 2.2 below) to students and closely observes student progress in this activity, nudging them towards the right direction when appropriate

  • Possible hints:

    • Find the more straightforward laws to connect first: log_a(1) = ? (soln: 0), then work your way from there.

    • There are two laws that both equate to x, so it is important to flag it when figuring out the puzzle configuration.

Student Action

• Students complete the provided worksheet (see Worksheet 2.2 below).

Differentiated Version

• Students may work in pairs/threes with the puzzles cf. individually if appropriate

• KhanAcademy also offers a recap on log properties (that includes quizzes with solutions and explanations) which students can explore.

Worksheet 2.2