assessment task cover sheet - robert cooper

Unit Co-ord./Lecturer Dr Helen ChickOFFICE USE ONLYAssessment received:

Tutor:(if applicable) Dr Helen Chick

Student ID 259143

Student Name Robert Cooper

Unit Code EMT525

Unit Name Teaching the Mathematics Curriculum 7-12

Assessment TaskTitle/Number Lesson Plan with Pedagogical Analysis / AT2

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Assessment Task Cover Sheet

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EMT525: Teaching the Mathematics Curriculum 7-12 Assessment # 2Robert Cooper

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Section A – Overview of Lesson

Lesson Title: Introduction to logarithms

Grade Level: 10 (extended) Duration: 90 minutes

Prior Knowledge:

Students can

Manipulate and simplify algebraic expressions using index laws including fractional

and negative indices

Solve algebraic equations using index laws

Objectives:

Students will

Apply the definition of a logarithm

Understand the relationship between exponential and logarithmic expressions

Use logarithmic laws to simplify expressions

Learn how logarithms can be used to solve exponential equations

Introduction (15 minutes):

Show some examples of equations where the unknown is an index. For example, the

equation: 2x = 8: how do we find x? Many students may be able to solve this by simple

inspection: It must be x = 3 because 23 = 8. Increase complexity of examples until solution

be inspection is apparent. Ask the class, how the equation can be re-arranged so that

unknown (x) can be moved away from being an exponent to being all by itself? No doubt, a

student will notice the underlined heading at the top of the whiteboard and say: ”Use

logarithms!” Write the logarithmic definition on the whiteboard and explain. For more

detail, refer to Section B.


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Main part of lesson (65 minutes):

Discussion with explanation and applications (10 minutes)

On the whiteboard, provide students with examples of expressions in both index and

logarithmic form – two numerical and two algebraic. In each case, ask the class, to identify:

(1) the logarithm (now to be called “log”), and (2) the base. Stress that a log is an index and

can also be called a “power” or “exponent”. Describe a real-life application (bacteria

growth in a pond).

Worked examples (10 minutes)

On the whiteboard, present two worked examples: writing an index equation in log form.

(eg. 6x=216 => log6 216 = x); and writing a logarithmic equation in index form. Following

this, evaluate two numerical log expressions including one as a fraction, e.g. log3 .

Student exercises, providing assistance (15 minutes)

Students to complete exercises from textbook: writing log expression in index form;

writing index expressions in log form; and evaluating logs in numerical form, using the

definition.

Working with logarithms instruction and worked examples (15 minutes)

Introduce logarithm laws 1, 2 & 3 i.e.:

loga x + loga y = loga (xy)

loga x - loga y = loga

loga xn = n loga x by comparing with the index form of the logs and applying associated

index laws .

Student exercises, providing assistance (15 minutes):

Textbook exercises, first applying each law separately and then combinations of the three.

Conclusion (10 minutes)

Re-iterate definition and ask the class if they can think of any other special cases where a

solution may not be possible. Can a or y be negative? Discuss cases where logs are

undefined, using comparison with index laws. What is log 1 for any base? (law 4). Assign

some further questions for homework that (1) involve questions similar to exercises

covered and (2) extend to incorporate log laws 5,6 & 7: i.e. loga a = 1 ; loga = - loga x ;

loga ax = x


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Section B – Teaching Activity

Introduction to Logarithms (whiteboard text in bold colour)

Teacher: We are going to have a look at logarithms and find out how they can be useful.

When we solve an equation we usually try to simplify things by getting what we don’tknow on one side by itself and what we do know on the other – which is often a number orsomething we know the value of. But what happens if the unknown is an index like this?

2x = 8 (and saying two to the power of something equals eight)

How do we solve this? Who knows what x is?

Student: Three

T: How do we know?

S: Because two to the power of three equals eight?

T: Yes. 2 times 2 times 2 equals 8 (from now on written 2 x 2 x 3 = 8, 23 etc.)

How about this one? (writes 3x = 243 and saying

S: 35=243, so x must equal 5

T: Good. How about this? (writes 4x = 17)

S: A bit more than 2?

T: Yes, but no matter how much we try, we can’t get the exact number by just guessing, sowe need to come up with a way of writing the exact value for x.

S: We use logarithms.

T: Yes. Logarithms are about what power we have to raise a number to, to get anothernumber. Here is the mathematical definition…

Loga y is an index, say x, such that ax = y

Read: “log base a of y… a to the power of x equals y.”

Loga y = x <--> ax = y (use different colours for a,y and x)

T: These two expressions are equivalent. x, or the log, tells us how many times we need tomultiply a to get y; or what power we need to raise a to, to get y. In both index and logform, a is the called the base, and x is the index. (use arrows pointing to the definition withthese descriptors). Can a be any positive number? (refer section H)

So for the first two equations in index form, we write the logarithmic form as:

23 = 8 -> log2 8 = 3


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35 = 243 -> log3 243 = 5

Now for the third example we write:

4x = 17 -> log4 17 = x or x = log4 17

Now log base 4 of 17 is a number, a real number and, in this form, an exact value.

We can use a calculator to work out that, x is approximately 2.04373 (5dp)

S: But my calculator gives a different answer!

T: Most calculators can’t evaluate base 4 logs, so there is another step required to covertthis log to base ten which calculators can evaluate. I’ll show you how to do that later.


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Section C – Pedagogical Analysis

Logarithms, even for advanced students, may appear daunting. Consequently, the logarithm

definition was not stated immediately. Rather, the students were provided with some

examples, in index form, where the unknown is a power. Saying “two to the power of

something…” emphasises that something needs to be solved. Students would find that such

equations could still be solved by inspection when the index was a small integer. The 3x =

243 example was chosen because it was at the “difficult” end of this “easy” category.

Similarly, 4x = 17 was chosen as students could estimate the answer to be slightly over two,

given they know that 42 = 16, however it would also be apparent that another strategy is

required to obtain an exact answer.

The logarithmic definition was stated in two formats, the first highlighting the fact that a

logarithm is an index (with students, hopefully, now comfortable with indices). The second

definition was provided immediately below the first to help students grasp the literacy

aspects within the first definition. The definition was also read out aloud straight after

writing to ensure correct terminology. At this point it may have been appropriate that to add

that logarithms are inverse to exponentiation. The log concept was repeated about three

other times around this stage as reinforcement: “what power we have to raise a number to,

to get another number”

Use of different colours for a, y and x helps to imply the interchangeability between

logarithmic and exponential forms.

The example of log417 ~ 2.04373 was provided to give the students some reassurance that a

concrete answer can be obtained with a calculator.


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Section D – Worked Example


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Section E – Student tasks / problems / exercises

1. Write the following in logarithmic form:

(a) 7x = 49

(b) p4 = 16

2. Write the following in index form:

(a) log3 x = 5

(b) log10 0.01 = -2

3. Evaluate the following logarithms:

(a) log2 128

(b) log10

4. If logx y = 2, find the value of logy x. Justify your answer.

5. Simplify: log2 9 + 2 log2 2 – log2 12

Tasks have been adapted from Maths Quest 10+10A for the Australian Curriculum(Boucher, Kempff, Elms, Bakogianis, Scott, Conner, & Cooper, 2012).


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Section F – Solutions to tasks / problems / exercises


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Section G – Pedagogical Analysis of Student Tasks

The student tasks are assigned in two session during the lesson – refer lesson plan.Questions 1 to 3 are a sample of the activities to be completed in the first session.Questions 1 and 2 are designed to build fluency with using the logarithm definition andwith transferring between logarithmic and exponential forms of numerical and algebraicexpressions without evaluation. The argument, y, together with base, a and index, x can bemixed up easily, so plenty of practice is helpful. By mixing up the position of the“unknown” and combining numerical and algebraic examples, together with decimals andnegative terms, the students are kept on their toes to minimize complacency. Question 3 isincluded for two reasons. Firstly it requires extended thinking to come up with 27 as beingequal to 128, together with the introduction of the very small fraction (1/100) in 3(b). Thesecond reason is that this question can be revisited later, after some logarithmic identitiesare introduced that may simplify such evaluation.

Question 4 provides further practice with the logarithmic definition – solution requires itsapplication in both directions. Furthermore, reinforcement of previous work with indices ispracticed – specifically fractional indices. Finally, the question elegantly highlights theinverse of nature of logarithms with respect to indices. Students may also think the answerto be very cool and may make them wonder if this observation would be repeated in allsuch cases.

Question 5 was chosen because it combines three logarithmic identities examined in onequestion. It can also re-inforce that multiple identities can be combined in one operationthrough formative assessment. Finally, the question can highlight that the final answer neednot just a number, but may be left as a simplified log.

Section H – Teacher Questions

To be asked during the worked example: Do log laws need to be applied separately or canthey be grouped together? Without stating an answer, the teacher can then test thehypothesis whilst comparing with related index laws. That is:

logx a – logx b + logx c logx is synonymous with x(p-q+r)

The result to learning is that students gain further understanding on how the logarithmicidentities are constructed.

Another question during the worked example: Rather than grouping the log terms first, canwe evaluate the logs we know as they arise? In this case, students can be asked to try it forthemselves: The middle term, ½ log216 can be evaluated immediately as ½ x 4 = 2,leaving the remainder to be simplified to 1 which results as 2+1 = 3. Again, by trying adifferent approach to solving the same problem, exposure is provided to new strategies.

The last question comes from the teaching activity: Can (the base), a be any positivenumber? A yes answer is incorrect and should be discussed further. A no answer leads tofurther questions as to which number (1) can’t it be. Regardless, it allows students tofurther appreciate the relationship between logarithmic and exponential expressions.


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Section I – Personal Development

It has been a long time since I have examined logarithms in detail - about 4-1 centuries ago,to be more precise. The topic was chosen for that very reason – to provide a personalchallenge and to also elicit some new pedagogical knowledge. To this end, I believe I haveachieved some success on both counts although to what extent remains to be seen. I havecertainly become acutely aware of the importance of clear expression with regard to bothlanguage and layout.

I recall from my own learning in mathematics, that the way a new concept is introduced iscritical in achieving deep understanding with the subject matter in the ensuing days andweeks. I feel I now have a sound understanding of the definition of a logarithm and how itcan be compared with exponential expression in an inverse relationship. I have also cometo appreciate the power (no pun intended) of logarithmic identities to simplify expressions.

I also have gained a considerable appreciation of the value of carefully articulatedquestions. It seems to allow students, in their responses, to build upon what they alreadyknow. I found this aspect of the assignment the most challenging and feel that I still havemuch to learn in this area.

As for this unit, EMT525, I found particular enlightenment with the strategies that candiagnose mathematical misconceptions, particularly with regard to the unit on decimals.What I need to work on, I feel, is how I can better address these misconceptions usingsound pedagogical strategies. I found the dual number line very useful for convertingbetween units and explaining ratios. The techniques presented there resonated well withme. Online activities and lectures were particularly enlightening. Tutor feedback fromdiscussions posted online, if present, may have been useful in consolidating specificpedagogical strategies.

I was particularly strong with maths at school and university but I feel this may maketeaching the material more difficult as I often relied on my methods when most otherswould listen to the teacher and do it the “proper way”. This said, I had some success atteaching algebra to grade 7s in my latest teaching practicum, employing some methodsdiscovered in this unit.


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References

Boucher, K., Kempff, D., Elms, L., Bakogianis, R., Scott, D., Conner, C. (2012). Maths

quest 10+10A for the Australian curriculum. Milton, Qld: John Wiley & Sons.

assessment task cover sheet - robert cooper

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