3 Day Lesson Plan



3 Day Lesson Plan

Introduction to Logarithms, their uses, and properties

Shane Randa

CI 403 M2

28 October 2010

Shane Randa

CI 403 M2

28 October 2010

3 Day Lesson Plan Schedule (Intro to Logarithms, their Purpose, and Properties

Day 1: Introduction to Logarithms (with Problamization)

Day 2: Properties of Logarithms (addition, subtraction, taking to a power

Day 3: Logarithmic functions, and change of base formula

(Day 4): 20 minute quiz on logarithm equations/properties (day 1 and 2)

Overview

For the next three days in this Algebra II class, we will be covering the topic of logarithms. We will be introducing what they are, and how we apply them to both mathematical expressions and equations. This topic is usually introduced first as a function, and then the properties are shown, and then students work with equations. However, I decided that I would try and reverse the order with this unit. By introducing logarithms first as a means to solve an exponential equation (because A LOG IS AN EXPONENT), hopefully the idea of using logarithms will be problematized in a much better way.

When I learned logarithms as a high school student, we only discussed the procedure of applying them in mathematical expressions, and their sole purpose was ignored. Logarithms had no meaning. However, in the real world, they can be applied in so many ways. Whether the temperature of a hot cup of coffee cooling off (which is NOT linear), how populations of people, plants, animals, or bacteria multiply, how the Richter scale is applied to earthquakes, or even how nuclear waste loses its radioactivity (Yucca Mountain is still a hot topic of debate in politics), this could all be discussed with this unit, and would be a great discussion after the Day 4 quiz to ease the anxiety of students. While I understand that there may not be enough time to really investigate a wide variety of real life examples extensively, students need to see a purpose for what they are doing. Besides, once students become familiar with how to work with logarithms algebraically, the movements of various logarithmic graphs should make more sense.

The opening day of the unit deals with the very basics of a logarithm. I show it as a way to find the precise exponent of a base that will give us a desired power. Students work through an exponential equation warm up sheet until they come to one they cannot solve. This is where we discuss how to do this, and introduce what logarithms are. This opening lesson is very fundamental, in that is stresses a lot of importance in how logarithms are read (“the log base b of x equals y” is the same thing as saying “y is the power of b that gets us x” or even “we multiply b by itself x times and then get y”). By reinforcing the very basics of what a logarithm is, it is the hope of the teacher that students will be better prepared to go more in depth with logarithms on days 2 and 3.

The connection between logarithms and exponents is also highly stressed on all three days. A logarithm is an exponent! Students will spend a lot of time going over how to rewrite exponents into logarithms, and rewriting logarithms into exponents. It is essential to have this skill in order to solve basic logarithmic equation, which students will do on both Day 1 and Day 2 homework, and on the quiz that opens up Day 4.

Rewriting exponents into logarithms (and vice-versa) is also practiced heavily on day 2. We will introduce the 3 arithmetic properties of logarithms, those being addition, subtraction, and the equivalency of a multiple of a log and the exponent inside of a log. The change of base formula is being saved for day three just because time is limited. Students will use the equivalent exponential forms of logarithms to prove each of these properties (two of them will be proved as a class, while the subtraction property will be proved in small groups). The proofs are not overly complicated, and are in Algebraic terms that I expect all students to be able to follow. In my classroom, I will expect my students to be aware of what is going on within the material, and not just to memorize the procedures and definitions. Discovering the properties in this way, although time consuming, will deepen students’ understandings of a logarithm.

Day three will involve the change of base formula, and students will also be investigating the logarithmic functions. By showing examples of logarithmic expressions that are impossible to evaluate mentally, students will see a need for the change of base formula (and a true need for the use of calculators), which will hopefully put extra importance on it. After discovering the change of base formula, students can now see the graph of any logarithmic function on their calculators, and begin to investigate properties and shapes of logarithmic curves (they can now input any logarithmic function, not just those of base 10).

This final day also includes some work with inverse functions. After all, students by this day will know that a logarithm is simply the inverse of the exponent, and they will see how a given logarithm curve is merely its corresponding exponential curve reflected over the line y = x.

Finally, there will be a 20 minute quiz to open up Day 4. This will assess students’ ability to understand what logarithms actually are, their ability to rewrite them as exponents, and their ability to write given logarithms into equivalent forms of logarithms.

Objective and Purpose

Mathematically, students will gain a greater understanding of inverse functions on Day 3. If they haven’t seen them yet, this would be a great introduction, otherwise it is still relevant supplementary material. Students may be learning about logarithms for the first time, but I do expect them to become affluent with expressing them in both plain English and their equivalent exponential form. This 3 day unit will also deepen students ability to solve equations through reasoning, since now students will have one more tool (logarithms) at their disposal.

This brings me to the main pedagogical purpose of this unit. Students must be able to take a given expression, whether it is logarithmic or exponential, and have the necessary skill set to rewrite that expression into numerous equivalent forms. Different forms will be necessary in different situations, and some are more useful than others to solve given problems. Students must be able to recognize when different forms should be used. For example, given a function like y = Log3 x, students should recognize that in order to plot it in a calculator (to visualize how the function looks), they must input it as its equivalent y = (Log x) / (Log 3) form. Basically, given one representation, students should easily be able to rewrite it as multiple equivalent representations, and see them as equivalent.

Days 2 and 3 also have another important pedagogical aspect, and that is reasoning and proof. The arithmetic properties, as well as change of base, when working with logarithms, are not just ‘taught’ to students. Through Algebraic proof, both as a class an in small groups, students will investigate why these properties hold true. I expect my students to have the abilities to truly investigate the world around them, and be able to form logical arguments to both fully convince themselves and others why things work (or don’t work) the way they do. Math is merely an ideal means to learn the logical thinking skills necessary in society. Because of all this, I also hope to reinforce my students’ ability to think critically.

Meeting the Needs of My Students

My class has 26 students, 4 of which are deemed ELL. Because of this, I have to mind the fact that certain expressions, phrases, and text will take much longer and be more difficult to understand for these students than others. Logarithms can be a very difficult topic to reinforce into students’ heads regardless of language ability, so it is important that the unit starts off slow, and then increases in pace as the foundation of understanding logarithms becomes stronger. Heavy emphasis is placed on “speaking exponents and logarithms” on Day 1. Besides just throwing up some gibberish, we want students to be able to say in plain English what they will be mathematically operating with for the remainder of this unit.

It is my hope that spending so much time learning how to communicate logarithms in multiple ways, will both reinforce the concept to my proficient English students, and keep my ELL students up to par while simultaneously providing an aid to better their English speaking abilities. The proofs on Day 2 will hopefully do something similar, as students will be required to prove one property in small groups, and thus must communicate with each other in terms everyone can comprehend.

Meanwhile, there are two students in my class who are allotted extra time for assessment by their IEP plans. This was due to their reading proficiency and differing levels of academic motivation. Besides the obvious of allowing these two students more time on the Day 4 quiz, there were a couple other areas of the unit where I have tried to address these issues. For one, the “speaking exponents” portion of Day 1 will hopefully reinforce what is meant in simple English by Log5 x = y. In this way, students will not be overwhelmed when they encounter this type of expression, and will be able to interpret it clearly. Reinforcing the idea of switching between equivalent forms of logarithmic expressions (Days 2 and 3) will hopefully accomplish the same (increasing mathematical literacy).

To address the differing degrees of motivation, I tried to include motivations and problematizations whenever possible. Day 1 has a great problematization, where students are unable to solve the final problem on the warm up without logarithms. On Day 3, students cannot graph many logarithm functions without the Change of Base formula (much less evaluate many logarithmic expressions). These are obviously not perfect examples of exciting motivation, but I still believe that we have a purpose to the mathematical topics being explored in these 3 upcoming days. In this way, I hope that each student doesn’t just consider what we are doing “busy work,” but understands that these topics connect to previous topics, and the need does exist for expressions and equations like these (even if they may specifically not use it in their lifetime).

Name: Anna Armamentos

Shane Randa

Zane Ranney

Date of Lesson: 9/30/2010

Grade Level: 9th or 10th

Course: Advanced Algebra 2

Allotted Time: 40 Minutes

Number of Students: 26

I. Goal:

• To gain a basic understanding of the logarithmic function and its purposes

II. Objectives:

• Students will recognize the relationship between exponential and logarithmic function

• Students will summarize in their own words the meaning of the logarithmic function

• Students will practice solving logarithmic equations

III. Prior Knowledge:

• Exponential Functions, and their graphs

• Solving Equations with exponents (such as 3^x = 27)

• Inverse functions

IV. Materials and Resources

• Graphing Calculators

• ELMO

• Warm Up

• Guided Notes Sheet

• Homework worksheet

V. Motivation and Warm Up

1. We will begin by giving the students a warm-up (about 5 minutes, and on page 11). During this time, the teacher(s) will walk around the room and check to make sure students are completing it appropriately. We don’t anticipate problems, since students have already reviewed the exponents in detail (and they are simple), but there will likely be problems on the last problem (see #3 below).

2. Students will use calculators to try and figure out the power of 10 that gives them 349.5. (x is approximately 2.54344718…)

3. Once the students look like they are finishing the warm-up, or look like they are about to give up and become distracted from math in general, Mr. Ranney will call the class back to attention and ask students to provide their answers to the warm-up. Mr. Randa will be at the ELMO plugging in the numbers that the students provide. Ms. Armamentos will be at the board writing down all student contributions. (about 2 minutes). We anticipate that students will do trial and error to get an answer closer and closer to the actual answer, and a few may reason that this problem is unsolveable.

4. Mr. Ranney will conduct the discourse with the students focusing on how to solve for x in the last example. The class will have a discussion about the last example, looking at what exactly it is asking, and how it can be solved (“Why is this harder?” “Can we find an exact value of x?”) (about 10 minutes)

VI. Lesson Procedure (steps 1-5 will take about 10 minutes, 6- will take about 5 minutes)

1. Mr. Ranney will finally show the students a “trick,” where he will show the students how to type in LOG(349.5) in their calculators to obtain the exact answer. Afterward, he will have the class check this answer in their calculators to prove it is correct.

2. Mr. Randa will then take over, and ask the students “So we used the LOG button in our calculators, and somehow it worked. What exactly did we just do to find our exact answer?”

3. Afterwards, Mr. Randa will instruct the students to refer to their guided note sheets (see page 12), and will formally define what a logarithm is, being the inverse of the exponent. We will first discuss the explicit definition (the implication statement on the notes sheet), stressing that b is the Base and x is the eXponent. In this way there can be a clear connection for the rationale of the variables (promotes the understanding level of thinking vs. just rote memorization). This can also help ELL students to make an easier connection to the material, since we do not anticipate that there will be too much difficulty with the alphabet, since the pronunciations are very similar to other Latin Languages.

4. I anticipate questions upon the rules for the base (b>0, b neq 1), which is given in the guided notes sheet. I will answer these questions by first discussing the case that log1(x) = y. In this case, we only get an answer if x = 1 (where then y can be any real number). But this would then not make a function, and I would say something like “we will apply these to functions in a couple days, so for now we will just ignore the case that b = 1.”

5. This is then related to the plain English definitions for a logarithm. I will ask a student to read the left hand side of the implication statement in words (should be something like “b to the power of x is y,” or “x is the power of b that gets us y.” This, of course, is the exact same way to read the right hand side of the implication, and I will highly stress this when we go over the bottom two fill-in-the-blank sentences. “Even though these two mathematical statements are written differently, we can read both of these in the exact same way.”

6. Finally, if time allows as a way to show an additional representation, I will start a discussion on the inverse relationship between logarithms and exponents. This is done by showing addition as the inverse of subtraction, division as the inverse of multiplication, and then logarithms as the inverse of exponents. If time allows, I could relate this with examples of functions (for example, the inverse of y=x+2 is x=y+2 (y=x-2).

a. Inverse of (Y=X+2) is (Y = X – 2)

b. Inverse of (Y=2X) is (Y = X/2)

c. Inverse of (Y= 8^(X)) is (Y = log8X)

d. For each of these, I will lead students with some discursive moves, for a. and b. especially. We could possibly have students write up on the board (1 for a., 1 for b.) what happens when we take the inverse (swap the variables). This way students are held more accountable to participate, and they get to hear the content represented a second way from their peers.

7. Ms. Armamentos will then transition into the “speaking exponents” versus “speaking logarithms” worksheet (see page 14). We will go through the first example as a class and then the students will work through the next one at their tables.

8. We will then discuss the third problem again where the students are asked to find b.

9. The students will then work on the last problems on the worksheet, while we walk around the class and observe their work.

VII. Closure

“So today we expanded our knowledge of exponential functions to logarithmic functions. We learned that the logarithmic function is simply the inverse of the exponential function, as well as how to use this fact in solving equations.”

VIII. Extension

We can discuss #5 in our lesson procedure if time allots. Additionally, we can hand out their homework assignment (see page 13) to begin during class. Otherwise it will be due as an assignment for the following day.

IX. Assessment

There is ample opportunity for formative assessment throughout our lesson. We begin by measuring the level of understanding of prior knowledge of exponents in the warm-up and its following discussion. Next, students are prompted to participate in a deep and meaningful discussion about the use of logs as well as its connection to their prior knowledge. Lastly, we will measure their comprehension of the day’s lesson through completion of the worksheet. Furthermore, we can determine the effectiveness of the lesson through grading their homework the following day.

X. Illinois Math Learning Standards

• 8.B.4a Represent algebraic concepts with physical materials, words, diagrams, tables, graphs, equations and inequalities and use appropriate technology.

o For this lesson, students must not only represent equations as words (Speaking Exponents), but they must also do the reverse and represent a sentence in words into an equation (during the main lesson and also on the Speaking Exponents sheet).

• 8.B.5 Use functions including exponential, polynomial, rational, parametric, logarithmic, and trigonometric to describe numerical relationships.

o Students will have to witness, understand, and then describe/explain how we can relate two different equations (one logarithmic, and one exponential) together. This is basically the entire portion of the lesson once the motivation is over.

XI. NCTM Process Standards

• Connections: to prior knowledge of exponential functions

o It is necessary for students to connect this previous knowledge to solve equations in logarithmic form. During Mr. Randa’s presentation, this connection will be first explained, and then during the Speaking Exponents in-class assignment, students will have to again tap into their prior knowledge.

• Communication: “Speaking exponential versus speaking logs”

o In the assignment named for this standard, students must communicate mathematical symbols into plain English. In this way, students can better understand what is exactly going on, and think above the “rote” level of Bloom’s Taxonomy.

XII. Common Core State Standards for Mathematics

• A-SSE: Write expressions in equivalent forms to solve problems

o This stems from the motivation of the lesson, as we will convert logarithmic functions into exponential ones in order to solve for the equation.

• A-REI: Reasoning with Equations and Inequalities

o Since we tend to represent logarithms in equation form (especially to drive the point home about what they exactly are when we relate them to an exponential equation), students are then forced to work through equations to deepen their understanding on the material (after the motivation).

(Cut in half)

Warm Up!

Solve for the value of x in the following equations.

1. 2x = 8

2. 3x=9

3. 4x=1/4

4. 4x=2

5. 10x=1000

6. 10x = 349.5

Warm Up!

Solve for the value of x in the following equations.

1. 2x = 8

2. 3x=9

3. 4x=1/4

4. 4x=2

5. 10x=1000

6. 10x = 349.5

[pic]

• A logarithm is the _________________ of an exponent.

• Make the Connection: This is the general formula that must always be true:

[pic]

Exponential Function Logarithmic Function

• b = base b>___, and b(____

• x = exponent (or power)

• logby = x can be read as:

▪ What _______ (or exponent) of _____will give you _____?

OR

▪ How many times do you ___________ b by itself to get _____?

Logarithm Homework

Rewrite the following expressions in logarithmic form:

1. [pic]= 16 3. [pic]

2. [pic] 4. [pic]= [pic]

Solve for the unknown variable:

1. [pic] 5. [pic]

2. [pic] 6. [pic]

3. [pic] 7.[pic]

4. [pic] 8. [pic]

For the following problems, solve for x. Afterwards, write down how we would read this in plain English (think of how we read it in class/ on the notes).

|Speaking Exponents |Speaking Logarithms |

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|[pic] |[pic] |

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|We can read this as “____________________________________” |We can read this as |

| |“____________________________________” |

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|[pic] | |

| |[pic] |

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|[pic] | |

| |[pic] |

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|We can read this as “____________________________________” | |

| |We can read this as “____________________________________” |

| | |

| | |

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| | |

|[pic] | |

| |[pic] |

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|[pic] | |

| |[pic] |

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Shane Randa

Lesson Plan: Day 2

~45 minutes of class time

Day 2: Properties of Logarithms

Goal: For students to investigate properties of logarithms and equations involving logarithms (addition, subtraction, and exponents of logs)

Objectives

1. Students should be able to identify the addition property of logarithms, that logb(xy) = logb(x) + logb(y). The same goes for the properties of subtraction of logs (division as one log), the exponent of a log (that exponent multiplied by the original log).

2. When given an equation with a sum, difference, or power of a logarithm(s) with the same base, students should be able to compute what the expression would be as one logarithm. Students will also be able to apply this to an equation.

3. Students can also be able to apply these properties in order to separate a complicated logarithm into a linear combination.

Prior Knowledge

• (See Day 1 Lesson Plan for all the prior knowledge students have already attained before this unit started)

• In addition, students have been recently introduced to logarithms and the motivation for using them (particularly unsolvable exponential equations).

• Students can describe a logarithmic expression or equation in words.

• Students can also solve a logarithmic equation when there is only a single logarithmic term and one variable (so the most basic of these equations) by converting the logarithmic equation into an exponential one.

Materials

• Graphing Calculators, since it is impossible to mentally calculate the results from applying the change of base formula (although we will throw in a couple examples where the base and the solution (x) are powers of 10, just so we can better assess that students are grasping this concept.

• Visual Aids, mainly the ELMO and some type of whiteboard/chalkboard.

• “Chain of Thought” notes sheet.

Intro/Motivation (< 5 min)

• We will open up the class by collecting homework from the previous day. The homework was assigned the day before as due today, so we expect students to have had it done. After students pass their homework up, I will open up by asking questions about the recent assignment.

o If any hands go up, I will gladly answer their question and work through a problem from the day before. It will be essential to stress the important concept from the day before (we can rewrite single log expressions as an exponential expression). “Given that ____ is a logarithm equation Francis, how should we go about solving this?” …and then work from there, hopefully with student led discourse.

• I will open up the lesson with a quick question. “So yesterday we opened up a new topic, which was an operation known as a logarithm. What previous operations did we relate it to, and how?”

• Hopefully, since the students just had an assignment about this, one of them will raise their hand and remind their peers that logarithms are related to exponents (and technically are exponents, but that’s for another day), and that any logarithm can be rewritten as an exponent, and vice-versa.

o If students are reluctant to speak, I could lead them a bit. “OK, I know it’s the morning, and a lot of you are still probably waking up, but think about it. How do we solve and equation like……(put on board) log7(49) = x?”

o At this point, I anticipate several students mentioning that we convert this into an exponential equation 7^x = 49, x =2. Again, students just did an assignment on this, and their prior knowledge leaves them familiar with exponents.

o “So as a class, when we see a logarithm, we can break down what it tells by converting it into ….what exactly? (response: “an exponent”)

• “Wonderful! So today, we are going to use that same notion of relating logarithms to exponents to investigate logarithms a little deeper than we did yesterday.” (hand out “Chain-of-Thought,” page 21)

Lesson Procedure (30-35 minutes)

• (once the sheets are passed out) “So in order to fill this chain-of-thought in, I am going to need four recorders up on the board. Would anyone like to get up and help me out this fine morning/afternoon?”

▪ Call on four students with their hands raised to come up to the board. I will try and call on students from different parts of the room if possible, and maybe call up one of the ELL students.

• (Once I have four students up at the board, give each of them their own share of the chalk/white board. I will move towards the back of the classroom, to both give the participating students the spotlight by taking attention away from me and have a better control of classroom management.)

• I will now go through filling this out as a class. “So for (1), can anyone at their desks tell me what could go on ___________’s portion of the wall?” (Wait for response) “Right _______!” We write x times y as a^12. Call on another student…”So ____, _____ said that we rewrite x times y as a^12 in the second blank. Why is that?” (if no response) “Can anyone else help _____ out?”

▪ I’m looking for a student to respond with the fact that when we multiply two exponents who have the same base together, we simply have an expression with the same base, and the power the sum of the other two powers

▪ If students are reluctant, I could lead them with something like “Think back to our lesson on exponents, what do we know about them?” or something like that.

• “Ok, thanks for your help (person who wrote (1) on the board!”

• “Now on to the next part in our chain-of-thought, how could we rewrite the last two members of the four part equation from (1)?” (if no response) “We did it yesterday?” I anticipate that once I say this, students will know right away to rewrite it as a log.

• For the last 3 parts (and the “similar to” portion), we complete the guided notes sheet in a similar way. For (4), I’m hoping that one of the students sees that 5+7 =12. If not, then I may have to give the class a slight nudge in the right direction, but it shouldn’t be too much.

• “Finally, for (5), what can we say about the logarithm expressions in the multi-step equation?” (wait up to five seconds)

▪ For this connection, if it is not seen right away (it being the addition of logarithm property), I cannot really anticipate students coming up with it after leading them along. So in this case I think it would be alright to just highlight both logarithmic expressions and show that they are equal. (I couldn’t really think of a better question to ask, and I know it isn’t the best one to ask).

• After showing the property in this case for (5), I will also define the property explicitly (with b as the base instead), and then draw the connection for the similar to (again stressing the exponential relationship we used to prove this property).

• “Ok, thank you so much volunteers! Now we have a few more properties to investigate, and I’ll go ahead and tell you right now that the next one is the subtraction property of logarithms.”

• “I need every one of you to take out your notebooks, and on a new page do another chain-of-thought for this property. Go ahead and get in groups of up to 3-4 if you want. For the opening statement, keep the same values of x and y. For your first step in the chain. Start it off with (x/y) = ____________, and then follow steps in the same way as the sheet we already completed.”

• For this, I simply walk around the room and help the groups if need be. I anticipate a few student misconceptions. It is possible that a few students may forget that a^7/a^5 = a^(7-5) = a^2. By reminding them that subtraction is just the opposite of addition, like division is the opposite of multiplication (for the exponents here especially), this should drive the point home.

• The reason I don’t offer a guided notes sheet for the rest of the lesson, is that I still want my students to develop good note taking habits. Being able to interpret what is and what is not important is a vital literacy skill that should be taught through ALL subjects. The guided note sheet at first gives students a sense of direction, and helps the ELL students better comprehend what is going on, but ALL students need to practice good note taking skills.

▪ Other anticipated problems include students trying to still add logarithms together (their final point for (5) may be something like loga(x/y) = loga(x) + loga(x). It is possible that they won’t understand how to apply the guided notes sheet to this case. I’m hoping that since students are doing this activity in groups, the probability of the whole group not seeing the connection is diminished.

• After it appears that most groups are getting the main idea (or about 7 minutes or so has gone by), I will bring the whole class back together.

• “So after the brief investigation, how could we state the subtraction property of logarithms?” (wait…)

▪ If a student comes up with an answer: “Who in this room agrees with ______?” (hands go up) “Who disagrees/” (if original answer was wrong, I anticipate several hands, otherwise there still may be a few) “and who just doesn’t want to vote?” (side joke)

▪ I will call on one of the people who disagreed. “_____, What makes you think it’s something else?” or “Why do you think _________’s answer is incorrect/”

▪ From here I could break off for a couple minute discussion (if we have time). Questions like “Can you explain why that is?” “Who wants to restate what ______ just said?” “What about what ______ said do you like?” are ideal.

• Once the subtraction property is defined. I will discuss the exponential property (loga(x^r) = r loga(x).)

• “We have one more property of logarithms to relate with exponents. This one we’ll do as a class, so I expect everyone to be taking important notes down.” (Whether or not I still need to scaffold this for my class will depend upon where I feel they are at with their study skills at this point).

• (I also hope by now we have laid a foundation for the connection of logarithms to exponents for various properties, so the last two should go by much faster)

• “We want to start with an exponent. This time b^y = x. Since this is an equation, what will happen if I raise both sides to the power of r?” From here I can lead class discourse, showing that (b^y)^r = x^r ( b^(yr) = x^r ( logb(x^r) = yr ( logb(x^r) = rlogb(x) (since y = logb(x) (This is the important point to drive home)

• Afterwards, we will work through several examples as a class. I will hand out the homework (page 24), which will be due the following day. I hope to do several examples, likely the more challenging ones (#5, #8, and #9) as a class if time allots.

▪ For #5, I want to stress that there is often more than one way to solve a problem. Many students want to only memorize a procedure (“just tell me what I have to do for these problems!”), but that bounds a student to lower level thinking. By stressing with this example that we can solve this equation in two ways (whether changing logb(25) to logb(5^2) to 2logb(5), or applying the logarithm subtraction property), students will realize that it is important to fully analyze the problem, and find a proper course of action(s). Without time to look at a problem like this, I anticipate that a few students (especially for the ELL students who may need more specific directions) may be confused as to what action to take.

▪ For #8, this just stresses the explicit definition of logarithmic addition. The bases must be the same to apply the logarithmic properties, and a few students may just become so used to the procedure that the fact that the two bases of the logs to be added aren’t the same. Again, this stresses the importance of thinking conceptually in addition to procedurally.

▪ For #9, this is another example where there is more than one way to go about solving. It would also be nice to use this problem as an example for solving equations with more than one logarithmic expression.

Closure

I will stop the main lesson when there is either five minutes left in class, or when I have gone over enough examples of the material and feel like my students have a good feel for the material. I think it is important to wrap up the content for the day, just so students remember what it is they actually learned and need to master.

▪ “Today we took our knowledge of logarithms a step further. Not only can we work with one logarithm in expressions/equations, but now we can work with multiple logarithms at once, whether we are adding them together, subtracting them, or raising them to a power.”

▪ (Pass out the assignment if I haven’t already) “For the rest of today, and at home on your own, completing this worksheet will better your abilities to work with logarithms. This of course helps us to better work with exponents, since as we saw yesterday, A LOGARITHM IS AN EXPONENT!”

▪ “For tomorrow we will look at one more property of working with logarithms. This final property is one that we will use for the many occasions that we can’t just mentally figure out the answer to a logarithm. Afterwards we will look at the graph of the logarithm function y = logb(x). So for tomorrow please remember to bring your graphing calculators to class.”

Extensions

▪ I plan on going over a few examples from the homework after going over the proofs of the three properties. The main reason for this, as explained before, is that I want to represent how we actually apply these properties, and when each one can and cannot be applied.

o I could work on additional examples as a class to show cases where we have more than one option. One example of this would be y = logb(1/3x), where we could also write it as y = logb((3x)^(-1)) or y = -logb(3x) or even y = -logb(x) – logb(3). I really want to stress to students that we oftentimes use two or even all three of these properties at once, and that we can represent certain expressions in a wide variety of ways.

▪ I do not anticipate the lesson finishing too early, since the proofs tend to eat up quite a bit of time. The homework worksheet should be enough where all students will be working to the bell.

Assessment

• Not only are we assessing the content of operations on logarithms during the lesson, but we are also assessing students’ abilities to reason with algebraic proof with today’s lesson. I can assess how students are following the addition property proof as an entire class.

o Can students explain the steps we used to arrive at our conclusion?

o Are they able to reproduce a similar algebraic proof for the property of subtraction of logarithms?

o To what extent are students feeling comfortable with working with the three properties? Are students able to analyze the way we use each of the properties and understand that there are examples where we can/must use multiple examples?

• Finally, the homework assignment (“Logarithm Properties”) I plan on giving out either towards the end of the day or as class is wrapping up will assess how students can work with each of the properties of logarithms. In addition, the assignment will also assess if students understand when we can and CANNOT use these properties (see #8).

Standards

NCTM Process Standards

Algebra: Students have to understand the relationship between exponents and logarithms in a very similar way to the day before. However, today they must recognize the combining and separation of exponents, and relate it to the combining and separating of logarithms. This is also done with the changing of mathematical symbols, another facet of the Algebra principle.

Reasoning and Proof: Students work as a class, and then in smaller groups, to prove all three properties of logarithms they learned today. This not only helps them to better relate logarithms to exponents, but also helps students understand why these properties work, and how they work. Simply memorizing these properties (as I had done as a high school student) does little to nothing as far as increasing the logical thinking capacity of students.

Communication: Students have to be able to relate the main points and ideas with the step-by-step proofs used in class. In this way, they must communicate mathematical ideas in verbal language.

Connections: Today students investigated further the relation between logarithms and exponents. This not only helps students to learn a new concept (logarithms), but stresses more importance on the previous topic of exponents, and forces students to further investigate exponents as well.

Common Core State Standards for Mathematics

A-SSE-Seeing Structure in Expressions: For the entire duration of this lesson (and the day before as well), students have to convert exponents into logarithms, logarithms into exponents, and logarithms into equivalent logarithms. These conversions happen during the introduction proof, the other two proofs, and basically every example on the homework.

F –LE Linear, Quadratic, and Exponential Models: Since we have to express exponential equations in terms of logarithms for solving equations, this satisfies part 4) of this standard. While we do not yet work with functions, as that is for the day following this lesson, the material from this lesson sets up for that.

Name_____________________

Chain of Thought Proof Activity

Properties of Logarithms

First, we are given that a^7 = x, and a^5 = y.

(1) xy = ______=_____ = xy …. ( …

… ((2) taking the last two expressions in log form gets us ___________

(3) Also, rewriting a^7 = x and a^5 = y into log form gets us ____________ and ___________

(4) How do we relate (2) and (3)? …(

…(5) … ( Therefore, we conclude that _________

in this problem, and ________________ overall.

**Similar to _________________________

Name_____________________

Chain of Thought Proof Activity

Properties of Logarithms

First, we are given that a^7 = x, and a^5 = y.

(1) xy = _a^7*a^5_=_a^12 = xy …. ( …

… ((2) taking the last two expressions in log form gets us _ loga(xy) = 12 _________

(3) Also, rewriting a^7 = x and a^5 = y into log form gets us _ loga(x) = 7 ___________ and _ loga(y) = 5

(4) How do we relate (2) and (3)? …( (12 = 5 +7 )

…(5) … ( Therefore, we conclude that _loga(x) + loga(y) = 7 + 5 = 12 = loga(xy) = loga(x) + loga(x) ________

in this problem, and _= logb(xy) = logb(x) + logb(x) ___ overall.

**similar to _b^(xy) = b^x * b^y

For subtraction: This is what should be in their notes.

First, we are given that a^7 = x, and a^5 = y.

(1) (x/y) = _(a^7)/(a^5)_=_a^2 = x/y …. ( …

… ((2) taking the last two expressions in log form gets us _ loga(x/y) = 2 _________

(3) Also, rewriting a^7 = x and a^5 = y into log form gets us _ loga(x) = 7 ___________ and _ loga(y) = 5

(4) How do we relate (2) and (3)? …( (2 = 5 -7 )

…(5) … ( Therefore, we conclude that _loga(x) - loga(y) = 7 - 5 = 2 = loga(x/y) = loga(x) - loga(x) ________

in this problem, and _= logb(xy) = logb(x) + logb(x) ___ overall.

**similar to _b^(x-y) = (b^x) / (b^y)

Name__________________________ Date__________

[pic]

Write each of the following as a single logarithm

1. 5 logb(x) + logb(y) 2. (1/2) logb(y) + logb(1)

3. (-3)logb(y) 4. logb(x+1) + logb(x-1)

5. logb(25) - logb(5) 6. logb(4) - (1/2)logb(2)

7. -logb(Z) + 4logb(y) + logb(3x) 8. logb(5x) + 2logp(1/5)

Solve the following equations

9. 3log3(9) + log3(3) = x 10. (-1/2) log2(1/16) = y

11. log8(10^x) = log9(25) 12. 7log7(y) - 9log7(y) = 2

Expand 13 and 14 as a sum or difference

13. logb(x^4y^2) 14. 2logb(sqrt(Z) * x^(-3))

Shane Randa

Lesson Plan: Day 3

~45 minutes

Day 3: Logarithmic Functions and Change of Base

Goal: 1) To investigate and analyze logarithmic functions, and the properties of them.

2) Have students understand the change of base formula and its purpose.

Objectives

• Given a logarithm expression or equation, where students are unable to evaluate/solve it mentally (ex, log3(6) = x), students can use the change of base formula to evaluate/solve what is given.

• Students should be able to define the change of base formula.

• Students will recognize when the change of base formula should be used, and create solvable equations with logarithms where it must be applied.

• Students will also recognize that the logarithm function y = logb(x) is the inverse to the function y = b^x.

• Students can also relate a given logarithm function to its inverse, by drawing them on the same graph (and vice versa).

Prior Knowledge:

• Students have all the given prior knowledge from the first two days of the lesson

• In addition, students have worked with addition of logs, subtraction of logs (with the same base of course), as well as multiples of logs.

• Students can relate functions to their graphs, and know how to read a graph on Cartesian coordinates.

• Students have seen exponential growth functions before, and can recognize the shape of an exponential function.

• Students can round a number to a given amount of decimal places properly.

Materials

• Each student needs to have a graphing calculator, both to operate the change of base formula, and for looking at exponential and logarithmic graphs.

o Either a class set of calculators will be provided, or hopefully students can have their own calculators (so they can do work on their own time).

• Graph paper, since each student will be plotting points and graphing curves. I would much rather have students use precise graphs to get a clearer picture of what is happening (as opposed to sloppy hand drawn ones), especially on the first day of looking at logarithmic functions.

• If students can’t acquire graphing paper, I can print off mass quantities for free at

• “Warm-Up Worksheet” (see page 32) and Homework sheet (page 35)

• To go along with the calculators, I will need some way to project an image of a calculator on screen. This could be with an ELMO, a SMART board, or some sort of projector.

• Students will be taking notes at one point, so they should have their notebooks.

Intro/Motivation

• When students walk into the classroom, I will have a warm up worksheet to hand to them. Students will get about 5 minutes from when the bell rings to complete as much as they can. Once the bell rings I will walk around and see how students are fairing and answer questions.

o The worksheet should be easy, since it is day one material on evaluating single logarithmic expressions. However, there are some logarithms that students will not be able to use without the change of base formula. This is what I am trying to motivate the use for. I anticipate a lot of questions about how to do this, and since students have their calculators I think they may try to evaluate them that way. I will tell students that they are fine using them, but I feel like many students will type in something like Log(4)8 as Log(4)8 or Log4(8), which would output 8*Log104.

o I am not grading this for correctness, so if students do this at first that is fine with me. We will correct this false notion shortly after.

o Students may also do what was done on Day 1, and simply narrow the answer down by guessing exponents of 4 that would yield 8. Of course, I could remind students here that while it works (to a point), this would defeat the purpose of logarithms and everything we’ve done up to this point would be useless.

• After the five minutes are up, or I notice that students have gone as far as they can go, and not really moving forward with the assignment, I will refocus the class.

• I will run through the answers that were not complicated, but skip 3, 4, 6, and 8 for now. For this part, I will just present any work needed on the whiteboard/chalkboard in front of the class (possibly rewriting the logarithms as exponents just as a means to further drive the main point).

• For the others, “Why don’t we just skip these for now since they are harder to find?” When we do get back to these, I will write the sentence on the bottom of the warm up page on the board (with blanks filled in).

• “The reason we say this is useful when the base is not 10, is only because typing in Log followed by a number, say 5, has the calculator perform Log105. Remember from two days ago, when the only exponential equation we could not solve had a base of 10, and we merely typed in Log on our calculators to solve. So basically if there is no base given, assume it is 10. This is why for number 7 the answer is 2, for those of you who just typed it into a calculator.”

o (Write statement Log x => Log10 x on the corner of the board, this way students can refer back to this point for the rest of the day)

o Students may ask why the common base is 10 (it would be ideal if my students could be that naturally inquisitive). See extensions if this happens.

Lesson Procedure

• Finally, we get to explaining just exactly what the Change of Base formula is, and how it is defined. Unlike the properties from Day 2, I will define this property first, and then dissect it with a brief Algebraic proof afterwards.

• “The Change of Base formula is very important to this topic, because with it we can find the exact answer to any logarithm with the help of our handy dandy calculators.” (While speaking, write up LogbA = (Log10 A)/(Log10 b) on the board)

• “To prove this, or show why this works, let’s start with the equation b^(Logb x = x). This is true since the power of 10 and the log will cancel out. Since both sides equal, we can take the log with base A of both sides, like so.”

o On the board, show the next step, where we have Loga( ) on both sides of the equation. Within the parentheses on each side, we have whatever was on that respective side of the equation.

• “Again, both sides are equal, so doing the same thing, like in this case, still makes both sides be equal. Now notice that here (point to left side), we have an exponent within this log (emphasize the power of b which is “Logb x”), how can we rewrite this whole Logarithm?”

o Hopefully, one of the students remembers that we can change the exponent to a multiple of logs, so in other words the LHS should look like this: (Logb x)(Loga b) = Loga x. This may also be a great opportunity for discourse, especially if students appear confused about why the power of 10 and the logarithm can cancel out.

• “All we have to do is take the log, base 10, which actually means we can just type in Log 7 if A was 7 (be sure to point to A on the board, and use hands or a laser pointer for emphasis. This way we are less likely to lose our students).”

o Work through a couple warm up examples as a class. Students will be asked to follow along with their notes. Since there really isn’t that much deep thought to this part of the lesson, there really aren’t many opportunities for discourse.

o We will first use the examples from the warm-up that weren’t able to be solved without the calculator. For example, Log4 8 could be used, and we would apply the Change of Base formula and get (Log 8)/(Log 4).

o I would also have students take out their calculators at this point. I will show how the formula is used in the calculator to evaluate an expression like this. We simply have to type in ( Log 8 ) / ( Log 4 ), and get our answer of 1.496. We will use 3 decimal places so the answer is more exact, and distinguishable from other answers.

o I will use the projector to show myself type this in. This way students have a clear example of how to do this. I will then ask students to do the next one (number 4). “Ok, try doing this for number 4, and raise your hand when you have an answer.” I will of course be alert and see if 1, my students are actually doing this, and 2, if they are understanding this.

o Do the same thing for 6 and 7, where students will do the problems on their own, once each student is done, come back as a class and discuss the answer. For 7, the answer will be 2, which in because 10^2 is 100. This just helps students to remember what we are actually doing.

• “Finally, we want to get Logb x by itself, so we divide both sides by Loga b, and viola (again, follow on the board)! Log base b of x is our change of base formula. Just remember that we use 10 for the ‘a’ in this case, just so we can actually use the calculator.”

• After looking at these examples, we will move on to functions. “What is so great about the change of base formula, as I had said before, is that now we can evaluate any log expression we want to. Now we are going to change things up, and move onto functions of logarithms.” (Pass out graph paper, or ask students to take out their own, depending on the class)

• “First things first, I want you guys to see what the graph looks like by plotting it and drawing it yourself. So on your own, just draw the graph y = Log2 x. First, as always, draw a table on the right side of the graph (also draw one on the board to model for the students). For example, if I plug in 1 for x, what will my y then be, _____(pick a student at random)?”

o Hopefully, students get right away that y would be 0, since 2^0 = 1. If not I will have to rewrite it as an exponent (but I do not anticipate this). Then I would just remind them that for each case they could rewrite the logarithm as an easy to understand exponent.

• “Great! So go ahead and find more point for the table, plot them, and see what our curve looks like. Please don’t use your calculators just yet. I actually want you to really understand why the graph looks the way it does.”

o During this time, as usual, I will walk around the room and make sure that students are following along as I had hoped. I anticipate possibly some trouble with negative values for x (which they should realize aren’t possible here).

• Once I see that students have it, I will bring the class back together, and have them hold their papers up. This is a quick way to assess the whole class on the drawing activity. I will walk around, looking at each one, and see that the curve is the right shape and point in the right direction, and also falls only in Quadrants I and II.

• From here, I will put up the image of the graph y = (log x) / (log 2) up on the projector. I will ask if anyone knows why I did this, and I’m hoping that one astute student will realize this is the Change of Base Formula version of our function. I would probably have another student re-explain it, just to assess if multiple students understand. If this doesn’t happen, then I will have to lead a bit more with the following question (“Think about what we learned in the first half of class. What does this fraction look like?” for example)

• Next comes to connection to inverse functions. I will also plot two more functions on the graph. Y = x and y = 2^x. This will create an image where y = x is the mirror that reflects the other two curves.

• “Notice that y = x almost looks like a mirror for our log and exponential equations. Remember that these two functions are inverses (see extensions) of each other. Since inverses switch x and y, the line where x equals y turns into a mirror.”

• “Now for another function, say I wanted to plot the graph y = -log8 x. How could I plot this?”

o There are multiple ways to do this. We could plot the graph of (-1)(log x)/(log 8). We could also convert the function to its equivalent form of y = log8 (1/x), and then plot y = (-1) (log (1/x)) / (log 8). The latter method does take longer, but I wouldn’t mind if a student saw the connection.

• Once we come to the idea as a class, I will type it in my calculator, so it projects to the whole class. Now students have seen both growing and shrinking examples of a logarithmic function.

• Finally, discuss the restrictions on the base. “One last thing guys. Remember the restrictions on the base of a logarithm function (or expression)?” (wait) If no response, “What can b NOT equal, and what must it be greater than?” Functions are the reason we keep those restrictions

o “While Log(-2) 4 makes sense, and would be 2, the function of y = Log(-2) x will NOT be a function we can work with. After all, the function would jump all over the place, since x = 1 (have it on the board so it can be referred to) would give us 0, yet -2 for x gets us 1, but 4 for x gets us 2. Logarithm functions only go in one direction, and this one would have to change directions.”

o For the base b equaling 1, realize that Log1 x means that x can ONLY be 1, since any power of 1 is also 1. Meanwhile, y can be anything, so our resulting function is x = 1, which is NOT a function at all!

Closure

o “I will give you the rest of the day to work on your assignment (35) for the day. It is a simple 2-sided worksheet, and will be due tomorrow. Also, don’t forget we have a quick quiz on solving logarithm expressions and equations tomorrow, and it will include the change of base formula as well.”

o Until time runs out (if there is any to begin with), have students work quietly on this assignment. If students seemed well behaved during the lesson, and there is 10 minutes left or more, I would allow them to work in pairs. Otherwise I would want to assess them individually, so working alone would be ideal for myself.

Extensions

• We could discuss as a class why the common base of a logarithm is 10 in math. There are several reasons for this, but the best answer to this question is the fact that engineers use log (Base 10) most commonly, and engineers are the same people who designed calculators (thank you Wikipedia).

o 10 is also a very simple number to work with when we factor numbers. Factoring 10 out of 120 yields 1.2 x 10, factoring it out of 286 yields us 10x 28.6. This factoring is simple (we just slide the decimal point), as opposed to factoring 7 out of 120 or 286.

o In this way, we can convert extremely large logarithms to much smaller logarithms very simply. I can put an example on the board such as Log1083,975, and show how this converts to Log10(8.3975 x 10,000) or Log10(10^4 x 8.3975) = Log10(10^4) + Log10(8.375) = 4 + Log108.375, which will be easier to work with (try doing THAT with 7).

o “Another consequence of this method is that we now know that Log1083,975 will be between 4 and 5, so we can easily get an idea of where our answer should be. I really think it’s really interesting that numbers that range so far, like 1 and 83,975, can appear very close with logarithms (0 and ~4.924).”

• If the day has enough time left when I get to this point, I could show the inverse relationship algebraically. This was briefly touched upon on Day 1, but showing it again may further reinforce the concept on inverse relationships and functions in general.

o A function’s inverse can be found by merely switching the x and y for each other, and then finding y by itself again. For example, in the case that y = x +2, the inverse is x = y + 2, or y = x – 2. So in the case of y = b^x, which implies logb y = x. The inverse would be logbx = y, our desired function. I would likely show various examples of taking inverses algebraically on the board, or even have students find inverses to various functions. It would be a great way to lead into a topic on inverse functions for a future lesson.

o The Extra Worksheet (on page 33) is a great way to practice the topic of inverse functions. I could probably squeeze in this worksheet no matter what, but I don’t want to spend a significant portion of the day discussing inverse functions, out of fear of overloading my students.

• I did not show many examples of a logarithm graph as a class, but I feel that if students see what the shape is supposed to look like (and even drew one themselves), then they should be ok with the assignment. However, if time allows, I could show more examples. Examples such as Log x^2, which is merely 2Log x, or Log2.5 x (to show that the base CAN be a noninteger) would suffice.

• For the restrictions on the base, go more in depth with the definition of functions. Explain how certain fractional powers of a negative number have no solution.

Assessment

Students will be informally assessed on their ability to operate single logarithms during the warm up worksheet. During the actual lesson, informal assessment takes place while we prove the change of base formula as a class. I will be checking to see if students can recall how to write a logarithm with an exponent in the correct equivalent form. When students are taught the change of base formula, they are expected to apply it to several problems from the warm up. Thus almost instantaneously I can see if students are following the correct procedure. Finally, students’ ability to understand a function, and graph the proper representation of a logarithmic curve will be assessed when students are introduced to the topic.

Formal assessment begins with the homework assignment (and the extra one if need be). Now I will be able to see if students can operate using the change of base formula, as well as recognize graphs of logarithmic functions, individually.

Standards

NCTM Process Standards

Number and Operations: During the extension about the reason for 10 being the common base, we discuss how applying the method of changing the expression to a sum of two logs (one a power of 10), we can be sure what two integers our answer will be between. This way students will be able to reason if the answer a calculator spits back at them is actually reasonable.

Algebra: Every single logarithm expression and equation we work with has the equivalent exponential representation, which helps to solve problems in many cases (also to see what’s going on). To master the lesson for the day, students must grasp the inverse relation between exponents and logarithms, and the relationship among different logarithm expressions (especially when dealing with functions and their graphs).

Reasoning and Proof: In the same way we proved algebraically the 3 arithmetic properties of logarithms on day 2, today we prove the change of base formula with another quick proof. Students will have to see why and how the thought pattern which concludes with the change of base formula actually makes sense.

Connections: Students must continue to relate logarithms to exponents in order to evaluate them properly. Here, students will compare logarithmic functions to their exponential function inverses.

Representations: After spending two days working with logarithmic expressions, students begin to investigate the functions of logarithms. In this way, students will begin to investigate how logarithms operate for multiple values of x. Students also work with the inverse function, and will see how these are represented as well.

Common Core State Standards for Mathematics

Seeing Structure in expressions (A-SSE): Students must rewrite logarithmic functions into their equivalent ‘Change of Base’ forms to graph them properly in their calculators.

Interpreting Functions (F-IF): Students begin to dig into the logarithmic function. The growth path (or shrink path) of this function is investigated, and students must also be able to convert any logarithmic function into an equivalent form to graph.

Name________________________ Date____________

Warm Up

Evaluate the following expressions, you may use a calculator if you wish

1. Log3 9 2. Log8 (1/32)

3. Log4 8 4. Log7 12

5. Log1.5 2.25 6. Log 50

7. Log 100 8. Log2 3

Did any of the previous problems present difficulty? If so, what made them harder?

STOP!

**FOR AFTER** To evaluate any Logarithm with a _(base) not equal to _10_, we can use the ___Change of base formula_____________________.

(Define it here)

Name____________________________ Date____________

Inverse Functions and the Logarithmic Function

For each of the following functions, find its inverse, by swapping x and y and solving for y.

1. y = 5x + 3 2. y = x^2 – 5

3. y = 9 4. y = sqrt(4x)

5. y = (1/2)^x + 5 6. y = (x^2)/3

7. y = x^2 + 2x + 4 8. y = 8x – 12

For each of the following graphs, draw the line y = x as a dotted line. Then draw the inverses of each curve. Answer the question that goes along with each graph as well.

HINT: It may be easier to find point of the first curve, and then draw those ‘points’ for the inverse curve (a point (2,0) on one curve goes to (0,2) on the inverse).

[pic] Do the two functions intersect?

If so, where?

Do the two functions intersect?

If so, where?

Name_______________________________ Date____________

Logarithm Change of Base

And some functions too

For the following problems, use change of base to compute the logarithm, if the solution exists. Round to the nearest 3 decimal places.

1. Log7 11 2. Log5 625

3. Log9 25 + Log9 4 4. Log12 13

5. Log 6 6. Log7.524

7. Log (-3) 8. –Log6 100

9. Log1/2 (7/8) 10. Loge 50

Write 2 Logarithmic expressions that can only be evaluated using the change of base formula.

Example: Log4 9 cannot be solved mentally because no simple power of 4 will yield 9.

I. II

For the following two logarithmic graphs, name 2 points each function crosses through. Use these points to deduce what each function should be. Finally, draw the inverse of the function, by reflecting it over the line y = x

[pic] [pic]

Point 1: Point 1:

Point 2: Point 2:

The function is: The function is:

Graph y = 4 Log3 x on the graph below. Be sure to label at least 4 points of the curve

[pic]

Shane Randa

Day 4 plan (just for first half of lesson): Quiz on day 1 and day 2 material

20 minutes allowed.

• I will collect the homework at the start of class (if there is any)

• Afterwards, as with every other day, I will give the students a chance to ask any questions they feel like asking (knowing this is quiz day, I anticipate several questions). I will answer questions, and from there I will cut off discussion and hand out the quiz. “It’s now time now for that exciting quiz. There are no calculators allowed for this portion. When you finish this page, turn it in up front, grab the second page, take out your calculators, and finish that one. You have 20 minutes to complete both parts of the quiz. As always, I expect all of you to remain silent until either time is up or everyone has finished their quiz. And……. Good luck to all of you!!”

o The students with IEPs for extended time for assessments are aware that in my class I allow them the extra time. So even though I said that every student has only 20 minutes, they know the policy as it applies to them. If they need more time when time is up, they then move to desks at opposite sides of the room in the back row (where nobody sits), and have an extra 5-10 minutes to complete their quizzes.

• During the allotted 20 minutes, I will occasionally walk around, just as a means to ensure that no one is cheating. Obviously, if any calculators are out, I make sure that student is on the second portion.

• When students have questions, I will typically not answer them, unless the question pertains to their interpretation of the question (especially if I write the problems in a way my ELL students cannot understand). Any mathematical questions should have been asked before, and it is not fair to lead one student further along than others.

• After the 20 minutes are up, I proceed with the lesson for the day……

Name__Answer Key____________ Date___________

Logarithm QUIZ #1 (part 1)

Solve for the variable in each of the following problems

1. [pic] 2. log9(0) = x

8=(-3) x does not have an answer

3. logb(1/625) = -4 4. log100(10^11)= x

b=5 (10^2)^x = 10^11, so x =11/2

Rewrite each of the following as one logarithm (if possible)

5. logb(x) + 7logb(M) 6. -2 logb(x+1) + 4 logb(x-1)

logb(xM^7) logb(((x-1)^4)/((x+1)^2))

7. (1/3) logb(x+2) – (1/2) logb(x+2) 8. 6 logb(x) - 9 logC(x^2)

logb(((x+2)^(1/3))/(sqrt(x+2))) bases not the same, not possible

For 9 and 10, rewrite the logarithm as a sum or difference

9. logb(x sqrt(x+1)) 10. logb((x^6)/(sqrt(x+4))

logb(x) + (1/2) logb(x+1) 6logb(x) - (1/2)logb(x+4)

**ok if they dont convert the exponents**

If we can always convert logarithms into an equivalent exponent form, why do we have/use logarithms? In other words, when does having an equation in exponential form do us no good?

When we cannot determine the variable (power of the base) that solves the equation. Think about the example from the first day we learned this. **Multiple correct answers exist**

Name__Answer Key__________ Date___________

Logarithm QUIZ #1 (part 2)

**With a calculator**

Apply the change of base formula for the following four problems to evaluate. (round to 3 decimal places)

1. Log9(7) 2. 2log14(5)

(log10(7))/( log10(9)) ~ .886 log10(5^2 or 25) / log10(14) ~ 1.220

SOLVE

3. -2 log7(12) = x^2

Log10(1/144)/ log10(7) = x^2 => x^2 = -2.554 => x has no real solution

**Students will get correct points for work and for conclusion

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