MATH 70 TEACHING IDEAS FOR CHAPTERS 2, 3, 4, 5, 6, and 8
Topics Sections Number of Weeks Ch. 2 Rates of Change/Linear Functions 2.6 & 2.7 2 Ch. 3 Systems of Equations, Systems of Inequalities 3.1&3.2 2 Ch. 4 Laws of Exponents and Logarithms 4.1-4.3,4.5, 4.7 2 Ch. 5 Exponential Functions 5.1-5.5 2.5 Ch. 8 Quadratic Functions and Equations 8.1-8.3 3.5 Ch. 6 Logarithmic Functions 6.1-6.4 3
Note: Some GTAs have found success teaching the chapters in non-chronological order. For example, some teach in the following order: chapters 2, 3, 8, 4, 5, 6. Others have taught in the order: 2,3,4,8,5,6.
PRELIMINARIES: DAY 1 Student Skills from preliminaries: Students should be introduced to function notation (since chapter 1 is omitted from their text)
Goals: Students start to work together in groups Review what we hope they know from Elementary Algebra Diagnose strengths and weaknesses
Get students used to the idea of working together in groups by doing a group activity the first day. Ask them to count off to form groups of 4 (If there are 40 students they should count to 10 to form ten groups. You may have some groups of 3. Try to avoid groups of 5, because they tend to spread out too far to work together.)
Once students are in groups they should introduce themselves and tell their group mates something about themselves, maybe where they went to school before SFSU or one "fascinating fact." Fascinating facts can be a good way to get people talking. If you want, you can have each group report out its most fascinating fact.
Once the groups have introduced themselves they should start to work in pairs and as a group to correct the errors on the "Math 60 Final."
Activity: Pre-test (A completed Math 60 final with student answers, containing mistakes)
1. Students have trouble with: Getting started. They have difficulty dealing with the material with no context for it. Also many have test anxiety, even if it is only diagnostic. This is a way to do a diagnostic test without testing. 2. Teaching activity: Have students work in pairs to correct student errors on a set of problems that cover the topics we hope they already know. Give each pair fo students one copy of the exam that includes an incorrect student response for each problem. They are to identify the error and show how to do the problem correctly. They can identify the error on the test sheet, but they should describe the mistake and rework the problem on their own separate sheet of paper.
3. If they do not have time to finish the assignment during class, assign the completion of the corrections as Homework and hand our extra copies so everyone has one.
4. Possible student reactions: Most students enjoy this type of trouble shooting, looking for errors in some one else's work.
Working together, students will be able to find many, but not all, of the errors. However, they may not be able to complete some problems correctly. Circulate as they are working, listening to their discussions and responding to their questions.
With help they will be able to complete these problems and by doing this have reminded themselves of what they know. You will also find out their strengths and weaknesses.
We anticipate that they will have most difficulty with graphing and with writing an algebraic expression to represent a problem.
Tell students they will need GRAPH PAPER for many assignments in this course. If students have made good progress on revising the test during class, consider assigning the writing of a math autobiography including: Their experience in learning mathematics elementary school to present Their feelings about mathematics Their most recent mathematics course and experience in it. Which math course they think they will take to fulfill the math requirement Whether they plan to take mathematics after that Their planned major CHAPTER 2 (2.6,2.7) Student Skills in Chapter 2: Understand what means to be a solution of a linear equation, graphically and algebraically. Be able to graph a linear equation efficiently (i.e. using slope and y-intercept). Know how to find a linear equation from a word problem and interpret slope and y- intercept with respect to the problem. Know how to find a linear equation from a data table.
Linear Functions All of what we need to review can be dealt with through selected problems from sections 2.6 and 2.7. Students worked with this material in Chapter 3 in Math 59 or 60, but graphing tends to be a weak or easily forgotten skill for many (depending on how they "learned" it).
Start by telling students that we will briefly visit chapter 2 in order to review the graphing of linear equations. Ask them what is a "linear equation?" See if they can come up with examples.
Idea for Day 1 Use the graphing calculator to relate the numbers in a linear equation to its graph. Students have probably memorized "m is the slope and b is the y-intercept" in y = mx + b, but the relationship is still worth revisiting.
Use Exploration 2.2B on pages 130-131 of the text and the TI-83 graphing calculators. It's best when students work in pairs, so you will need 20 graphing calculators. Have the groups share their summaries with the whole class.
Select problems from the exercises page 116: Problems 87, 88, 90-96, and 100 (#100 is a good lead into the next lesson).
Sets of graphing calculators are located in the Math Mail Room under the mailboxes. They are in plastic suitcases, one large one with 20 calculators and two small ones with 10 each. When you use them, please pick them up immediately before your class and return them immediately after. If new batteries are needed please let us know.
Idea for Day 2 The purpose today is to calculate slope as a measure of the rate of change or steepness of a line between two points on a graph. Students should start from the graph and reach their own generalizations about vertical to horizontal change.
You could start with students’ methods for writing the linear equations for the four data sets in problem 100 that show a linear relation. Students should be able to calculate the slopes from the tables In two cases the y-intercept is given. In the other two students could project backwards one step to get the y-intercept What if you can determine the slope, but it’s not so easy to see the y-intercept? Problem 101, page 117 parts (a) and (b) are straightforward examples where the student can use the graph to determine the slope and then use one of the points to calculate the y-intercept. Work through the process with students using the graph.
REFRAIN FROM TELLING STUDENTS HOW TO FIND THE SLOPE BY WRITING THE RATIO OF THE DIFERENCE IN Y VALUES OVER THE DIFFERENCE IN X'S. They have been told this formula before, and that's probably why they don’t know what it means. They need to use the graph and figure out this formula for themselves.
Assign pairs or groups to work on problems 97, 99, 102, 135, 138, 139, 143, 146 As you circulate encourage students to describe their methods for finding the slope. Toward the end of the class time, lead a whole class discussion about slope and what it measures and find out from the students their methods for figuring out the slope.
Finish as homework.
(Optional) Idea for Day 3 Use one of the problems in Aerobics 2.9, page 102. You will need to make a copy of the graph on an overhead transparency or get to class early and put the graph on the board. If you do it on the board, it will help to have a meter or yard stick to approximate a line of best fit. Work with the class to develop a linear equation to represent the line of best fit.
Then put students into groups of four. Each group will work on a different problem and prepare a poster for their problem on large poster graph paper (available in the copy room). Each poster should include a graph and a linear equation to represent their problem (even if the problem doesn’t specifically ask for one.) along with their answers to the questions. As groups finish they tape up their posters, and each group explains their poster to the rest of the class.
Possible problems: The other Aerobics problem, 154, 155, 156, 157, 158, 160, 162, 163, 164 on pages 123-125. For homework, look ahead to Chapter 3. CHAPTER 3 (3.1, 3.2, 3.3) Student Skills from Chapter 3: Be able to solve a system of linear equations using substitution, elimination and graphs. Be able to graph a system of linear inequalities.
On completing Chapter 3 students should be able to write and solve systems of linear equations and inequalities in two variables and be able to explain how the equations and their solutions are related to their tables, graphs, or problem situations. They
Solving Systems of Equations and Inequalities The focus in Chapter 3 should be on Sections 3.2 and 3.3 solving systems of equations using tables, graphs, and equations and graphing systems of inequalities. Many students when reminded will be able to solve systems of equations, and they should be able to graph linear equations and inequalities. Working together in groups they will come up with different methods.
We suggest using a problem from 3.1 (see Idea for Day 3), but you do not need to “cover the material” in 3.1 as it should be easily handled in the context of doing the problems. Definitely SKIP solving systems of three equations in three variables and ALL of section 3.4.
Idea for Day 1 Use an introductory problem that can be solved with a table or graph to lead into the more efficient method of solving systems of equations.
We recommend problem: #14 from the Exercises on page 187. Give students the first part of the problem (without any of the question parts, a, b, c's etc.) and ask what they might do to figure out which is the best deal. If they do not know how to start, suggest that they make tables and possibly graphs. Expect some creative answers and be sure to be supportive of their ingenuity. But say that you want to focus on the table approach.
Students should work in groups of 3 or 4 and share the work of making tables. Circulate to see how the groups are doing. When they have made enough progress to see when the salaries are about the same, lead a whole class discussion on the importance of that point and ask for some ideas on some other ways to find it. If any group came up with a graph, they should put it on the board. Some may come up with equations. Put these on the board and ask for a way to use the equations to solve the problem (NOTE: problem 15 asks for equations and graphs, but don’t assign this at first. Maybe assign it after the discussion).
Assign Aerobics 3.1 without explanation, for students to complete in class in pairs and the heating systems (the text example for this section). Work on these in groups depending on time. Possible HW assignment: Exercises 6, 11, 12, 13, 15 (pages 185-187)
Idea for Day 2 Many students, when reminded, will be able to solve systems of equations. Working together in groups they will come up with different methods. Some will remember or partially remember addition to eliminate a variable, others might prefer substitution. Your goal is to find out what they know, refresh their memories, and build their repertoire of methods.
Start class by putting students into new groups of four by having them count off. Be sure they introduce themselves. Then have them do four problems from the Aerobics sections: 3.2a problems 3a and 4a and 3.3b problems 1a and 1b. I think I would just get to class a few minutes early and put these on the board.
Circulate as the groups work on these and as problems are finished ask groups to put them on the board. Try to find groups that have done them in different ways so there will be several solutions for the same problem. Then when all are done have groups present and explain their methods and discuss efficiency and choice of method.
After this discussion and summary of alternative methods Assign: Exercises 17, 19, 20d, h, i, 23, 24. NOTE: in #24 the questions of no solutions and many solutions comes up. Wait until the next day, if they come with questions, to discuss this.
Idea for Day 3 Introduce guess and check as a method for solving word problems and setting up equations.
A good example is Problem 21 from the Exercises page188.. Work through the problem guessing and checking until you end up with a table like the one below. You may or may not get to the correct solution, but be sure to make at least three guesses. Notice that in the following example the second guess is not well thought through in advance. This is on purpose to emphasize that there is no “bad” guess. Every guess gives us some information. .
Safe Risky Amount Amount Total money Check Investment Investment earned in earned with earned $100 safe invest. risky invest. 1000 1000 40 80 120 Too much 800 1200 32 96 128 Too much 1200 800 48 64 112 Too much 1500 500 60 40 100 √ x 2000-x .04x .08(2000-x) 100 x y .04x .08y 100
Once you have done three or four guesses start with a variable in the guess column. When you move to the second column ask students how to fill it in. Some may say 2000-x, but others may say y. Go with their preference, and then come back later and do it the other way.
Show how the equations come directly from the table as either .04x + .08(2000-x) = 100 And the other way .04x + .08y = 100 and x + y = 2000 Actually you’ll have to talk about where the second equation comes from.
Follow this with group work on problems 22 and 26 and allow time for some discussion of these before the end of class. Be sure to circulate to see what students are getting and what they are having trouble with.
Possible assignment in addition to 21, 22, 26: Exercises 27, 38, and 42 pages 188-190.
Idea for Day 4 Inequalities Again, students worked on these kinds of problems before. Different students will remember different details. Rather than bore them by talking about what they already know, give them some problems to work on in groups so you and they can find out what they don’t know and concentrate on that. They will be able to answer many of each others questions.
NOTE: You can skip the business of writng “compound inequalities.” This level of symbolic detail is not necessary at this point, and for some obscures the main issues. Those who will use it later will figure it out (as you did).
Start with group work on Problem 58 (leave off the compound inequality part) from page 192. If many groups have the same question, call the whole class together to discuss just that question. As groups finish, they should work on problems 55, 56, 59, 60, 65.
Possible Assignment in addition to 58, 55, 56, 59, 60, 65 Aerobics 3.3, exercises 1-4, page 173-174.
Alternative to Day 4 Or in addition, as a group assessment Linear Programming Problem Several are available: Cookies, The Toy Factory, Sandy Dandy Dune Buggies. One of these could be used as a group assessment for students work on this chapter.
AN EXPLORATION TO USE BETWEEN CHAPTERS 3 AND 4 The following Exploration requires the use of graphing calculators. It should be done in groups or pairs in class. It provides an opportunity for students to review or learn a lot of the details and language of functions as they do the investigation. Start with the exploration and answer questions about definitions such as “domain” and “range” as the come up while students are working. MATH 70 Exploring Functions
As you begin investigating functions, it is important that you understand what is expected when you are asked to sketch a graph. To sketch a graph means you show the approximate shape of the graph in the correct location with respect to your axes and that you clearly label all key points.
1. Consider the following equation: y = - 1 Use a graphing calculator to help you to answer each of the following questions completely.
a) Sketch the graph from your graphing calculator.
b) What are all the possible values of x? Are there any values that will not work for x? What is the largest value you can use for x? Explain. Describe the domain for this function.
c) Does the graph ever cross the horizontal line y = 50? How about y = 500? How do you know?
d) What is the smallest possible value of y? What are all of the possible values of y? Describe the range for this function.
e) Does the line y = x intersect the graph? How could you find the point of intersection?
f) Use the grapher to estimate the point(s) of intersection.
2. Use the grapher and sketch the graph of each of the following functions on graph paper. Clearly describe the domain and range for each.
2 a) g(x) = √(100 – x )
2 b) h(x) = x – 7
c) y = √ (x + 6) + 2 CHAPTER 4 (4.1-4.3, 4.5, 4.7) Student Skills for Chapter 4: Be able to simplify expressions using rules of exponents Be able to explain rules of exponents based on the definition. Be able convert between scientific and decimal notation (optional). Understand definition of logarithm in base 10 and be able to convert between exponential and logarithmic equations. Be able to solve simple log equations in one variable (4.7).
Note: Some instructors choose to wait and teach 4.7 (intro to logs) in conjunction with chapter 6.
The Laws of Exponents and an Introduction to Logarithms On completing Chapter 4, students should know and be able to use and explain the laws of exponents and have a beginning understanding of an exponent as a logarithm.
(4.1) Measuring Time and Space This section is about scientific notation. You have three choices: Do Exploration 4.2 of Kepler's Law, do some routine Aerobics problems and exercises, or skip the section entirely, and start with section 4.2.
Exploration 4.2, page 252-253 has students working with calculators, in groups to discover Kepler's Law through examining patterns some of which require the use of scientific notation. This could be an interesting class activity to motivate the use of exponential notation.
Alternatively you could start by having student groups work on Algebra Aerobics 4.1, 1-9, page 205 and follow that with a discussion and summary.
Possible assignment: Exercises selected from 1-12 or 18-20 pages 237-239.
(4.2) Simplifying Expressions with Positive Integer Exponents Laws of exponents are difficult because they seem so deceptively simple, and yet students often don't know how to use them when they need them. Instead of telling them the rules, try this. Put the following multiple choice questions (it's not a quiz) on a half-sheet (students should work in pairs or groups of three on this so you can copy one-half sheet for each pair or triad) or on the board. Have students work together to agree on the correct choice and then to write a justification for their choice. Saying they remember a rule is not enough. They need to explain why that rule works. There are lots of rules, why is that particular rule the correct choice.
Select the correct answer for each expression and explain why that should be the correct simplification. Just writing down a rule is not enough. You need to show why that rule would make sense for this problem.
1. a2a3 = 2. (a2)3 = 3. 2a-1 = a) a6 a) a6 a) –2a b) a5 b) a5 b) 1/2a c) 2a5 c) a8 c) 2/a
4. (8a6)/(2a2) 5. a0 = 6. (-2a2b2)3 = d) 4a3 a) 0 a) 1/(2a6 b6) e) 4a4 b) 1 b) -6a6 b6 f) 6a4 c) a c) -5a3 b5 d) -8a6 b6
As groups finish select groups to put their explanations on the board and present them. Then finish with a summary of the rules and reasons for them generated from the whole class. Assign a small sample of exercises from 21-45 pages 239-241.
(4.3) Simplifying Expressions with Negative Integer Exponents There are several ways to get students to come up with definitions for a0 and a-1 Ideas for the zero power:
1. Paper folding is a good way to start. Ask students to take a piece of 8.5x11 paper and fold it once, then ask how many regions there are. Fold it again and ask how many regions, keep folding and recording # of regions. They could do this in pairs with one student folding and the other making a table to record (# folds, # regions). Then, after three or four folds go back to the original unfolded piece of paper and ask if I had not folded it, e.g. 0 folds gives me how many regions?
2. Consider powers of 2 (or some other number like 3 or 5) in a table, starting with 32 or 64. (A vertical table is better in class, but this fits better here)
64 32 16 8 4 2 1 26 25 24 23 22
Ask students to continue the table. They will come up with 20 = 1. Then ask them to continue the top row and speculate on what the pattern implies for the bottom.
3. Have students carefully graph the (reasonable) values in the table.
4. Consider using the eating cake and drinking metaphor in Ask Alice. Some students prefer this explanation to all others.
5. LAST, after all are somewhat convinced that the definitions they are coming up with for a0 and a-1 make some sense, ask students to make an argument for their definitions based on comparing the extension of the division law for positive exponents they came up with earlier with the basic definition of exponents and division. In other words a3 /a5 = a3-5 = a-2 But also a3 /a5 = aaa/aaaaa = 1/a2 so a-2 = 1/a2
See if students can write a similar argument for a0 Assign pairs to work on the Aerobics in 4.3 page 215 to check their understanding plus problems from 46-56 to practice.
(4.4) Converting Units (optional) Converting units is a practical skill that makes more sense when you really care about the outcome and is easier to learn in context. But it may be worth a day if you are not feeling pressed for time.
On page 217 the “something to think about” brings up the often confusing question of two alternative ways to look at conversion factors. For problems that involve a lot of conversion factors, a paper manipulative can be very useful. Here’s what I mean Suppose you knew that light travels at 3x108 meters per second, and you wanted to find out how many kilometers light would travel in one year. So you need to convert from meters to kilometers and from seconds to years. So tear up a bunch of slips of paper. On the front of one put 1000m/1km and on the back put 1km/1000m. On others put 60sec/1min on one side and 1min/60sec on the other, then on another 60min/1hr and 1hr/60min etc for days and years. Once you have the 5 slips of paper front and back, you can just write down the original amount and start placing the paper slips so all but km/year cancel
The above example came from example 4 in the text (page 217) Do several others with students making and using their own manipulatives to set them up, before you show the problem on the board. They need to try them out to see how helpful they can be for organizing and thinking about conversion problems.
Problem # 9 in the Aerobics, page 217, might be another one to try in class or one from the exercises: 77, 82, 89 pages 243 and 244. Take the assignment from the easier Aerobics, page 217, and choose a few exercises from those under section 4.4 on pages 242-243.
(4.5) Simplifying Expressions with Fractional Exponents This is all about roots and their representation as fractional exponents. Our goal is a better understanding of roots, a more intuitive sense for approximations, and an appreciation of the use of fractions as exponents to represent them instead of those root signs. Students will need calculators.
Start by having students pretend that their square root button and all other roots or exponent buttons are broken. They can only use the basic operations +, –, x, /
Start by asking students to approximate to the nearest thousandth: Things such as √90, the cube root of 70, the cube root of –241, the fourth root of 161. Try it yourself. Using a calculator you can do it fairly quickly AND you get a good sense of size. Then use the calculator to consider (√90)(√90) = and others.
Without a calculator: have students graph y = √x by calculating outputs and plotting points (x, y) for the following x-values: 0, 1, 4, 9, 16 .25, 2.25, 6.25, -1 Then have them use their graph to estimate √3.5 and check it with the calculator.
Then put up a bunch of problems like a1/2 x a1/2 and 251/2 x 251/2 and 91/2 x 91/2 And ask what students think the fraction should be interpreted to mean based on the basic laws of exponents already developed. Show others such as 81/3 x 81/3 x 81/3
Use a cube that is 3x3x3, like a Rubix cube and start by asking for the volume then work backwards from 27 to represent one side as 271/3 = 3 and then the area of one face as 272/3 = 271/3 x 271/3 = 9
Most problems are practice. Choose some from Aerobics 4.5a, page 222 and Exercises 92, 93, 95, 97, 98, 99, 101 on page 245. Do not spend time on simplifying radicals in 4.5b.
(4.6) Skip this section
(4.7) Logarithms Base 10 This is meant to be just a bare introduction to logarithms so don’t let them panic! Sneak up on logs with a silent board game.
As students are coming in to class put the following table up on the board. Start with the x values in no particular order and wait until someone suggests you put them in order. Be very clear that this game is to be played SILENTLY. Do not allow talking or explaining in any other way. Start by filling in a couple of y-values, then offer the chalk to students, motion for a volunteer to come up and fill in just one number. If it is correct, nod and smile and offer the chalk to another. If it is incorrect, frown and erase. Students should add numbers for both x and y. Don’t call on the same person more than once and keep it going until most seem to have it figured out. You might want to try a second game using 3 or 10 as a base.
x 2 8 4 16 1 1/2 64 y 5
Have students copy the table (they may want to put it in order) and make a careful graph of the table.
Now in groups of three or four, have students fill in the following table using guess and check to find the exponent they’ll need on 10 to the nearest thousandth. This is a good problem for a group to divide up the labor and share results so each person only does two of them. Show them how to use the calculator to make closer and closer guesses to the answer for 2 or 3 and how they can use the exponent key to verify that 10.301 ~ 2 x 1 2 3 4 5 6 7 8 9 10 y .301 .478
So this function means that the exponent base 10 for 2 is .301 We might have called it exp10(2) = .301 but for some reason mathematicians decided to call it log10(2) = .301 Using exp10(2) = .301 should help to make the meaning of “log” clearer. After they have completed the table they should carefully draw the graph. If there’s time here’s another table they can try: x 10 20 30 40 50 60 70 80 90 100 y
Suggested problems: Page 235 Aerobics 4.7b #2 and #3 and Exercises 4.7 pages 248-249 choose from 133-139, 142, 147. CHAPTER 5 (5.1-5.6) Student Skills for Chapter 5: Recognize difference between linear and exponential data from a table or graph. Using percentages in exponential growth and decay. Understand asymptotic behavior of exponential functions. Recognize linear and exponential growth/decay from a situation or word problem.
Exponential Functions On completion of this chapter students should be able to write equations for exponential functions to represent a variety of growth and decay situations, and they should be able to make the transitions from table to graph to equation and vice versa with facility.
(5.1 and 5.2) Exponential Growth (This may be a short lesson-maybe to follow the return of a test on the last unit or to combine with 5.2)
Start with problem 12 (page 301) The King and the grains of rice on the checkerboard. Students should work in pairs to make a table and a graph for this function.
Then continue with problems like #18 on page 302 to look at the comparison of exponential to linear growth. Lead a class discussion on constant rates of change verses increasing rates after students have completed work on this problem.
For an assignment there are lots of practice problems in Aerobics 5.2, page 265, and you can select a few problems from 19-29 pages 302 and 303.
(5.3) Exponential Decay Penny Lab Each group gets 100 pennies in a box or bag. Dump all of them. Remove all the tails. Next round, same thing. Keep track of how many are left after each dumping. Either merge class data, or each group graphs own data. Display a graph or merged graphs and with the class construct the table for the expected “halving” function and create the equation by using x as the next table entry to get f(x) = 100(.5)x x f(x) 1 50 100(.5)1 2 25 100(.5)2 x 100(.5)x
Ask students how many coins there were to start with, for 0 flips to reinforce the meaning of (.5)o in relation to the formula. Also you could where the graph would go if there were a previous flipping (before 0).
Have students complete #31 (page 303) to practice writing an equation and then put them to work on the Chlorine problem (#80, p.308)
Be sure to comment on the Limitations of the Exp Growth Model. (See page 258-259) For the Chlorine problem asking when the value is in reasonable range is a nice twist. It is good to give problems where students have some space to think. Work with the groups on this and summarize when most have completed it.
From pages304-306 assign 37 and 39 or ones like them and select some practice from Exercises 30-36. Be sure they explain their answer to #37.
Root Beer Lab : Exponential v Linear decay models.
(5.4) The Graphs of Exponential Functions Instead of talking about the effects of the parts of an exponential equation on its graph have students use the graphing calculators to do the following.
Target Game with graphing calculators. Warm-up with the graph of y = 5x. and introduce the Window and Trace features.
Then they graph y=7(3x) and see if they can verify that (3, 189) is on the graph. Introduce the Zoom feature to get close to (3, 189).
Then ask them to find a graph for an equation of the form y= Cax which goes through (4,4) within two decimal places. They or you can make up more targets. Ask students: What happens when a gets bigger? When C gets smaller, etc.? Can you hit any target?
Mystery Problem. Each group puts in 4 graphs on a graphing calculator. They also give the 4 formulas that go with the graphs. Then trade calculators with another group. The other group must tell which formula goes with which graph without peeking at the "y=" window.
The above problem is like #51 p.305; they could do #47, 50, 51, or 53 for HW.
(5.5) Exponential Growth or Decay Expressed in Percentages (Note: the book plays with some language about growth rate vs. growth factor. This is NOT NECESSARY. It would be best to avoid those practice problems. You might just use the term “multiplier” to refer to the base (e.g. the growth or decay factor) of the exponential function.
Give students the following problem to start the discussion on percent increase: Suppose back when you were 8 years old you had found an unmarked envelope on the playground and it contained $5000. Nobody had claimed it and your parents suggested that you put it in the bank where it would earn 4.5% interest, compounded annually, until you needed it for college. If the year you needed it was this year, how much money did you have? Use this example to discuss two different ways of calculating the amount at the end of the first year, and discuss which is more efficient and how the Distributive property is the reason they are the same. Then have students solve the problem (there should be many different answers and it might be interesting to compare differences for different numbers of years) Then ask them to figure out when the money will double? They must do this by guessing and checking.
Propose a second problem for percent decrease: Prof. Fernandez bought a new Honda Element last summer for $19,900. One source said the car would depreciate at the rate of 20% per year. At that rate, how much will the car be worth in five years? Again start by having students compute percent decrease for one year in two ways. Then have them complete the problem. Then ask them to figure out when the car will worth only half of what she paid. Again they should use a table or guess and check.
Don’t tell them about the rule of 70. If they bring it up, suggest they try it for problems they have already done to verify it.
#114, page 314, is a good in-class problem for groups (Millionaire). #120, page 315, might be good for HW.
(5.6) Examples of Exponential Growth and Decay This section is full of applications. It provides an opportunity assess students understanding and at the same time expose them to a wide variety of these applications.
Suggested activity: Group poster presentations. Select a variety of problems from those remaining in the chapter. Assign one per group of four. The assignment for each group is to complete the problem and then display their solution on a poster. (using big poster paper and marking pens-we have them). On completion the posters will be displayed around the room and each group will explain theirs.
Each group poster can be assessed according to criteria you have spelled out ahead of time: Appropriate solution method(s), Accuracy, Clarity of communication of solution, completeness, etc.
Some possibilities: Aerobics 5.5, #4, 1, or from the exercises: 77, 78, 79, 80, 98, 111, 113, 115 CHAPTER 6 – LOGARITHMIC FUNCTIONS (6.1-6.4) Student Skills from Chapter 6: Understand the rules/properties of logs and be able relate them to rules of exponents. Use definition of log to solve simple logarithmic equations in one variable. Use logarithms to solve exponential equations. Students should be introduced to e and the natural logarithm.
(6.1) Using Logarithms to Solve Exponential Equations This section could take 3 days, with a short first day.
(6.1a) Estimating Solutions to Exponential Equations: This section starts off with problems similar to the ones students were doing at the end of Chapter 4. To introduce logarithms base 10, ask students to estimate an answer to 10^? = 3 using a guess and check approach with a calculator, where they are only allowed to use the exponent buttons. We call this a (common) logarithm.
Aerobic 6.1a could be used as a “warm-up” or saved for use as homework. Problems 1-5, page 361 are also possible practice.
(6.1b) Rules for Logarithms: It is probably not important for students to memorize the laws of logarithms. It would be nice if they could tell real laws from fakes. To practice this, use these problems or ones you make up:
log(A * B) = log A + log B log (A + B) = (log A) + B log A^B = B log A log (5^2) = (log 5)^2 log 1 = 0 log 0 = 1 log 10^A = A log A - log B = (log A)/(log B)
Give them the above statements. Can they disprove the wrong ones and justify the right ones? Let them use a calculator to try out various numbers first. Can they do it without a calculator?
(Optional) Show the proof for the multiplication law and then see if groups can replicate the argument for the division law or the power rule.
Aerobics 6.1b can be homework or problems selected from 7—15 pages 361-362.
(6.1c) Using Logarithms to Solve Exponential Equations: Suppose you had received an inheritance of $5,000 when you were 8 years old and your parents had insisted that you put it into a savings account where it would earn 3.5% interest, compounded annually. Furthermore they would not allow you to take it out until the amount had doubled. How old would you be at that time? Would it help to pay your college tuition? Have students set up an exponential equation and then use a table, graph or guess and check to solve. When they have come up with an answer ask if they’d like to learn a more efficient method. Then introduce the method of taking the log base 10 of both sides of the equation and solving from there.
Select some in-class problems for group work from pages 362-363 , 24-32 (you might assign one to each group to develop as a poster problem for presentation). There are also plenty of practice problems to choose from for homework in Aerobics 6.1c and problems 16-20. or all Aerobics 6.1a, b, c can be used as homework. Problems 11, 14, and 15 should be in class problems, if assigned. The rest of 1-20 is a lot of practice-select some. Groups could work on one or two of 27-32 in class, with one or two others assigned.
(6.2) Base e and Continuous Compounding This can be a one day section. To introduce e, use the compounding interest problem with $1000 and 100% interest over 20 years. (See the handout "Compounding Interest" you can use one copy per group as an in-class problem solving assignment.)
Aerobics 6.2, problems 1-7, page 338-339, can be homework (It recaps work in class). The problems 33-50 provide lots of practice with interest rates. Select a few.
(6.3) The Natural Logarithm This can be one day. Define the natural log, ln, and say we choose e as the universal base because it works well in calculus. It crops up in other surprising places, like the compound interest.
(AN ASIDE for YOUR INTEREST) Hat check problem. (If you have n people who check their hats, and they get their hats back at random, what is the probability that no one gets their own hat back? As n gets bigger, the answer approaches 1/e. See
This section in the book is really more practice with laws of logarithms, this time in base e. A more valuable use of class time would be to bring in the graphing calculators and have students do Exploration 6.1 page 373.
Problems 51-74 page 365-366, are all practice. Maybe use of a few and assign some problems from previous sections.
(6.4) Logarithmic Functions Another one-day section. Use the decibels discussion pg 348-349 and example 8 pg 350 (three airplanes) to introduce logarithmic functions.
Aerobics 6.4, problems 1-3 and 5-8, page350 can be part of homework. Problems 83, 84, 85, page 367 are based on the decibels discussion. Problem 81, page 367 is good practice. (6.5) Skip this section
(6.6) Skip this Section CHAPTER 8 –QUADRATIC FUNCTIONS AND EQUATIONS (8.1-8.3) Student Skills for Chapter 8: Be able to find vertex and intercepts of quadratic functions. Be able to use the quadratic formula to solve quadratic functions. Use quadratic functions to model and solve max/min problems. Know how to complete the square to solve quadratics and/or to prove quadratic formula (optional).
(8.1) Polynomials These activities may take two days. The are a substitute for Section 8.1 in the text, which is interesting but not related to the rest.
Polynomial Functions Exploration: Start with a calculator exploration of patterns for polynomial functions of degree 2, 3, 4, 5 (instead of looking at the graphs on p. 400)
See if students can guess some of the properties on p.401. In particular, as you plug in big negative & positive numbers, you either get both HUGE positive numbers or HUGE negative numbers. Why from the formula?
There are several possible ways to have student do this exploration, ranging from very structured to unstructured. A structured approach appears below.
Use the graphing calculator to graph and then sketch the graph of each of the following (All will show well in a standard window). Discuss the graphs with your group. What relationships do you notice between the graphs and their equations? As students work with these ask them to Identify the x- and y- intercepts. (You can use the ones that are easily factorable later to illustrate the zero product rule).
2 1. f(x) = 3 x – 2x – 8
2 2. g(x) = –x + x + 6
3 3. h(x) = x – 5x
3 2 4. F(x) = 2x + x – 6x
3 2 5. G(x) = x + x – 6x
4 2 6. j(x) = x – 5x + 4
4 3 2 7. k(x) = –x – 3x + 10x
5 3 8. p(x) = x – 5x + 4x 5 4 3 2 9. s(x) = x + 5x + 5x – 5x – 6x
6 4 2 10. q(x) = .2x –3x + 11x – 7
An unstructured approach would be to write just a few examples of polynomial functions of degree 2, 3, 4, 5 and then direct students to make up others to graph and look for patterns.
Zero Product Rule: It's not hard for students to discover this rule themselves. See Zero Products Activity below. This can be a class activity or assigned as homework to be discussed at the beginning of the next class.
1. Write down at least five number pairs (v,w) to solve the equation vw = 50. (Keep in mind that v and w do not have to be whole numbers, and don't even have to be positive.) 2. Do the same for the equation vw = 100. 3. Do the same for the equation vw = 0. 4. Discuss your results. How does Question 3 compare to the first two examples? Why is it so different? 5. Write down at least five number pairs to solve the equation (r - 2)(s + 1) = 100. 6. Do the same for the equation (r - 2)(s + 1) = 0.
Define and blow trumpets over the ZERO PRODUCT RULE.
Ask students to graph certain equations in factored form such as y= (x-3)(x+5) and find their x- intercepts. They should see a pattern. What is the pattern? They then should explain why the pattern exists in terms of the zero product rule.
If you used the structured version of the graphing activity, ask students to factor the polynomials in 1-8 and discuss the factors and their relation to the the x-intercepts. And/or ask them to create polynomials with roots or x-intercepts at 3, 4 and 5. Then do some more and graph them.
Then ask why we care about the "zeros" or x-intercepts or roots of polynomials. Why don't we care about where a polynomial is equal to 7? Help them to see that the answer is that all we've got to work with is the zero-product rule.
The zero product rule is a very handy way to find a zero for a quadratic (or higher) by getting a zero of a simpler linear function. That's why we like to take any problem where a quadratic (or any polynomial! or product of functions!) equals something and change it to a quadratic = 0.
Problems. Page 523, problems 105, 107, 109, 110, 111, 112 might be used for homework.
(8.2) Finding the Vertex: Transformations of y = x2 The properties described in Sections 8.2a and 8.2b can be conjectured by students based on their exploration of relationships between the graph and the equation using the graphing calculator. The attached Parabola Investigation (including HW) will provide a more structured approach where the equation is in graphing (or vertex) form, or you can use a less structured exploration of 2 the function in standard form y = ax + bx + c, described as follows.
Have students explore graphs of quadratics and ask them what happens to the ends. Do they have ones where one side goes down and the other up? If not, why not? They should be able to reason from the formula that that won't happen.
Recall from polynomial investigation that there is supposed to at most one hump. Help them see there is exactly one and it is called the vertex. Some vertices are at the top (called maxima) and some are at the bottom (called minima). Have them play with the graphing calculators to find out which f(x) = ax2+bx+c have maxima and which have minima. Where is the vertex in relation to the intercepts? Notice symmetry.
Instead of showing them, give the students (in groups, in class) the problem on page 446 Example 4 (before they read about it) on maximizing area. Ask them to make a table to find the maximum and then to graph it by hand,. They can also write an equation and check the graph of their equation against their graph from the table.
Select from problems 26-36, pages 514-515. PARABOLA INVESTIGATION
As you use the graphing calculator to investigate each question be sure to keep track of all the equations you try along with their resulting graphs. What doesn't work in one case may be the key to another question that will come later.
a) Graph the parabola y = x2. Make an accurate sketch of the graph. Be sure to label any important points on your graph. In addition to x- and y- intercepts be sure to label the lowest point which is called the vertex.
b) Find a way to change the equation to make the same parabola open downward. The new parabola should be congruent (the same shape and size) to y = x2, with the same vertex, except it should open downward so its vertex will be its highest point.
Record the equations you tried, along with their results. Write down the results even when they are wrong—they may come in handy below.
c) Find a way to change the equation to make the y = x2 parabola stretch vertically (it will appear steeper). The new parabola should have the same vertex and orientation (i.e., open up) as y = x2.
Record the equations you tried, along with their results and your observations.
d) Find a way to change the equation to make the y = x2 parabola compress vertically (it will appear as if the points in y = x2 move toward the x-axis).
Record the equations you try, their results, and your observations.
e) Find a way to change the equation to make the y = x2 parabola move 5 units down. That is, your new parabola should look exactly like y = x2, but the vertex should be at (0, -5).
Record the equations you try, along with their results. Include a comment about moving the graph up as well as down.
f) Find a way to change the equation to make the y = x2 parabola move 3 units to the right. That is, your new parabola should look exactly like y = x2, except that the vertex should be at the point (3, 0). If you need an idea to get started recall one of the graphs you did for homework.
Record the equations you try, along with their results. Tell how to move the parabola to the left as well as how to move it to the right.
g) Find a way to change the equation to make the y = x2 parabola move 3 units to the left and stretch vertically, as in part (c). Your new parabola might look like y = 4x2, except that the vertex should be at the point (-3, 0).
Record the equations you try, along with their results. Comment about how to move the parabola to the left as well as how to move it to the right.
h) Finally, find a way to change the equation to make the y = x2 parabola vertically compressed, open down, move 6 units up, and move two units to the left. Where is the vertex of your new parabola?
Record the equations you try, their results, and your comments on how each part of the equation affects its graph.
For each equation below, predict the vertex, the orientation (open up or down?), and tell whether it is the same as a vertical stretch or compression of y = x2. Before using the graphing calculator, sketch a quick graph based on your predictions.
a) y = (x + 9)2 b) y = x2 + 7 c) y = 3x2
2 2 d) y = e) y = -(x - 7)2 + 6 f) y = + 1
g) y = 2(x + 3)2 - 8 h) 4y = -4x2 i) y = 4x - 4
j) Check your predictions for the equations in parts (a) through (i) on your graphing calculator. If you made any mistakes, correct them and briefly describe why you made the mistake (what incorrect idea you had). Then make a neat and accurate graph for each function. (8.2c) Standard to Vertex Form and vice versa (Optional) This section includes completing the square or locating the vertex by calculating x = –b/2a. Develop a method with the whole class.
Practice problems can be selected from Aerobics 8.2c or page 515-516 37-44. Groups might work on some from 49-58 on pages 516 and 517.
(8.3) Finding Horizontal Intercepts of a Quadratic
You have some options. -You can prove the Quad Formula (using completing the square). -Or bring students along by showing them how to complete different squares. -Or give them a choice of seeing the proof or not.
Factoring itself isn't that important, though they should be able to multiply binomials (distributive law). Use a picture of multiplying as area for a rectangle with dimensions (Ax+B) by (Cx+D) instead of just FOIL. Then you can set up completing the square by working 2 backwards from (for example) x + 6x + 4 by showing a square with dimensions (x + 3) by (x + 3) and that our polynomial needs +5 to complete the area of the square. So adding 5 will “complete the square” and subtracting 5 will allow them to keep an equivalent expression. They 2 also need to see one like x + 4 x + 6 where there are extras and 2 x + 5 x + 3 .
2 Once they have completed the square they may be able to derive a formula for solving x + bx + c, and then you can lead them through the derivation for Quadratic Formula
The Really Important Part: Students should be able to write the factored form using the quadratic formula and for any quadratic, no matter what the roots are. Be sure to do examples in class and check that what you get from the formula makes the equation 0 and gives the x-intercepts of the equation. (That is much more important than having seen the proof.)
Discriminant is NOT a topic to spend time on; it will show up anyway when you use the QF. You need to say something about when the part under the radical (then you can say, otherwise known as the discriminant) is negative what happens.
(8.4), (8.5), and (8.6) are not part of the course.