Instructor Notes For Module 3 - Kent State University
MATH 10775 Pathways Algebra for Calculus PLUSFall 2017Instructor Notes for Module 3Skill DayThe focus today is a review of basic factoring skills, including quadratic trinomials, difference of squares, sum ordifference of cubes, factoring by grouping. Also included is factoring out the GCF, including binomial GCFs.Suggested order of lesson:1. Work through several examples of each type of factoring with students. Worked examples are available onthe Blackboard site. Be sure to include sum or difference of cubes, then factoring a binomial GCF.2. Ask students to work on the odd numbered problems on the “Factoring Gateway Practice” worksheet onBlackboard.HW due next class: Paper and Pencil, to hand in: finish odd-numbered (even?) problems on the worksheet; and 51– 56 in the Pathways workbook, p. 261; IMathAS: PC8M3 I1This module introduces the notion of function.From Marilyn Carlson: The purpose of this investigation is to support students in acquiring the ability to imagine theco-variation of two quantities in context (imagine how one quantity changes as another changes), and to makemeaningful connections between a problem context, numerical data, a formula and a graph.There are several ways of approaching this lesson and I recommend the following. Before class, cut four equal sizesquares from the corners of 35- 40 pieces of 8.5” by 11” sheets of paper, so that each student in your class can haveone. Use a variety of cut sizes, but I’d suggest using 8 of each of the following: 1” cutouts; 1.5” cutouts, 2”, 3”, 4”.By cutting the paper yourself ahead of time, you save class time and still allow students the opportunity to have ahands-on experience. The paper cutter in the copy room is ideal for this chore.Learning outcomes. Students will:Explain how the size of the cut affects the volume of the box.Write an expression for the length of a box with a cutout of size x;Write an expression for the width of a box with a cutout of size x;Write an expression for the height of a box with a cutout of size x;Write an expression for the volume of a box with a cutout of size x;Indicate the largest and smallest cut possible for a 8.5” by 11” piece of paper (i.e begin to work with thenotion of domain in context).Discuss, either by using a table, an interactive graph, or a formula, how the volume of the box changes as xIncreases;Use and interpret function notation in the context of the box problem.Suggested order of lesson:1. As students enter the classroom, ask them to pick the piece of paper that they think will create a box that willhold the most popcorn. You can have the papers sitting in piles on the front desk.2. Have students fold up their papers and create a box with an open top.3. You might show the “Paper and Box” app from the first Power Point in this section and have students discusshow the size of the cut affects the amount of popcorn the box will hold (volume).4. You might then show the “Volume with Cubes” app and discuss more formally the notion of volume.5. Ask students to work through problem #1 – 3 on pp. 81 – 83. Discuss and answer student questions as you go.
MATH 10775 Pathways Algebra for Calculus PLUS6.7.Fall 2017Introduce function notation as given in the box on p. 83 and discuss the notion of domain.You might review the ideas of dependent and independent variables and, as students work through #3, writethe input and output both as points: 1.5,66 or size of cutout, Volume of box and using functionnotation: f 1.5 66 or f size of cutout volume of box8.If you have time, show students the “Graphs” app with Power Point 1. This is a great tool for discussing theideas in this section, as well as introducing the ideas of max or min value of a function.The IMathAS homework deals mainly with function notation, so it’s a good idea to assign the paper and pencilproblems below as well. IMathAS includes problems using interval notation, which may be new for some students.I’ve posted a handout on Blackboard as a guideline for those students.HW due next class: Paper and Pencil, to hand in: #6-#9 on pp. 85 (review skills); IMathAS: HW8 and PC9M3 I2 Functions and domains of functions.Pre-class assignment: PC9 on IMathAS consists of the following Investigations in the workbook: M3 I2: #1 and #6.You might take student answers to the pre-class assignment and use them as a springboard for class discussion.This section continues the work on functions, focusing on function notation and domains.Learning outcomes. Students will:Be able to accurately use and interpret function notation in a variety of settings.Be able to correctly identify the domain of a function, whether the function is given as a table of values, areal-world scenario, or an abstract formula.Be able to write a function for a linear real world scenario.Use interval notation accurately to describe the domain of a function.Be able to find the vertical and horizontal intercepts of a function, given its formula or graph.Suggested order of lesson:1. Ask students to work on #2 on p. 87 in small group. Discuss, emphasizing function notation and also theapplied domain.2. Work through #3 together with students, using interval notation for the domain.3. Work through #6. You may have to cut short #3 in order to get to #6, but you do need to discuss theseproblems today. The notion of domain is not an easy one for students, so be sure to get to this problem!HW due next class: Paper and Pencil, to hand in: #6 on p. 90; IMathAS: HW9 and PC10M3 I3 More with function notationThe Pre-class assignment for this section (PC10) on IMathAS consists of problems similar to (but NOT identical to) #5,#1, #3 in the workbook: You might take student answers to the pre-class assignment and use them as a springboardfor class discussion.Learning outcomes. Students will:Same is in M3 I2 above.Suggested order of lesson:1. As students to work through #1,#3 - #5 then discuss.HW due next class: IMathAS: HW10 and PC11
MATH 10775 Pathways Algebra for Calculus PLUSFall 2017Skill DayThe focus today is a review of basic arithmetic of polynomial and rational expressions and functions.Suggested order of lesson:1. Quickly review addition and subtraction of polynomials and squaring a binomials, since this material is quiteeasy for students. You might ask students to come to the board and work in pairs as you give them thesereview problems.2. Cubing a binomial using the binomial theorem may be new to some students, so you might take a bit moretime with this skill.3. Work through a couple examples of simplifying rational expressions and addition and subtraction of rationalexpressions. You might choose even numbered problems from the “Rational Expressions – GATEWAYpractice” worksheet on Blackboard. The Pathways workbook also has good examples on p. 260.4. Ask students to work the odd numbered problems on the worksheet and finish for homework.HW due next class: Paper and Pencil, to hand in: finish odd-numbered problems on the worksheet; Also, #39 – 50 asneeded in the Pathways workbook, p. 260. IMathAS: PC8M3 I6 Exploring the Difference QuotientThe Pre-class assignment for this section (PC11) on IMathAS consists of the following Investigations in the workbook:p. 109: #1 and #2a.This section seems to follow M3 I3 quite well, with its emphasis on function notation. Be sure to develop the notionof difference quotient as the slope of the line containing two points of a function that may be, but is not necessarily,linear.There is an extra practice worksheet on Blackboard.Learning outcomes. Students will :Given a graph of a non-linear function, explain the meaning of the difference quotient between two points onthe graph;Given the graph of a non-linear function, explain the meaning of the variable h in the difference quotient;Compute the difference quotient for a linear, quadratic, cubic, and simple rational function, given thefunction formula.Suggested order of lesson:1. Ask students to work through #1 on p. 109, then #5 on p.110. Discuss.2. Do several traditional difference quotient problems, i.e. given a quadratic, cubic, or rational function, find theformula for the difference quotient. After finding each formula, you might ask students to find the slope ofthe secant line two ways: using two points on the function and using the formula for difference quotient.HW due next class: IMathAS: HW11 and PC12; paper and pencil to be handed in: as many from the worksheet asyou deem appropriate. You might inform students that you will be giving them a quiz during the next class and theproblem will be taken right from the worksheet.
MATH 10775 Pathways Algebra for Calculus PLUSFall 2017M3 I4 Function CompositionWe have two days for this sectionThe Pre-class assignment for this section (PC12) on IMathAS consists of problems similar to (but NOT identical to#1and #7 on pp. 95 and 99 respectively.Learning outcomes. Students will :1. Compose two functions given a real-world scenario with two different functions represented by tables ofvalues and interpret the result;2. Compose two functions given the graphs of those two functions;3. Compose two functions given function formulas;4. Discuss restrictions on the domain of a composition of two functions.DAY 1I like to start with a problem from the homework section in the workbook for this section:Suggested order of lesson:1. Ask students to work on #43 on p. 121 either in small groups or together as a class with you. You might askthem questions such as:What is the meaning of g(11)? g(13)? g(9)?What is the meaning of f(50)? What about g f 50 ?2.3.4.5.Give the definition of composition of functions.Work through together with students: #5 on p. 97, then #6 on p. 98.Time permitting, run through #7 with studentsWork through #8. This is important! You need to get to this problem, perhaps doing even more plug andchug type so they see it before doing the homework.If necessary, work quickly through #7 so that you get to #8. You should do at least one of each type, though:composition given table of values, a real world scenario, graphs, and formulas.Be sure to call your students’ attention to the boxed definition on p. 29 in the textbook and to the worked exampleson the following pages.HW due next class: IMathAS: HW12.DAY 2The focus on Day 2 is using composition of functions defined symbolically in context and finding the domain of acomposition.Suggested order of lesson:1. Ask students to work on #1 on p. 101 either in small groups, then discuss. Be sure to emphasize the domainquestions in parts d) and e).2. Work through #3 with students, then relate the notion of domain of a composition when functions are givenby a formula. You will need to create a couple examples like #7 and #8 on the IMathAS HW12B so thatstudents are able to do the homework.3. Work through #4 and #5 with students and be sure to discuss the domain of the composition.HW due next class: IMathAS HW 12B and PC 13
MATH 10775 Pathways Algebra for Calculus PLUSFall 2017M3 I5 Inverse Functions: Reversing the ProcessThe Pre-class assignment for this section (PC13) on IMathAS consists of #5 in the workbook: p. 106.Learning outcomes. Students will be able toVerify that two functions are inverses of each other;Determine if a given function has an inverse function;Given the graph of a one to one function, sketch its inverse;Given the formula of a one to one function, find its inverse.I start this lesson by discussing how to “undo” a simple calculation, like adding 10 to a number. Most students willrespond “subtract 10.” I use function notation to write these functions: f x x 10 and g x x 10 focusing onthe fact that “the function f takes the input and adds 10 to it. The function g takes an input and subtracts 10 from it.”I then have students make a table of values for f and g, asking them to use the output of f as the input of g. Theyquickly notice that the output of g is simply what they started with, i.e. the input of f.I then give the definition of inverse functionsI have an application problem I’ve used for years to develop the distance formula. It has to do with the distance a ballwas thrown on a football field, given the location of the quarterback and receiver. I’ve posted this problem onBlackboard. Alternately, you might use #1 on p. 49. I would simply ask students to find the distance from the houseto the cell tower, though, and not drag it out as it is in the workbook. Guide students carefully through the derivationof the general formula using the Pythagorean Theorem. They need some practice here. Problems #86 – 88 on p. 77are good for that.Suggested order of lesson:1. I start this lesson by discussing how to “undo” a simple calculation, like adding 10 to a number. Moststudents will respond “subtract 10.” I use function notation to write these functions: f x x 10 andg x x 10 focusing on the fact that “the function f takes the input and adds 10 to it. The function g takes5.6.an input and subtracts 10 from it.” I then have students make a table of values for f and g, asking them touse the output of f as the input of g. They quickly notice that the output of g is simply what they started with,i.e. the input of f. You might do a few more simple examples, e.g. multiplying and dividing by 3, etc.Ask students to work through #4 on p. 106. Discuss the same pattern as above, noting that the inputs of theoriginal function are the outputs of its inverse and vice versa.Work through #6 with students.I then give the definition of inverse function as described on p. 38 in the text, though I have not yetmentioned the notion of one to one functions.Student practice – worksheet to verify if two functions are inverses of each other (on Blackboard).Practice – given the sketch of a one-to-one function, sketch its inverse (worksheet on Blackboard).7.Ask students to sketch the graph of the function y x 2 , then sketch its inverse. What’s the problem with the8.inverse function? Students readily see that the inverse is not a function and then we discuss one to onefunctions and how we can test if the inverse of a function is also a function (horizontal line test).At this point, students are ready for the procedure of finding the inverse of a function given its formula.2.3.4.We have two class days for this section and we will definitely need it!HW due next class: One day assign HW 14 on IMathAS and the other can be to complete the worksheet onBlackboard.
The focus today is a review of basic factoring skills, including quadratic trinomials, difference of squares, sum or difference of cubes, factoring by grouping. Also included is factoring out the GCF, including binomial GCFs. Suggested order of lesson: 1. Work through several examples of
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