general mathematics teaching guides

Von Christopher G. Chua, LPT, MST [Education Program Supervisor, DepEd, Eastern Samar Division]

DOLORES NATIONAL SENIOR HIGH SCHOOL Dolores, Eastern Samar, Philippines

[SY 2016-17] General Mathematics Teaching Guides

General Mathematics Teaching Guides •••

General Mathematics Teaching Guides Von Christopher G. Chua, LPT, MST [Education Program Supervisor, DepEd, Eastern Samar Division]

Introduction The teaching guides included in this document have been designed for teachers who have been assigned to teach the first core Mathematics subject in the senior high school, General Mathematics. It has been patterned with the curriculum guide mandated by the Department of Education with each plan explicitly stating the content, the content standards, the performance standards, and the learning competencies together with their distinct codes. The codes follow a format such as M11GMIa-1 where the first letter stands for Mathematics, 11 for the grade level, GM for General Mathematics as course, the roman numeral I for the quarter and in this case the first, the lowercase letter for the week where a is for the first week, and finally the last Arabic number is the unique code for each learning competency. Instead of creating the guides on a daily basis, the author has decided to adapt a weekly frame to ascertain that all tasks in the week are coherent and promote continuity of learning. The procedures, however, are detailed per unit of instructional time, that is, one hour sessions with four sessions per instructional week. The guides follow the general format of a lesson plan and include five major parts: the objectives and the learning competencies, the subject matter together with the references and the essential ideas; the daily instructional procedure; student assessment complete with the required rubrics; and the assignment. These guides have been designed for ease of use and to provide other teachers with ideas for activities. The author takes into account both the availability of learning materials that are easily accessible even to teachers in the most remote areas of the country and the instructional innovations provided by today’s technology. By elucidating a variety of activities, teachers may opt to choose the more convenient and appropriate for their learning environment. Since this is a work-in-progress, the author also made sure to include important notes after the actual implementation of each teaching guide. This serves as evaluation of the effectiveness of the material and will be used to further improve the guides constructed. It is therefore highly recommended that teachers who have opted to use these guides should contact the author to contribute to the development of these teaching aides. Contact numbers are available on the attached author’s page.

Course Overview •••

The Senior High School (SHS) curriculum includes two core subjects in Mathematics that students in the eleventh grade are required to take regardless of their chosen SHS career track and strand. These two core subjects are General Mathematics and Statistics and Probability with the former being included in the first term of the school year followed by the latter in the second term. The General Mathematics course is outlined with fifty-one (51) competencies in distributed over a period of eighty (80) hours during the whole term. It has no prerequisite course and is divided into three central Math areas: (a) Functions and their graphs; (b) Basic business mathematics; and (c) Logic.

Introduction  1 Von Christopher G. Chua


Learning Principles in the K to 12 Basic Education Program [excerpt from K to 12 Curriculum Guide in Mathematics (2013)]

The framework is supported by the following underlying learning principles and theories: Experiential and Situated Learning, Reflective Learning, Constructivism, Cooperative Learning and Discovery and Inquirybased Learning. The mathematics curriculum is grounded in these theories. Experiential Learning as advocated by David Kolb is learning that occurs by making sense of direct everyday experiences. Experiential Learning theory defines learning as "the process whereby knowledge is created through the transformation of experience. Knowledge results from the combination of grasping and transforming experience" (Kolb, 1984, p. 41). Situated Learning, theorized by Lave and Wenger, is learning in the same context in which concepts and theories are applied. Reflective Learning refers to learning that is facilitated by reflective thinking. It is not enough that learners encounter real-life situations. Deeper learning occurs when learners are able to think about their experiences and process these, allowing them the opportunity to make sense of and derive meaning from their experiences. Constructivism is the theory that argues that knowledge is constructed when the learner is able to draw ideas from his/her own experiences and connect them to new ideas. Cooperative Learning puts premium on active learning achieved by working with fellow learners as they all engage in a shared task. The mathematics curriculum allows for students to learn by asking relevant questions and discovering new ideas. Discovery Learning and Inquiry-based Learning (Bruner, 1961) support the idea that students learn when they make use of personal experiences to discover facts, relationships, and concepts. A complete copy of the curriculum guide for General mathematics may be found as part of the appendices.

Learning Principles  2 Von Christopher G. Chua


Functions Week One, Functions and Their Graphs Content Standards: The learner demonstrates understanding of key concepts of functions. Performance Standards: The learner is able to accurately construct mathematical models to represent real-life situations using functions. Learning Competencies: Over the course of one week, the learner is expected to (1) represent real-life situations using functions, including piece-wise functions [M11GM-Ia-1]; (2) evaluate a function [M11GM-Ia-2]; (3) perform addition, subtraction, multiplication, division, and composition of functions [M11GM-Ia-3]; and (4) solve problems involving functions [M11GM-Ia-4] Learning Materials: In order to develop the targeted competencies, the following materials are needed: (a) chalkboard and chalk; (b) LCD projector and laptop or in the absence, visual materials. Expected Outputs: In order to assess the attainment of the learning competencies targeted, students will be required to undertake two group performance tasks and two written works. Procedure (Teacher’s Activity)

INTRODUCING FUNCTIONS Day One [Target M11GM-Ia-1] Represent real-life situations using functions, including piece-wise functions. Other Specific Objectives: The learner is also expected to attain at least 75 percent proficiency in the following objectives: (a) explain the concept of functions in comparison to relations; and (b) compare the different types of functions learned in the previous grades 1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competency (and other learning outcomes).

Essential Ideas How does it function?

••• A function can be thought of as a correspondence from a set X of real numbers x to a set Y of real numbers y, where the number y is unique for a specific value of x. It is the set of ordered pairs of numbers (x, y) in which no two distinct ordered pairs have the same first number. The set of all admissible values of x is called the domain of the function, and the set of all resulting values of y is called the range of the function (Leithold, 1996). A piecewise-defined function is one that is defined by more than one expression. These expressions or pieces are determined by restrictions in the domain. This function is also called a split function because of the behavior of its graph.

3. Facilitate activation of prior knowledge and motivation at the same time. Accomplish this by doing the initial activity described below.

Exploration Activity: Will the relationship function? The teacher may choose to relay the story in different ways but the use of visual materials is highly suggested. Characters and even the actual situation may be altered to suit the interest of the learners. In this case, the story is relayed through a slideshow presentation (https://prezi.com/q3a5mylivqfc/kasi-nga-senior-na-sila/). At the end of the presentation, a question is posted for students to discuss. This will lead the class to enrich their understanding of relations and functions. At this point, it is necessary that the teacher directs the discussion so that the class is able to process the following ideas: (1) a relation is any set of ordered pairs; (2) not all relations are functions; and (3) only those whose domain do not include an abscissa that is shared by two or more ordinates are considered functions.

Functions  3 Von Christopher G. Chua


Once upon a time there were seven senior boys: Peter, James, John, Matthew, Paul, Andrew and Mark. Close as they were, they did almost everything together. So when they learned about the school holding a Valentine’s Ball, they decided to find themselves some date.

Peter

Janna

James

Valerie

John

Marcia

Incidentally, there were also seven beautiful senior girls who were in the same class the boys were: Janna, Valerie, Marcia, April, May June, and Julie. Our boys each had a crush on the girls and they have decided to ask them out for dates. Of the boys, Peter, James and John have something in common. They always believe that honesty and trust make any relationship strong. Peter asked Janna out for the date, James chose Valerie and Marcia was asked by John.

April Mark

May June

Mark on the other hand thinks he needs to be on top of the pack. He likes to collect then select. To make sure he gets first date, he asked April, May and June without the three girls knowing that all of them are being asked out by the same guy.

Matthew

Matthew, Paul and Andrew are very competitive and they all liked Julie so they decided to ask her out and agreed that whoever she picks, the other two would accept defeat.

Andrew

Paul

Julie

There are three kinds of relationships described in our story but would all of these function? 4. Proceed by asking students the following questions. a. What type of functions do you remember from your previous Math subjects? b. Can you describe these functions? 5. Initiate discourse regarding equations being used as representations of functions. As initial example, present the situation stated below.

The flag down rate for metered taxis in Metro Manila is now down to PhP 30.00. For every minute, the fare goes up by PhP 5.00.

What are the two related variables mentioned in the situation presented? The two related variables are time of travel and taxi fare. Which variable is dependent and which is independent? Taxi fare is dependent upon the time of travel which is the independent variable. How much will a passenger pay if he rode the taxi for 20 minutes? 𝑃ℎ𝑃 5.00 (20 𝑚𝑖𝑛) ( ) + 𝑃ℎ𝑃 30.00 = 𝑃ℎ𝑃 130.00 𝑚𝑖𝑛

What equation will best represent fare (F(t)) as a function of time (t) in minutes? 𝐹(𝑡) = 5𝑡 + 30

6. Utilize the other examples below to demonstrate the process. The same questions should be asked in order to acquaint the students with the systematic way of representing functions through equations.

For every box of cookies, she sells in a month, Anna donates a peso to the Bantay Bata Foundation. This and the PhP 45.00 she saves every month are put into the foundation’s bank account.

What are the two related variables mentioned in the situation presented? The two related variables are number of boxes of cookies sold and donation. Which variable is dependent and which is independent? Anna’s donation depends on the number of boxes of cookies she sold in a month. How big of a donation will the foundation receive from her this month if she was able to sell 243 boxes of cookies this month? (243)(𝑃ℎ𝑃 1.00) + 𝑃ℎ𝑃 45.00 = 𝑃ℎ𝑃 288.00

What equation will best represent the relationship between the two variables described? 𝐷(𝑐) = 𝑐 + 45


General Mathematics Teaching Guides ••• Mang Juan, who owns the biggest meat shop in the market has offered a sale on pork. A kilo costs PhP 160.00 but if you buy more than 3 kilos, each kilo is priced at PhP 155.00.

What are the two related variables mentioned in the situation presented? The price of the meat is related to the number of kilos bought. Which variable is dependent and which is independent? The number of kilos determines the price of the pork sold. How much will you pay for five kilos of pork? (5)(𝑃ℎ𝑃 155.00) = 𝑃ℎ𝑃 775.00

What equation will best represent the relationship between the two variables described? 𝑃(𝑘) = {

160𝑘, 𝑘 ≤ 3 155𝑘, 𝑘 > 3

By this point, the teacher should describe piecewise functions as compared to other functions. Also known as split functions, these functions contain multiple expressions called pieces that are used depending on restrictions. The next situation provides another example.

For Valentine’s day, SSG officers launched a fund-raising program called “Dinner for a Cause”. Each ticket is worth PhP 120.00. If one buys five tickets, he only needs to pay PhP 550.00. If he buys more than five tickets, the price per ticket goes down to PhP 105.00

What are the two related variables mentioned in the situation presented? The amount and the number of tickets bought. Which variable is dependent and which is independent? The number of tickets bought determines how much one needs to pay. How much will I save if I buy 7 tickets in one than just buying 5 tickets then another 2? [(2)(𝑃ℎ𝑃 120.00) + 𝑃ℎ𝑃 550.00] − [(7)(𝑃ℎ𝑃 105.00)] = 𝑃ℎ𝑃 55.00

What equation will best represent the relationship between the two variables described? 120𝑡, 𝑡 < 5 𝑃(𝑡) = { 550, 𝑡 = 5 105𝑡, 𝑡 > 5

The following questions should be raised to aid generalization of the concept targeted: (a) Describe a function in mathematics; (b) How do we represent functions through equations; (c) What is a piecewise function? How is it different from the other types of function you have previously encountered?

Similar situations and problems may be taken from The Calculus 7 by Louis Leithold (7th Edition). Use some of these to further test students’ understanding.

PERFORMANCE TASK NO. 1

Evaluation/Assignment (Performance Task) Functioning through Pieces Divide the class into groups of four. Each group shall be asked to determine one situation that produces a piecewise function. Through a written report, the group should be able to narrate the situation, identify the variables, and finally, represent it through a function. The rubric below should guide the teacher in evaluating the output of the students in this performance task Criteria

Quality of the situation presented

Predictors Doing okay

Can do better The situation is neither novel nor related to the immediate environment of the student. (3 credit)

Meeting expectations

The situation presented is either novel or related to the immediate environment of the student. (4 points)

The situation reflects both novelty and real-life environment. (5 points)




Clarity and coherence of work

The situation lacks clarity and is deemed confusing. (2 point)

The situation is understandable but may be improved through paraphrasing. (3 points)

Situation was narrated clearly and is coherent with the objective of the task. (4 points)

Correctness of the function presented

Function is entirely incorrect based on the narrated problem. (2 point)

Function may be further improved through simplification. (4 points)

Function is correct. (5 points)

Group dynamics

Members worked independently or more than one did not help with the task at hand. (2 point)

One member clearly did not participate. (3 points)

The project reflects collaboration among members of the group. (4 points)

Promptness in Submission

Task completed past the deadline. (no credit)

Score Interpretation

7 points or less

Task completed on or before the set deadline (2 points) 8 to 14 points

15 to 20 points

Highest Possible Score: 20 points (2 per item) Passing Score: 15 points (75 percent)

EVALUATING FUNCTIONS

Day Two [Target M11GM-Ia-2] Evaluate a function 1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competency.

3. Facilitate activation of prior knowledge by using the following questions for recapitulation. a. What is a function? How is it different from a mathematical relation? b. What are peicewise-defined functions? c. How are functions usually represented?

4. Present the problem stated here: Mark has an internet shop as business. He charges PhP 20.00 for every hour that a customer uses a computer. Being the wise businessman that he is, he always gets 10% of his daily income and saves it to pay for electric charges. He also subtracts PhP 180.00 per day for his shop’s monthly rent. The rest of the amount is the café’s income for the day. a.

What are the two variables that may be determined from the problem? Which of these two is the independent variable? [The computer shop’s income is dependent on the number of hours of computer use per day] b. Represent the café’s daily income through an equation. [𝑓(𝑥) = 20𝑥 − (2𝑥 + 180) 𝑜𝑟 𝑓(𝑥) = 18𝑥 − 180] c. How much would the café’s income be if it raked 80 hours’ worth of income? [PhP 1,300.00] 107 hours? [PhP 1,786.00] d. What is least number of hours of computer use in the shop so that Mark gets his daily return of investment? [10 hours] Ask students to present their solutions for c and d on the board to be used for discussion. 5. Explain the process of evaluating functions based on the students responses to the problem stated above. To better illustrate this process, use the concept of the function machine.

INPUT (independent variable, x)

FUNCTION MACHINE 𝒇(𝒙)

OUTPUT (dependent variable, Y)



6. Discuss the next couple of examples to the class. Let f be the function with domain all real numbers x and defined by the formula, 𝒇(𝒙) = 𝟑𝒙𝟑 − 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟕. Find f(2) iand f(-2).

𝑓(𝑥) = 3𝑥 3 − 4𝑥 2 − 3𝑥 + 7 𝑓(2) = 3(2)3 − 4(2)2 − 3(2) + 7 𝑓(2) = 3(8) − 4(4) − 3(2) + 7 𝑓(2) = 24 − 16 − 6 + 7 𝒇(𝟐) = 𝟗

If x represents the temperature of an object in degrees Celsius, then the temperature in degrees Fahrenheit is a function of x, 𝟗 given by 𝒇(𝒙) = 𝒙 + 𝟑𝟐.

Water freezes at 0°C and boils at 100°C. What are the corresponding temperatures in °F? 9 𝑓(0) = (0) + 32 = 𝟑𝟐℉ 5 9 𝑓(100) = (100) + 32 = 𝟐𝟏𝟐℉ 5

𝟓

𝑓(𝑥) = 3𝑥 3 − 4𝑥 2 − 3𝑥 + 7 𝑓(−2) = 3(−2)3 − 4(−2)2 − 3(−2) + 7 𝑓(−2) = 3(−8) − 4(4) − 3(−2) + 7 𝑓(−2) = −24 − 16 + 6 + 7 𝒇(−𝟐) = −𝟐𝟕 Aluminum melts at 660°. What is its melting point in °F? 9 𝑓(660) = (660) + 32 = 𝟏𝟐𝟐𝟎℉ 5

For the more advanced students the next problem may be given as way of enrichment: Given 𝑓(𝑥) = 2𝑥 2 + 3𝑥 − 9, evaluate 𝑓(𝑥 − 3) 𝑓(𝑥) = 2(𝑥 − 3)2 + 3(𝑥 − 3) − 9 𝑓(𝑥) = 2(𝑥 2 − 6𝑥 + 9) + 3(𝑥 − 3) − 9 𝑓(𝑥) = 2𝑥 2 − 12𝑥 + 18 + 3𝑥 − 9 − 9 𝒇(𝒙) = 𝟐𝒙𝟐 − 𝟗𝒙 7. The following question should be raised to aid generalization of the concept targeted: How do we explain the process of evaluating a function? 8. Evaluate the function given at the values of the independent variable stated. a. 𝑓(𝑥) = 2𝑥 2 − 7𝑥 + 9; 𝑓𝑖𝑛𝑑 𝑓(3) 𝑎𝑛𝑑 𝑓(−5) [f(3)=6; f(-5)=94] b. 𝑔(𝑥) = 𝑥 − 8⁄2𝑥 + 1 ; 𝑓𝑖𝑛𝑑 𝑔(10) [2/21] c. ℎ(𝑥) = 10𝑥 5 − 𝑥 4 + 3𝑥 3 − 7𝑥 2 − 10𝑥 + 1; 𝑓𝑖𝑛𝑑 ℎ(0)𝑎𝑛𝑑 ℎ(−1) [h(0)=1; h(-1)=-10]

WRITTEN WORK NO. 1

Evaluation (Pen and Paper Test) Evaluate the following functions at the given value of the independent variable. 1. 𝑓(𝑥) = 8𝑥 − 11; 𝑓𝑖𝑛𝑑 𝑓(−5) [29] 2. 𝑓(𝑥) = 7𝑥 3 − 𝑥 + 3; 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑓(1) [9] 3.

𝑣

3

𝑚(𝑣) = 𝑣2 −3𝑣+2 ; 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑚(3) [2]

𝑟 + 7, 𝑟 ≤ 5 4. 𝑔(𝑟) = { ; 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑟(5)? [12] 19 − 𝑟, 𝑟 > 5 𝑟, 𝑟 < −3 5. 𝑔(𝑟) = { 𝑟 2 , 𝑟 = −3 𝑟(−2)? [3] 1 − 𝑟, 𝑟 > −3 Highest Possible Score: 15 points Passing Score: 11 points (75 percent)

Scoring Guide: Each item is good for three (3) points. Give no point for no answer. One point should be given for correct substitution only. Three points is credited to correct answers.

ALGEBRA OF FUNCTIONS Day Three [Target M11GM-Ia-3] Perform addition, subtraction, multiplication, division, and composition of functions. 1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge by using the following questions for recapitulation. a. From our what we have discussed, what do you already know about functions? b. How does the evaluation of a function work? 4. Divide the class into four sets. These sets will not necessarily work with each other but each set will be given different tasks. General Instruction: Given the expressions, 𝑥 2 − 8𝑥 − 20 and 𝑥 − 10, perform the operation assigned to your set.


General Mathematics Teaching Guides ••• Set A: Addition [(𝑥 2 − 8𝑥 − 20) + (𝑥 − 10) = 𝒙𝟐 − 𝟕𝒙 − 𝟑𝟎] Set B: Subtraction [(𝑥 2 − 8𝑥 − 20) − (𝑥 − 10) = 𝒙𝟐 − 𝟗𝒙 − 𝟏𝟎] Set C: Multiplication [(𝑥 2 − 8𝑥 − 20)(𝑥 − 10) = 𝑥 3 − 8𝑥 2 − 20𝑥 − 10𝑥 2 + 80𝑥 + 200 = 𝒙𝟑 − 𝟐𝟖𝒙𝟐 + 𝟔𝟎𝒙 + 𝟐𝟎𝟎] Set D: Division 𝑥 2 − 8𝑥 − 20 (𝑥 − 10)(𝑥 + 2) [( )= = 𝒙 + 𝟐] 𝑥 − 10 (𝑥 − 10) 5. Ask one student from every set to discuss his/her answer to class. Let the other students judge the solution and open up the solutions for discussion.

𝑓(𝑥)

ቐ

‫۔‬ ۖ ‫ە‬

‫ۓ‬ ۖ

6. In order to introduce the notation for the operations on functions, represent the given expressions as functions, that is, Given the expressions, 𝑥 2 − 8𝑥 − 20 and 𝑥 − 10

𝑔(𝑥)

From the solutions of the students that are written on the board, use the function notation to define each operation as presented under “Essential Ideas” 7. Proceed with composition of functions. Have the students look at the following functions,

𝑓(𝑚) = 𝑚2 − 3𝑚

𝑔(𝑚) = 𝑚 − 5

𝑔(𝑓(𝑚)) = 𝑚2 − 3𝑚 − 5

Essential Ideas Operating Functions

••• Definition of the Sum, Difference, Product, and Quotient of Two Functions: Given two functions, f and g, (i) their sum denoted by f + g, is the function defined by (f + g)(x) = f(x) + g(x); (ii) their difference, denoted by f – g, is the function defined by (f - g)(x) = f(x) - g(x);

Ask the following questions: What do you notice about the three functions? Can you see any relationship existing among the functions given? [Expected response: The value of n in g(n) when replaced by f(m) will give us the third function. Evaluating g(n) at f(m) will result to g(f(m).] Explain that the third function is called a composite function of the first two functions. Show the process of obtaining the third function from the first two using the correct notation for composite functions.

(iii) their product denoted by f · g, is the function defined by (f · g)(x) = f(x) · g(x); (iv) their quotient denoted by f/g, is the function defined by (f / g)(x) = f(x) / g(x)

Definition of a Composite Function

(𝑔 ∘ 𝑓)(𝑚) = 𝑔(𝑓(𝑚)) 𝑔(𝑓(𝑚)) = (𝑚2 − 3𝑚) − 5 (𝑔 ∘ 𝑓)(𝑚) = 𝑚2 − 3𝑚 − 5 8. The following question should be raised to aid generalization of the concept targeted: What are the five operations on functions? State each in the general function notation. 9. Work with the following functions: 𝑓(𝑥) = 𝑥 3 − 1 𝑔(𝑥) = 𝑥 2 + 2𝑥 + 1 ℎ(𝑥) = 𝑥 + 1 a. (𝑓 + 𝑔 + ℎ)(𝑥) [𝒙𝟑 + 𝒙𝟐 + 𝟑𝒙 + 𝟏] b. (𝑓 − ℎ)(𝑥) [𝒙𝟑 − 𝒙 − 𝟐] 𝑓⋅𝑔 c. ( ) (𝑥) [(𝑥 3 − 1)(𝑥 + 1) 𝑜𝑟 𝒙𝟒 + 𝒙𝟑 − 𝒙 − 𝟏] ℎ d. (𝑔 ∘ ℎ)(𝑥) [𝑥 2 + 2𝑥 + 1 + 2𝑥 + 2 + 1 = 𝒙𝟐 + 𝟒𝒙 + 𝟓] For the more advanced students the next problem may be given as way of enrichment: With the same functions what is (ℎ ∘ 𝑓)(3𝑥)? [(ℎ ∘ 𝑓)(𝑥) = 𝑥 3 ; 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 (ℎ ∘ 𝑓)(3𝑥) = 𝟐𝟕𝒙𝟑 ]

Given the two functions f and g the composite function, denoted by 𝒇 ∘ 𝒈 is defined by (𝒇 ∘ 𝒈)(𝒙) = 𝒇(𝒈(𝒙) And the domain of 𝒇 ∘ 𝒈 is the set of all numbers x in the domain of g such that g(x) is on the domain of f. (Leithold, 1996)


General Mathematics Teaching Guides ••• Assignment (Pen and Paper Test)

WRITTEN WORK NO. 2

Three functions are defined as follows: 𝑚(𝑥)

= −2𝑥 + 7; 𝑛(𝑥) = 𝑥 4 + 𝑥 2 − 2; 𝑝(𝑥) =

𝑥+1

.

𝑥−3

Define/derive the following functions: Scoring Guide: 1. (𝑚 + 𝑛)(𝑥) [𝑥 4 + 𝑥 2 − 2𝑥 + 5] Each item is good for three (3) points. 2. (𝑛 − 𝑚)(𝑥) [𝑥 4 + 𝑥 2 + 2𝑥 − 9] 2 Give no point for no answer. −2𝑥 +5𝑥+7 3. (𝑚 ⋅ 𝑝)(𝑥) [ ] One point should be given for correct function notation. 𝑥−3 Two for correct substitution of the functions involved. 𝑥+1 𝑝 4. ( ) (𝑥) [ ] 2 Three points is credited to correct answers. 𝑚 −2𝑥 +13𝑥−21 5. (𝑚 ∘ 𝑛)(𝑥) [−2𝑥 4 − 2𝑥 2 + 11] 6. (𝑚 − 𝑛)(−2) [−(−2)4 − (−2)2 − 2(−2) + 9 = −𝟕] Highest Possible Score: 18 points Passing Score: 14 points (78 percent) FUNCTIONS AS MATHEMATICAL MODELS

Day Four [Target M11GM-Ia-4] Solve problems involving functions 1. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss items that need to be discussed. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge by using the following questions for recapitulation. a. From our what we have discussed, what do you already know about functions? b. What are the different operations on functions? Describe each. 4. Discuss the five suggestions for solving problems involving functions as mathematical models according to Leithold (1996). 5. Demonstrate the process of solving the following problems using functions as mathematical models. PROBLEM NO. 1 A wholesaler sells a product by the kilo (or fraction of a kilo). If not more than 10 kilograms are ordered, the wholesaler charges PhP 200.00 per kilo. However, to invite large orders, the wholesaler charges only PhP 180.00 per pound if more than 10 kilograms are ordered. Find a mathematical model expressing the cost of the order as a function of the amount of the product ordered. In the given situation, the cost of an order is dependent on the number of kilos of the product. Represent these two variables by letting C(x) be the cost of ordering x kilos of the product. Since the cost is calculated differently when not more than 10 kilos is ordered and when the order exceeds 10 kilos, the function is a piecewise-defined function. For orders not exceeding 10 kilos, 𝐶(𝑥) = 200𝑥 For orders more than 10 kilos, 𝐶(𝑥) = 180𝑥 𝐶(𝑥) = {

200𝑥, 180𝑥,

𝑖𝑓 0 ≤ 𝑥 ≤ 10 𝑖𝑓 𝑥 > 10

Essential Ideas Making Problem Solving less of a problem

••• Suggestions for Solving Problems Involving a Function as Mathematical Model (Leithold, 1996) 1. Read the problem carefully so that you understand it. Make up a specific example that involves a similar situation in which all the quantities are known. Another aid is to draw a picture. 2. Determine the known and unknown quantities. Represent the independent variable and the function that is obtained. 3. Write down any numerical fact known about the variable and the function value. 4. From the information in step 3, determine two algebraic expressions for the same number, one in terms of the variable and one in terms of the action value. From these two expressions, form an equation that defines the function 5. After applying the mathematical model to solve for the unknown quantities, write a conclusion that answers the question of the problem. Be sure this contains the correct unit of measure.


General Mathematics Teaching Guides ••• Determine the total cost of an order of 9.5 kilos and of an order of 10.5 kilos. For 𝑥 = 9.5 𝑘𝑖𝑙𝑜𝑠, 𝐶(9.5) = 200(9.5) = 1900 For 𝑥 = 10.5 𝑘𝑖𝑙𝑜𝑠, 𝐶(10.5) = 180(10.5) = 1890 PROBLEM NO. 2 A cardboard box manufacturer wishes to make open boxes from rectangular pieces of cardboard with dimensions 10 in. by 17 in. by cutting equal squares from four the four corners and turning up the sides. Find a mathematical model expressing the volume of the box as a function of the length of the side of the square cut out. Since the length of the side of the squares is relative, we represent it with x and shall determine the volume of the rectangular open boxes represented by V(x). The volume of a rectangular prism is length x width x height. Therefore, 𝑉(𝑥) = (10 − 2𝑥)(17 − 2𝑥)𝑥 Simplified, 𝑽(𝒙) = 𝟒𝒙𝟑 − 𝟓𝟒𝒙𝟐 + 𝟏𝟕𝟎𝒙 6. To serve both as application and performance task, assign the activity described below. Evaluation/Assignment (Performance Task) In groups of five, students will need to solve any two of the five problems stated below. Answers should be written on a whole sheet of paper.


1. An ice cream vendor makes a profit of 𝑃(𝑥) = 7𝑥 − 525 when selling 𝑥 scoops of ice cream per day. How many scoops of ice cream should be sold for break-even sales (𝑃(𝑥) = 0)? How much profit will the vendor earn for selling 235 scoops of ice cream? 2. A senior high school student earns income through encoding for which she charges PhP 10.00 a page. However, she gives a 5 percent discount if the encoding job exceeds 20 pages. Represent how much she charges per encoding job as a function of the number of pages per job. How much would she earn for encoding 29 pages? 3. A cellular phone company estimates that if it has 𝑥 thousand subscribers, then its monthly profit is 𝐸(𝑥) = 671𝑥 − 53,032. How many subscribers are needed for a monthly profit of PhP 766,259.00? How much will the company earn if it has 13,799,000 subscribers in one month? 4. The regular adult admission price to an evening performance at a cinema is PhP 300.00 while the price for children under 12 years of age is PhP 200.00 and the price for senior citizens (60 or older) is PhP 225.00. Find a mathematical model expressing the price as a function of the person’s age. How much will one pay for 7 tickets if two of these are for children, one for a senior, and the rest are for regular adults? 5. The cost of a cellular phone call for a telecom is at PhP 6.00 for the first minute and PhP 4.50 for every minute after the first. How much would an eight-minute long call cost you? Express the relationship between the two variables through a function. Highest Possible Score: 20 points (10 per item) Passing Score: 15 points (75 percent) Find time to discuss the answers to the problems. As a means of wrapping up the topics discussed over the week, use the following questions for generalization: What are functions? How are functions different from relations? How are they evaluated? When performing operations with functions, how would you describe each process? Why is there a need to study functions? What is the advantage of knowing how to create mathematical models through functions?



Rational Functions Week Two, Functions and Their Graphs Content Standards: The learner demonstrates understanding of key concepts of functions. Performance Standards: The learner is able to accurately formulate and solve real-life problems involving rational functions. Learning Competencies: Over the course of one week, the learner is expected to (1) represent real-life situations using rational functions [M11GM-Ib-1]; (2) distinguish rational function, rational equation, and rational inequality [M11GM-Ib-2]; (3) find the domain and range of a rational function [M11GM-Ib-5]; (4) determine the: (a) intercepts, (b) zeroes, and (c) asymptotes of rational functions [M11GM-Ic-1]; (5) graph rational functions [M11GM-Ic-2]; and (6) represent a rational function through its: (a) table of values (b) graph, and (c) equation [M11GM-Ib-4] Important Note: For an improved continuity of competencies under the same content standard, the learning competencies have been rearranged but the codes have been maintained for reference. Learning Materials: In order to develop the targeted competencies, the following materials are needed: (a) chalkboard and chalk; (b) LCD projector and laptop or in the absence, visual materials; (c) calculators Expected Outputs: In order to assess the attainment of the learning competencies targeted, students will be required to undertake two group performance tasks and one written work. Procedure (Teacher’s Activity)

RATIONAL FUNCTIONS Day One [Target M11GM-Ib-1 and M11GM-1b-2] Represent real-life situations using rational functions and distinguishes rational function, rational equation, and rational inequality Other Specific Objective: The learner is also expected attain at least 75 percent proficiency in comparing rational functions with other types of functions.

Essential Ideas Polynomial versus Rational

••• An algebraic function is one formed by a finite number of algebraic operations on the identity function and a constant function. These algebraic operations include addition, subtraction, multiplication, division, raising to powers, and extracting roots. If a function f is defined by 𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0

where 𝑎0 , 𝑎1 , … , 𝑎𝑛 are real numbers and n is a nonnegative integer, then f is called a polynomial function of degree n. If a function can be expressed as the quotient of two polynomial functions, it is called a rational function. (Leithold, 1996) Said differently, r is a rational function if it is of 𝑝(𝑥)

the form, 𝑟(𝑥) = 𝑞(𝑥), where p and q are polynomial functions.

1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competencies (and other learning outcome). 3. Facilitate activation of prior knowledge with these questions: a. What are functions? b. What are the different types of functions? Differentiate them. 4. Present the following couple of sets of functions to class. Ask the students what they think was the basis/rule for grouping the functions. Exploration Activity: Grouped how? Group A Group B 6 𝑓(𝑥) = 5𝑥 − 2 𝑓(𝑥) = 2 2𝑥 − 3 𝑔(𝑥) = 𝑥 + 3𝑥 + 8

Rational Functions  11 Von Christopher G. Chua

General Mathematics Teaching Guides ••• 2 5 𝑥 + 8𝑥 3 − 𝑥 − 18 3 𝑥3 + 𝑥2 − 𝑥 − 1 𝑚(𝑥) = 10

𝑥−5 𝑥+2 12𝑥 + 8 ℎ(𝑥) = 2 𝑥 +𝑥+1 𝑥3 + 𝑥2 − 𝑥 − 1 𝑚(𝑥) = 𝑥−1 [Expected response: Functions in Group B are those that have variables as denominators while those in Group A don’t.] Teacher’s Question: One of these two groups of functions is composed of rational functions. Which of the two groups do you think is it? Explain the basis for your answer. ℎ(𝑥) =

𝑔(𝑥) =

5. Use students responses in the previous item to differentiate polynomial and rational functions. If necessary present some other examples to deepen understanding.

6. The concept of functions, equations, and inequalities are not entirely new to the students at this

level. Use their prior knowledge and ask them to construct concept maps by groups of threes. You may choose to implement a standard form of the concept map such as the one provided below. Exploration Activity: Know your Circles… Sort out the keywords/ mathematical statements provided by writing them inside the appropriate circles found in the diagram below.

(1)


(3)

• • • • • • • • • •

(2)

KEYWORDS: Function Equation Inequality Dependent and independent variables Equal sign Greater or lesser Shaded graphs Graphs formed by lines and curves One to many One to one

• 𝑦=

𝑥

𝑥−3

• 𝑥𝑦 − 7𝑦 = 9 − 2𝑥 2 • 𝑦≥ 6𝑥−7

Key to Correction: EQUATION Equal Sign, Graphs formed by lines and curves, one to many, 𝑥𝑦 − 7𝑦 = 9 − 2𝑥

FUNCTION Dep Ind var, one to one,

𝑦=

INEQUALITY Greater or lesser, shaded graphs, one to many, 𝑦 ≥

2 6𝑥−7

𝑥

𝑥−3

Supplement students’ responses in the activity by discussing their answers as a class. Have them defend their answers and ask them provide other mathematical statements that conform to each kind determined above.

7. Ask the following questions for wrap-up: a. What are rational functions? b. How would you determine if a functional is rational and not polynomial? c. How would you differentiate rational equations, rational functions, and rational inequalities?


General Mathematics Teaching Guides ••• 8. Are the following functions rational? a.

𝑓(𝑥) =

b.

𝑔(𝑏) =

c.

𝑚(𝑡) =

3𝑥 2 +5𝑥−7

[R]

𝑥 𝑏5 −𝑏4 +8𝑏3 −3𝑏2 +𝑏 𝑡−13 9

13

𝑏

[P, the function may be simlplified by division]

[P, no variable in the denominator]

d.

𝑘(𝑟) = + 5𝑟 − 7𝑟 2 [R]

e.

𝑓(𝑥) = (6𝑥 − 1)−2 [R]

𝑟

DOMAIN AND RANGE

Day Two [Target M11GM-Ib-5] Find the domain and range of a rational function.

SOLVING RATIONAL FUNCTIONS

1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge with these questions: a. What are rational functions? b. What is meant by the domain of a function? the range of a function? 4. The first half of the discussion shall be spent on inquiry – one where students will be asked a series of questions and they have to answer as a class. Their answers to the questions will help them come up with the general idea of how to determine restrictions in the allowable values of the independent variable, 𝑥. What follows would be a detailed simulation of the class discussion facilitated by the teacher. Student Activity Teacher Activity (Answers should vary. Adjust accordingly) I would like to ask somebody to provide an example [Gives a rational function. In this case, let us assume of a rational function, one which has a linear 𝑥+4 expression as its denominator. Any volunteer? that the function is 𝑓(𝑥) = ] 𝑥−3 (Call one student.) Let’s consider the function given by (name of student). Can 𝑥 take any number for its value? [Students provide suggestions] (For every answer, ask the student why s/he thinks that 𝑥 cannot take 3 for its value because it will make the the number should be a restriction for 𝑥. Do this until denominator zero. If this happens, the value of 𝑦 the correct answer is suggested and properly becomes undefined. explained.) So how would you describe the domain of the function 𝑓(𝑥)? The domain is the set of all real numbers except 3. (Call another student.) Correct. {𝑥|𝑥 ∈ ℝ, 𝑥 ≠ 3} Can anybody state this in set notation? (Call another student.) [Gives the second rational function. In this case, let us Let’s discuss another function. Who can give me another rational function but this time, with a quadratic expression as denominator? (Call another student.) Good. We shall do the same with this function. What values of 𝑥 will make the function undefined? How do we express the domain of the function in set builder notation?

assume that the function is 𝑓(𝑥) =

5 ] 𝑥 2 +4𝑥+3

It is preferable to consider examples with denominators that are factorable. If it happens that the student comes up with an expression with irrational roots, use the appropriate method to identify its zeroes. We cannot take the values of -3 and -1 for 𝑥. {𝑥|𝑥 ∈ ℝ, 𝑥 ≠ −3, −1}

5. Next, we discuss the how to define the range of a rational function. For this process, there is a need for students to be taught how to manipulate the function in order to find the restrictions. Important Note: The teacher should plan the functions to be discussed very carefully as many functions would require higher mathematical skills. Restrict examples depending on the capacity and readiness of the students. Use the examples previously provided by the students. The complete solutions are shown below.


General Mathematics Teaching Guides ••• Emphasize here that the main objective in order to define the range is to express the function explicitly in 𝑥 in terms of 𝑦.

𝒇(𝒙) = 𝑦= 𝑥 − 3 (𝑦 =

𝑥+4 𝑥−3 𝑥+4 )𝑥 − 3 𝑥−3

𝒙+𝟒 𝒙−𝟑

𝒇(𝒙) =

Change the function notation into the dependent variable y. Multiply both sides by the denominator of the right side.

𝑦=

5 2 𝑥 + 4𝑥 + 3

𝑥 2 + 4𝑥 + 3 =

𝑥 2 + 4𝑥 + 4 =

Simplify

𝑥𝑦 − 𝑥 = 3𝑦 + 4

Isolate all terms with 𝑥 on the left side of the equation and those without it to the right side.

𝑥 2 + 4𝑥 + 4 =

𝑥(𝑦 − 1) = 3𝑦 + 4

Factor out 𝑥 on the left side.

(𝑥 + 2)2 =

Divide both sides of the equation by the other factor.

3𝑦 + 4 𝑦−1

{𝒚|𝒚 ∈ ℝ, 𝒚 ≠ 𝟏}

Finally, determine the restriction through set builder notation.

𝑥=

5 𝑦

5 +4−3 𝑦

𝑥𝑦 − 3𝑦 = 𝑥 + 4

𝑥=

𝒙𝟐

5 +1 𝑦

𝟓 + 𝟒𝒙 + 𝟑

Change the function notation into the dependent variable y.

Multiply both sides by the denominator and divide both sides by 𝑦 Transpose the constant, complete the square and add the third term of the perfect square trinomial to the right side of the equation. Simplify. Express the perfect square trinomial as a square of a binomial. Extract the roots, rationalize, and transpose the constant.

5 +1 𝑦

√5𝑦 + 𝑦 2 −2 𝑦

{𝒚|𝒚 ∈ ℝ, −𝟓 ≥ 𝒚 > 𝟎}

Finally, determine the restriction through set builder notation.

Ask the following questions for wrap-up. a. What is meant by the domain of a function? The range of a function? b. How do we define the domain of a rational function? c. Describe the process of determining the restrictions for the range of a rational function. Have the students perform a drill on the targeted competency. Let them find the domain of the following rational functions. 2𝑥−1

a.

𝑓(𝑥) =

b.

𝑔(𝑥) = 2 − 𝑥+1 [𝑥 ≠ −1]

c.

ℎ(𝑥) =

d.

𝑥+1

[𝑥 ≠ −1]

2𝑥 2 −1

𝑟(𝑥) =

𝑥 2 −1

3

3𝑥−2

− 𝑥 2 −1 [𝑥 ≠ ±1]

2𝑥2 −1 𝑥2 −1 3𝑥−2 𝑥2 −1

2

[𝑥 ≠ ±1, ] 3

INTERCEPTS, ZEROS, ASYMPTOTES

Day Three [Target M11GM-Ic-1] Determine the: (a) intercepts, (b) zeroes, and (c) asymptotes of rational SOLVING functions. RATIONAL FUNCTIONS 1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge with these questions: a. How would you describe the domain and range of a rational function? b. What do you think happens to the graph of the rational function at the values for which it becomes undefined?

The second question should allow students to make assumptions in relation to the competency targeted. Take note and emphasize their hypotheses and have them look after the discussion.


General Mathematics Teaching Guides ••• 4. Project (or post on the board) the graph of 𝑥+2 𝑓(𝑥) = as shown at 𝑥−2

the right.

Point out the intercept, zero, and the asymptotes of the graph of the function. Define each of these terms as they relate to the graph.

Essential Ideas Intercepts, Zeros, Asymptotes

horizontal asymptote zero

••• intercept

5. Ask the following questions: a. It is obvious that the zero of the 𝒙+𝟐 function is a point Graph of 𝒇(𝒙) = vertical 𝒙−𝟐 asymptote on the x-axis so the y-coordinate is zero. How do we look for its xcoordinate? [substitute zero to the value of 𝑦 in the function then solve for the value of 𝑥.] 𝑥+2 0= ; 𝑥 + 2 = 0; 𝒙 = −𝟐 𝑥−2 b. How do we determine the intercept? [substitute zero to the value of x in the function then solve for the value of 𝑦.] 0+2 2 𝑦= ;𝑦 = ; 𝒚 = −𝟏 0−2 −2 c. How do we determine the vertical asymptote based on the equation? The vertical asymptote of the graph is at 𝑥 = 2. How is this related to the domain of the function? [The function is undefined for 𝑥 = 2. It is a restriction for the domain.]

A point at which the graph crosses the y-axis is called a y-intercept, and a point at which it crosses the x-axis is called an x-intercept. The x-coordinate of an xintercept is sometimes called a zero of the function since the function has a zero value there. The line 𝑥 = 𝑐 is called a vertical asymptote of the graph of a function 𝑦 = 𝑓(𝑥) if as 𝑥 → 𝑐 − or as 𝑥 → 𝑐 + , either 𝑓(𝑥) → +∞ or 𝑓(𝑥) → −∞. The line 𝑦 = 𝑐 is called a horizontal asymptote of the graph of a function 𝑦 = 𝑓(𝑥) if as 𝑥 → −∞ or as 𝑥 → ∞ , either 𝑓(𝑥) → 𝑐.

Let us find out what happens to the function as our xcoordinate gets closer and closer to 2. Complete the table below. 𝒙 𝒇(𝒙) 𝒙 𝒇(𝒙) 1 -3 3 5 1.5 -7 2.5 9 1.9 -39 2.1 41 1.99 -399 2.01 401 1.999 -3,999 2.001 4,001 1.9999 -39,999 2.0001 40,001 What happens to 𝑓(𝑥) as we get closer and closer to 2 from the left? [The function gets lesser and lesser values.] From the right? [The function increases.] Teacher: This is the reason why the graph has an asymptote at that portion. As we get closer to the value of 2, we get larger and larger values or smaller and smaller values but we will never get an actual value of the function when 𝑥 = 2. d. How do we determine the horizontal asymptote of the graph of the function? Do you think this may be related to the restriction in our range? [The function has a horizontal asymptote at 𝑦 = 1 since there is no value for 𝑥 that would give us a function value of 1.] 𝑥+2 𝑥+2 𝟐𝒚 + 𝟐 𝑓(𝑥) = ; 𝑦= ; 𝑥𝑦 − 2𝑦 = 𝑥 + 2; 𝑥𝑦 − 𝑥 = 2𝑦 + 2; 𝑥(𝑦 − 1) = 2𝑦 + 2; 𝒙 = ; 𝒚≠𝟏 𝑥−2 𝑥−2 𝒚−𝟏 Use the following statements for wrap-up: a. Define the following: intercept, zero, asymptote. b. How do we find the intercept of a rational function? c. How do we determine the zero(s) of a rational function? d. What process can we do to identify the location of the asymptotes of a rational function?


General Mathematics Teaching Guides ••• Allow students to work with a similar task together with a partner. Give the function 2𝑥 𝑔(𝑥) = for which they have to identify 𝑥+3

the intercept(s), the zero(s), and the asymptotes. [The intercept and zero are the same. The graph passes through the origin. The vertical asymptote is at 𝑥 = −3 while the horizontal asymptote is 𝑦 = 2] Show the graph as a means to confirm the answers of the students. Discuss in class.

WRITTEN WORK NO. 3

Assignment (Pen and Paper Test) Identify the intercept(s), the zero(s), and the asymptotes of the rational functions.

Graph of 𝒇(𝒙)

=

𝟐𝒙 𝒙+𝟑

1

1. 𝑓(𝑥) = [𝑛𝑜𝑛𝑒, 𝑛𝑜𝑛𝑒, 𝑥 − 𝑎𝑥𝑖𝑠, 𝑦 − 𝑎𝑥𝑖𝑠] 𝑥

2. 𝑔(𝑥) = 3. 𝑚(𝑥) =

3𝑥 𝑥−5

Scoring Guide: For every item, give one point each for the correct intercept and the correct zero, two points for each correct asymptote. Solutions are necessary.

[0, 0, 𝑥 = 5, 𝑦 = 3]

2𝑥−7 7𝑥−1

1

2

7

7

[7, 7, 𝑥 = , 𝑦 = ]

Highest Possible Score: 18 points Passing Score: 14 points (78 percent)

GRAPHS OF RATIONAL FUNCTIONS Day Four [Target M11GM-Ic-2 and M11GM-Ib-4] Graph rational functions and represent a rational function through its: (a) table of values (b) graph, and (c) equation.

1. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss items that need to be discussed. 2. Present the lesson and the targeted competencies. 3. Facilitate activation of prior knowledge by using the question for recapitulation. a. What are the three significant features of the graphs of rational functions that we discussed? Describe each of those. 4. Discuss the five steps in graphing rational functions. Provide 𝑥+2 the function, 𝑓(𝑥) = as initial example. 𝑥−2

If necessary, use another function, 𝑔(𝑥) =

4−𝑥 𝑥+1

as second

example. For every step, ask students to participate in order to give them the chance to assess the steps on their own. In constructing the graphs, start by locating the asymptotes, the intercept, and the zero. Then use the table to determine the behavior of the graph as it gets closer to the vertical asymptote.

Essential Ideas Graphing Rational Functions

••• To graph 𝑓(𝑥) =

𝑝(𝑥) 𝑞(𝑥)

where p

and q have no common factor other than 1, 1. determine x-intercepts by solving 𝑓(𝑥) = 0. 2. determine the y-intercept by evaluating 𝑓(0). 3. determine the vertical asymptotes by solving 𝑞(𝑥) = 0 4. determine the horizontal asymptote (if any) by dividing by the highest power of x in the denominator. 5. plot additional points choosing at least one value of x from each interval determined by the xintercepts or vertical asymptotes


General Mathematics Teaching Guides ••• Example, 𝒇(𝒙) =

Procedure STEP ONE. Determine xintercepts by solving 𝑓(𝑥) = 0. STEP TWO. Determine the yintercept by evaluating 𝑓(0). STEP THREE. Determine the vertical asymptotes by solving 𝑞(𝑥) = 0 STEP FOUR. Determine the horizontal asymptote (if any) by dividing by the highest power of x in the denominator.

𝒙+𝟏 𝒙−𝟑

𝑥+1 0= ; 𝑥 + 1 = 0; 𝒙 = −𝟏 𝑥−3 0+1 1 𝑓(0) = =− 0−3 3 𝑥 − 3 = 0; 𝑥 = 3 𝑥 1 1 + 1+ 𝑥 𝑥 𝑥; 𝑦 = 1 𝑓(𝑥) = = 𝑥 3 3 − 1− 𝑥 𝑥 𝑥 𝑥 𝑓(𝑥)

𝑥<3 1.5 2 2.5 -1.25 -3 -7 Decreasing towards 3

𝑥>3 3.5 4 4.5 9 5 3.67 Increasing towards 3

STEP FIVE. Plot additional points choosing at least one value of x from each interval determined by the x-intercepts or vertical asymptotes

Example, 𝑔(𝑥) =

Procedure STEP ONE. Determine xintercepts by solving 𝑔(𝑥) = 0. STEP TWO. Determine the yintercept by evaluating 𝑔(0). STEP THREE. Determine the vertical asymptotes by solving 𝑞(𝑥) = 0 STEP FOUR. Determine the horizontal asymptote (if any) by dividing by the highest power of x in the denominator.

4−𝑥 𝑥+1

4−𝑥 0= ; 4 − 𝑥 = 0; 𝒙 = 𝟒 𝑥+1 4−0 𝑔(0) = =4 0+1 𝑥 + 1 = 0; 𝑥 = −1 4 𝑥 4 − −1 𝑔(𝑥) = 𝑥 𝑥 = 𝑥 ; 𝑦 = −1 𝑥 1 1 + 1+ 𝑥 𝑥 𝑥 𝑥 𝑔(𝑥)

𝑥 < −1 -2.5 -2 -1.5 -4.33 -6 -11 Decreasing towards -1

𝑥 > −1 -0.5 0 0.5 9 4 2.33 Increasing towards -1

STEP FIVE. Plot additional points choosing at least one value of x from each interval determined by the x-intercepts or vertical asymptotes


General Mathematics Teaching Guides ••• To serve both as application and performance task, assign the activity described below. Evaluation/Assignment (Performance Task) In groups of seven, students will need to graph the following rational functions by following the steps previously discussed. Ask one representative from each group to pick a number that would determine the function they will have to work with. This will ensure fairness in the assignment. 𝑥−2

GROUP A. 𝑓(𝑥)

= 2𝑥−3

GROUP B. 𝑓(𝑥)

2𝑥−1

=

GROUP E. 𝑔(𝑥) GROUP F. 𝑔(𝑥)

𝑥+2

= =

3𝑥

GROUP G. ℎ(𝑥)

−2

GROUP H. ℎ(𝑥)

GROUP C. 𝑓(𝑥)

= 𝑥−5

GROUP D. 𝑔(𝑥)

= 3−𝑥

3𝑥−1 𝑥 3−2𝑥 𝑥+1 5𝑥

= 𝑥+5 𝑥+2

= 2𝑥+3


Use the rubric below as guide in evaluating students’ outputs. Criteria

Can do better



Correctness and accuracy in following graphing procedure.

At least two of the steps were not properly carried out (3 points)

One of the steps in the procedure was incorrectly done (4 points)

All steps in the procedure were followed and properly employed (5 points)

Quality of Graph

The graph was not accurately sketched with two or more features not characterized (3 point)

One important feature of the graph is not correct (4 points)

The graph is correct (5 points)

Cleanliness and completeness of output.

Erasures are evident in the work submitted (no credit)

Output is free of erasures and unnecessary writings (2 points)

Group dynamics



Time Management

Task completed past the intended time. (no credit)


8 points or less


Task completed on or before the intended time (3 points) 9 to 16 points

17 to 20 points

Find time to discuss the graphs constructed by the students. As a means of wrapping up the topics discussed over the week, use the following questions for generalization: a. What are rational functions? How are they different/ related to polynomial functions? b. How do you determine if a function is rational in nature? c. What is meant by the intercept of the graph of a rational function? How do we locate the intercept of the graph of a rational function? d. Describe asymptotes? e. What is the significance of restrictions in the domain and range of a rational function? f. How do we graph a rational function?



Rational Equations and Inequalities Week Three, Functions and Their Graphs Content Standards: The learner demonstrates understanding of key concepts of functions.

Essential Ideas Sign Diagrams

Performance Standards: The learner is able to accurately formulate and solve real-life problems involving rational functions.

••• Steps for constructing a sign diagram for a rational function:

Learning Competencies: Over the course of one week, the learner is expected to (1) solve rational equations and inequalities [M11GM-Ib-3]; and (2) solve problems involving rational equations and inequalities [M11GM-Ic-3]

Suppose r is a rational function.

Important Note: For an improved continuity of competencies under the same content standard, the learning competencies have been rearranged but the codes have been maintained for reference.

1. Place any value excluded from the domain of r on the number line with an “!” above them.

Learning Materials: In order to develop the targeted competencies, the following materials are needed: (a) chalkboard and chalk; (b) LCD projector and laptop or in the absence, visual materials; (c) calculators

2. Find the zeros of r and place them on the number line with the number “0” above them.

Expected Outputs: In order to assess the attainment of the learning competencies targeted, students will be required to undertake one group performance task and one written work.

3. Choose a test value in each of the intervals determined in steps 1 and 2.

Procedure (Teacher’s Activity) SOLVING RATIONAL EQUATIONS Day One [Target M11GM-Ib-3] Solve rational equations and inequalities

4. Determine the sign of r(x) for each test value in step 3, and write that sign above the corresponding interval.

1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge with this question: a. How would you compare functions, equations, and inequalities? 4. Discuss in detail the solutions for solving the following rational equations and inequalities.

Emphasize that in solving rational equations, the technique is to rid the equation of denominators. Multiplying both sides of the equation by the LCD’s will do the job. Once this is done, the rational equation should then be polynomial and would be easier to solve. There is a need for the student to be proficient in factoring and understanding how to get the zeros of polynomial functions. For this, the teacher might need to reintroduce the rational zeros theorem, and to some extent if necessary, the Descartes’ Rule of Signs. 𝒙−𝟏=

𝒙 + 𝟒𝟑 𝒙−𝟕

𝒙𝟐

𝟒𝟐 =𝒙 + 𝟐𝒙 − 𝟐𝟗

(𝑥 − 1)(𝑥 − 7) = 𝑥 + 43

42 = 𝑥(𝑥 2 + 2𝑥 − 29)

𝑥 2 − 8𝑥 + 7 = 𝑥 + 43

𝑥 3 + 2𝑥 2 − 29𝑥 + 42 = 0

𝑥 2 − 9𝑥 − 36 = 0

(𝑥 − 3)(𝑥 + 7)(𝑥 − 2) = 0

(𝑥 − 12)(𝑥 + 3) = 0

𝑥 − 3 = 0; 𝑥 + 7 = 0; 𝑥 − 2 = 0

𝑥 − 12 = 0; 𝑥 + 3 = 0

𝒙 = 𝟑; 𝒙 = −𝟕; 𝒙 = 𝟐

𝒙 = 𝟏𝟐; 𝒙 = −𝟑

Rational Equations and Inequalities  19 Von Christopher G. Chua

General Mathematics Teaching Guides ••• 2𝑥 3 − 𝑥 2 − 𝑥 = 0

𝒙𝟑 − 𝟐𝒙 + 𝟏 𝟏 = 𝒙−𝟏 𝒙−𝟏 𝟐

𝑥(2𝑥 + 1)(𝑥 − 1) = 0

𝑥 3 − 2𝑥 + 1 1 ( ) ∙ 2(𝑥 − 1) = ( 𝑥 − 1) ∙ 2(𝑥 − 1) 𝑥−1 2

𝑥 = 0; 2𝑥 + 1 = 0; 𝑥 − 1 = 0 𝟏 𝑥 = 𝟎; 𝒙 = − ; 𝒙 = 𝟏 𝟐

2𝑥 3 − 4𝑥 + 2 = 𝑥 2 − 3𝑥 + 2

When dealing with inequalities, it would be unwise to multiply both sides with the LCD as we don’t know exactly if the expression is positive or negative. If it is negative, the inequality symbol would have to change. Therefore, the best way to solve a rational iunequality is to transpose all terms on the left side of the equation and simplify it there by unifying the fraction. 𝟐𝒙 + 𝟏𝟕 𝒙𝟑 − 𝟐𝒙 + 𝟏 𝟏 For the succeeding three > 𝒙+𝟓 ≥ 𝒙−𝟏 3 2 𝒙+𝟏 𝒙−𝟏 𝟐 examples on solving rational 3 2𝑥 + 17 inequalities, it is important to 𝑥 − 2𝑥 + 1 1 − (𝑥 + 5) > 0 − 𝑥+1≥0 𝑥+1 explain comprehensively 𝑥−1 2 3 2𝑥 + 17 (𝑥 + 5)(𝑥 + 1) the need for sign diagrams. 2(𝑥 − 2𝑥 + 1) − 𝑥(𝑥 − 1) + 2(𝑥 − 1) − >0 ≥0 𝑥+1 𝑥+1 The process of constructing 2(𝑥 − 1) one is explained under 2𝑥 + 17 𝑥 2 + 6𝑥 + 5 2𝑥 3 − 𝑥 2 − 𝑥 − >0 ≥0 “Essential Ideas.” 𝑥+1 𝑥+1 2𝑥 − 2 −𝑥 2 − 4𝑥 + 12 2𝑥 3 − 𝑥 2 − 𝑥 = 0 >0 𝒙−𝟑 𝑥+1 ≤𝟎 1 𝑥(2𝑥 2 − 𝑥 − 1) = 0 𝒙+𝟐 𝑥 2 + 4𝑥 − 12 <0 𝑥(2𝑥 + 1)(𝑥 − 1) = 0 𝐼𝑓 𝒙 = 𝟑, 𝑓(𝑥) = 0 𝑥+1 !

(+)

0 (+)

(-)

-2

3

2𝑥 − 2 ≠ 0; 𝒙 ≠ 𝟏

𝑥 2 + 4𝑥 − 12 <0 𝑥+1

𝑥 + 2 ≠ 0; 𝒙 ≠ −𝟐

(𝑥 − 2)(𝑥 + 6) = 0; 𝒙 ≠ 𝟐, −𝟔 𝑥 + 1 ≠ 0; 𝒙 ≠ −𝟏

(−𝟐, 𝟑)

(-) 0

(+)

-6

!

(-)

-1

0 (+)

(+)

0

(-)

-1/2

0 0

(+)

!

(+)

1

𝟏 (−∞, − ] ∪ [𝟎, 𝟏) ∪ (𝟏, +∞) 𝟐

2

(−∞, −𝟔) ∪ (−𝟏, 𝟐)

The following question should be raised to aid generalization of the concept targeted: Briefly, how would you explain the process of solving rational equations and inequalities? As exercise, solve the rational equations and inequalities. For the inequalities, express answer in interval notation. 𝑥 𝟔 a. =3 [𝑥 = 15𝑥 + 12; −14𝑥 = 12; 𝒙 = − ]

b. c. d.

5𝑥+4 3𝑥−1 𝑥 2 +1 1 𝑥+2 𝑥

𝟕

=1

2

[3𝑥 − 1 = 𝑥 + 1; −𝑥 + 3𝑥 − 2 = 0; 𝑥 2 − 3𝑥 + 2 = 0; (𝑥 − 2)(𝑥 − 1) = 0; 𝒙 = 𝟐, 𝟏 ]

≥0

𝑥 2 −1

2

[𝑥 + 2 ≠ 0; 𝑥 ≠ −2 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡, 𝑦 =

>0

1 2

(−𝟐, +∞)]

[𝑥 2 − 1 ≠ 0; 𝑥 ≠ ±1, 𝑧𝑒𝑟𝑜: 𝑥 = 0, 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡: 𝑦 = 0, (−𝟏, 𝟎) ∪ (𝟏, +∞)]

WRITTEN WORK NO. 4

Assignment (Pen and Paper Test) Solve the following rational equations and inequalities. 1. 2. 3. 4.

2𝑥−7 𝑥+3

=2

1

𝑥 2 −3

1

+ 𝑥−3 = 𝑥 2 −9 𝑥+3 𝑥 2 −𝑥−12 𝑥 2 +𝑥−6

>0

3𝑥 2 −5𝑥−2 𝑥 2 −9

<0

Highest Possible Score: 20 points Passing Score: 15 points (75 percent)


General Mathematics Teaching Guides ••• Day Two [Target M11GM-Ic-3] Solve problems involving rational equations and inequalities

PROBLEM SOLVING

1. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss items that need to be discussed. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge by using the following questions for recapitulation. a. How would we differentiate rational equations from rational inequalities? b. Wxplain the technique in solving rational equations and inequalities. 4. Discuss the following problems involving rational equations and inequalitites to the class. Given a cost function 𝑪(𝒙), which returns the total cost of producing x items, the average cost function, ̅(𝒙) = 𝑪(𝒙) computes the cost per item when 𝒙 items are produced. Suppose the cost 𝑪, in pesos, to 𝑪 𝒙 produce 𝒙 cellphone protective cases for a local retailer is 𝑪(𝒙) = 𝟖𝟎𝒙 + 𝟏𝟓𝟎, 𝒙 ≥ 𝟎. Find an expression for the Solve 𝐶̅ (𝑥) < 100 and interpret. Determine the behavior of 𝐶̅ (𝑥)as 80𝑥 + 150 average cost function, 𝐶̅ (𝑥). 𝑥 → ∞ and interpret. < 100 𝟖𝟎𝒙 + 𝟏𝟓𝟎 ̅ (𝒙) 𝑥 𝑪 𝑥 ̅(𝒙) = 𝑪 ,𝒙 > 𝟎 80𝑥 + 150 1 230 𝒙 − 100 < 0 𝑥 10 95 80𝑥 + 150 − 100 100 81.50 <0 𝑥 1000 80.15 −20𝑥 + 150 <0 10000 80.015 𝑥 100000 80.0015 𝑥 > 0, 𝑧𝑒𝑟𝑜: 𝑥 = 7.5 As 𝑥 → ∞, 𝐶̅ (𝑥) gets closer and ( 7.5, ∞) closer to 80. This means that the In the context of the problem, average cost per case is always solving 𝐶̅ (𝑥) < 100 means we are greater than PhP 80.00 but is trying to find how many systems we need to produce so that the average approaching this amount as more cost is less than PhP 100.00 per case. and more cases are produced. Our solution tells us that we need to produce more than 7.5 cases to achieve this but it doesn’t make 1 sense to produce just of a case so 2 our final answer should be 8. A box with a square base and no top is to be constructed so that it has a volume of 1000 cubic centimeters. Let 𝒙 denote the width of the box in centimeters. Express the height ℎ in Solve ℎ(𝑥) ≥ 𝑥 and interpret. Express the surface area 𝑆 of the box centimeters as a function of as a function of 𝑥 and state the 1000 1000 the width 𝑥 and state the applied domain. ≥ 𝑥; −𝑥 ≥0 applied domain. 𝑥2 𝑥2 1000 − 𝑥 3 1000 ≥0 𝑆(𝑥) = 𝑥 2 + 4 ( ) 2 The formula for the volume of 𝑥 𝑥 2 4000 𝑥 ≠ 0; 𝑥 ≠ 0 a rectangular prism such as 𝑆(𝑥) = 𝑥 2 + Solving for the zero of the function, 𝑥 the box is 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙𝑒𝑛𝑔𝑡ℎ× 1000 − 𝑥 3 𝑤𝑖𝑑𝑡ℎ×ℎ𝑒𝑖𝑔ℎ𝑡. With both the =0 Domain: (0, ∞) 𝑥2 volume and the width given, 3 1000 − 𝑥 = 0 the equation is, 𝑥 3 = 1000 1000 = 𝑥 2 ℎ 𝑥 = 10 1000 ℎ(𝑥) = 2 (−∞, 0) ∪ (0,10) Solution set: 𝑥 Therefore the domain of the function is 𝑥 ≠ 0 and it should But since 𝑥 represents measurement, it cannot take a also be greater than zero. negative quantity, (0,10) (0, ∞) This means that the width of the box can take any measurement as long as it is less than 10.



The following questions should be raised to facilitate abstraction: (a) What challenges do you think would you have whenever you encounter word problems involving rational functions? (b) What can be done to minimize the difficulty these challenges impose?

A television set costs PhP 27,400.00 and a yearly cost of electricity of PhP 550.00. a. Determine the total annual cost for a television set that lasts for 10 years. Assume that the cost includes electricity and depreciation. (𝐷𝑒𝑝𝑟𝑒𝑐𝑖𝑎𝑡𝑖𝑜𝑛 =

𝑐𝑜𝑠𝑡 𝑜𝑓 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒 𝑢𝑠𝑒𝑓𝑢𝑙 𝑙𝑖𝑓𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠

)

b. Write a function that gives the annual cost of a television set as a function of the number of years. c. Determine the asymptotes of the function. Explain the meaning of the horizontal asymptote in terms of the television set. Evaluation/Assignment (Performance Task) In groups of three, students will need to either find a problem related to rational equations and inequalities or concoct one of their own. They will also need to solve this problem and have their solutions submitted for evaluation. In order to check that the solution to the selected problem is not available from the same source as the problem, references need to be indicated. Use the rubric below as guide in evaluating students’ outputs.


Criteria

Can do better

Level of Difficulty

Problem selected was easy (1 points)

Clarity, completeness, and conciseness of problem statement

The problem lacks some important information (1 point)

Student’s understanding of the problem

Translation and representation of variables is incorrect (1 point)

Correctness of solution presented

The solution contains at least two errors (3 point)

Group dynamics


Time Management



10 points or less



Problem selected was average in terms of difficulty (2 points) All information needed to solve the problem is present but the manner by which it was stated may be redundant or even misleading (2 points)

Problem selected was difficult (3 points)

Problem statement is short and clear (3 points)

Mathematical sentence constructed from the problem is correct (3 points) The solution contains one error (4 points)

The solution is correct (5 points)


The project reflects collaboration among members of the group. (3 points) Task completed on or before the intended time (3 points)

11 to 16 points

17 to 20 points

Make sure to return the outputs with the appropriate notations.



Inverse Functions Week Four, Functions and Their Graphs Content Standards: The learner demonstrates understanding of key concepts of inverse functions, exponential functions, and logarithmic functions. Performance Standards: The learner is able to accurately apply the concepts of inverse functions, exponential functions, and logarithmic functions to formulate and solve real-life problems with precision and accuracy. Learning Competencies: Over the course of one week, the learner is expected to (1) represent real-life situations using one-to-one functions [M11GM-Id-1]; (2) determine the inverse of a one-to-one function [M11GM-Id-2]; (3) represent an inverse function through its (a) table of values, and (b) graph [M11GM-Id-3]; (4) find the domain and range of inverse functions [M11GM-Id-4]; and (5) graphs inverse functions [M11GM-Ie-1] Learning Materials: In order to develop the targeted competencies, the following materials are needed: (a) chalkboard and chalk; (b) LCD projector and laptop or in the absence, visual materials; (c) calculators; (d) worksheets Expected Outputs: In order to assess the attainment of the learning competencies targeted, students will be required to undertake two group performance tasks and one written work. Procedure (Teacher’s Activity) ONE-TO-ONE FUNCTIONS AND Day One [Target M11GM-Id-1 and THEIR INVERSE M11GM-1d-2] Represent real-life situations using one-to-one functions and determine the inverse of a one-to-one function.

1. Do routines and other preparatory activities for five minutes. 2. Present the lesson and the targeted competencies (and other learning outcome). 3. Facilitate recall with the following questions: a. What are relations? b. What are the three possible correspondences of ordered pairs for relations? c. Which of these correspondences are functions? Explain

Essential Ideas Stick-to-one

••• A function f is said to be oneto-one if f matches different inputs to different outputs. Equivalently, f is one-to-one if and only if whenever 𝑓(𝑐) = 𝑓(𝑑), then 𝑐 = 𝑑. Horizontal Line Test: A function f is said to be oneto-one if and only if no horizontal line intersects the graph of f more than once. Suppose f and g are two functions such that (1) (𝑔 ∘ 𝑓)(𝑥) = 𝑥 for all 𝑥 in the domain of f and (2) (𝑓 ∘ 𝑔)(𝑥) = 𝑥 for all 𝑥 in the domain of g then f and g are inverses of each other and the functions f and g are said to be invertible. Properties of Inverse Functions: • The range of f is the domain of g and the domain of f is the range of g. • 𝑓(𝑎) = 𝑏 if and only if 𝑔(𝑏) = 𝑎 • (𝑎, 𝑏) is on the graph of f if and only if (𝑏, 𝑎) is on the graph of g Uniqueness of Inverse Functions: There is exactly one inverse function for f denoted by 𝑓 −1 (read as f-inverse)

4. Ask the class to determine which of the following functions are one-to-one. If a function is not oneto-one, they need to provide two ordered pairs that belong to the function but share ordinates. (1)𝑓(𝑥) = 3𝑥 + 4 (2) 𝑔(𝑥) = 𝑥 2 (3) ℎ(𝑥) = 2𝑥 3 2 (5)𝑔(𝑥) = 4 (4) 𝑓(𝑥) = 7 − 2𝑥 (6) 𝑚(𝑥) = |𝑥 − 1| 𝑥 (7)

(8)

Inverse Functions  23 Von Christopher G. Chua

General Mathematics Teaching Guides ••• Answers to the exploration activity in the previous page: (1) one-to-one; (2) No. (−2,4)𝑎𝑛𝑑 (2,4); (3) one-toone; (4) one-to-one; (5) No. (−1,2)𝑎𝑛𝑑 (1,2); (6) No. (−2,3)𝑎𝑛𝑑 (4,3); (7) No. (−1,3)𝑎𝑛𝑑 (7,3); (8) one-to-one

5. Consider two of the functions from the previous activity, specifically, 𝑓(𝑥) = 3𝑥 + 4 and 𝑔(𝑥) = 𝑥 2 . Allow the students to construct tables of values for both functions, such as the ones that follow. 𝑓(𝑥) = 3𝑥 + 4 𝑔(𝑥) = 𝑥 2 𝑥 𝑥 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 𝑓(𝑥) 𝑔(𝑥) -5 -2 1 4 7 10 13 9 4 1 0 1 4 9 Ask one student to remind the class why the two are functions? [No two ordered pairs share the same domain] Teacher: Let us say we want to interchange the values of the dependent and the independent variables in both functions we have just constructed tables for. 𝑓′(𝑥) =____________ 𝑔′(𝑥) = _____________ 𝑥 𝑥 -5 -2 1 4 7 10 13 9 4 1 0 1 4 9 𝑓(𝑥) 𝑔(𝑥) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Teacher: What can be said about the resulting tables? [Interchanging the values in 𝑓(𝑥) resulted to another function which cannot be said (is not the case) for 𝑔(𝑥)] Teacher: Who can tell us what expressions represent the ordered pairs in the last two tables? [𝑓′(𝑥) =

𝑥−4 3

𝑔′ (𝑥) = √𝑥]

Teacher: The last two relations are the inverses of the functions we started with. Note that not all functions have a function for its inverse as in the case of 𝑔(𝑥). Only one-to-one functions have inverses that are also functions. These are called invertible. 6. How do we obtain the inverse of a one-to-one function? For simple functions such as 𝑓(𝑥) = 3𝑥 + 4, working backwards is one technique. For example, if 𝑥 = 2, to get the value of 𝑓(2), we multiply by 3 first then add to 4 to get an answer of 10. Starting with 10 in order to obtain 2, we subtract by 4 then divide by 3 instead, hence, 𝑓 −1 (𝑥) =

𝑥−4 3

.

Teacher: Can you do the same with 𝑓(𝑥) = −2𝑥 + 7? [𝑓 −1 (𝑥) =

𝑥−7 −2

or 𝑓 −1 (𝑥) =

7−𝑥 2

]

7. Discuss the second, more procedural method for getting the inverse of a function with the two examples below. Example 1: 𝒇(𝒙) = 𝟖𝒙 − 𝟑

Procedure

𝑦 = 8𝑥 − 3

Temporarily replace the function notation by the dependent variable 𝑦

𝑥 = 8𝑦 − 3

Interchange 𝑥 and 𝑦

𝑥 = 8𝑦 − 3 −8𝑦 = −𝑥 − 3 𝑥+3 𝑦= 8

Solve for 𝑦 in terms of 𝑥.

𝒇−𝟏 (𝒙) =

𝒙+𝟑 𝟖

Revert back to function notation, this time, using the notation, 𝑓 −1 (𝑥).

Example 2: 𝒈(𝒙) = 𝑦=

𝟔 𝒙+𝟑

6 𝑥+3

6 𝑦+3 6 (𝑦 + 3) (𝑥 = ) (𝑦 + 3) 𝑦+3 𝑥𝑦 + 3𝑥 = 6 𝑥𝑦 = −3𝑥 + 6 −3𝑥 + 6 𝑦= 𝑥 𝑥=

𝒈−𝟏 (𝒙) =

−𝟑𝒙 + 𝟔 𝒙

The following questions should be raised to aid generalization of the concept targeted: What are one-to-one functions? Inverse functions? How do we determine the inverse function of a one-toone function?


General Mathematics Teaching Guides ••• As exercise, students need to determine the inverse function of the following one-to-one functions. 1−2𝑥 1−5𝑥 a. 𝑓(𝑥) = [𝑓 −1 (𝑥) = ] 5

b. 𝑚(𝑥) = 1 − c. 𝑘(𝑥) =

2𝑥−1 3𝑥+4

2

4+3𝑥 5

[𝑚−1 (𝑥) = −

[𝑘 −1 (𝑥) = −

4𝑥+1 3𝑥−2

5𝑥−1 3

]

]

Assignment. Research and then describe ways by which we can prove that two known functions 1

are inverses of each other. Use the function, ℎ(𝑥) = √3𝑥 − 1 + 5 and its inverse, ℎ−1 (𝑥) = (𝑥 − 5)2 + 3

1 3

, 𝑥 ≥ 5. Then, provide another example (a function and its inverse) to further demonstrate the

method(s) you discussed.


Criteria Generalizability of the method(s) presented Clarity of explanation/ description

Application of the method to the enforced example Application of the method to the student-made example More than one correct method is presented References are properly cited Score Interpretation


Can do better The method presented does not apply to all inverse functions (3 points) The description lacks some important information and is generally unclear (3 point) The process is entirely incorrect but a solution I at least provided following the suggested method (1 point) The functions provided are not even correct inverses of each other (1 point)

Meeting expectations The method presented applies to all inverse functions (5 points)

Description is stated clearly (5 points)

There are at most two errors in the process following the suggested method (2 point)

The solution is flawless (3 points)

The functions are inverses but the solution contains errors (2 points)


(additional 1 point)

(additional 1 point) 8 points or less

9 to 14 points

15 to 18 points

REPRESENTING INVERSE FUNCTIONS Day Two [Target M11GM-Id-3 and M11GM-Id-4] Represent an inverse function through its (a) table of values, and (b) graph; and find the domain and range of inverse functions.

1. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss the item if it is deemed necessary. The purpose of the assignment was for students to find out through individual research that using composition of functions, we can verify if two functions are actually inverses of each other. Expect that some students were unable to dig this up in their assignment so have those who found out discuss the process in class. The correct solution is given as follows: 1

1

The functions ℎ(𝑥) = √3𝑥 − 1 + 5 and ℎ−1 (𝑥) = (𝑥 − 5)2 + , 𝑥 ≥ 5 are inverses of each other if and only if 3 3 (ℎ ∘ ℎ−1 ) = (ℎ−1 ∘ ℎ)(𝑥) = 𝑥. Hence, to check, (ℎ ∘ ℎ−1 )(𝑥) = ℎ(ℎ−1 (𝑥)) (ℎ−1 ∘ ℎ)(𝑥) = ℎ−1 (ℎ(𝑥)) 1 1 2 1 1 ℎ−1 (ℎ(𝑥)) = (√3𝑥 − 1 + 5 − 5) + −1 (𝑥)) 2 3 3 ℎ(ℎ = √3 ( (𝑥 − 5) + ) − 1 + 5 3 3


General Mathematics Teaching Guides ••• 1 1 2 (√3𝑥 − 1) + 3 3 1 1 −1 ℎ (ℎ(𝑥)) = (3𝑥 − 1) + 3 3 1 1 −1 ℎ (ℎ(𝑥)) = 𝑥 − + 3 3 ℎ−1 (ℎ(𝑥)) = 𝑥

ℎ(ℎ−1 (𝑥)) = √(𝑥 − 5)2 + 1 − 1 + 5 ℎ(ℎ−1 (𝑥)) = √(𝑥 − 5)2 + 5 ℎ(ℎ−1 (𝑥)) = (𝑥 − 5) + 5 ℎ(ℎ−1 (𝑥)) = 𝑥

ℎ−1 (ℎ(𝑥)) =

The solutions in the previous page show that the two functions are inverses of each other. 2. Present the lesson and the targeted competency. 3. Facilitate activation of prior knowledge by using the following questions for recapitulation. a. What are one-to-one functions? b. What are inverse functions? Do all functions have inverse functions? c. How do we determine the inverse function of a one-to-one function? 4. Divide the class into ten groups for this topic’s activity. Each group shall be assigned to just one function and must accomplish the following: (a) determine the inverse function of the function they are assigned to; (b) construct a table of values composed of four ordered pairs for their function; and(c) graph he function they were assigned to. For the purpose of presentation, students will need the graph their function on a meter’s length of acetate (all groups should have the same size) the teacher shall provide a standard measure of the Cartesian plane on a manila paper.

acetate over manila paper

When graphing students needs to place their acetate above this manila paper and plot their points, sketch their graph using permanent markers. Ask the groups working with the first five functions to use blue ink and red for the other five. The reason for this is the graphs need to be compared by putting one graph over the other after they have been individually presented. The assigning of functions may be done at random through fishbowl. These functions are the following: 𝑓(𝑥) = 3𝑥 + 1 𝑓(𝑥) = 2𝑥 − 3 𝑓(𝑥) = 𝑥 2 , 𝑥 > 0 𝑥−1 𝑥+3 𝑓(𝑥) = 𝑥 + 4 𝑓(𝑥) = √𝑥 𝑓(𝑥) = 𝑓(𝑥) = 3 2 A maximum of 5 minutes may be alloted for the undertaking of this task. 𝑓(𝑥) = 𝑥 − 4

𝑓(𝑥) = 𝑥 3 3

𝑓(𝑥) = √𝑥

Objective: To show and conclude that the graphs of two inverse functions are symmetric with respect to the graph of the identity function, 𝒚 = 𝒙. Function, 𝒇(𝒙)

𝑥−4

Inverse Function 𝒇−𝟏 (𝒙)

𝑥+4

Table of Values

𝑥 𝑓(𝑥) 𝑥 𝑓 −1 (𝑥)

-1 -5 -5 -1

0 -4 -4 0

1 -3 -3 1

Graph (𝒇(𝒙); 𝒇−𝟏 (𝒙); 𝒚 = 𝒙)

2 -2 -2 2



3𝑥 + 1

𝑥−1 3

𝑥 𝑓(𝑥) 𝑥 𝑓 −1 (𝑥)

-1 -2 -2 -1

0 1 1 0

1 4 4 1

2 7 7 2

2𝑥 − 3

𝑥+3 2

𝑥 𝑓(𝑥) 𝑥 𝑓 −1 (𝑥)

-1 -5 -5 -1

0 -3 -3 0

1 -1 -1 1

2 1 1 2

√𝑥

𝑥 𝑓(𝑥) 𝑥 𝑓 −1 (𝑥)

0 0 0 0

1 1 1 1

2 4 4 2

3 9 9 3

𝑥 𝑓(𝑥) 𝑥 𝑓 −1 (𝑥)

-1 -1 -1 -1

0 0 0 0

1 1 1 1

2 8 8 2

𝑥 2, 𝑥 > 0

𝑥

3

3

√𝑥

Guide Questions (distribute or reveal these questions AFTER the group finished their graphs) Give the groups another five minutes to discuss answers to the questions below before using them as guide for the analysis of the activity. 1. 2.

What can be said about the domain and range of a function and its inverse function? [The domain of a function is the range of its inverse and vice versa] How is the slope of the graph of a linear function related to the slope of the graph of its inverse? [The slope of a linear function’s graph is the reciprocal of the slope of the graph of its inverse function]


General Mathematics Teaching Guides ••• 3.

Did you notice any pattern or relationship between the graphs of each pair of inverse functions? Is it possible to determine the graph of a function from its inverse? Discuss. [The graphs of inverse functions are symmetric with respect to the graph of the identity function, 𝒚 = 𝒙]

5. Ask the groups to present their graphs to class. The presentation should only cover how they went about their table of values and then the graph itself. Each group should only consume 2 minutes. Then, pair up the inverse functions and compare their graphs by putting one acetate over the other. Do this for all pairs. With the aid of the guide questions enumerated above, facilitate a class discussion revolving around the outputs of the students. Emphasize the pattern graphs of inverse functions have. As an aid for abstraction, ask the same questions used as guide questions in the activity. However, in this case, the class should have already agreed on generalized answers to these questions.

Graph the function, 𝑓(𝑥) = −

𝑥−3 3

and its inverse.

Show that the graphs are symmetric with respect to the graph of the identity function. Also, determine the domain and range of these two functions.


Evaluation. Use the rubric below to score students’ output in the activity. Criteria

Can do better



Correctness of graph presented

The graph is incorrect (2 point)

The graph is correct (5 points)

Group dynamics



Time Management



3 points or less


Task completed on or before the intended time (2 points) 4 to 7 points

8 to 10 points

2

Given the function, 𝑓(𝑥) = 𝑥 − 1, determine its inverse function and graph both functions. Show 3

WRITTEN WORK NO. 5

that the two functions are symmetric with respect to the identity function’s graph. Criteria Accuracy of each graph

Can do better

Correctness of the inverse function derived.

The graph constructed is not correct. (1 point each) At least two ordered pairs in the table are incorrect (1 point each) The inverse function identified is incorrect (1 point)


5 points or less

Correctness of table of values


Meeting expectations The graph constructed is correct. (3 points each)

One value in the table is not correct (2 points each)

All values are correct (3 points each) The correct inverse function has been determined (3 points)

6 to 10 points

11 to 15 points



Exponential Functions Week Five, Functions and Their Graphs Content Standards: The learner demonstrates understanding of key concepts of inverse functions, exponential functions, and logarithmic functions. Performance Standards: The learner is able to accurately apply the concepts of inverse functions, exponential functions, and logarithmic functions to formulate and solve real-life problems with precision and accuracy. Learning Competencies: Over the course of one week, the learner is expected to (1) represent real-life situations using exponential functions [M11GM-Id-1]; (2) determine the inverse of a one-to-one function [M11GM-Id-2]; (3) represent an inverse function through its (a) table of values, and (b) graph [M11GM-Id-3]; (4) find the domain and range of inverse functions [M11GM-Id-4]; and (5) graphs inverse functions [M11GM-Ie-1] Learning Materials: In order to develop the targeted competencies, the following materials are needed: (a) chalkboard and chalk; (b) LCD projector and laptop or in the absence, visual materials; (c) calculators; (d) worksheets Expected Outputs: In order to assess the attainment of the learning competencies targeted, students will be required to undertake two group performance tasks and one written work. Procedure (Teacher’s Activity) ONE-TO-ONE FUNCTIONS AND Day One [Target M11GM-Id-1 and THEIR INVERSE M11GM-1d-2] Represent real-life situations using one-to-one functions and determine the inverse of a one-to-one function.

8. Do routines and other preparatory activities for five minutes. 9. Present the lesson and the targeted competencies (and other learning outcome). 10. Facilitate recall with the following questions: d. What are relations? e. What are the three possible correspondences of ordered pairs for relations? f. Which of these correspondences are functions? Explain




(8)

Exponential Functions  29 Von Christopher G. Chua



𝑥−4 3

𝑔′ (𝑥) = √𝑥]


𝑥−4 3

.


𝑥−7 −2

or 𝑓 −1 (𝑥) =

7−𝑥 2

]


Procedure

𝑦 = 8𝑥 − 3


𝑥 = 8𝑦 − 3


𝑥 = 8𝑦 − 3 −8𝑦 = −𝑥 − 3 𝑥+3 𝑦= 8


𝒇−𝟏 (𝒙) =

𝒙+𝟑 𝟖



𝟔 𝒙+𝟑

6 𝑥+3


𝒈−𝟏 (𝒙) =

−𝟑𝒙 + 𝟔 𝒙



General Mathematics Teaching Guides ••• As exercise, students need to determine the inverse function of the following one-to-one functions. 1−2𝑥 1−5𝑥 d. 𝑓(𝑥) = [𝑓 −1 (𝑥) = ] 5

e. 𝑚(𝑥) = 1 − f.

𝑘(𝑥) =

2𝑥−1 3𝑥+4

2

4+3𝑥 5

[𝑚−1 (𝑥) = −

[𝑘 −1 (𝑥) = −

4𝑥+1 3𝑥−2

5𝑥−1 3

]

]



1 3
















9 to 14 points

15 to 18 points

REPRESENTING INVERSE FUNCTIONS Day Two [Target M11GM-Id-3 and M11GM-Id-4] Represent an inverse function through its (a) table of values, and (b) graph; and find the domain and range of inverse functions.

6. Do routines and other preparatory activities for five minutes. Collect the assignment and discuss the item if it is deemed necessary.



Exponential Functions Week Five, Functions and Their Graphs Content Standards: The learner demonstrates understanding of key concepts of inverse functions, exponential functions, and logarithmic functions. Performance Standards: The learner is able to accurately apply the concepts of inverse functions, exponential functions, and logarithmic functions to formulate and solve real-life problems with precision and accuracy. Learning Competencies: Over the course of one week, the learner is expected to (1) represent real-life situations using exponential functions [M11GM-Ie-3]; (2) distinguishes between exponential function, exponential equation, and exponential inequality [M11GM-Ie-4]; (3) solve exponential equations and inequalities [M11GM-Ie-f-1]; (4) represents an exponential function through its: (a) table of values; (b) graph; and (c) equation [M11GM-If-2] Learning Materials: In order to develop the targeted competencies, the following materials are needed: (a) chalkboard and chalk; (b) LCD projector and laptop or in the absence, visual materials; (c) calculators; (d) worksheets Expected Outputs: In order to assess the attainment of the learning competencies targeted, students will be required to undertake two group performance tasks and one written work. Procedure (Teacher’s Activity) INTRODUCTION TO EXPONENTIAL Day One [Target M11GM-Id-1 and FUNCTIONS M11GM-1d-2] Represent real-life situations using one-to-one functions and determine the inverse of a one-to-one function.

15. Do routines and other preparatory activities for five minutes. 16. Present the lesson and the targeted competencies (and other learning outcome). 17. Facilitate recall with the following questions: g. What are relations? h. What are the three possible correspondences of ordered pairs for relations? i. Which of these correspondences are functions? Explain




(8)




𝑥−4 3

𝑔′ (𝑥) = √𝑥]


𝑥−4 3

.


𝑥−7 −2

or 𝑓 −1 (𝑥) =

7−𝑥 2

]


Procedure

𝑦 = 8𝑥 − 3


𝑥 = 8𝑦 − 3


𝑥 = 8𝑦 − 3 −8𝑦 = −𝑥 − 3 𝑥+3 𝑦= 8


𝒇−𝟏 (𝒙) =

𝒙+𝟑 𝟖



𝟔 𝒙+𝟑

6 𝑥+3


𝒈−𝟏 (𝒙) =

−𝟑𝒙 + 𝟔 𝒙



General Mathematics Teaching Guides ••• As exercise, students need to determine the inverse function of the following one-to-one functions. 1−2𝑥 1−5𝑥 g. 𝑓(𝑥) = [𝑓 −1 (𝑥) = ] 5

h. 𝑚(𝑥) = 1 − i.

𝑘(𝑥) =

2𝑥−1 3𝑥+4

2

4+3𝑥 5

[𝑚−1 (𝑥) = −

[𝑘 −1 (𝑥) = −

4𝑥+1 3𝑥−2

5𝑥−1 3

]

]



1 3
















9 to 14 points

15 to 18 points


general mathematics teaching guides

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