Mathematics I. Quarter 2 : ALGEBRAIC EXPRESSIONS AND FIRST-DEGREE EQUATIONS AND INEQUALITIES IN ONE VARIABLE

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1 Mathematics I Quarter : ALGEBRAIC EXPRESSIONS AND FIRST-DEGREE EQUATIONS AND INEQUALITIES IN ONE VARIABLE Hello guys!!! Let s have a great time learning Mathematics. It s time for you to discover the language of ALGEBRA. This quarter will feed your mind with about Algebraic Expressions and First- Degree Equations and Inequalities in One Variable. Enjoy the activities while you explore, firm up what you will discover, deepen your learning and transfer through real-life applications. Module.1 : Algebraic Expressions This module is about algebraic expressions. As you go through the activities/exercises, you will develop skills in identifying constants, variables, terms, numerical expressions and algebraic expressions; translating verbal phrases into algebraic expressions and vice-versa; evaluating algebraic expressions. More importantly, you will find out how useful algebraic expressions are in representing patterns and relationships that will guide one in understanding how certain problems can be solved. 1

2 EXPLORE Your Understanding activities that will Warm up your brain cells! Let us begin with exploratory activities that will introduce you to the basics of algebraic expressions Activity 1 MEAN IT A. Identify the meaning of the following signs and symbols. Write your answer in the box provided for each.

3 Easy, isn t it? Those symbols are commonly used in public places. Now let s try some math symbols. B. Write the meaning of the following mathematical symbols. > + = 1 y 1. Why are symbols important?. What are the advantages of using symbols instead of words (say for example, road signs)? 3

4 In mathematics, what are the most common symbols that you can recall aside from those given in Activity 1? List them down. Classify the mathematical symbols as operation symbols, number symbols, relation symbols, etc. RELATION SYMBOLS NUMBER SYMBOLS OPERATION SYMBOLS 4

5 Activity How Do You Group Us? Given the symbols below, which should be grouped together? 8 b -5 y a n z Regroup the symbols in the box into two. Write your answers below. How did you come up with such grouping? Activity 3 Represent Me Below are two fruit baskets that contain mangoes. How do you represent the number of mangoes in the first basket? How about the number of mangoes in the second basket? How will you represent the total number of mangoes in these two baskets? 5

6 Basket 1 Basket Activity 4 We Belong Together! Given the following expressions, examine them carefully and look for expressions with something in common. Group them into two. x 5 n + 8 x + y (5-4)(+1) - a (a ) 3 6 (4 + 1) 10x 7x (l + w) 3 + (4 ) (x 4) + (x+3) Group 1 Group Explain how you come up with such groupings. 6

7 Activity 5 PHRASE ME What words should you look for to translate these operations into verbal phrases? The first one is done for you to serve as an example. plus increased by more than + added to the sum of - or or / FIRM UP Your Understanding Now let s keep going! Enjoy learning more and more about algebraic expressions. Here are enabling activities that will help you. Constants and Variables Arithmetic is concerned mainly with the study of the structure, operations and applications of whole numbers and positive rational numbers whether in the form of fraction, decimal or percent. Algebra, on the other hand, is concerned with the study 7

8 of the variables represented by letters and the operations relating these variables. These variables are symbols for numbers from the simple set of counting numbers to other sets of real numbers The essence of algebra lies in representing quantities as symbols other than numerals. This is the advantage of applying algebra and not arithmetic alone in solving practical problems. These different symbols are grouped into expressions, which in turn bring meaning to equations and inequalities. Observe how the symbols are grouped together Why? What do you call these symbols? b y a n z Why? What do you call these symbols? Does each symbol in the first row have a fixed value? Does each symbol in the second row have a fixed value? Every symbol that has no fixed value and stands for a number is called a placeholder symbol. In arithmetic, students meet problems like =, 3 x =, 1 = 3, n = 8. Algebra uses x, n, y, or any letter to represent numbers. A letter that is used as a placeholder symbol that has no fixed value is called a variable while a symbol that has a fixed value is called a constant. Example: In Activity 3, you can easily represent the number of mangoes in the first basket by the constant 8. Notice that the second basket is covered, hence the number of mangoes is an unknown quantity which can be presented by a variable, n. 8

9 8 mangoes n mangoes CONSTANT VARIABLE The total number of mangoes can be represented by the expression8 n. Numerical Expressions and Algebraic Expressions Why are these expressions grouped this way? x 5 (a ) 3 - a x + y n + 8 (x 4) + (x+3) (l + w) 10x 7x + 1 Why? What do we call these expressions? 6 3 (5-4)(+1) 6 (4 + 1) 3 + (4 ) Why? What do we call these expressions? Do the expressions in each group have something in common? If yes, what do the expressions in the first group have that the expressions in the second group do not have? A mathematical phrase that contains a variable is an open phrase. A number phrase or numerical expression is an expression that does not contain a variable. It is also referred to as a numerical expression. The verbal phrase a certain number added to 5 is translated to an open phrase n 5 where n stands for a certain number. The verbal phrase seven added to 5 is translated to a number phrase 5 7. Expressions like 8, 1, 5, and 0 are some number phrases for the number 10. Expressions like x, 5 a, 8 n, l w are examples of open 9

10 phrases. Another name for open phrase is algebraic expression. An expression composed of constants, variables, grouping symbols, and operation symbols, is called an algebraic expression. Examples: Activity 6 Numerical Expressions: 5 7 Algebraic Expressions: 5 4x 3 4 3xy x y m 6m 19 Classify the following expressions as numerical or algebraic. 1. 6x 4y m 3 n 7. 17a b (9 ) w 3 j 7u x 1m a 10. 5m 4 54 p In the algebraic expression - 7x 1, 7x and 1 are the terms of the expression. In the term 7x, 7 is called the numerical coefficient while x is the literal coefficient. The numerical coefficient of a term is written before the literal coefficient. A term is an indicated product or quotient of coefficients. The term 1 is the constant term, which does not have any indicated literal coefficient. Terms in an algebraic expression are separated by the plus (+) or minus (-) sign. Examples: Algebraic Expression Term/s Numerical Coefficient Literal Coefficient Variable/s Constant/s 5 4x 5 4x 4 5 x x 4 and 5 3xy 3xy 3 xy x and y 3 m 1 x y 4 6m x 1 x y y 4 4 m 1 m 6 m 6 m x 1 y and 4 m 6, and 19 10

11 Activity 7 For each of the following algebraic expressions, identify the variable/s, constant/s and term/s. For each of the terms, identify the numerical coefficient and the literal coefficient. mp x abc xy 18y 3. 9 w 8. 79b 1c 0. 7a 4 10r 4. 8m m n n x Translating English Phrases into Algebraic Expressions and Vice Versa English phrases involving quantities can be translated into algebraic expressions. Constants and variables together with symbols of operations and relations are important in translating English phrases into algebraic expressions. The symbols of operations and relations used in algebra are as follows: Symbol Meaning + addition, plus, increased by, added to, the sum of, more than - subtraction, minus, decreased by, subtracted from, less than, diminished by, ( ) multiplication, times, multiplied by /, division, divided by, ratio of, the quotient of = equals, is equal to < is less than > is greater than is less than or equal to is greater than or equal to is not equal to 11

12 Study how each of the following English phrases is translated into an algebraic expression. English Phrase Algebraic Expression 1. Sum of two numbers a b. Twice a certain number n 3. Difference of eight and thrice a certain number 8 3b 4. Quotient of a number and three diminished by two x 3 5. Square of the sum of a number and two. x Any letter or variable can represent the unknown number. In item #1, a and b are the symbols used to represent the said numbers. We add them because of the word sum, hence the use of the plus sign (+). In item #, the variable n represents the number. The phrase is translated as the product of and n. In item #3, the variable b represents the number and is used as a coefficient of 3. Their product is subtracted from 8 because of the word difference. In item #4, the variable x represents the number with 3 as its denominator because of the word quotient. The phrase diminished by is translated as. In Item #5, the variable x represents the number added to. Because of the word square, the quantity x is raised to the second power. An algebraic expression can also be translated into an English phrase. The translations may vary but still have the same meaning. Examples: 3 m thrice a number three times a number the product of a number and three x 4 twice a number diminished by four the difference between twice a number and four four subtracted from twice a number 5y+3 five times a number increased by three the sum of five times a number and three three more than five times a number 1

13 Activity 8 A. Translate each English phrase into an algebraic expression. English Phrase 1. A number y increased by four. Three times a number m decreased by 6 3. Nine added to the quotient of m and five Algebraic Expression B. Translate each algebraic expression into an English phrase. Algebraic Expression 4. 8n 1 x y x 3 English Phrase Evaluating Algebraic Expressions Can you think of an instance when substitution is done? In a basketball game, a better player usually replaces a player who does not perform well, or maybe one player needs some rest so another player has to come in. Replacing one player by another player is called substitution. In Algebra, we replace a variable with a number. This is called substituting the variable. To evaluate an algebraic expression, replace the variable by a number or substitute a number for the variable and simplify the expression. Evaluating an algebraic expression means obtaining or computing the value of the expression where value/s of the variable/s is/are assigned. Examples: 1. Evaluate y + 3 when y = 3 Solution: = (3) + 3 Substituting 3 for y = Multiplying and 3 = 9 Adding 6 and 3. Evaluate 3(a + 4) + (a ) when a = 6 Solution: = 3(6 + 4) + (6 ) Substituting 6 for a = 3(10) + 4 Computing and 6 - = Multiplying 3 and 10 = 34 Adding 30 and 3. Evaluate (x + 4) + 3(y 3) when x = - 3 and y = 5 Solution: = (-3 + 4)+ 3(5 3) Replacing x by 4 and y by 5 = (1) + 3() Computing and 5-3 = + 6 Computing (1) and 3() = 8 Computing

14 4. Evaluate (x 3) - y + y when x = - 6, y = 3 Solution: = [(-6) 3]-(3)+(3) Substituting 6 for x and 3 for y = [-1 3] (9) Computing (-6); -(3) and (3) = Computing -1 3 and (9) = 8 Computing Activity 9 Evaluate each of the following expressions at the given value(s) of the variable(s). Algebraic Expression Values of the variables 1. 7a + 3b a = -4 b =. 4xy - (x + y) y x = 3 y = 4 3. (3y + y) + y y = (m n) + (6m n) m = 6 n = (4x y) + 3(x - y) x = -3 y = 6 DEEPEN Your Understanding Get ready to take on more challenges to your mathematical thinking and reasoning in relation to the lessons. Activity 10 Do you remember me? I pioneered the use of symbols in representing numbers and I m quite well known for that. Answer the matching test to know me. 14

15 Match each English phrase in Column A with its corresponding algebraic expressions in Column B. Write the letter of your answer in the blank provided before the item number. Read the word formed by the letters to identify the person above. Answer Column A Column B 1. The square of six minus the number x a) ab 0. Two more than six times the number n d) 6 x 3. Three times a number x decreased by five h) 5 3m 4. The quotient of sixteen and the number n i) 6n 9 m n 6. Twenty added to the product of the numbers a and b o) 3x 5 7. Nine increased by the quotient of the numbers m and n 16 p) n 8. The product of twelve and the number y divided by seven n s) 4m 3 1y 9. Eight subtracted from the sum of eleven and the number y t) 7 10.The sum of product of four and the number m and the u) y 11 8 quotient of the number n and three 5. Five times the product of the number m and three n) Activity 11 A. Assign a variable then write an algebraic expression for each English phrase. 1. five more points than the Alaska Aces. Arthur s age seven years ago 3. nine decimeters less than the length 4. fifteen years younger than James 5. twice the number of messages in Ayel s cellular phone B. Use algebraic expressions to represent the following situations: 6. Mabel has thrice as much money as James. If James has d pesos, represent the amount of money that Mabel has. 7. In a basketball game, Bryan scored xy points. Shawn scored six less than twice Bryan s points. How will you express the number of points scored by Shawn? 8. Ronald and his friends bought four large drinks and two boxes of popcorns before watching a movie. If a box of popcorn costs Php30 and a large drink costs p pesos, write an algebraic expression for the amount of money they spent on snacks. C. If y represents the number of songs in Gudy s MP3 player, translate the following algebraic expressions into English phrases. 9. y y 15

16 11. y 16 y D. Evaluate each algebraic expression if x 3, y 1 and z 8. 7y x 14. 4z 5 3x z y TRANSFER Your Understanding It s time to demonstrate what you have learned. Answer the questions/ do the activities that follow and compile them in your portfolio. Activity 1 A. Why are variables used in algebraic expressions? B. Compare and contrast algebraic expressions and numerical expressions. You may use examples to explain your answers. C. Compare the English phrases eight more than eight times as many. D. Linnea wrote an algebraic expression for the phrase a number subtracted from 1 Is her answer correct? If not, explain why and correct it. 16

17 Am I correct? n 1 E. Think of real-life situations and use algebraic expressions to represent/translate them. F. In Geometry, the expression 180 n is used to find the sum of the degree measures of the interior angles of a regular polygon with n sides. Find the sum of the number of degrees of all the interior angles of an eight-sided picture frame. 17

18 Answers Key Module.1: Algebraic Expressions Activity 1 A. NO U TURN MALE RESTROOM NO PARKING FEMALE RESTROOM STOP NO SMOKING B. > greater than + addition = equals 1 twelve y a number, y 18

19 Activity b a z y n The first group are all numbers with fixed values (constant), while those in the second group are either letters (variables) or symbols for numbers of unknown values. Activity 3 8 mangoes n mangoes Activity 4 Group 1 Group (5-4)(+1) 6 (4 + 1) (4 ) x 5 x + y n a (a ) 3 10x 7x + 1 (l + w) (x 4) + (x+3) 19

20 The first group are expressions that are numerical, while the second group are expressions that contain variables. Activity 5 minus decreased by less than - subtracted by the difference of subtracted from multiplied by times or the product of divided by the ratio of or / the quotient of the square root of 0

21 Activity 6 1. algebraic 6. numerical. algebraic 7. algebraic 3. numerical 8. numerical 4. algebraic 9. numerical 5. algebraic 10. algebraic Algebraic Term/s Numerical Literal Variable/s Constant/s Expression Coefficient Coefficient x 3.14x x x and. 3abc 3abc 3 abc a, b and c w 9 9 w 1 w w 9 Activiy 7 1

22 4 4. 8m m n 8 m 8 m m 4 n 1 m 4 n n n n 4 4 mp 6. 4 mp 1 3 mp xy 7 xy xy 18y 3 18y 18 3 y 8. 79b 1c 0. 7a 79 b 79 b 1c 1 c 0.7a 0. 7 a 10r 10r r x x 9 x m and n 8 and 4 n m and p and 1 3 x and y 7 and 18 a, b and c 79, 1 and 0. 7 r 10 9 x 15 and 9 Activity 8 A. Translate each verbal phrase into an algebraic expression. English Phrase Algebraic Expression 1. A number y increased by four y 4. Three times a number m decreased by 6 3m 6 3. Nine added to the quotient of m and five m 9 5 B. Translate each algebraic expression into a verbal phrase. Algebraic Expression English Phrase 4. 8n 1 Twelve subtracted from eight times a number n. The product of eight and a number n, decreased by twelve 5. x y Twice the sum of two numbers x and y 6. x 3 The square of a number x, increased by three Three more than the square of a number x Activity 9

23 1. 7a + 3b if a = -4 and b =. 4xy - (x + y) y if x = 3 and y = 4 = 7(-4) + 3() = 4(3)(4) (3+4) (4) = = = - = (3y + y) + y if y = (m n) + (6m n) if m = 6and n = -4 = [3(-5) + (-5)] + (-5) = 3[6 (-4)] + [6(6) (-4)] = [-15 5] +5 = 3[6 + 4] + [36 (-4)] = [-0] + 5 = 3[10] + [-9] = = 30 9 = 5 = 1 5. (4x y) + 3(x - y) if x = -3 and y = 6 = [4(-3) 6] + 3[(-3) 6] = [4(9) 6] + 3[-3 6] = [36 6] + 3[-9] = 6 18 = - 1 Activity d. i 3. o 4. p 5. h 6. a 7. n 8. t 9. u 10. s Activity 11 A. 1. Let a be Alaska Aces points a 5 five more points than the Alaska Aces 3

24 . Let a be Arthur s present age a 7 Arthur s age seven years ago 3. Let l be the length l 9 nine decimeters less than the length 4. Let j be Jame s age j 15 fifteen years younger than James 5. Let m be the number of messages in Ayel s cellular phone m twice the number of messages in Ayel s cellular phone B. C d 7. xy p Twenty-five more than the number of songs in Gudy s MP3 player 10. Five times the number of songs in Gudy s MP3 player 11. Sixteen less than the number of songs in Gudy s MP3 player 1. The number of songs in Gudy s MP3 player, divided by ten D. 7y 13. 8, if x 3, y 1 and z 8 x 7 y x \ z 5 3x, if x 3, y 1 and z 8 4z 5 3x z y, if x 3, y 1 and z 8 9z y

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