Algebra Terminology Part 1

Transcription

1 Grade 8 1

2 Algebra Terminology Part 1 Constant term or constant Variable Numerical coefficient Algebraic term Like terms/unlike Terms Algebraic expression Algebraic equation Simplifying Solving TRANSLATION Grade 8 2

3 Algebra Learning algebra is a little like learning another language. In fact, algebra is a simple language, used to create mathematical models of real-world situations and to handle problems that we can't solve using just arithmetic. Rather than using words, algebra uses symbols to make statements about things. In algebra, we often use letters to represent numbers. Since algebra uses the same symbols as arithmetic for adding, subtracting, multiplying and dividing, you're already familiar with the basic vocabulary! Grade 8 3

4 Language of algebra Real Numbers In algebra, we work with the set of real numbers, which we can model using a number line. Real numbers describe real-world quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal. Grade 8 4

5 Language of algebra The first step in learning to "speak algebra" is learning the definitions of the most commonly used words. Algebraic Expressions An algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign. Algebraic expression: 3x 9 + 2y 7n + 5 In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has five terms, 3x, -9, 2y, -7n & 5. Terms may consist of variables and coefficients, or constants. variable 3n 6x + 7 numerical coefficient constant term Grade 8 5

6 Language of algebra Variables In algebraic expressions, letters represent variables. These letters are actually numbers in disguise. In this expression, the variables are k & z. We call these letters "variables" because the numbers they represent can vary that means, we can substitute one or more numbers for the letters in the expression. Coefficients Coefficients are the number part of the terms with variables. In 3n - 6n +7, the coefficient of the first term is 3. The coefficient of the second term is -6, and the third term is the constant, 7. variable 9k 2z 5 numerical coefficient constant term If a term consists of only a variable, its coefficient is 1, for example, x by itself means 1x. Grade 8 6

7 Language of algebra Constants Constants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes, since there are no variables in the term that can change its value. In the expression 3n - 6n + 7 the constant term is 7. variable 7p + j numerical coefficient constant term Grade 8 7

8 Concept Attainment a + a = 2a a + a = 2 4b + 2b = 6b 4a + 4 = 8a 6r 3r = 3r 6r 3r = 3 1k 8k = - 7k k 8k = - 7 1x + 4x = 5x x + 4x = 5 2n + 4 = 2n + 4 2n + 4 = 6n 2a + 2a = 4a 4n + 2n = 6 7r 3 = 4 3t 2 = t x + 5x = 6x - 8b + 5b = - 3b 5d 4 = 1d 4x 2 = 4x 2 Grade 8 8

9 Think about what the examples in the YES column have in common! Now turn to someone sitting next to you and share your ideas about the YES column. Share your thoughts with the class! Grade 8 9

10 Algebraic Expressions Can only put together like terms Unlike terms cannot be added or subtracted Grade 8 10

11 Algebra & Perimeter Create an algebraic expression to represent the perimeter of each figure. P = P = Grade 8 11

12 Algebra & Perimeter Create an algebraic expression to represent the perimeter of each figure. P = P = Grade 8 12

13 Simplifying Algebraic Expressions You can only combine LIKE term 3x + 7-2x p + 8p

14 Algebra & Perimeter 2x + 7 4x 5 3n 1 n 2n + 3 Perimeter = Combine like terms! Grade 8 14

15 Algebra & Perimeter 2x + 7 4x 5 3n 1 n 2n + 3 P = Grade 8 15

16 Simplifying Algebraic Expressions Addition and Subtraction A sum or difference can be simplified by adding or subtracting like terms, such as the constant terms together and the terms with the same variable together. Example 3n n + 6 Bring like terms together 3n + 2n n + 7 6x + 4 4x 2 6x 4x x + 2 Grade 8 16

17 Practice with ALGEBRAIC EXPRESSIONS #1] x + 2x + 4 #2] 3b 4 + 2b + 8 #3] 5a a #4] n + 3n 7a #5] - 8v + 3 4v 5 #6] 5a 3a Grade 8 17

18 1. Choose any number 2. Multiply it by 5 3. Add the number of wheels on a car 4. Double the result 5. Add the number of eggs in a dozen 6. Add Divide by 10 Tell me the final result and I ll tell you the original number you picked.

19 Translating Words into an Algebra Expression Here is a statement the sum of three times a number and eight How would you translate it to algebra? In this expression, we don't need a multiplication sign or brackets. Phrases like "a number" or "the number" tell us our expression has an unknown quantity, called a variable. In algebra, we will use letters to represent numbers. Grade 8 19

20 Algebraic Translation Operation Algebraic Expression Read Addition n + 5 Subtraction n 5 Multiplication 5n Division n 5 n 5 n plus five the sum of n and 5 n increased by 5 5 more than n n - 5 the difference of n and 5 n decreased by 5 5 less than n five n the product of 5 and n 5 multiplied by n n over 5 the quotient of n and 5 n divided by 5 one fifth of n Exponentiation n 2 n squared the square of n n with exponent 2 Grade 8 20 n to the second power

21 Read these and translate to ALGEBRA 1) The sum of a number and 10 2) The quotient of 4 and a number 3) The difference of 9 and b 4) The sum of double a and 8 5) The square of the difference of a and 2 Grade 8 21

22 6) The sum of a number and 7. 7) The difference between 12 and a number. 8) The product of a number and 9. 9) The quotient of 24 and a number. 10) The sum of double m and triple x. 11) Triple a number less the square of c. 12) Quadruple n divided by half s. Grade 8 22

23 Difference between LESS & LESS THAN five less a number 5 x five less than a number x 5 Grade 8 23

24 Practice with ALGEBRAIC EXPRESSIONS #1] - 6x 2x + 9 #2] - 4b 6b + 2b #3] 3a 7 8a + 4 #4] 6n + 2a 7a #5] - v + 2 7v 9 #6] - a + 4a 1 4 2a Grade 8 24

25 Algebraic Expressions Reading a problem and creating an algebraic expression. A hockey player accumulates 40 minutes in penalties. If the player then receives: 1 minor penalty, the total will be ( ) min 2 minor penalties, the total will be ( ) min 3 minor penalties, the total will be ( ) min s minor penalties, the total will be ( s) Grade 8 25

26 Algebraic Expressions Reading a problem and creating an algebraic expression. There are 162 games in a major-league baseball season. If, after the season opener, the team has played: 1 game, (162 1) games remain to be played 2 games, (162 2) games remain to be played 3 games, (162 3) games remain to be played y games (162 y) games remain to be played Grade 8 26

27 Algebraic Expressions Reading a problem and creating an algebraic expression. The jackpot in a lottery is $ , if there is 1 winning ticket, the jackpot is worth $ winning tickets, each jackpot is worth $ winning tickets, each jackpot is worth $ x winning tickets, each jackpot is worth $ x Grade 8 27

28 Patterns & Expressions Here is a made of toothpicks How many toothpicks are needed to make: 2 s? 3 s? If the # of s is represented by the variable t, give an expression to calculate the number of toothpicks needed. 4 s? Grade 8 28

29 Patterns & Expressions Use the expression to find the number of toothpicks needed to make: t ) 18 s 2) 39 s 3) 107 s Grade 8 29

30 Patterns & Expressions Here are some s made from toothpicks. How many toothpicks are needed to add 1? How many toothpicks are needed to build? 1 2 s 5 s 8 s If the # of s is represented by c, give a formula to calculate the # of toothpicks. Grade 8 30

31 Patterns & Expressions Use the expression to find the number of toothpicks needed to make: 3 c + 1 1) 16 s 2) 35 s 3) 129 s Grade 8 31

32 Simplifying Algebraic Expressions Addition and Subtraction Adding a whole number means adding each of its parts; (3n + 2) + (2n + 3) 3n n + 3 Bring like terms 3n + 2n together 5n + 5 Remove brackets, rewrite expression Subtracting a whole number means subtracting each of its parts; (7n + 6) (4n + 3) Remove brackets, rewrite expression 7n + 6 4n 3 Bring like terms together 7n 4n n + 3 Grade 8 32

33 Find the sum or difference for each 1. (2a + 6) + (3a 4) Remove brackets & rewrite expression Bring like terms together 2. (6n 5) (4n + 4) 3. (12s 7) (9s 5) Grade 8 33

34 Remember: When you add or subtract algebraic expressions you only combine like terms! Multiplication & Division You can MULTIPLY or DIVIDE unlike terms! 5a 2 = 2y - 3= - 4x - 6= 6a 2 = - 40y/ - 4 = 15x 3 = Grade 8 34

35 Area & Perimeter 2a Perimeter = 4 4 2a Area = length width Grade 8 35

36 Area & Perimeter Perimeter = 6 8 3x Area = b h 2 5x Grade 8 36

37 Simplifying Algebraic Expressions Multiplication and Division Multiplying a whole number means multiplying each of its parts 4(3n + 2) 4 3n n + 8 Dividing a whole number means dividing each of its parts; (8n + 6) or 2 8n n + 3 (8n + 6) 2 Grade 8 37

38 Find the product/quotient for each: 6(3a + 5) (18s 12) 6-3(5x 4) [3(6n + 3)] 9 Grade 8 38

39 Let s Look at MORE Than 1 set of BRACKETs 3(2x 5) + 4(x + 6) 7(x 4) + 5(3x 9) 6(8x 6) 3(5x 8) Grade 8 39

40 Equation An equation is a mathematical sentence containing an equal sign. It tells us that two expressions mean the same thing, or represent the same number. An equation can contain variables and constants. Using equations, we can express math facts in short, easy-to-remember forms and solve problems quickly. Here are examples of equations: 3z + 2 = 14 x 9 = 20 6p + 2 = 32 The most important skill to develop in algebra is the ability to TRANSLATE a word problem into the correct equation, so that you can solve the problem easily. Grade 8 40

41 Equations Let's try a few examples: 1) A number n times 3 is equal to ) Ten less than four times a number equals seventy. 3) Lori worked fifteen hours, she earned $120.00, write an equation to represent her earnings. Grade 8 41

42 Equations 4) Nine more than five times a number is twenty four. 5) Four times a number decreased by seven equals seventeen. 6) Three times a number plus five equals five times a number less nine. Grade 8 42

43 Level I Equations When solving a Level I equation, we use the opposite operation that is displayed to determine the value of our variable (letter). For example, addition is the opposite operation of subtraction, and multiplication is the opposite operation of division. There are 4 kinds of Level I equations that we will be solving. Level I (addition) x + 5 = x = 4 V x + 9 = x = -15 n + 7 = 18 Grade 8 43

44 Level I Equations Level I (subtraction) x 9 = x = 15 V x 4 = - 8 x 11 = 18 x 9 = - 5 Grade 8 44

45 Level I Equations Level I (multiplication) 6x = x = - 7 Validate 6-7 = -42 To solve a Level I multiplication equation we use division to help us find the value of x. 4x = 48-8x = x = 55 Grade 8 45

46 Level I Equations Level I (division) x = x 1= 4-9 x = - 36 Validate = -9 In division we use a special type of opposite operation called cross-multiplication. x = 7 12 x = - 6 x =

47 Level I Equations 1) x + 7 = 12 2) x 6 = 11 3) x = 9 6 4) 7x = 35 5) x + 10 = -5 6) x = Grade 8 47

48 Level II Equations A Level II equation will require you to do two opposite or inverse operations to solve for the variable (letter). Always do the opposite of any addition or subtraction first, then proceed to do the inverse operation of any multiplication or division. E.g.: 5x 9 = Now put 3.4 into the equation & calculate! 5x = 17 x = x = 3.4 Validate = 17 9 = 8 E.g.: x + 7 = x 1 = x = 11.5 Validate = = 9.3 Grade 8 48

49 Level II Equations 1) 3x + 2 = 14 3) x + 3 = ) 4x + 8 = 4 Grade 8 49

50 Level II Equations 4) 5x - 3 = 18 6) x = ) - 13x + 3 = 6 Grade 8 50

51 Level III Equations A Level III equation is one with more than one group of variables. To solve you must first gather like terms on each side of the equal sign. Example #1: 4x 6 = 2x x = 16 2x = x = 8 Example #2: 3x + 2 = x x = 2x x = x 7 2x 2x x x Validate = = = -8.5 Validate = 32 6 = = = 26 2x = - 7 2x = x = Grade 8 51

52 #1) 4x + 3 = 9x + 7 Level III Equations #3) 7x 3 2x = 8 #2) 7x 10 = 3x + 14 Grade 8 52

53 Level III Equations #4) 3x + 8x = 44 #5) - 2x - 8x = #6) 7x + 6 = 12 Grade 8 53