Teacher Road Map for Objective in Student Friendly Language: Today I am: sorting equation cards So that I can: determine when an equation is true or false. I ll know I have it when I can: write my own always true, sometimes true or never true equations. Flow: o o Students examine work done by a fictional student to determine any errors. Student sort equation cards into three categories Always True, Sometimes True and Never True and then write their own equations. Big Ideas: o o This lesson introduces the concept of algebraic equations and their truth value. An equation with variables can be viewed as a question asking for which values of the variables will result in true number sentences when those values are substituted into the equation. College Prep Adjustments: o There are a multitude of exercises in the Homework Problem Set. You might want to use some of them for a game on the 2 nd day if you think your students need another day of practice. CCS Standards: o A-REI.3 Solve linear equation and inequalities in one variable, including equations with coefficients represented by letters. 139
Student Outcomes Students understand that an equation is a statement of equality between two expressions. When values are substituted for the variables in an equation, the equation is either true or false. Students find values to assign to the variables in equations that make the equations true statements. Exploratory Exercise 1. Read over Rita s answers to the six statements below. Was she correct every time? Circle any incorrect answers given by Rita. Name: Rita Answer True or False for each statement below. A. The president of the United States is a United States citizen. True B. The president of France is a United States citizen. False C. 2 + 3 = 4 + 1 True D. 6 2 = 2 6 True E. 2 4 = True 4 2 F. 3 6 = 2 5 False 2. What makes a statement true? How can you prove that a statement is false? Responses will vary. A statement is true if it is true for all values. To prove a statement false you need a counterexample. 140
Hold a general class discussion about Exercise 1C and 1D. Be sure to raise the following points: One often hears the chime that mathematics is a language. And indeed it is. For us reading this text, that language is English. (And if this text were written in French, that language would be French, or if this text were written in Korean, that language would be Korean.) A mathematical statement, such as 2 + 3 = 1 + 4, is a grammatically correct sentence. The subject of the sentence is the numerical expression 2 + 3, and its verb is equals or is equal to. The numerical expression 1 + 4 renames the subject (2 + 3). We say that the statement is true because these two numerical expressions evaluate to the same numerical value (namely, five). The mathematical statement 2 + 3 = 9 + 4 is also a grammatically correct sentence, but we say it is false because the numerical expression to the left (the subject of the sentence) and the numerical expression to the right do not evaluate to the same numerical value. 3. For each statement below, determine if the statement is True or False. Be prepared to support your answer. A. (5 + 2) 2 = 5 2 + 2 2 B. 3 2 + 4 2 = 5 2 C. 6 + 3 = 5 + 4 False 49 29 True 25 = 25 True 9 = 9 D. 680 (520 12) = 12 (520 680) E. 3x + 6 = 9 F. 2x + 7 = 2x True, by the associative and True if x = 1 False, any value of commutative properties x will make this false. 4. An open sentence is one in which there might be one or more values that will make the sentence true but all others will make it false. Open sentences are also called algebraic equations. Which sentences in Exercise 3 are algebraic equations? E and F are algebraic equations. 141
You will need a set of Algebraic Equations Cards. You may wish to have one set for each student to cut up and glue into their table. The set of Algebraic Equations Cards are at the end of these teacher notes (page 147). 5. Sort the cards and determine in which category the algebraic equation belongs. One example of each has been done for you. You will not have enough algebraic equations to fill in this entire table. One Solution No Solutions This statement is NEVER true! An Infinite Number of Solutions This statement is ALWAYS true! 5x + 3 = 2x + 12 True when x = 3 5x+ 8 = 10x+ 4 True when x = 4/5 2x+ 8= 6x+ 4 True when x = 1 7 + x = 12 True when x = 5 2x = 6 True when x = 3 2x + 12 = 2x + 3 2x + 12 = 2(x + 6) 6x+ 12 = 6( x+ 3) 3( x+ 3) = 3x+ 9 4x+ 15 = 2(2x+ 5) 5x+ 14 = 5( x+ 2) + 4 5x+ 10 + 3x= 8x+ 12 3x+ 5 + 7x= 2(5x+ 2) + 1 8x+ 3 = 4(2x+ 1) 7x+ 14 = 7( x+ 2) 6. Write your own equations one for every category. Then check your answers with your partner. Answers will vary. 142
Lesson Summary A number sentence is a statement of equality between two numerical expressions. A number sentence is said to be true if both numerical expressions are equivalent (that is, both evaluate to the same number). It is said to be false otherwise. True and false are called truth values. An algebraic equation is a statement of equality between two expressions. Algebraic equations can be number sentences (when both expressions are numerical), but often they contain symbols (variables) whose values have not been determined. When algebraic equations contain a symbol whose value has not yet been determined, we use analysis to determine whether: a. The equation is true for all the possible values of the variable(s), or b. The equation is true for a certain set of the possible value(s) of the variable(s), or c. The equation is never true for any of the possible values of the variable(s). Closing: Consider the following scenario. Julie is 333333 feet away from her friend s front porch and observes, Someone is sitting on the porch. Given that she did not specify otherwise, we would assume that the someone Julie thinks she sees is a human. We cannot guarantee that Julie s observational statement is true. It could be that Julie s friend has something on the porch that merely looks like a human from far away. Julie assumes she is correct and moves closer to see if she can figure out who it is. As she nears the porch, she declares, Ah, it is our friend, John Berry. Often in mathematics, we observe a situation and make a statement we believe to be true. Just as Julie used the word someone, in mathematics we use variables in our statements to represent quantities not yet known. Then, just as Julie did, we get closer to study the situation more carefully and find out if our someone exists and, if so, who it is. Notice that we are comfortable assuming that the someone Julie referred to is a human, even though she did not say so. In mathematics we have a similar assumption. If it is not stated otherwise, we assume that variable symbols represent a real number. But in some cases, we might say the variable represents an integer or an even integer or a positive integer, for example. Stating what type of number the variable symbol represents is called stating its domain. Exit Ticket (5 minutes) 143
Name Date Exit Ticket 2. Consider the following equation, where aa represents a real number: aa + 1 = aa + 1. Is this statement a number sentence? If so, is the sentence true or false? 3. Suppose we are told that bb has the value 4. Can we determine whether the equation below is true or false? If so, say which it is; if not, state that it cannot be determined. Justify your answer. bb + 1 = bb + 1 4. For what value of cc is the following equation true? cc + 1 = cc + 1 144
Exit Ticket Sample Solutions 1. Consider the following equation, where aa represents a real number: aa + 11 = aa + 11. Is this statement a number sentence? If so, is the sentence true or false? No, it is not a number sentence because no value has been assigned to aa. Thus, it is neither true nor false. 2. Suppose we are told that bb has the value 44. Can we determine whether the equation below is true or false? If so, say which it is; if not, state that it cannot be determined. Justify your answer. bb + 11 = bb + 11 False. The left-hand expression has value 44 + 11 = 55 and the right-hand expression has value 22 + 11 = 33. These are not the same value. 3. For what value of cc is the following equation true? cc + 11 = cc + 11, if we let cc = 33. cc + 11 = cc + 11 Homework Problem Set Sample Solutions 1. Determine whether the following number sentences are true or false. A. 4 + 8 = 10 + 5 False B. (71 603) 5876 = 603 (5876 71) True C. (7 + 9) 2 = 7 2 + 9 2 False D. ππ = 3.141 False E. (4 + 9) = 4 + 9 False 145
2. Circle all of the following that are algebraic equations. Underline any that are number sentences. A. 3.1xx 11.2 = 2.5xx + 2.3 B. 10ππ 4 + 3 = 99ππ 2 C. ππ + ππ = 2ππ D. 1 2 + 1 2 = 2 4 E. 79ππ 3 + 70ππ 2 56ππ + 87 = 60ππ + 29 928 ππ 2 3. In the following equations, let xx = 3 and yy = 2. Determine whether the following equations are true, 3 false, or neither true nor false. A. xxyy = 2 True B. xx + 3yy = 1 True C. xx + zz = 4 Neither true nor false D. 9yy = 2xx True E. yy xx = 2 False F. 2 xx yy = 1 False 4. Name a value of the variable that would make each equation a true number sentence. For example, Let ww = 2. Then, ww 2 = 4 is true. There might be more than one option for what numerical values to write. Feel free to write more than one possibility. Warning: Some of these are tricky. Keep your wits about you! A. Let x = 5. Then, 7 + xx = 12 is true. B. Let r = 6. Then, 3rr + 0.5 = 37 is true. 2 C. mm 3 = 125 is true for m = -5. D. A number xx and its square, xx 2, have the same value when x = 1 or x = 0. 146
E. The average of 7 and nn is 8 if n = - 23. F. Let a be any real number. Then, 2aa = aa + aa is true. G. qq + 67 = qq + 68 is true for no values of q. 5. For each of the following, assign a value to the variable, xx, to make the equation a true statement. A. (xx + 1)(xx + 2) = 20 for x = 3 or x = -6 B. (dd + 5) 2 = 36 for d = 1 or d = -11 C. 1+xx 1+xx 2 = 3 5 for x = 2 D. 1 xx = xx 1 if x = 1. E. xx + 2 = 9 x = 7 F. xx + 2 2 = 9 x = -13 G. 12tt = 12 t = -1 H. 12tt = 24 t = 2 I. 1 bb 2 = 1 4 b = 6 J. 1 2bb 2 = 1 4 b = -1 K. xx + 5 = xx + 5 x = 0 L. (xx 3) 2 = xx 2 + ( 3) 2 x = 0 M. xx 2 = 49 There is no real value of x to make this true. 6. Fill in the blank with a variable term so that the given value of the variable will make the equation true. A. xx + 4 = 12; xx = 8 B. 2xx + 4 = 12; xx = 4 7. Fill in the blank with a constant term so that the given value of the variable will make the equation true. A. 4yy 0 = 100; yy = 25 B. 4yy _24 = 0; yy = 6 C. rr + 0 = rr; rr is any real number. D. rr 1 = rr; rr is any real number. 147
8. Generate the following. Answers will vary. Sample responses are given. Note: These sample responses will be used in the Exploratory Exercise in Lesson 11. A. An equation that is always true 2x + 4 = 2(x + 2) C. An equation that is never true x + 3 = x + 2 E. An equation that is true when yy = 0.5 2y + 1 = 0 B. An equation that is true when xx = 0 x + 2 = 2 D. An equation that is true when tt = 1 or tt = 1 t 2 = 1 F. An equation that is true when zz = ππ z π = 1 148
Algebraic Equation Sorting Cards 5x+ 8 = 10x+ 4 2x+ 8= 6x+ 4 3( x+ 3) = 3x+ 9 6x+ 12 = 6( x+ 3) 5x+ 14 = 5( x+ 2) + 4 7 + x = 12 4x+ 15 = 2(2x+ 5) 3x+ 5 + 7x= 2(5x+ 2) + 1 5x+ 10 + 3x= 8x+ 12 7x+ 14 = 7( x+ 2) 8x+ 3 = 4(2x+ 1) 2x = 6 Algebraic Equation Sorting Cards 5x+ 8 = 10x+ 4 2x+ 8= 6x+ 4 3( x+ 3) = 3x+ 9 6x+ 12 = 6( x+ 3) 5x+ 14 = 5( x+ 2) + 4 7 + x = 12 4x+ 15 = 2(2x+ 5) 3x+ 5 + 7x= 2(5x+ 2) + 1 5x+ 10 + 3x= 8x+ 12 7x+ 14 = 7( x+ 2) 8x+ 3 = 4(2x+ 1) 2x = 6 Algebraic Equation Sorting Cards 5x+ 8 = 10x+ 4 2x+ 8= 6x+ 4 3( x+ 3) = 3x+ 9 6x+ 12 = 6( x+ 3) 5x+ 14 = 5( x+ 2) + 4 7 + x = 12 4x+ 15 = 2(2x+ 5) 3x+ 5 + 7x= 2(5x+ 2) + 1 5x+ 10 + 3x= 8x+ 12 7x+ 14 = 7( x+ 2) 8x+ 3 = 4(2x+ 1) 2x = 6 149
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