teacher road map for lesson 10: true and false equations · · 2016-11-07teacher road map for...

Hart Interactive – Algebra 1 M1 Lesson 10 ALGEBRA I

Teacher Road Map for Lesson 10: True and False Equations

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 Students examine work done by a fictional student to determine any errors. o Student sort equation cards into three categories – Always True, Sometimes True and Never

True and then write their own equations.

Big Ideas: o This lesson introduces the concept of algebraic equations and their truth value. o 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 2nd 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.

Lesson 10: True and False Equations

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This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This file derived from ALG I-M1-TE-1.3.0-07.2015

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.


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.

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.

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 44 2= True

F. 3 • 6 = 2 • 5 False


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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 = 52 + 22 B. 32 + 42 = 52 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.


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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

2x + 12 = 2x + 3 2x + 12 = 2(x + 6)

5 8 10 4x x+ = +

True when x = 4/5 6 12 6( 3)x x+ = + 3( 3) 3 9x x+ = +

2 8 6 4x x+ = +

True when x = 1 4 15 2(2 5)x x+ = + 5 14 5( 2) 4x x+ = + +

7 12x+ =

True when x = 5 5 10 3 8 12x x x+ + = + 3 5 7 2(5 2) 1x x x+ + = + +

2 6x =

True when x = 3 8 3 4(2 1)x x+ = + 7 14 7( 2)x x+ = +

6. Write your own equations – one for every category. Then check your answers with your partner.

Answers will vary.


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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)


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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


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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

√𝒄𝒄 + 𝟏𝟏 = √𝒄𝒄 + 𝟏𝟏, if we let 𝒄𝒄 = 𝟑𝟑.

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 = 72 + 92

False

D. 𝜋𝜋 = 3.141

False

E. �(4 + 9) = √4 + √9

False


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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. 10𝜋𝜋4 + 3 = 99𝜋𝜋2

C. 𝜋𝜋 + 𝜋𝜋 = 2𝜋𝜋

D. 12 + 1

2 = 24

E. 79𝜋𝜋3 + 70𝜋𝜋2 − 56𝜋𝜋 + 87 = 60𝜋𝜋 + 29 928𝜋𝜋2

3. In the following equations, let 𝑥𝑥 = −3 and 𝑦𝑦 = 23. Determine whether the following equations are true,

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. 𝑦𝑦𝑥𝑥 = −2

False F.

−2𝑥𝑥𝑦𝑦 = −1

False

4. Name a value of the variable that would make each equation a true number sentence. For example,

Let 𝑤𝑤 = −2. Then, 𝑤𝑤2 = 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 = 372 is true.

C. 𝑚𝑚3 = −125 is true for m = -5 .

D. A number 𝑥𝑥 and its square, 𝑥𝑥2, have the same value when x = 1 or x = 0 .


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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+𝑥𝑥1+𝑥𝑥2 = 3

5 for x = 2

D. 1𝑥𝑥 = 𝑥𝑥

1 if x = 1 .

E. 𝑥𝑥 + 2 = 9 x = 7

F. 𝑥𝑥 + 22 = −9 x = -13

G. −12𝑡𝑡 = 12 t = -1

H. 12𝑡𝑡 = 24 t = 2

I. 1

𝑏𝑏−2 = 14

b = 6

J. 1

2𝑏𝑏−2 = −14

b = -1

K. √𝑥𝑥 + √5 = √𝑥𝑥 + 5 x = 0

L. (𝑥𝑥 − 3)2 = 𝑥𝑥2 + (−3)2 x = 0

M. 𝑥𝑥2 = −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.


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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)

B. An equation that is true when 𝑥𝑥 = 0 x + 2 = 2

C. An equation that is never true x + 3 = x + 2

D. An equation that is true when 𝑡𝑡 = 1 or 𝑡𝑡 = −1 t2 = 1

E. An equation that is true when 𝑦𝑦 = −0.5 2y + 1 = 0

F. An equation that is true when 𝑧𝑧 = 𝜋𝜋 zπ = 1


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Algebraic Equation Sorting Cards

5 8 10 4x x+ = + 2 8 6 4x x+ = + 3( 3) 3 9x x+ = +

6 12 6( 3)x x+ = + 5 14 5( 2) 4x x+ = + + 7 12x+ =

4 15 2(2 5)x x+ = + 3 5 7 2(5 2) 1x x x+ + = + + 5 10 3 8 12x x x+ + = +

7 14 7( 2)x x+ = + 8 3 4(2 1)x x+ = + 2 6x =


5 8 10 4x x+ = + 2 8 6 4x x+ = + 3( 3) 3 9x x+ = +

6 12 6( 3)x x+ = + 5 14 5( 2) 4x x+ = + + 7 12x+ =

4 15 2(2 5)x x+ = + 3 5 7 2(5 2) 1x x x+ + = + + 5 10 3 8 12x x x+ + = +

7 14 7( 2)x x+ = + 8 3 4(2 1)x x+ = + 2 6x =


5 8 10 4x x+ = + 2 8 6 4x x+ = + 3( 3) 3 9x x+ = +

6 12 6( 3)x x+ = + 5 14 5( 2) 4x x+ = + + 7 12x+ =

4 15 2(2 5)x x+ = + 3 5 7 2(5 2) 1x x x+ + = + + 5 10 3 8 12x x x+ + = +

7 14 7( 2)x x+ = + 8 3 4(2 1)x x+ = + 2 6x =


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teacher road map for lesson 10: true and false equations · · 2016-11-07teacher road map for...

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