Patterns and 1 Equations - Houston Independent School ... number systems and their properties ......

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1

Unit

1

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Patterns and Equations

Essential Questions

How are patterns, equations, and graphs related?

Why are the properties of real numbers important when solving equations?

Unit OverviewInvestigating number systems and their properties will help you establish a good foundation for learning Algebra 1. Th rough the activities in the unit, you will analyze, describe, and generalize patterns using tables, rules, equations, and graphs. You will also write and solve equations and inequalities to solve real-world problems.

Academic VocabularyAdd these words and others you encounter in this unit to your Math Notebook.

expression variable distributive property equation

solution compound inequality absolute value equation absolute value inequality

These assessments, following Activities 1.3, 1.5, and 1.7, will give you an opportunity to demonstrate what you have learned about patterns, properties, equations, and inequalities.

Embedded Assessment 1

Multiple Representationsof Data p. 25


Properties and SolvingEquations p. 39


Inequalities and Absolute Value p. 57

EMBEDDED ASSESSMENTS

1-2_SB_A1_1-0_SE.indd 11-2_SB_A1_1-0_SE.indd 1 1/7/10 11:57:26 AM1/7/10 11:57:26 AM

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2 SpringBoard® Mathematics with Meaning™ Algebra 1

Write your answers on notebook paper. Show your work.

1. 2 __ 3 + 4 __ 5

2. What condition must be met before you can add or subtract fractions?

3. Jennifer is checking Megan’s homework. Th ey disagree on the answer to 2 2 __ 3 · 2 2 __ 5 . Jennifer says the product is 4 4 ___ 15

and Megan says it is 6 2 __ 5 . Who has the correct answer? Explain how she arrived at

that correct product.

4. A piece of lumber 2 1 __ 4 feet long is to be cut into 3 equal pieces. How long will each

piece of cut wood be? Give the measure-ment in feet and in inches.

5. Arrange the following expressions in order from least to greatest.a. 4 - 6 b. -4 + 6 c. -4 - 6

6. Which is greater? Justify your answer. a. -8 + 3 b. -8 × 3

7. Which of the following are equal to 14.95?a. 2.3 × 6.5 b. 21.45 - 6.5c. 8.32 + 6.63

8. Th e Venn diagram below provides a visual representation of the students in Mr. Griffi n’s class who participate in music programs aft er school. What does the diagram tell you about the musical involvement of Student B and Student G? Explain how you reached your conclusion.

Students whoplay piano

Students inthe band

B

D

A

F

C

E

G

UNIT 1

Getting Ready


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Unit 1 • Patterns and Equations 3

My Notes

ACTIVITY

1.1Investigating PatternsFor Love of the GameSUGGESTED LEARNING STRATEGIES: Close Reading, Shared Reading, Predict and Confirm

Level 9

Stu

dio

sTh e SpringBoard Informer is pleased to bring you an exclusive sneak peek at Level 9 Studios’ upcoming multiplayer game, Modus Operandi. With the game set for a December release, Level 9’s lead designer, Floyd Castle, sat down with us for a one-on-one interview.

SBI: Before we get into the game, tell us a little about the company itself. Is there any signifi cance to the name?

Castle: Th ere is, actually. I played a ton of video games while I was growing up. I’d spend hours and hours trying to fi gure out how to beat certain levels. Th at all changed when I took Algebra.

SBI: Algebra?

Castle: At fi rst, Algebra didn’t interest me that much because it just seemed like a bunch of steps strung together that I was supposed to memorize. I preferred video games because I felt challenged to seek out a pattern and felt confi dent in knowing that there was oft en more than one way to complete a level. Th en I made the connection between my approach to a particular level in a video game and the process of completing an Algebra problem, and the rest is history. Level 9 Studios was born.

SBI: You started a video game design company in your Algebra class?

Castle: [laughs] Well, not really, but I did design the logo in study hall one day. Th e logo represents a symbol of that connection I made, and it has become the basis for the design principles of our games.

CONNECT TO LANGUAGE ARTSLANGUAGE ARTS

Modus Operandi is a Latin expression meaning method of procedure. The expression is often used in abbreviated form (MO) to describe a person’s habits or manner of working.


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My Notes

Investigating Patterns ACTIVITY 1.1continued For Love of the GameFor Love of the Game

SUGGESTED LEARNING STRATEGIES: Create Representations, Look for a Pattern, Think/Pair/Share, Group Discussion, Group Presentation

Level1 2 3 4 5 6 7 8 9

1. Investigate the patterns in the logo by completing the table below:

Level # of Squares Added Perimeter Total Area

1 1 4 123456789

2. Describe any patterns you notice in the columns and rows from the table in Item 1.


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My Notes

ACTIVITY 1.1continued

Investigating Patterns For Love of the GameFor Love of the Game

SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Activating Prior Knowledge, Create Representations

3. Use the patterns you described in Item 2 to predict the values for a tenth level in the design.

SBI: OK, I see that the logo has plenty of diff erent patterns in it, but how does that translate to game design?

Castle: Take the Perimeter column, for instance. In Modus Operandi, a character can regenerate health points at each level. Th e number of health points regenerated at each level follows the same pattern as the numbers in the Perimeter column. One problem we had was that characters in our game can reach up to level 50. Rather than writing a program that does repeated calculations, we decided to come up with an expression for fi nding the number of health points regenerated at any level.

4. How does the perimeter of each level compare to the number of the level itself?

5. Using the variable L to represent the level number of a character in the game, write an expression that could be used to determine the number of health points (perimeter) regenerated at any level in the game.

6. Use the expression you wrote in Item 5 to determine the number of health points regenerated for a character at level 50.

ACADEMIC VOCABULARY

An expression is a mathematical phrase that uses numbers or variables, or both. A variable is a letter or symbol used to represent one or more numbers.


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My Notes


SUGGESTED LEARNING STRATEGIES: Create Representations

7. Th e second column of the table, # of Squares Added, represents the number of gold coins acquired at each level.

a. Explain how the number of squares added at each level is related to the level number.

b. Use the variable L, the level number of a character in the game, to write an expression for fi nding the total number of squares added at each level.

c. Use Levels 3, 6, and 8 to verify that your expression is correct.

8. Use the expression you found in Item 7(b) to determine the number of gold coins acquired at the following levels.

a. Level 15 b. Level 36 c. Level 50

9. Look at the fourth column that shows the Total Area for each level.

a. Explain how the values in the column relate to the level number.

b. Use the variable L to write an expression for fi nding the total area of the logo based on the level number.


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My Notes




SBI: So you can use tables and expressions to show the same information.

Castle: Don’t forget graphs! Tables, expressions, and graphs are among the multiple approaches for dealing with a problem.

10. Graph the data from the table on the appropriate grid.

a.

b.

c.

y

x

# of

Squ

ares

Add

ed

Level Number1 2 3 4 5 6 7 8 9

y

x

# of

Squ

ares

Add

ed

Level Number1 2 3 4 5 6 7 8 9

y

x

Peri

met

er

Level Number1 2 3 4 5 6 7 8 9

y

x

Peri

met

er

Level Number1 2 3 4 5 6 7 8 9

y

x

Tota

l Are

a

Level Number1 2 3 4 5 6 7 8 9

y

x

Tota

l Are

a

Level Number1 2 3 4 5 6 7 8 9


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My Notes


SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Visualization

11. Do any of the graphs you made on the previous page show a linear pattern? Explain.

SBI: You spoke earlier about multiple approaches. Are there other ways the logo represents this?

Castle: Good question. Take another look at the second column in the table, # of Squares Added. Th ese numbers represent the number of gold coins acquired at each level. We already have one expression for fi nding the number of coins at each level. See Item 7(b). You can look at the logo in a diff erent way to fi nd an equivalent expression. Th e following diagram depicts the fi rst four levels of the logo as individual fi gures.

12. Beginning with the second level, what kind of shape appears to be “missing” from each fi gure.

13. How does your answer to Item 12 compare to the values in the Total Area column?

Equivalent expressions have the same value.

The graph of a linear pattern has points that lie along a straight line. The graph of a nonlinear pattern has points that do not lie along a straight line.

MATH TERMS


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My Notes




14. Explain how you could use the values in the Total Area column to generate the number of squares added in a given level.

15. Use the expression you wrote in Item 9(b) for the Total Area column and your answer to Item 13 to generate another expression for the number of squares added in a given level.

16. Use the new expression to fi nd the number of squares added in Level 6 and Level 15. Compare these values to the ones you found using the table in Item 1 and the expression in Item 8.


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My Notes


SUGGESTED LEARNING STRATEGIES: Close Reading, Shared Reading

SBI: Some of your previous titles have received mixed reviews from the gaming community. How are you addressing this in your new game?

Castle: Game design is an interesting fi eld of study. As gamers, we want to be challenged, but we also need to feel as though we’re being rewarded appropriately for our eff orts. From a development standpoint, that’s easier said than done. Our fi rst game, Realm of Empires, had a great story. On the other hand, it was too easy, and players felt they didn’t get their money’s worth. Our last game, Rube Goldberg’s Lab, ended up being too complex and alienated many casual gamers. With Modus Operandi, we think we’ve hit the sweet spot.

SBI: I’m guessing there’s something in the logo that explains that too?

Castle: Of course! Part of achieving the appropriate balance involves two variables, time and rewards. For instance, we know we want it to take a certain amount of time (T) to reach the maximum level or complete the game. We also know that early levels need to be completed or gained quickly, with the later levels becoming more complex and taking longer to complete. To balance the additional time and complexity of each level, the rewards should also increase accordingly.

CONNECT TO HISTORYHISTORY

Rube Goldberg (1883–1970) was an American cartoonist. He is most famous for his drawings of very complex-looking machines that performed very simple tasks. There are Rube Goldbergcontests that challenge high school students to build their own complex machines for performing simple tasks.


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My Notes



SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Think/Pair/Share

17. Consider the following graphs:

Graph A Graph B

a. Which graph refl ects the intended design model for earning levels in the game? Explain your reasoning.

b. When graphed, which columns from the table in Item 1 would produce the graphs that are similar in shape to the ones above? Explain your reasoning.

18. Compare and contrast the process of designing and/or playing a video game to completing a math problem.


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CHECK YOUR UNDERSTANDING

Write your answers on notebook paper or grid paper. Show your work.

Use the diagram and table of values to answer 1 through 5.

1 1 42 4 83 8 124 12 165 16 20

1. Explain any connections you notice between the pattern and the table of values.

2. Describe any patterns you see in the table of values.

3. Based on your investigation of the two representations, how could you label the columns in the table of values? Explain your reasoning.

4. Explain two diff erent ways to determine the next two rows in the table of values.

5. Which relationships between columns in the table of values could be represented by a graph with a shape similar to the one below? Explain your reasoning.

6. MATHEMATICAL R E F L E C T I O N

Explain how creating multiple representation

of the same situation can be helpful in solving problems.


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My Notes

ACTIVITY

1.2Real NumbersGet RealSUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Predict and Confirm, Quickwrite, Think/Pair/Share, Self/Peer Revision, Interactive Word Wall

Your teacher is going to give you a card with six numbers on it.

1. Write the numbers from your card below.

2. Do the numbers you received look like they will be good numbers? Why or why not?

Th e numbers on the cards belong to sets of numbers you have studied: natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

3. List some examples and non-examples below for the set of natu-ral numbers.

Examples of natural numbers Non-examples of natural numbers

4. Aft er reviewing your examples and non-examples above, write your own defi nition for natural numbers.

5. List examples and non-examples of whole numbers below and use your list to write what a whole number is.

CONNECT TO LANGUAGELANGUAGE

Criterion (plural: criteria) is a stan-dard on which a decision is based.


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My Notes

Real NumbersACTIVITY 1.2continued Get RealGet Real

6. What is the relationship between natural numbers and whole numbers?

7. Which Venn diagram below shows the relationship between the sets of natural and whole numbers? Fill in your choice with the names of these sets of numbers and explain why you chose the diagram you did.

8. List examples and non-examples of the set of integers.Consider these lists to write your own defi nition of an integer.

9. What do you notice about the relationship between whole numbers and integers?

A Venn diagram is a graphic organizer used to represent relations between sets and operations with sets.

MATH TERMS

SUGGESTED LEARNING STRATEGIES: Quickwrite, Activating Prior Knowledge, Create Representations, Look for a Pattern, Interactive Word Wall


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My Notes


Real NumbersGet RealGet Real

You have probably noticed that trying to fi t the names of the number sets into a Venn diagram can be diffi cult. To save time and space, mathematicians have given the sets of numbers abbreviations. Th ese abbreviations are:

Natural numbers: NWhole numbers: W

Integers: Z

10. Redraw the Venn diagram you chose in Item 7 below. Add the set of integers to this Venn diagram to show how these number sets are related. Use the abbreviations to represent each set of numbers.

Later, you will perform an activity with the numbers on the card your teacher gave you. Numbers from diff erent sets will have diff erent point values. But fi rst you will perform a practice round of the activity using the numbers listed below. Notice that they are all natural numbers, whole numbers, and/or integers.

17 0 −4 33 −11 −24

11. Place the numbers from the list in the appropriate space in the second column and compute your score.

Scoring Criterion My numbers thatfi t the criterion Score

1 __ 4 point for each natural number 17, 33 1 __ 2

3 points for each odd integer

1 __ 2 point for each whole number less than 20

4 points for each integer greater than -5

2 points for each whole number less than 1

SUGGESTED LEARNING STRATEGIES: Graphic Organizer, Look for a Pattern, Activating Prior Knowledge

WRITING MATH

It is obvious why N and W repre-sent natural and whole numbers, respectively. The abbreviation Z for integers is not quite as obvi-ous. Z represents the integers because the term “Zahlen” means “number” in German.


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My Notes


SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Predict and Confirm, Quickwrite, Activating Prior Knowledge, Interactive Word Wall

12. Choose six numbers that will give you a higher score than the score you got on the previous page.

Your numbers:

Scoring Criterion My numbers thatfi t the criterion Score

1 __ 4 point for each natural number

3 points for each odd integer

1 __ 2 point for each whole number less than 20

4 points for each integer greater than -5

2 points for each whole number less than 1

13. By how much did your score increase?

14. How did understanding the number sets and subsets help you to make good choices when picking your numbers?

Th e abbreviation for the set of rational numbers is Q, which stands for quotient.

15. List examples and non-examples of the set of rationalnumbers.

16. Why is the term ratio a good word to use when describing rational numbers?

A set A is a subset of a set B if every element of A is an element of B. Every set is a subset of itself.

MATH TERMS


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My Notes

ACTIVITY 1.2continuedGet RealGet Real

SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Self/Peer Revision, Graphic Organizer, Look for a Pattern, Activating Prior Knowledge, Interactive Word Wall

Real Numbers

17. Considering this abbreviation and your examples and non-examples, write your own defi nition for a rational number.

18. In the space below, redraw your Venn diagram from Item 10.

a. Add to the diagram to show how rational numbers are related to natural numbers, whole numbers, and integers.

b. Why did you choose to place the rational numbers in the diagram where you did?

19. Another set of numbers you have studied is the set of irrational numbers, which are numbers that cannot be writ-ten as the ratio of two integers. Irrational numbers include π, the square root of any nonnegative number that is not a perfect square, and decimals that do not repeat or terminate.

a. Give an example of a square root that is irrational and a decimal that is irrational.

b. Why do you think these numbers are considered to beirrational?

c. Add the irrational numbers to your Venn diagram in Item 18. Use Ir as an abbreviation for irrational numbers.


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My Notes


SUGGESTED LEARNING STRATEGIES: Graphic Organizer, Look for a Pattern,

All of the numbers discussed in this activity are examples of real numbers, which are all rational numbers and all irrational numbers combined.

20. Add the set of real numbers to your Venn diagram in Item 18.

21. Use the numbers on the card your teacher gave you. Complete the table with the numbers from your card and compute your score.

Criterion My numbers thatfi t the criterion Score

2 points for each rational number that is not an integer

4 points for each real number

6 points for each even whole number

3 points for each negative rational numbers

4 points for each positive irrational number

WRITING MATH

The set of real numbers is abbreviated by the letter R.


Write your answers on notebook paper.

Write one example for each criterion. If not possible, write “not possible.”

1. A rational number that is not a whole number.

2. An integer that is not a natural number. 3. A natural number that is not a whole

number. 4. An irrational number.

5. A whole number that is not a natural number.

6. A real number that is not a rational number.

7. A rational number that is not an integer. 8. An integer that is a real number. 9. MATHEMATICAL

R E F L E C T I O N Describe situations that you encounter daily for

at least four of the sets of numbers in this activity.


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My Notes

ACTIVITY

1.3Properties of Real NumbersProperties by the PoundSUGGESTED LEARNING STRATEGIES: Create Representations, Activating Prior Knowledge, Think/Pair/Share, Interactive Word Wall

Th e local girls’ track team is strength training by lift ing weights. One of the runners puts weights on each end of the weightlift ing bar while she is getting ready to bench press.

1. On one end she places a ten-pound plate and then a fi ve-pound plate. On the other end, she places fi rst a ten-pound plate and then a fi ve-pound plate. Does the bar have the same amount of weight at each end? Explain using a diagram in the My Notes section.

In symbols, this situation described above is written as a + b = b + a, where both a and b are real numbers. Th is is a statement of the Commutative Property.

2. What does the commutative property state?

3. Demonstrate the commutative property of addition by writing an example using:

a. negative integers

b. rational numbers represented as fractions

Another runner needs to increase the amount of weight on the bar for another lift . On each side she places three diff erent plates: one fi ve pound plate, one ten pound plate, and one twenty pound plate as shown in the diagram below.

20 2010 10

5 5

To ensure that the weight on both sides is the same, she wants to fi nd the total weight on both sides. For the left side of the bar, she used the expression (5 + 10) + 20 and on the right side of the bar, she used the expression 5 + (10 + 20).

4. Use the order of operations to evaluate each expression and compare the results.

Order of Operations:

1st: Evaluate expressions within grouping symbols.

2nd: Evaluate powers and radicals.

3rd: Do all multiplication and division from left to right.

4th: Do all addition and subtraction from left to right.


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My Notes

Properties of Real Numbers ACTIVITY 1.3continued Properties by the Pound Properties by the Pound

Th e expression (5 + 10) + 20 = 5 + (10 + 20) can be written in symbols as (a + b) + c = a + (b + c), where a, b, and c are real numbers. Th is is an example of the associative property.

5. What does the associative property state?

6. Demonstrate this property of addition by writing an example using:

a. negative integers

b. rational numbers represented as decimals

7. Th e properties you have been using were described as properties of addition. Determine if these properties also apply for the operations that follow. If the property applies, then write an example that demonstrates it. If not, write a counterexample.

Commutative Property Associative PropertySubtraction

Multiplication

Division

Th e identity properties apply to both addition and multiplication.

8. Use what you know about identity properties to complete these statements. Th en state the identity properties in words.

a. a + = a, and + a = a.

Identity Property of Addition:

b. x · = x, and · x = x.

Identity Property of Multiplication:

c. Explain whether the identity properties apply to division and subtraction. Include an example or counterexample with each explanation.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representations, Interactive Word Wall, Group Presentation

A counterexample is an example or case that proves a conjecture or theory wrong.

MATH TERMS


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My Notes


Properties of Real Numbers Properties by the Pound Properties by the Pound

Each real number has an associated number called its additive inverse. All but one real number has an associated number called its multiplicative inverse or reciprocal. Th e Additive Inverse Property states that a number added to its additive inverse gives a sum of zero. In symbols, the additive inverse of a is –a: a + (–a) = 0, and –a + a = 0

9. Write an equality showing a number and its additive inverse.

Th e Multiplicative Inverse Property states that a number multiplied by its multiplicative inverse yields a product of one. In symbols, themultiplicative inverse of a is 1 __ a : a · ( 1 __ a ) = 1 and 1 __ a · a = 1

10. Write an equality showing a number and its multiplicative inverse.

11. Which real number does not have a multiplicative inverse? Explain.

TRY THESE A

Complete the following number sentences by fi nding each:

a. Additive inverse

5 + = 0 -b + = 0 4 __ 5 + = 0

b. Multiplicative inverse

6 · = 1 2 __ t · = 1 0.23 · = 1

12. Th e distributive property can be written in symbols as a(b + c) = a(b) + a(c). Th e factor a is said to be “distributed” to

both addends in the parentheses.

a. Describe the order of operations used on the left side of the equation a(b + c) = a(b) + a(c) in terms of a, b, and c.

b. Describe the order of operations used on the right side of the equation a(b + c) = a(b) + a(c) in terms of a, b, and c.

c. What is true about the two sides of the equation?

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation, Activating Prior Knowledge, Quickwrite, Think/Pair/Share

ACADEMIC VOCABULARY

distributive property

The distributive property can also be written as (b + c)a = ba + ca.


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My NotesMy Notes


TRY THESE B

Use the distributive property to rewrite and evaluate each of the following expressions.

a. 5 (-7 + 12) b. 1 __ 2 ( 7 __ 3 + 5 __ 3 ) c. -3(-24 + 16)

d. (2 + 9) 5 e. 4 (x + 2y)

13. Does the distributive property apply to operations other than multiplication and addition? Write an example or a counterexample for each of the following.

a. Does multiplication “distribute” over subtraction?

b. Does division “distribute” over addition?

A set has closure under an operation if, when an operation is performed on members of that set, the result also is included in the set.

14. Th e set of whole numbers is closed under addition because the sum of any two whole numbers is also a whole number. Write an equality that demonstrates the closure of the whole numbers under addition.

15. Not all number sets are closed under an operation. Write an equality that demonstrates that natural numbers are not closed under subtraction.

16. Determine for which operations the sets of numbers listed are closed. Use whole numbers in the fi rst table and integers in the second table.

Operation Example or Counterexample

Closed or Not Closed Operation Example or

CounterexampleClosed or

Not ClosedAddition 1 + 4 = 5 Closed Addition

Subtraction SubtractionMultiplication Multiplication

Division Division

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Activating Prior Knowledge, Group Presentation


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My Notes


Properties of Real Numbers Properties by the Pound Properties by the Pound

17. Th ere are properties of equality that can make it easier to solve equations and to work with proofs in geometry. Th e weights that the track team is lift ing can be used to illustrate these.

a. Th e refl exive property states that a = a.20 20=

Tell what the refl exive property means in your own words using at least one example.

b. Th e symmetric property states if a = b, then b = a.

Tell what the symmetric property means in your own words using at least one example.

c. Th e transitive property states if a = b and b = c, then a = c.

What does the transitive property mean? Explain in your own words using at least one example.

18. If 3x + 2 = y and y = 8 what property allows you to conclude that 3x + 2 = 8?

20 2010 10 10 10

20 2010 10 10 10

205 5 5 5 5 5 5 51010 1010 20205 5 5 5 5 5 5 51010 1010 20

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Presentation

CONNECT TO FITNESSFITNESS

The illustrations of different kinds of plates on a barbell in this activity are meant to demonstrate the properties of equality. If you were actually placing plates on a barbell at a gym, however, you would use exactly the same plates, placed in the same order from each end.


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My Notes


19. If 10 = y, which property allows you to conclude that y = 10?

20. Do each of these properties apply to inequalities? Use a numerical example or counterexample to explain.

a. refl exive

b. symmetric

c. transitive:




1. Identify the property illustrated in each of the following equations.a. 5(2 + 3) = (2 + 3)5b. 25 · (0.04) = 1c. 4 + (9 + 3) = (9 + 3) + 4d. (x + 3)4 = 4x + 12

2. Write an equality that illustrates the associative property of multiplication.

3. Write an equality that illustrates the additive inverse of 3.

4. Th e expressions below are the result of distributing multiplication over addition. Rewrite each to show the expression prior to distributing.a. 3(x) + 3(8)b. −10y + xyc. 6w + 9z

5. Using the table below, determine for which operations the set of natural numbers is closed or not closed.

Operation Example or Counterexample

Closed or Not Closed

AdditionSubtraction

MultiplicationDivision

Identify the property illustrated in each of the following equations.

6. y = y 7. 10 · 5 = 5 · 10 8. If y = 11 and y = 3x + 2, then 11 = 3x + 2. 9. If 7 + x = 12, then 12 = 7 + x 10. MATHEMATICAL

R E F L E C T I O N Which of the properties of operations of real

numbers is most useful to you? Explain using more than one example.


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Embedded Assessment 1 Use after Activity 1.3.

Multiple Representations of DataSHOOTING STARS

Th e Gemenids meteor shower occurs each December. Paul and Linda are interested in astronomy, so they began to count meteors aft er sunset and recorded their observations. Paul and Linda wondered how many meteors they would have seen if they continued counting.

Hour After Sunset Number of Meteors Observed During the Hour

1 9

2 21

3 33

1. Use Paul and Linda’s data. Predict the number of meteors that they would have seen in the fourth hour, fi ft h hour, and sixth hour. Explain your predictions.

2. Write an expression that models their data.

3. Professional astronomers at a local observatory were counting meteors at the same time. Th ey collected data that were more extensive. In the fi rst hour aft er sunset, they recorded 22 meteors. In the second hour, they recorded 40 meteors; in the third hour, they recorded 54; in the fourth hour, they recorded 64. As the night continued in the fi ft h hour, they recorded 70; in the sixth hour, they recorded 72; in the seventh hour, they recorded 70. Finally in the eighth hour, they recorded 64 meteors. Th is graph shows the data.

5 10

1020304050607080

Num

ber o

f Met

eors

Hour After Sunset

y

x

Using the astronomers’ data, predict the number of meteors observed in the tenth hour.

4. To compare and contrast two sets of data, it is helpful to have the data represented in the same way. For example, represent both sets as graphs, both as tables, both as expressions, and so on.

a. An expression for the astronomers’ data is -2x2 + 24x. Compare this with the expression you wrote for Paul and Linda’s data. What do the expressions tell you about the data?

b. Choose another representation for the two sets of data. Compare and contrast those representations.

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Multiple Representations of DataSHOOTING STARS

Exemplary Profi cient Emerging

Math Knowledge# 5, 6

Student provides correct mathematical justifi cation for use of natural numbers (5) and the correctness or incorrectness of expression (6).

Student provides both justifi cations but only one is complete and correct.

Student provides at least one justifi cation, but it is not complete and correct.

Problem Solving# 1, 3

Student gives three correct predictions from Paul and Linda’s data (1) and one correct prediction from the astronomer’s data (3)

Student gives three correct predictions out of the four.

Student gives at least two predictions.

Representations# 2, 4b

Student gives a correct expression to model the data (2) and provides an additional correct representation for comparison (4b).

Student provides both representations, but only one is completely correct.

Student provides at least one of the two representations, but it is not completely correct.

Communication# 1, 4a

Student gives a reasonable explanation of predictions (1) and gives plausible conclusions about what the expressions indicate about the data (4a).

Student gives explanations for both, but only one of the explanations is complete without any errors.

Student gives at least one of the two explanations, but it is not correct and complete.

5. Why are only natural numbers used for recording the two sets of data?

6. Would it be correct to state that an expression for the astronomers’ data is 24x - 2x 2? Why or why not?

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ACTIVITY

1.4Solving Equations with ModelsWhat’s My Number?SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Guess and Check, Look for a Pattern, Quickwrite, Think/Pair/Share

Let’s play “What’s My Number?”

1. Determine my number, and explain how you came up with the solution.



4. How did fi nding the number in Item 3 compare to fi nding the numbers in Items 1 and 2?

5. Determine my number and explain how you came up with the solution.

If you multiply 4 times the sum of 3 and me, and subtract 8 from that value, I am 12! What’s My Number?


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Solving Equations with Models What’s My Number?What’s My Number?


6. How did fi nding the number in Item 5 compare to fi nding the numbers you found in Items 1 through 3?

7. Write each of the “What’s My Number?” problems as an equation.

You have probably seen diff erent methods to solve equations. Diff erent methods are useful in diff erent situations.

The Undoing MethodTh e undoing method “undoes” or reverses the order of operations in an equation. To solve an equation using the undoing method:• Create a fl owchart to show what happens to the variable. Follow

the order of operations. • On the line below, work backward by doing the inverse

operations.

EXAMPLE 1

Solve 5(x + 30) - 18 = 17 using the undoing method. Check your solution.

Step 1: Create a fl ow chart to show what happens to the variable.

x x + 30 5(x + 30) 5(x + 30) – 18

17

=

+ 30 × 5 – 18

SUGGESTED LEARNING STRATEGIES: Create Representations, Think/Pair/Share, Group Presentation

ACADEMIC VOCABULARY

An equation is a mathematical statement that shows that two expressions are equal. A solution is any value that makes an equation true when substituted for the variable.


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SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Group Discussion, Group Presentation, Work Backward

Step 2: Work backward, showing the inverse operations.

x x + 30

7 35

5(x + 30) 5(x + 30) – 18

17

=

+ 30

+ 18

× 5 – 18

– 23– 30 ÷ 5

Step 3: Check your work by substituting -23 in the original equation. 5(x + 30) - 18 = 175(-23 + 30) - 18 = 17 5(7) - 18 = 17 35 - 18 = 17 17 = 17 Th is is a true statement, so the solution checks.

Solution: x = -23

TRY THESE A

a. Solve the “What’s My Number?” problem from Item 5 using the undoing method. Check your solution.

Solve each equation using the undoing method.

b. 20 = 9k + 2 c. 0.25x - 6 = 2

d. ( 1 __ 2 m + 3 ) - 5 = 2 e. 27 = 9 __ 2 (t - 3)

f. 5(x- 7) + 2 __________ 8 = 4 g. 12x ____ 5 + 18 = 36

8. Try to solve this “What’s My Number?” problem using the undoing method. What problems do you run into?

If you multiply me by 7, subtract 9, and add twice me to that value, I am 90! What’s My Number?


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My Notes

Solving Equations with Models ACTIVITY 1.4continued What’s My Number?What’s My Number?

SUGGESTED LEARNING STRATEGIES: Create Representations, Group Presentation

Algebra-Tiles MethodTh e undoing method can be eff ective to solve equations that have only one variable term. When equations become more complex, you may want to use another method, such as using algebra tiles, to solve.

Th e tile for +1 is +1 , and the tile for -1 is −1 .Th e tile for x is x .Th e tile for -x is −x .

Th e pairs (1 and -1; x and -x) are the additive inverses of each other. Th ey are called zero pairs since adding them together yields 0.

EXAMPLE 2

Solve 2x - 5 = 3 using algebra tiles.

Step 1: Represent the equation using a model.

=x−1

−1 −1 +1 +1

+1−1 −1

x

Step 2: Add fi ve +1 tiles to each side of the equation and use zero pairs to isolate the x-tiles.

=

x

x−1

+1

−1 −1 +1 +1

+1+1

+1

+1 +1

+1−1 −1

+1 +1

+1 +1

Step 3: Group the tiles in rows equal to the number of x-tiles

=

x

x

+1 +1 +1 +1

+1 +1 +1 +1

From the groupings, you can see that each x-tile is equal to 4.

Solution: x = 4

TRY THESE B

a. Solve this problem using algebra tiles: If you multiply 2 times the sum of 2 and me, I am 10! What’s My Number?

You can represent the distributive property with algebra tiles.

2(x + 2) would be shown as 2 groups of tiles, each group containing x + 2 tiles.

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My Notes



SUGGESTED LEARNING STRATEGIES: Create Representations, Group Discussion

TRY THESE B (continued)

Use algebra tiles to solve each equation. You can use the My Notes section to draw pictures representing the problems.

b. 4 + 5t = 11 + 4t c. 14 + 3n = 2n d. 2(y + 1) – 3 = –7

e. 4x + 2 = –22 f. –5 = 9 + y g. 2y + 3 + y = 12

9. Solve these two “What’s My Number?” problems. Which of the methods you have seen so far did you use? Why? What are some of the drawbacks of the two methods?

a. If you multiply me by 25 and subtract 37 from that value, I am 113!

b. If you multiply me by 1 __ 2 and subtract 3 __ 4 from that value, I am 3 1 __ 4 !

Balancing MethodTh e balancing method is useful when the undoing method or algebra tiles are too cumbersome. Th is method is also called the symbolic method or solving equations using symbols. Keep the ideas from the other two methods in mind as you use this new method.

EXAMPLE 3

Solve the equation 3x + 90 + 2x = 360 using the balancing method. Check your solution.

Step 1: Apply the commutative 3x + 90 + 2x = 360 property. 3x + 2x + 90 = 360Step 2: Combine like terms. 5x + 90 = 360Step 3: Apply the subtraction 5x + 90 − 90 = 360 − 90 property of equality. 5x = 270Step 4: Apply the division property 5x ___ 5 = 270 ____ 5 of equality. x = 54Step 5: Check by substitution. 3(54) + 90 + 2(54) = 360 162 + 90 + 108 = 360 360 = 360Solution: x = 54.

The Properties of Equality state that you can perform the same operation to both sides of an equation without affecting the solution.

Addition PropertyIf a = b, then a + c = b + c.

Subtraction PropertyIf a = b, thena − c = b - c.

Multiplication PropertyIf a = b, thena · c = b · c.

Division PropertyIf a = b, thena ÷ c = b ÷ c, when c ≠ 0.

Like terms have the same variable(s) raised to the same power(s).

MATH TERMS


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My Notes

Solving Equations with Models ACTIVITY 1.4continued What’s My Number?What’s My Number?

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Activating Prior Knowledge, Group Discussion

TRY THESE C

a. Solve the “What’s My Number?” from Item 5 using the balancing method.

Solve each equation using the balancing method.

b. −2x − 6 − 3x = 1 c. 4(x − 2) − 5x = 10

d. 12d + 2 − 3d = 5 e. 8 = 3 ( 1 __ 2 w − 4 ) + 8

f. 20x − (3 − 5x) = 22 g. 7.4p − 5.3 − 9.2p = −2.6

10. How is the balancing method similar to the undoing method and the algebra tiles method? How is it diff erent?

11. You have learned to solve equations using guess and check, algebra tiles, the undoing method, and balancing method. Solve the following equations using any method you have learned.

a. 2x − 1 + 6x = 87 b. (x − 4)8 = −16 c. 1 __ 3 x + 7 = 27



1. Solve this equation using guess and check.67 = 13 + x

2. Solve this equation using algebra tiles.6 = 14 − 2x

3. Solve this equation using undoing.2(x − 3) + 7 = 23

4. Solve this using the balancing method.−1 = 9 − 2 __ 3 x

5. Use the distributive property to rewrite the following.a. 5(x − 3) b. −2(x − 5) c. (7 − x)8

6. Solve this equation using the any method. Explain why you chose the method you did.

3x + 7 − 9x = 2(8 − 5)


Which method of solving equations are you most

confi dent in using? Which method do you feel you still have questions about?

You can solve equations using a graphing calculator.

TECHNOLOGY


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ACTIVITY

1.5Solving Multi-Step EquationsField Trip FundraiserSUGGESTED LEARNING STRATEGIES: KWL, Question the Text, Create Representations, Quickwrite

Th e Future Engineers of America Club (FEA) wants to raise money for a fi eld trip to the science museum. Th ey will make and sell custom photo buttons and will sell them for two dollars. Th ey have found two companies to make the buttons, but the production costs are diff erent.

Picture Buttons will charge $125 for set-up and $0.15 per button. Buttons for You will charge $75 for set-up and $0.40 per button.

Cost of x buttons produced by Picture Buttons125 + 0.15x

Cost of x buttons produced by Buttons for You75 + 0.40x

To help decide which company to use, club members want to determine how many buttons they would have to sell for the production costs to be the same.

1. Write an equation that makes the production costs of the two companies equal.

2. Th e numbers in the equation from Item 1 make it diffi cult to solve by modeling or mental math. Solve the equation showing each step. Provide an explanation for each step.

Equations Explanationoriginal equationMultiply each side by 100.

3. What is the meaning of the value you got for x in the equation above?

The Multiplication Property of Equality states that an equation is still true after both sides are multiplied by the same number.

For example:

0.2 = 0.2 0.2 · 100 � 0.2 · 100

Both sides are still equal.


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My Notes

Solving Multi-Step EquationsACTIVITY 1.5continued Field Trip FundraiserField Trip Fundraiser

4. Th e FEA club estimates they will sell more than 200 buttons. Make a recommendation to the club explaining which company would be the better choice.

5. Each button will sell for $2.00. Th e revenue for selling x buttons at $2.00 each is 2x. Write and solve an equation to fi nd the break even point for the button fundraiser using the company you recommended to the FEA club.

6. How much profi t will the FEA club earn on sales of 250 buttons if they use the company you recommended?

7. Th e Future Engineers of America Club treasurer was going back through the fundraising records. On Monday, the club made revenue of $90 selling buttons at $2.00 each. One person sold 20 buttons, but the other person selling that day forgot to write down how many she sold. How many buttons did the other person sell?

CONNECT TO BUSINESSBUSINESS

Revenue is the amount of money made selling a product. Profi t is earnings after costs are subtracted from the revenue.

Profi t = Revenue - Cost

The break even point occurs when revenue equals cost.

Revenue = Cost

SUGGESTED LEARNING STRATEGIES: Create Representations, Quickwrite

The formula for fi nding profi t is P = R - C, where P represents the profi t, R represents the revenue, and C represents the cost.

This formula is an example of a literal equation. You can solve the literal equation for R to get a formula for fi nding revenue:

P = R - CP + C = R - C + CP + C = RR = P + C


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Solving Multi-Step EquationsField Trip FundraiserField Trip Fundraiser

8. On the day of the FEA fi eld trip, two other school clubs, the Environmental Club and the Drama Club, are also planning fi eld trips. Each group had to report the number of people going on their fi eld trip so the cafeteria would know how many meals to make that day. Th e cafeteria manager knows 105 students will be absent for fi eld trips. Th e principal knows that there are twice as many students in the Drama Club as in the FEA club and 10 more students in the Environmental Club than in the Drama Club. Write and solve an equation to fi nd out how many students will be on the FEA fi eld trip.

9. At the museum, the FEA students want to see the show at the planetarium. Tickets cost $11 for each student who is a member of the museum’s frequent visitor program and $13 for each student who is not a member. If all of the students (use your answer from the previous problem) see the show, it will cost $209. Write and solve an equation to fi nd out how many of the FEA club members are also members of the museum’s frequent visitor program.

10. Betsy is a member of the FEA club. She read a book for the entire time that the bus was on the highway on the way to and from the museum. She has a deal with her mother that if she reads for 4 hours every week, then she can use the Internet for that same amount of time on the weekend. On the way to the museum there was fog, so the bus driver had to reduce his speed on the highway to 45 miles per hour, but the fog had cleared by the time they drove back so he was able to go 55 miles per hour on the way back. Th e fog caused the highway portion of trip to the museum to take 12 minutes longer on the way there than it took on the way back. If the highway route was exactly the same on both parts of the trip and they did not hit any traffi c, for how long can Betsy tell her mother that she was reading?

SUGGESTED LEARNING STRATEGIES: Close Reading, Shared Reading, Questioning the Text, Marking the Text, Create Representations, Group Presentation

If the total number of people in a room is 25, and the number of males in the room is x, the number of females in the room can be expressed in terms of x with the expression 25 - x.


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My NotesMy Notes

Solving Multi-Step EquationsACTIVITY 1.5continued Field Trip FundraiserField Trip Fundraiser

Some equations require multiple steps to solve them effi ciently.

EXAMPLE

Solve the following equation using symbols.

3x - 2(x + 3) = 5 - 2x

Th e Explanation column shows the steps.

Equations Explanation3x - 2 (x + 3) = 5 - 2x Original equation3x - 2x - 6 = 5 - 2x Distributive property.

x - 6 = 5 - 2x Combine like terms.x + 2x - 6 = 5 - 2x + 2x Add 2x to both sides.

3x - 6 = 5 Combine like terms.3x - 6 + 6 = 5 + 6 Add 6 to both sides.

3x = 11 Combine like terms.

3x ___ 3 = 11 ___ 3 Divide both sides by 3.

x = 3 2 __ 3

Solution: x = 3 2 __ 3

TRY THESE

Solve the following equations using symbols. Check your answers using a diff erent method that you have learned.

a. -(5x + 3) - 4x = 8 + 6x

b. 4(x - 3) = 2(x + 2)

c. 6x - 2(x + 3) = 5x - 1(x + 2)

SUGGESTED LEARNING STRATEGIES: Note Taking, Think/Pair/Share

Solving equations using symbols is the same as using the balancing method.

Equations can have no solutions, one solution, or more than one solution.

CONNECT TO APAP

The process of solving multi-step equations is applied to literal equations in calculus and AP Statistics.


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SUGGESTED LEARNING STRATEGIES: Group Presentation

11. Th ere are many diff erent methods to solve any given equation. Solve the following equations and compare your method to the method of other students in the class.

a. 5x + 8 = 3x - 3 b. 2(4y + 3) = 16

c. 2 __ 3 p + 1 __ 5 = 4 __ 5 d. 3 __ 4 a - 1 __ 6 = 2 __ 3 a + 1 __ 4

12. Find solutions to each of the following three equations.

a. 3(2z + 4) = 6(5z + 2)

b. 3(x + 1) + 1 + 2x = 2(2x + 2) + x

c. 8b + 3 - 10b = -2(b - 2) + 3


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SUGGESTED LEARNING STRATEGIES: Predict and Confirm, Look for a Pattern, Think/Pair/Share, Graphic Organizer

You should have had some surprising results to the equations in Item 12. Some equations cannot be solved (they will never be true) while others have infi nitely many solutions (they are true for all numbers).

13. Which of the following equations will have no solutions and which will be true for all real numbers? Explain.

a. 3x + 5 = 3x b. 4r - 2 = 4r - 2

14. Create another equation that will have each of the following as its solution:

a. One solution b. No solution

c. Infi nitely many solutions d. A solution of zero

Use notebook paper to write your answers. Show your work.

Solve the following equations using symbols. Check your answer using a diff erent method.

1. 6x + 3 = 5x + 10 2. 6 + 0.10x = 0.15x + 8 3. 5 - 4x = 6 + 2x 4. 9 - 2x = 7x 5. 2(x - 4) + 2x = 4x - 1 6. 1 __ 2 x - 3 = 3 __ 2 x + 4

On-the-Go Phone Company has two monthly plans for their customers. Th e table shows the cost in dollars for x minutes on each plan.

EZ PAY Plan 0.15x40 TO GO Plan 40 + 0.05x

7. Determine the number of minutes that will make the two plans equal.

8. Which plan should you choose if you only want 200 minutes per month? Explain.


What have you learned about solving equations

as a result of this activity?



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Embedded Assessment 2 Properties and Solving Equations

CHECKING GROUP WORK

1. Is 9 a solution of 2x - 12 = 18? Justify your answer mathematically and in words.

2. One group turned in the following solution to an equation but noted that the answer they found did not check. Find the errors in their work and explain what they did wrong. Give the correct solution.

33 = 3 (6 + x) Original equation 33 = 3 (x + 6) Commutative Property 33 = 3x + 6 Distributive Property33 - 6 = 3x + 6 - 6 Subtraction Property of Equality 27 = 3x + 0 Additive Inverse 27 = 3x Identity Element of Addition 3x = 27 Symmetric Property

1 __ 3 · 3x = 1 __ 3 · 27 Multiplication Property of Equality 1 · x = 9 Multiplicative Inverse

x = 9 Identity Element of Multiplication

Check: 33 = 3 (6 + x) 33 ? 3 (6 + 9) 33 ? 3 (15) Since x = 9 does not check, it is not 33 ≠ 45 a solution.

3. Another group felt that their work was correct but was not sure they had provided the correct reason for each step. Look over the work below and explain to the group which reasons to change.

9 - x = (3x + 3) - 14 Original equation 9 - x = 3x + (3 - 14) Distributive Property 9 - x = 3x - 11 Simplify 9 - x + 11 = 3x - 11 + 11 Addition Property of Equality 9 + 11 - x = 3x + 0 Commutative Property 20 - x = 3x Simplify, Identity Element of Addition 20 - x + x = 3x + x Addition Property of Equality 20 + 0 = 4x Identity Element of Addition 20 = 4x Additive Inverse 4x = 20 Symmetric Property 1 __ 4 · 4x = 1 __ 4 · 20 Multiplicative Inverse 1 · x = 5 Multiplication Property of Equality

x = 5 Identity Element of Multiplication

Use after Activity 1.5.

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Properties and Solving EquationsCHECKING GROUP WORK

4. Consider the equation 3x + 6 = 2 (8 - x). • Solve this equation for x. • Provide an explanation or justifi cation for your work. • Check by substituting your answer back into the original equation and

showing that it simplifi es to a true statement.


Math Knowledge# 1, 2, 3, 4

The student:• Correctly

determines whether 9 is a solution of the equation (1)

• Determines the property that is applied incorrectly; correctly solves the equation (2)

• Correctly identifi es the incorrect reasons (3)

• Solves the equation correctly; check of work is correct (4)

The student attempts all four items and answers at least three correctly.

The student attempts at least three of the items and answers at least two parts correctly.

Communication# 1, 2, 3, 4

The student:• Correctly explains

or shows mathematically whether 9 is a solution of the equation (1)

• Correctly explains or shows why the property was not used correctly (2)

• Correctly explains which reasons are incorrect (3)

• Correctly explains or gives a complete and correct justifi cation for the work (4)

The student provides three completely correct explanations.ORExplanations for all four questions are incomplete but contain no mathematical errors.

The student provides at least two completely correct explanations.OR Attempts three or more explanations that are incomplete or contain mathematical errors.


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ACTIVITY

1.6Solving InequalitiesPhysical Fitness ZonesSUGGESTED LEARNING STRATEGIES: Shared Reading, Questioning the Text, Think/Pair/Share, Group Presentation

Spartan Middle School students participate in Physical Education testing each semester. In order to pass, 12- and 13-year-old girls have to do at least 7 push-ups and 4 modifi ed pull-ups. Th ey also have to run one mile in 12 minutes or less.

You can use an inequality to express the passing marks in each test.

The graph of an inequality in one variable is all the points on a number line that make the inequality true.

MATH TERMS

To verify a solution of an inequality, substitute the value into the inequality and simplify to see if the result is a true statement.

Push-Up, p Modifi ed Pull-Up, m One-Mile Run, r

verbal at least 7 push-ups

at least 4 pull-ups

12 minutes or less

inequality p ≥ 7 m ≥ 4 r ≤ 12

Graph 8 9 10 11765 5 6 7 8432 10 11 12 13987

1. Why do you think the graphs of push-ups and pull-ups are dotted but the graph of the mile run is a solid ray?

2. Jamie ran one mile in 12 minutes 15 seconds, did 8 push-ups and 4 modifi ed pull-ups. Did she pass the test? Explain.

Th e solution of an inequality in one variable is the set of numbers that make the inequality true.

3. Use the table below to fi gure out which x-values are solutions to the equation and which ones are solutions to the inequality. Show your work in the rows of the table.

x-values Solution to the equation?2x + 3 = 5

Solution to the inequality?2x + 3 > 5

1 2-1 08.5


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Solving InequalitiesPhysical Fitness ZonesPhysical Fitness Zones

4. How many solutions are there to the equation 2x + 3 = 5? Explain.

5. Are 2 and 8.5 the only solutions to the inequality 2x + 3 > 5? Explain.

6. Would 1 be a solution to the inequality 2x + 3 ≥ 5? Explain.

Here are the number line graphs of two diff erent inequalities.

7. Compare and contrast the two inequalities and graphs that are shown above.

Similarities Diff erences

0 1–1–2−3

x < 3

2 3 0 1–1–2–3 2 3

x ≥ –2

0 1–1–2−3

x < 3

2 3 0 1–1–2–3 2 3

x ≥ –2

SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share


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My Notes 8. Th ink about why the graphs are diff erent.

a. Why is one of the graphs showing a solid ray going to the left and the other graph showing a solid ray going to the right?

b. Why does one graph have an open circle and the other graph a fi lled-in circle?

TRY THESE A

Graph each inequality on a number line.

a. x < -2 b. x ≥ 5

9. Chloe and Charlie are taking a trip to the pet store to buy some things for their new puppy. Th ey know that they need a bag of food that costs $7, and they also want to buy some new toys for the puppy. Th ey fi nd a bargain barrel containing toys that cost $2 each.

a. Write an expression for the amount of money they will spend if the number of toys they buy is t.

b. Chloe has $30 with her and Charlie has one-third of this amount with him. Use this information and the expression you wrote in part (a) to write an inequality for fi nding the number of toys they can buy.

Th ere are diff erent methods for solving the inequality you wrote in the previous question. Chloe suggested that they guess and check to fi nd the number of new toys that they could buy.

10. Use Chloe’s suggestion to fi nd the number of new puppy toys that Chloe and Charlie can buy with their combined money.

SUGGESTED LEARNING STRATEGIES: Quickwrite, Create Representations, Guess and Check



WRITING MATH

An open circle represents < or > inequalities, and a solid circle represents ≤ or ≥ inequalities.


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Charlie remembered that they could use algebra to solve inequalities. He imagined that the inequality symbol was an equal sign. Th en he used equation-solving steps to solve the inequality.

11. Use Charlie’s method to solve the inequality you wrote in Item 9b.

12. Did you get the same answer using Charlie’s method as you did using Chloe’s method? Explain.

TRY THESE B

Solve and graph. Remember to substitute some sample answers back into the original inequality to check your work.

a. 3 + 4x < 7 b. 2(x - 3) + x ≥ 6 c. 5 > 2 + 2 __ 3 x

Chloe liked the fact that Charlie’s method for solving inequalities did not involve guess and check, so she asked him to show her the method. She suggested that they solve the following inequality:

Charlie showed Chloe the work below for -2x - 4 > 8:

-2x - 4 > 8-2x - 4 + 4 > 8 + 4

-2x > 12 -2x ____ -2 > 12 ___

-2 x > -6

When Chloe went back to check the solution by substituting a value for x back into the original inequality, she found that something was wrong.

13. Confi rm or disprove Chloe’s conclusion by substituting values for x into the original inequality.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Look for a Pattern, Think/Pair/Share


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Solving Inequalities Physical Fitness ZonesPhysical Fitness Zones

SUGGESTED LEARNING STRATEGIES: Quickwrite, Identify a Subtask

Chloe tried the problem again but used a few diff erent steps.

-2x - 4 > 8-2x + 2x - 4 > 8 + 2x

-4 > 8 + 2x-4 - 8 > 8 - 8 + 2x

-12 > 2x

-12 ___ 2 > 2x ___ 2

-6 > x

Chloe concluded that x < -6

14. Is Chloe’s conclusion correct? Explain.

15. Explain what Chloe did to solve the inequality.

Charlie looked back at his work. He said that he could easily fi x his work by simply switching the inequality sign.

16. What do you think about Charlie’s plan? Explain.

Although all of these methods worked, Charlie and Chloe wanted to know why they were working.


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Solving Inequalities ACTIVITY 1.6continued Physical Fitness ZonesPhysical Fitness Zones


Here is an experiment to discover what went wrong with Charlie’s fi rst method. Look at what happens when you multiply or divide by a negative number.

Directions Numbers Inequality

Pick two diff erent numbers. 2 and 4 2 < 4

Multiply both numbers by 3. 2(3) and 4(3) 6 < 12

Multiply both numbers by -3. 2(-3) and 4(-3) -6 > -12

17. Try this experiment again with two diff erent numbers. Record your results below. Compare your results to the rest of your class.

18. What happens when you multiply by a negative number?What happens when you divide by a negative number?

19. How does this aff ect how you solve an inequality?

EXAMPLE 1

Solve and graph: -3x + 5 ≤ 20 Step 1: Subtract 5 from both sides.

-3x + 5 - 5 ≤ 20 - 5-3x ≤ 15

Step 2: Divide both sides by -3. Remember to reverse the inequality sign.

-3x ____ -3 ≥ 15 ___

-3

x ≥ -5Solution: x ≥ -5

TRY THESE C

Solve and graph.a. 3 - 4x ≤ 11 b. 6 - 3(x + 2) > 15 c. 2(x + 5) < 8(x - 3)

0 1-1-2-3-4-5-6-7−8

x ≥ -5

2


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SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Interactive Word Wall, Look for a Pattern

Compound inequalities are two inequalities joined by the word and or by the word or. Inequalities joined by the word and are called conjunctions. Inequalities joined by the word or are disjunctions. You can represent compound inequalities using words, symbols or graphs.

20. Complete the table. Th e fi rst two rows have been done for you.

Verbal Description

Some Possible Solutions Inequality Graph

all numbers from 3 to 8, inclusive

3.5, 4, 4 1 __ 3 , 5, 6, 7.9, 8 x ≥ 3 and x ≤ 8 7 865432 9

all numbers less than 5 or greater than 10

-2, 0, 3, 4, 4.8, 10 3 __ 4 , 11 x < 5 or x > 10 8 976543 10 11

all numbers greater than -1 and less than or equal to 4

all numbers less than or equal to 3 or greater than 6

21. Compare and contrast the graphs for conjunctions and disjunctions.

Similarities Diff erences

ACADEMIC VOCABULARY

compound inequality


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SUGGESTED LEARNING STRATEGIES: Questioning the Text, Activating Prior Knowledge, Create Representations, Group Presentation

EXAMPLE 2

Spartan Middle School distributes this chart to students each year.

Age Mile Run (min:sec) Push-Ups Modifi ed Pull-Ups

Boys Girls Boys Girls Boys Girls

12 8:00–10:30 9:00–12:00 10–20 7–15 7–20 4–13

13 7:30–10:00 9:00–12:00 12–25 7–15 8–22 4–13

Write and graph a compound inequality that describes the push-up range for 12-year-old boys.

Step 1: Choose a variable.Let p represent the number of push-ups for 12 year old boys.

Step 2: Determine the range and write an inequality.Th e push-up range is 10 ≤ p ≤ 20.

Solution: Th e compound inequality is 10 ≤ p ≤ 20. Th e graph is shown above.

TRY THESE D

Write and graph a compound inequality for each range or score.

a. the push-up range for 13 year old boys

b. the pull-up range for 13 year old girls

c. the mile run range for 12 year old girls

d. the mile run range for 13 year old boy

e. a score outside the healthy fi tness zone for girl’s push-ups

9 10 11 12 13 14 15 16 17 18 19 20 219 10 11 12 13 14 15 16 17 18 19 20 21

WRITING MATH

The compound inequality “p ≥ 10 and p ≤ 20” can also be written as “10 ≤ p ≤ 20” because the two inequalities p ≥ 10 and 10 ≤ p are equivalent.

CONNECT TO APAP

In upper-level mathematics classes, inequalities are expressed in interval notation. The interval notation for x > 3 is (3,∞).


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SUGGESTED LEARNING STRATEGIES: Note Taking, Create Representations, Think/Pair/Share

22. Why are individual dots used in the graphs for Example 2 and some of the graphs in Try Th ese D?

To solve a conjunction, break the compound inequality into two parts and solve each part. Th e solution of the conjunction will be the solutions that are common to both parts.

EXAMPLE 3

Solve and graph the conjunction: 3 < 3x - 6 < 8

Step 1: Break the compound inequality into two parts.3 < 3x - 6 and 3x - 6 < 8

Step 2: Solve and graph 3 < 3x - 6. 3 < 3x - 6 3 + 6 < 3x - 6 + 6 9 < 3x 3 < x or x > 3 Step 3: Solve and graph 3x - 6 < 8. 3x - 6 < 8 3x − 6 + 6 < 8 + 6 3x < 14

x < 4 2 __ 3 0 1 2 3 4 5 6

Step 4: Determine what is common to the solutions of each part. What is common in Steps 2 and 3?Solution: 3 < x < 4 2 __ 3

0 1 2 3 4 5 6

TRY THESE E

Solve and graph each conjunction.

a. −1 < 3x + 5 < 6 b. 2 < x __ 3 − 5 < 6

To solve a disjunction, solve and graph each part. Th e solution of the disjunction will be all the solutions from both parts.

0 1 2 3 4 5 60 1 2 3 4 5 6


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EXAMPLE 4

Solve and graph the compound inequality: 2x − 3 < 7 or 4x − 4 ≥ 20.

Step 1: Solve and graph 2x − 3 < 7. 2x − 3 + 3 < 7 + 3 2x < 10 x < 5

Step 2: Solve and graph 4x − 4 ≥ 20. 4x − 4 ≥ 20 4x − 4 + 4 ≥ 20 + 4 4x ≥ 24

x ≥ 6Step 3: Combine the solutions. Solution: x < 5 or x ≥ 6

TRY THESE F

Solve and graph each compound inequality.

a. 5x + 1 > 11 or x − 1 < −4 b. −5x > 20 or x + 12 ≥ 7

SUGGESTED LEARNING STRATEGIES: Note Taking, Create Representations, Think/Pair/Share, Group Presentation



Solve and graph each inequality on a number line.

1. 3x − 8 + 4x > 6

2. 5 < 3x + 8

3. 4 − 2(x + 1) < 18

4. 2 __ 3 ≥ 1 __ 6 − 2x

5. −2 < 2(x + 5) ≤ 7

6. 5x ≥ 14 −2x or 3(x + 3) ≤ 6

7. Correct the mistakes in the problem. 2x + 5 − 6x > 25 −4x + 5 > 25 −4x > 20 x > −5

-8-9-10 -7 -6 -5 -4

8. Explain why you reverse the inequality when you multiply or divide both sides of an inequality by a negative number.

9. What is the largest number in the solution set of x < 3?


Describe the diff erences between solving and

graphing a conjunction and a disjunction.

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8


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ACTIVITY

1.7Absolute ValueStudent DistancesSUGGESTED LEARNING STRATEGIES: Summarize/Paraphrase/Retell, Vocabulary Organizer, Create Representations, Think Aloud

As part of an activity, Ms. Patel asked some of the students in her Algebra class to stand in positions along a number line created on the classroom fl oor. Th e students’ positions matched points on a number line as shown.

1. Use the number line to write each student’s distance from 0 next to their name. For example, Tania is 2 units away from 0. Israel’s distance from 0 is also 2 units even though he is at -2.

Derrick Laura Kia Israel Mara Antwan Tania Nick Sam

Th e absolute value of a number is the distance from 0 to the number on a number line. Using absolute value notation, Mara’s distance is �-3� and Antwan’s distance is �3�. Since Mara and Antwan are each 3 units from 0, �-3� = 3 and �3� = 3.

2. Write each person’s distance from 0 using absolute value notation.

Absolute value equations can represent distances on a number line.

3. Th e locations of the two students who are 5 units away from 0 are the solutions of the absolute value equation �x� = 5. Which two students represent the solutions to this equation?

0-2-4-6-8-10

Der

rick Kia

Mar

a

2 4 6 8 10

Isra

el

Laur

a

Tani

a

Antw

an

Nic

k

Sam

0-2-4-6-8-10

Der

rick Kia

Mar

a

2 4 6 8 10

Isra

el

Laur

a

Tani

a

Antw

an

Nic

k

Sam

ACADEMIC VOCABULARY

An absolute value equation is an equation involving the absolute value of a variable expression.

READING MATH

Read �-3� as “the absolute value of negative three.”

51-56_SB_A1_1-7_SE.indd Sec1:5151-56_SB_A1_1-7_SE.indd Sec1:51 1/7/10 12:01:48 PM1/7/10 12:01:48 PM

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Absolute ValueACTIVITY 1.7continued Student DistancesStudent Distances

Absolute value equations can also represent distances between two points on a number line.

4. In the student line, which two people are 4 units away from 1? Mark their location on the number line below.

From Item 3, you know that the equation �x� = 4 represents the numbers located 4 units away from 0. Th e equation �x| = 4 can also be written as �x - 0� = 4, which shows the distance (4) away from the point (0). In Item 4, you were looking for the numbers located 4 units away from 1. So you can write the absolute value equation �x - 1� = 4 to represent that situation.

5. What are two possible values for x - 1 given that �x - 1� = 4? Explain.

6. Use the two values you found in Item 5 to write two equations showing what x - 1 could equal.

7. Solve each of the two equations that you wrote in Item 6.

Th e solutions to this equation represent the two points on the number line that are 4 units from 1.

8. How do the solutions to the equation above relate to your answer for Item 4?

-4 -3-5 -2 -1 0 1 2 3 4 5-4 -3-5 -2 -1 0 1 2 3 4 5

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite, Create Representations, Work Backward


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Absolute Value Student DistancesStudent Distances

9. Write an absolute value equation and draw a number line that shows the answer to each question.

a. Which two points are 2 units away from 0?

b. Which two points are 5 units away from −2?

c. Which two points are 3 units away from 4?

10. Solve these absolute value equations.

a. �x� = 10 b. �x� = −3

c. �x + 2� = 7 d. �2x − 1� = 5

11. Which people in the line up are 3 or fewer units from 0?

12. Show the portion of the number line that includes numbers that are 3 or fewer units from 0.

0-2-4-6-8-10

Der

rick Kia

Mar

a

2 4 6 8 10

Isra

el

Laur

a

Tani

a

Antw

an

Nic

k

Sam

0-2-4-6-8-10

Der

rick Kia

Mar

a

2 4 6 8 10

Isra

el

Laur

a

Tani

a

Antw

an

Nic

k

Sam

SUGGESTED LEARNING STRATEGIES: Create Representations, Visualization, Guess and Check, Think/Pair/Share, Group Presentation, Role Play

If there are no values of x that make an equation true, the equation has no solution.


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Absolute ValueACTIVITY 1.7continued Student DistancesStudent Distances

Th e graph you created in Item 12 can be represented with an absolute value inequality. Th e inequality |x| ≤ 3 represents the numbers on a number line that are 3 or fewer units from 0.

13. Circle the numbers shown below that are solutions of |x| ≤ 3. Explain why you chose those numbers.

−1 3 0.5 4 −3.1

14. How many solutions does the inequality |x| ≤ 3 have?

15. If you were to write a compound inequality for the graph of |x| ≤ 3 that you sketched in Item 12, would it be an “and” or an “or” inequality? Explain.

16. Write a compound inequality to represent the solutions to |x| ≤ 3.

17. What numbers are more than 4 units away from 3 on a number line? Show the answer to this question on the number line.

ACADEMIC VOCABULARY

absolute value inequality

SUGGESTED LEARNING STRATEGIES: Shared Reading, Create Representations, Guess and Check, Visualization, Quickwrite, Group Presentation, Activating Prior Knowledge

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


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Absolute Value Student DistancesStudent Distances

SUGGESTED LEARNING STRATEGIES: Visualization, Activating Prior Knowledge, Quickwrite, Think/Pair/Share, Group Presentation

Th e absolute value inequality |x − 3| > 4 represents the situation in Item 17. A “greater than” symbol indicates that the distances are greater than 4.

18. Circle the numbers below that are solutions to the inequality |x − 3| > 4. Explain why you chose those numbers.

7.1 7 0 −2 6.9 100

19. If you were to write a compound inequality for the graph of |x − 3| > 4 that you sketched in Item 17, would it be an “and” or an “or” inequality? Explain.

20. To solve |x − 3| > 4 for x, you need to write the absolute value inequality as a compound inequality.

a. Based on the graph from Item 17, the expression x − 3 could either be greater than 4 or less than −4. Write this statement as a compound inequality.

b. Solve each of the inequalities you wrote in Part a. Graph the solution.

21. Write an absolute value inequality and make a graph that represents the answer to each question.

a. What numbers are less than 2 units from 0?

b. What numbers are 4 or more units away from 0?

c. What numbers are 4 or fewer units away from –2?


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Absolute ValueStudent DistancesStudent Distances


SUGGESTED LEARNING STRATEGIES: Think/Pair/Share

22. Solve each absolute value inequality by writing it as acompound inequality and then solve for x. Graph each solution on a number line.

a. |x| > 5

b. |x − 3| ≤ 7




Solve each equation.

1. |2x − 4| = 5

2. |x| − 3 = 7

Write an equation or inequality that represents each question. Th en graph the answer on a number line.

3. What numbers are 3 units away from −1 on a number line?

4. What numbers are 5 or fewer units away from 0?

5. What numbers are more than 3 units away from 5?

Solve each inequality and graph the solution on a number line.

6. |x − 4| ≤ 2

7. | 2 __ 3 x + 5 | > 4


Create a graphic organizer that compares

and contrasts the following equation and inequalities.

|x - 3| = 6 |x - 3| ≤ 6 |x | -3 > 6


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Embedded Assessment 3 Inequalities and Absolute Value

A HEALTHY POOL

For safe swimming, pool water chemistry needs to be maintained at proper levels. For example, “free available chlorine” should never fall below 2.0 ppm (parts per million) and the “combined available chlorine” levels should be less than 0.2 ppm.

1. Write and graph an inequality for the recommended levels of free available chlorine, f.

2. Write and graph an inequality for the recommended levels of combined available chlorine, c.

3. Joe tested his pool water and got these results: f = 2.0 ppm and c = 0.4 ppm. Explain whether or not the pool water falls within the proper levels.

In addition to chlorine, other chemicals also help maintain proper levels. Joe’s pool water test kit includes the following label. Use the table to answer the questions below.

Safe Pool Chemical Levels

Chemical Recommended Level

Free available chlorine, ppm 2.0 – 4.0

Combined available chlorine, ppm < 0.2

pH level 7.5 (plus/minus 0.3)

Total alkalinity, ppm 80 – 100

Total dissolved solids, ppm not to exceed 1500 greater than at pool start-up

Calcium hardness, ppm 200 – 400

Cyanuric acid, ppm 30 – 50

(adapted from data from www.healthypools.org)

4. Write and graph on a number line the compound inequality that represents the recommended levels of free available chlorine, f.

5. When Joe fi lled the pool for the summer, the total dissolved solids measured 500 ppm.

a. Write an inequality to fi gure out if the total dissolved solids s in his pool are at the recommended level.

b. Give an example of a safe level of dissolved solids s.

6. Th e inequality |x − 7.5| ≤ 0.3 determines proper pH levels. Solve this inequality and interpret the solution.

Use after Activity 1.7.


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Inequalities and Absolute ValueA HEALTHY POOL


Math Knowledge# 1, 2, 4, 6

The inequalities are written correctly (1, 2, 4) and the pH inequality is solved correctly (6).

All four inequalities are attempted, and at least two are written correctly.

The student attempts to write at least two of the inequalities and at least one is written correctly.

Problem Solving# 3, 5a

The student correctly states whether the pool water’s chlorine falls within proper levels (3) and the dissolved solids inequality is written correctly (5a).

The student attempts both items but is able to answer only one of the questions correctly.

The student attempts to answer both items but is unable to answer at least one question correctly.

Representations# 1, 2, 4

The graphs are correct based on the inequalities written (1, 2, 4).

All three inequalities are graphed, but only two of the graphs are correct based on the inequalities that were written.

The student attempts to graph at least two inequalities, and at least one is graphed correctly based on the inequalities written.

Communication# 3, 5b, 6

The student:• Gives a correct

explanation for the chlorine levels in the pool water (3).

• Gives a correct example of a safe level of dissolved solids (5b),

• Interprets answer correctly independent of the solution’s correctness (6).

At least two of the explanations, examples, or interpretations are correct and complete.ORExplanations, examples and/or interpretations are incomplete for all three items but contain no mathematical errors.

The student provides explanations, examples, or interpretations for two of the items and gives one complete and correct responseORExplanations, examples and/or interpretations are incomplete for all three items and contain mathematical errors.


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UNIT 1Practice


ACTIVITY 1.1

1 3 3 32 6 12 33 9 27 34 12 48 3

1. Describe any patterns you notice in the table of values.

2. Assume the pattern in each column continues. If you were given a random number, how would you determine if the number would appear in one of the columns?

3. Based on your investigation of the table of values, how could you label the columns? Explain your reasoning.

4. a. Write an expression that represents this relationship between numbers in the same rows in columns 1 and 2.

b. Explain how you could use this expression to determine the value in row 45 of column two.

5. Create a graphical representation of the relationship between two columns in the table.

ACTIVITY 1.2

For each number, identify each set for which it is a member. Use the symbols N, W, Z, Q, Ir, and R to represent each set.

N = natural numberW = whole numberZ = integerQ = rational numberIr = irrational number R = real number

6. -5 7. 0 8. 2 2 __ 5 9. √

__ 4 10. 0.55 11. √

__ 6

Identify the most specifi c set of real numbers to which all numbers in each given set belongs.

12. {5, 0, 12} 13. {1.5, -3, √ ___

16 } 14. Copy this table. Put an “x” in the box for

each set to which the number belongs.

N W Z Q Ir R

0.55

6

√ _

8

√ __

81

-3

0. _

4

5 2 __ 5

ACTIVITY 1.3

Identify the property illustrated by each of the following equations

15. 3x + 6y = 3(x + 2y) 16. 3x + 6y = 6y + 3x 17. 3x · 1 = 3x 18. (3x)y = 3(xy)

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UNIT 1 Practice


19. 4y + 0 = 4y 20. 3 and -6 are integers, and their product,

-18, is also an integer. 21. 12 · ( 1 ___ 12 ) = 1

22. If x = 8 and 8 = y, then x = y. 23. If 6 = k, then k = 6.

ACTIVITY 1.4

24. Is 7 a solution of 5x - 3 = 12? Justify your answer using multiple methods.

25. Th e perimeter of a triangle ABC is 54. If the triangle has side lengths AB = 3x, BC = 4x, and AC = 5x, fi nd the length of each side.

Solve each equation.

26. 4 ( 1 __ 4 x - 2 ) = 0

27. 6x + 2(x + 6) = 28 28. 16 - (x + 2) = 23 29. -3(x - 5) = 51 30. 36 = (4 - x)3 31. x - 2 = 31 32. 10x + 1 = 61 33. 62 = 9x - 10 34. 2.6 = 0.3x + 0.4x + 0.5 35. x + 0.04x = 26

ACTIVITY 1.5

In Items 36–40, solve the equations using symbols. Check the answer using a different method.

36. 8x + 5 = 3x + 15 37. 3x + 11 = 2x - 5 38. 6x - 9 = 8x + 11 39. 0.5x - 3.5 = 0.2x - 0.5 40. 6 - 2(x + 6) = 3x + 4 41. Provide a reason for each step in solving the

equation shown below. 2(x - 1) - 3(x + 2) = 8 - 4x 2x - 2 - 3x - 6 = 8 - 4x

-1x - 8 = 8 - 4x-1x + 4x - 8 = 8 - 4x + 4x

3x - 8 = 8 3x - 8 + 8 = 8 + 8 3x = 16

3x ___ 3 = 16 ___ 3

x = 5 1 __ 3

ACTIVITY 1.6

42. What is the solution of the inequality 3 __ 4 x - 8 > -4?

A. x > 3 B. x > 5 1 __ 3 C. x > 9 D. x > 16


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UNIT 1Practice


43. What is the solution of the inequality 2(x + 1) - 3(x - 5) > 17?

A. x < 0 B. x > 0 C. x < 6 D. x > 6 44. Which inequality describes the graph?

A. x < -2 or x ≥ 3 B. -2 ≤ x < 3 C. x ≤ -2 or x > 3 D. -2 < x ≤ 3 45. Which inequality is equivalent to

-1 ≤ 3(x - 2)-4x ≤ 9? A. -1 ≤ 7x - 6 ≤ 9 B. -1 ≤ 3x + 6 - 4x ≤ 9 C. -1 ≤ -x - 6 ≤ 9 D. -1 ≤ -7x + 6 ≤ 9 46. Name the property that justifi es each step in

the solution of this inequality. 1 __ 2 (3 - 2x) > 6x + 5 3 - 2x > 12x + 10 Step 1 -2x > 12x + 7 Step 2 -14x > 7 Step 3 x < -

1 __ 2 Step 4

ACTIVITY 1.7

47. What is the solution set of the equation |2x - 5| = 11?

A. {3, 3} B. {-3, 8} C. {-8, 3} D. {-8, 8} 48. What is the solution of the inequality

|x - 3| ≤ 2? A. x ≤ 5 B. x ≥ 1 C. 5 ≤ x ≤ 1 D. 1 ≤ x ≤ 5

49. Which absolute value inequality describes the graph?

A. |x| < 4 B. |x| ≤ 4 C. |x| > 4 D. |x| ≥ 4 50. To build certain machine part to the

manufacturer’s specifi cations, its length must be 10 cm plus or minus 0.05 cm. Write an absolute value inequality that represents the possible lengths l that meet these specifi cations.

-2 -1-4 -3 0 21 43-2 -1-4 -3 0 21 43

-2-4 0 2 4-2-4 0 2 4


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UNIT 1 Reflection

62 SpringBoard™ Mathematics with Meaning™ Algebra 1

An important aspect of growing as a learner is to take the time to refl ect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning.

Essential Questions

1. Review the mathematical concepts and your work in this unit before you craft thoughtful responses to the questions below. Support your responses with specifi c examples from concepts and activities in the unit.

1 How are patterns, equations, and graphs related?

2 Why are the properties of real numbers important when solving equations?

Academic Vocabulary

2. Look at the following academic vocabulary words:

expression equation absolute value variable solution equation distributive property compound inequality absolute value

inequality Choose three words and explain your understanding of each word and why

each is important in your study of math.

Self-Evaluation

3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each.

Unit Concepts

Is Your Understanding Strong (S) or Weak (W)?

Concept 1

Concept 2

Concept 3

a. What will you do to address each weakness?

b. What strategies or class activities were particularly helpful in learning the concepts you identifi ed as strengths? Give examples to explain.

4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world?

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Unit 1

Math Standards Review

1. A formula for computing a value s is s = mx + my ________ wz , where

m, x, y, w, and z are positive integers. An increase in which variable would result in a corresponding decrease in s?

A. mB. xC. yD. z

2. If x = 10, what is the value of x (x - 1) + x (x + 1) _________________ x ?

3. If 2x + 4 = 10, what is the value of the expression-4x + 20?

1. Ⓐ Ⓑ Ⓒ Ⓓ

2.

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3.

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Math Standards ReviewUnit 1 (continued)

4. Jason wants to earn money raking leaves. He buys a rake that costs $20.00. He plans to charge $6.00 for each lawn he rakes.

Part A: Write an equation to represent M, the amount of Jason’s profi t, if the number of lawns he rakes is r.

Part B: How can you determine the number of lawns Jason will have to rake in order to earn a profi t of $100.00?

Read

Explain

Solve

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Patterns and 1 Equations - Houston Independent School ... number systems and their properties ......

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