AF U1 L1 Expressions and Properties SJ

SolvingOne-VariableEquations

AIIFAlgebra II Foundations

Student Journal

Table of Contents

Lesson Page Lesson 1: Expressions and Properties ...............................................................................................................1

Lesson 2: Solving Equations by Using Properties .........................................................................................17

Lesson 3: Order of Operations..........................................................................................................................26

Lesson 4: Solving Equations Using Order of Operations .............................................................................44

Lesson 5: Solving One-Variable Inequalities..................................................................................................57

Lesson 6: Solving Absolute Value Equations and Inequalities....................................................................74

Lesson 7: Ratios, Proportions, and Percent of Change .................................................................................87

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Solving One-Variable Equations Lesson 1: Expressions and Properties Page 1

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Lesson 1: Expressions and Properties Activity 1 In this course, you will develop skills and concepts that lead to an understanding of algebraic concepts, such as unknowns, algebraic expressions and equations, graphing algebraic functions, and applying algebra to represent real-world situations. Before we jump into these concepts, let’s make a few links to prior understanding. Remember, integers are the positive and negative natural numbers including zero. You can represent integers on a number line by placing points above each number as shown below. 1. What do the arrows on the number line represent? Representing integers in relation to natural numbers and whole numbers with a Venn-Diagram may look like the figure to the right. 2. Place 10 other integers on the Venn-Diagram in

the appropriate location. There are many ways to represent an individual integer. We generally represent an integer with a positive or negative number, but we can also represent them with words, scenarios, graphs, and even tiles. Explore the following example of the integer negative 7. Number: –7 Tiles: Words: Negative seven Scenario: “It’s minus seven degrees Fahrenheit outside.” Drawing or Graph: 3. Pick a different integer other than a negative seven and represent it with the following methods. a. Words: b. Number: c. Scenario: d. Tiles: e. Drawing or Graph:

–8 –7 –6 –5 –4 –3 –2 –1 0 1 2

0 1 2 3 4–1 –2 –3 –4

integers –1

–2–3

–5–4

whole numbers 0

1 2 3 4 5 natural numbers

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4. Use some, or all, of the following numbers and operations to create a numerical expression and

represent it with symbols, words, tiles, and a scenario. 7, –9, 3, +, –, •, ÷, (, ) 5. Write an algebraic expression, words, and a scenario for the tile representation below. 6. Write an algebraic expression for the statement below and describe the unknown quantities. Then,

represent the scenario with tiles. "$15 more than Tanya."

7. Write an expression and a scenario for the following statement. 20% of 80 8. Translate “The quotient of x and seven” into an algebraic expression. 9. Translate “Thirty yards less than the length” into an algebraic expression. 10. A 24-ounce jar of apple juice is poured into two different sized containers. Write the algebraic

expression for the amount of juice poured into the smaller container, in terms of the larger container, if g ounces were poured into the larger container.

Numerical Expression – Algebraic Expression –


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11. Write an algebraic expression, words, and a scenario, for the tile representation below.

12. Write an algebraic expression for the scenario below and describe the unknown quantities. Include a

tile representation. Three years older than Susan.

13. Write any algebraic expression of your choice, and then represent it with multiple methods. 14. For expressions, the parts that are added are called terms. For example, 6x and 9 are the terms for the

expression 6x + 9. What are the terms for each of the expressions below? a. 3x + 5 b. 8 + (–5) c. 5x + 9x 15. In a term, the number multiplied by a variable is called a coefficient. A term that is just a number is

called a constant. For example, 6 is the coefficient of the term 6x and 9 is the constant in the expression 6x + 9. Determine the coefficients and constants for each of the expressions below.

a. 3x + 5 b. 8 + (–5) c. –7x + 9x 16. If two terms are the same except for their coefficients, the terms are said to be like terms. For example,

8x and –9x are like terms and 8x and 5y are not like terms. Two constants are also like terms. Determine which terms below are like terms.

a. 3x b. 8 c. 5x d. 9y e. –10x f. –4y g. –9 h. 1 i. x j. y k. 0.5x l. 0.3

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Activity 2

Additive Inverse Property For any number a, a + (–a) = 0.

1. Think of four other ways to represent the additive inverse property by completing each exercise below. a. Words: b. Tiles: c. Symbols:

d. Scenario: We also know that adding zero to any number does not change the value of the number. This is called the identity property of addition. 2. Think of four different ways to represent the identity property of addition by completing each exercise

below. a. Words: b. Tiles: c. Symbols:

d. Scenario:

3. In the past, you’ve learned certain rules for adding integers. In your group, experiment with the

following calculations. Use tiles and the concept of zero pairs as needed, then write the rule(s) in your own words. Complete more examples if needed.

a. –8 + 5 b. –8 + (–5) c. 8 + 5 d. 8 + (–5)

e. Rule(s) for adding any integer to another integer:

4. Use tiles to add the following like terms, then explain if your rule for adding integers also works for adding like terms.

a. –6x + 4x b. –6x + (–4x) c. 6x + 4x d. 6x + (–4x)

e. Do rules for adding integers work for adding like terms? Explain.


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5. A statement you might hear a mathematician say is, “Subtracting is the same as adding the opposite.” With symbols, the mathematician’s statement might look like a – b = a + (–b). Complete each example below with a calculator to examine each statement, and then write your conclusion as to whether you agree or disagree with this statement.

a. 7 – 15 b. 7 + (–15) c. 23 – 17 d. 23 + (–17) e. –7 – (–6) f. –7 + 6 g. Conclusion: 6. How might you use tiles to represent the idea that subtracting is the same as adding the opposite? Be

prepared to share how your group represents this idea with tiles. 7. Study the following properties and determine a way to show the property using tiles. For each

property, draw a sketch of what you did with the tiles. a. Commutative Property of Addition: You can add numbers in any order. For example,

4 + 5 = 5 + 4 or a + b = b + a. b. Associative Property of Addition: The sum of three or more numbers does not depend on how

they are grouped. For example, 3 + (4 + 5) = (3 + 4) + 5 or a + (b + c) = (a + b) + c.

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8. Sometimes people forget what to do with the sign when multiplying or dividing integers.

a. Complete the following calculations with a calculator, then write a rule for the sign of the product when multiplying integers.

–9(6) –9(–6) 9(6) 9(–6)

Rule for Multiplying Integers:

b. Complete the following calculations with a calculator, then write a rule for the sign of the quotient when dividing integers.

72/9 –72/9 –72/(–9) –72/(–9)

c. Rule for Dividing Integers: 9. What value can you multiply any number by and the result is the same number? Show an example

with numbers, then show another example with variables. This is called the Multiplicative Identity Property.

10. What value can you multiply any number by and the result is always zero? Show an example with

numbers, then show another example using a variable. This is called the Multiplicative Property of Zero.

11. What value can you multiply any number by and the result is the opposite number? Show an example

with numbers , then show another example using a variable. This is called the Multiplicative Property of Negative One.


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12. The concept of multiplication can relate to multiple groups of objects. For example, the product of 3 and 4 can be thought of as 3 groups of 4 objects or 4 groups of 3 objects. See the figures below.

Describe how this concept matches the Commutative Property of Multiplication, which states that you

can multiply numbers in any order. For example, 5(7) = 7(5) or ab = ba. 13. Many calculators accept the use of parenthesis ( ) when entering numeric expressions. Use your

calculator to show that the statements below are true. a. 3 •(4 • 5) has the same value as (3 • 4)• 5 b. 0.5• (4 • 0.6) has the same value as (0.5 • 4) •0.6

The property that you just modeled is the Associative Property of Multiplication: The product of three or more numbers does not depend on how they are grouped. For example, 3 (4 • 5) = (3 • 4) 5 or a(bc) = (ab)c.

14. Study the method of adding the fractions 1/2 and 3/5 shown below and use the method to add the

fractions 1/8 and 3/7.

( )( )

( )( )+ = + = + =

1 5 3 21 3 5 6 112 5 2 5 5 2 10 10 10

a. Write a real-world situation that represents adding your two fractions. b. If you had to generalize the method you used to add two fractions, how might you do that if the

fractions were ab

and cd

?

( ) =3 4 12

3 Groups of 4 4 Groups of 3

( ) =4 3 12

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15. The method for multiplying rational numbers in fraction form is to multiply the numerators and multiply the denominators. Two examples are given below, one is with specific numbers and the other is a general example with variables to represent all numbers.

( )( )

⎛ ⎞ = =⎜ ⎟⎝ ⎠

2 42 4 83 5 3 5 15

⎛ ⎞ =⎜ ⎟⎝ ⎠

a c acb d bd

, when ≠ 0b and ≠ 0d

Use the rule above to determine when each condition below could be true. Then, give one example for

each condition.

a. A positive rational number times a positive rational number gives a product that is smaller than either positive rational number.

b. The product of two positive rational numbers is larger than either rational number. c. The product of two positive rational numbers gives a value that is between each rational

number.

16. We know that multiplying reciprocals, also known as multiplying inverses, gives a value of 1. For

example, ⎛ ⎞ = =⎜ ⎟⎝ ⎠

5 6 30 16 5 30

. This mathematical truth is also called the Multiplicative Inverse Property.

Show that this property is true by multiplying 4/5 and 5/4. a. If you had to generalize the method you used to multiply two reciprocals, how might you do

that if the fractions were b/c and c/b? b. How might you use your calculator to show that multiplying two reciprocals gives a value of 1?


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17. You have probably heard a mathematics teacher say, “To divide by a fraction, multiply by the reciprocal.” Using the dry–erase board, your calculator, or any other tool, prepare a short presentation on how you could show this fact to be true. You may want to include an example with variables to

show how to divide ab

by cd

.

18. Represent the Distributive Property with tiles and symbols for the following expressions. Draw a

sketch of each representation. a. ( )− +2 3 5x Tile Symbol b. ( )−3 7x Tile Symbol c. ( ) ( )+ −3 2 3 2x Tile Symbol d. ( ) ( )− +4 4 4x Tile Symbol

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19. You can represent the Distributive Property of multiplication over addition with the symbols in an equation as shown below.

( )+ = +a b c ab ac

a. Write a definition of the Distributive Property of multiplication over addition in your own words.

b. Give an example of the Distributive Property of multiplication over addition with numbers

and/or variables. 20. Show how you might represent the Distributive Property of multiplication over subtraction with

symbols in an equation. a. Write a definition of the Distributive Property of multiplication over subtraction in your own

words. b. Give an example of the Distributive Property of multiplication over subtraction with numbers

and/or variables.


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21. The term Distributive Property is related to the words distribute, distribution, and distributor. Study the following definitions, and then explain how these definitions are similar to, or different from, the mathematical term, Distributive Property.

• Distribute: The action of sharing something among a number of recipients.

For example, the political candidates will distribute the fliers throughout the neighborhood.

• Distribution: The way in which something is shared in a group or spread over an area. For example, changes to the wilderness affected the distribution of its wildlife.

• Distributor: The agent or company that supplies goods to stores and other businesses. For example, the wholesale distributor shipped the computer monitors to all the stores in the region.

Explanation:

22. The Distributive Property can be used to multiply calculations in your head. For example, if you

bought six cans of soda that cost $0.98 each, you could represent $0.98 with $1.00 – $0.02 then multiply by 6 to determine the total cost of $5.88.

6(1.00 – 0.02) = 6.00 – 0.12 = 5.88

a. Use this method to determine the total price of purchasing eight jars of salsa that cost $0.96 each.

Describe how you completed the calculation in your head.

b. Use this method to determine the total price of purchasing seven cans of chili that cost $1.07 each. Describe how you completed the calculation in your head.

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Activity 3 In this activity, you will investigate writing equivalent expressions. An example of equivalent expressions is 3 + 7 and 2(5). There are various ways you can show that these two expressions are equivalent. Study the three different representations below. Numerically: Each expression has a value of 10 so they are equivalent: 3 + 7 = 10 and 2(5) = 10 Scenario: A woman that had three one-dollar bills in her purse and added another seven one-dollar bills to her purse would have the same value as a person who had two five-dollar bills in his or her wallet. Tiles: 3 unit tiles added to 7 unit tiles is the same as 2 sets of 5 unit tiles. 1. Create two equivalent numerical expressions. Each numerical expression should have different

numbers and different operations. 2. Write an equivalent numerical expression, using multiplication or division, for the expression 12 – 36. It is also possible to create equivalent algebraic expressions. For example, the expression 2x + 3 is equivalent to the expression 5x – 3x + 1 + 2. With tiles this might look like. . . 3. Write an equivalent algebraic expression for 6x + 3x – 8 – 9. Use tiles if needed. 4. Draw an equivalent tile representation for the expression 3x – x + 4 – 2.

Equivalent Expressions

Two expressions that have the same value.

is the same as

is the same value as


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5. Write, as a scenario, an equivalent numerical expression for, “I had 20 video arcade tokens and then used 10 of the tokens playing video arcade games.”

6. Write, as a scenario or real–world situation, an equivalent expression for, “One hundred fifty fewer cell

minutes than the available total.” 7. Write an equation for the following tile representation.

8. For the question, “What is 40% of 90?”, what are the important words that will be used to write an

equation to represent the question? Explain each use of the important words you chose. a. Write the equation which represents the question.

9. Draw a tile representation of the equation 3x – 4 = 2.

=

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Practice Exercises 1. Use some, or all, of the following numbers and operations to create a numerical expression and

represent it with symbols, words, tiles, and a scenario. 6, –7, 12, 1, 15, –9, +, –, •, ÷ 2. Represent the scenario below with an expression. "The temperature dropped 8°F."

3. Write an algebraic expression, words, and a scenario for the following tile representation. 4. Translate “The product of x and nine” into an algebraic expression and then write an equivalent

algebraic expression. 5. Write an equivalent expression for 24 + 36 using multiplication. 6. Write an equation for, “50 is what percent of 200?” 7. Write, in words, the following formula for the perimeter of a rectangle: P = 2l + 2w, where l represents

the length of the rectangle and w represents the width of the rectangle.


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8. Determine which property or rule is represented in each equivalent situation below. a. ( ) ( ) ( )5 4 8 5 4 5 8− + = − + b.

c. 11 55 55 11+ = +

d. The area of a rectangle can be determined by multiplying the length times the width or by multiplying the width times the length.

e. An accountant decided to add all the numbers in a list by adding groups of numbers that made a value of 10 first as shown below.

( ) ( ) ( )7 6 5 4 3 5 7 3 6 4 5 510 10 10 30

+ + + + + = + + + + += + + =

f. ( )21 17 21 17− − = − + −

g. 9 3 9 210 2 10 3

⎛ ⎞÷ = ⎜ ⎟⎝ ⎠

9. Use the distributive property to change the following expressions. Note: You may need to change a

subtracting to adding the opposite. a. ( )+5 7x b. ( )+2 2x c. ( )+a b c d. ( )−2 8x e. ( )− −2 5x f. ( )−a b c

=

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Outcome Sentences I didn’t know that zero pairs The distributive property now makes sense to me because All of these properties Using tiles really Reviewing expressions I didn’t know that expressions An equivalent expression is For me, converting words to expression and equations is ______________________________________ because

Solving One-Variable Equations Lesson 2: Solving Equations by Using Properties Page 17

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Lesson 2: Solving Equations by Using Properties Activity 1 1. Give examples of how substitution can be used in the world around us. Give as many examples as you

can. The first example is given. 2. The Substitution Property of Equality states that if a = b, then a can be substituted for b in any

expression or equation and b can be substituted for a in any expression or equation. Give examples of how substitution can be used in mathematics. Give as many examples as you can. The first example is given.

3. The formula for the area of a rectangle is A = lw. Give three solutions for the length, l, and width, w,

that would make an area of 36 square meters. 4. What value x is a solution to the equation x + 6 = 10? Explain your answer.

Solution to an Equation:

Substitution in the World Around Us

Substitute Teacher

Substitution In Mathematics If the side of square is 5 cm then the area is determined by substituting 5 for s in the formula A=s2.

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5. What values of x are solutions to the equation x + 6 = 6 + x? Explain your answer. 6. What values of c are solutions for c + (–c) = 0? Explain your answer. 7. Write an equation that represents the circumference, C, of the circle based on the radius. If the

circumference is 314 square feet, determine the solution for the radius that makes the equation true. 8. Write a scenario, or real–world example, and a matching equation that has a solution of t = 5 hours. 9. Write a real-world example and matching equation that has a solution of p equals $25. 10. The area of a triangle is one-half the base times the height. Write a formula for the area of a triangle and

give three solutions for the height, h, and base, b, that would make an area of 100 square meters. 11. Write a percent real-world example and matching equation that has a solution of $5.


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Activity 2 The Addition Property of Equality states that if the same number is added to both sides of an equation, then both sides of the equation remain equal. Using symbols, the following statement would be, if a = b, then a + c = b + c. Two examples of this property are shown below. Numbers: If –3 + 5 = 2, then –3 + 5 +4 = 2 + 4 Tiles: Use this information as an example and complete Exercises 1 through 3. 1. In your own words, write a statement that represents the Subtraction Property of Equality. a. Write the statement with symbols. b. Give an example with numbers. c. Give an example with tiles. 2. In your own words, write a statement that represents the Multiplication Property of Equality. a. Write the statement with symbols. b. Give an example with numbers. 3. In your own words, write a statement that represents the Division Property of Equality. a. Write the statement with symbols. b. Give an example with numbers.

= =

= =

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4. Will these four properties of equality always work for any type of number, such as integers, rational numbers, or real numbers? If not, give an example.

5. For each of the equations below, determine which property you could use to solve for the unknown?

a. + = −8 11x b. − =7 9x

c. = −7 98x d. = −135x

e. Use the mathematical properties of equality to solve the four equations. Show your work and check the solutions.

i. ii. iii. iv.

6. For the solved equation below, state which property of equality was used for each step. + =

+ ==

4 8 202 5

3

xx

x

a. The same equation could have been solved in a different way. State which property of equality was used for each step shown below.

4 8 20

4 123

xxx

+ ===

b. Describe why you think either method shown above worked to solve 4x + 8 =20.


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7. Solve − =2 35x and state which properties of equality were used for each step in the solution.

8. Sam solved the following equation incorrectly. Determine the mistake, explain how to fix it, and then

solve the equation correctly.

( )

8 56

6 8 6 56

8 308 8 30 8

38

x

x

xx

x

− =

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =− + = +

=

9. The area of a rectangle is 96 square inches and the length is 12 inches. What is the width of the

rectangle? Use the rectangle area formula of A = lw to solve for the width, show your work, check your solution. Describe how you used mathematical properties to solve and check this problem.

10. Write and solve an equation for the following percent problem.

25% of what is $20 11. The distance formula for traveling at a constant rate is d = rt where d is the distance traveled, r is the

rate, and t is the time of travel. At what rate must you travel to go 300 hundred miles in 6 hours? 12. The solution to a problem is 21. Write a real-world application and equation that results in the given

solution. Show that your equation has the given solution.

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13. Write an equation for the scenario below, and then solve the equation. Describe the properties you used to solve the equation.

“Ron, who had $40, had five times the sum of Nick's amount of money and $4. How much money did

Nick have?” 14. You have learned that you can add, subtract, multiply, or divide the same quantity to both sides of an

equation to solve it. Can you do the same with squaring or taking the square root of both sides of an equation? Look at each example below and determine if you can. If you can, write an explanation.

a. = 5x b. = 8x c. =2 81x d. =2 36x Explanation: 15. Study the following two different categories, then create a definition for the Symmetric Property of

Equality.

Examples of the Symmetric Property of Equality

Non-Examples of the Symmetric Property of Equality

= −5 11 6 and − =11 6 5

+ =25 2 27 and = +27 25 2

= 12x and =12 x

=a b and =b a

= −5 11 6 and + =5 6 11

+ =25 2 27 and + =20 7 27

+ = +3 4 4 3

=a b and + = +a c b c


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Practice Exercises 1. Before pulling away from the curb, a cab driver charges an initial fee of $2.00,

then an additional $1.50 for each one mile traveled. Determine how far a customer traveled if her fee was $32. Write an equation that represents the fee based on how far, in miles, a customer travels.

2. Apex Recycling Company pays $1.35 for every pound of copper. Write an equation for the amount of

money the recycle company pays for copper. Whenever Anthony remodels a house, he collects the old copper pipes. He sold a large pile of copper to

Apex Recycling Company and they paid him $58.05. How many pounds did Anthony recycle? 3. Hannah works in sales for a large company that sells treadmills to fitness centers. She earns $550.00 for

each fitness center that purchases a treadmill from her company plus another $75.00 for each treadmill she sells. Write an equation that represents the amount of money Hannah earns for each fitness center. Determine how many treadmills she must sell if her goal is to earn $1,150.00 per fitness center.

4. The height of a tree and the thickness (diameter) of its trunk are generally related to the tree’s age.

Assume that the height and thickness of a particular type of tree can be described by the following equations:

h = 0.4a + 1, where h is height in meters and a is age in years. t = 0.4a – 2.5, where t is thickness in centimeters and a is age in years. a. Calculate the age of the tree if it has a height of 30 meters. b. Calculate the age of a tree if the thickness is 25.5 centimeters.

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5. Many stores and banks have coin machines where people can pour in their loose change – pennies, nickels, dimes, and quarters – and receive a receipt to turn in for the equivalent amount in dollar bills. Write an equation representing the amount of money the coin machine will print a receipt for if you pour in p pennies, n nickels, d dimes, and q quarters.

Use your equation to solve the following: a. If Kyle pours 7 nickels and 17 dimes into the coin machine, how many quarters does he need to

also pour in to have a total of $11.05?

b. If Kelly has 30 dimes, no pennies, and 44 quarters, how many nickels does she have if her total is $15.15?

6. How can mathematical properties be used to solve equations and real–world applications? 7. Create a graphic organizer of all the properties and terms that have been developed in Lessons 1 and 2

of this unit so far.


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Outcome Sentences Using properties to solve equations I now understand that The equality property of division is still hard to understand because The four properties of equality make solving equations Substitution now makes sense because

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Lesson 3: Order of Operations Activity 1 The diagram below represents a series of operations with numbers. Each operation is completed before going to the next operation. For example, if the input is 14, first add 2 to get 16, then divide by 8 to get 2, then subtract 3 to get 1− , then multiply by 10 to get −10 , then subtract 7 to get an output of −17 . 1. Determine the outputs for the following inputs that match the diagram of operations and numbers

above. Check the outputs with your partner. a. Input: 62 Output: ________ b. Input: 22 Output: ________ 2. Determine the inputs for the following outputs that match the diagram above. Check your inputs with

your partner. a. Output: 63 Input: ________ b. Output: 13 Input: ________ c. What was different about solving this exercise as compared to Exercise 1? 3. Use the diagram below and experiment with inputs or outputs to complete the exercises below.

a. Write one example of a positive integer input that gives a negative integer output.

b. Write one example of a negative integer output that comes from a negative integer input.

c. Determine a whole number input that gives a whole number output.

d. Determine a whole number output that has a whole number input. The input and output values should be different than in Part C.

Input Output +2 ÷8 –3 ●10 –7

Input Output ÷5 +1 ●2 –15 ●7

Solving One-Variable Equations Lesson 3: Order of Operations Page 27

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4. Describe the strategies that you used in Exercise 3. Will these strategies always work? Explain your answer.

5. If doing is completing the operations in order from the input to the output, describe what undoing

would be. 6. Based on the numbers and operations displayed at the front of class, determine the inputs or outputs as

directed by your teacher. Record your responses below.

Input Output

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Activity 2 Knowing the order of operations is important so that everyone will calculate the same way. There are four main operations: addition, subtraction, multiplication, and division. Study and answer the following exercise related to money to see how important the order of operations is for real-world mathematics. 1. Camille is an accountant and charges $50.00 for the first time she meets with a new client. She then

charges $25.00 per hour after that to prepare a tax return. She met recently with a new client, and then she worked an additional 2 hours to complete his tax returns. Two different billing agents in her company calculated the total fee in two different ways. Which calculation below is correct? In your own words, explain why one person may have calculated it incorrectly?

a. 50 25 2 $150+ • = b. 50 25 2 $100+ • = To investigate the order of operations in more detail, let’s complete the activity Operation Order. Cut out the Operation Order arrows at the end of the activity. Place the arrows in a pile. The object of the activity is to determine which operations need to be completed in order and which operations do not. Operation Order 2. Select any number from –10 to +10 and write it in the INPUT section below. Continue to use this input

number for the entire activity. Pick two arrows from the pile that have the addition operation. Write the operations in the two arrow blocks in the diagrams below in the order you chose them. Determine the value of the output and write it in the OUTPUT section.

b. Switch the order of operations and write them in the two arrow blocks below. Using the same input value, determine the output and write it in the OUTPUT section.

c. Were the output values for both conditions the same or different? Explain why they may have

been the same or different.

Input Output

Input Output


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3. You just investigated completing two addition operations in different order; now continue investigating the combination of other operations in different orders. Pick arrows from your pile as needed to investigate each exercise below. Make sure to determine the output based on the same input using the operations in different order.

a. Pick two subtraction operations from your pile and determine the two outputs based on

completing the operations in a different order. b. Pick two multiplication operations from your pile and determine the two outputs based on

completing the operations in a different order. c. Pick two division operations from your pile and determine the two outputs based on

completing the operations in a different order.

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output

Page 30 AIIF

d. Pick an addition operation and a subtraction operation from your pile and determine the two outputs based on completing the operations in a different order.

e. Pick a multiplication operation and division operation from your pile and determine the two

outputs based on completing the operations in a different order. f. Pick an addition operation and multiplication operation from your pile and determine the two

outputs based on completing the operations in a different order. g. Pick an addition operation and division operation from your pile and determine the two

outputs based on completing the operations in a different order.

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output

Input Output


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h. Pick a subtraction operation and multiplication operation from your pile and determine the two outputs based on completing the operations in a different order.

i. Pick a subtraction operation and division operation from your pile and determine the two

outputs based on completing the operations in a different order. 4. Record your findings for Exercises 2 and 3 in the table below by listing which combinations of

operations where order mattered and which combinations where order did not matter when determining the output value.

Combinations Where Order Doesn’t Matter Combinations Where Order Matters

Input Output

Input Output

Input Output

Input Output

Page 32 AIIF

5. You have probably memorized the acronym PEMDAS to help you complete the order of operations, where P stands for parenthesis (or other grouping symbols), E stands for exponents, M stands for multiplication, D stands for division, A stands for addition, and S stands for subtraction. In your own words, describe how Exercises 2 and 3 relate to the acronym PEMDAS.

Because there are so many different ways to combine the operations with numbers, mathematicians use grouping symbols such as parenthesis to make sure some operations occur before others. This is represented by the P in PEMDAS. Other grouping methods that can be classified under P are brackets and braces, such as 4 [ 3 + 2 ] and –2 { 6 – 3 }. Square roots and, in some cases, division also imply grouping. For example, in the

expression 4 12+ the addition should be completed before the square root, and in the expression 17 52− the

subtraction should be completed before division. Because there are other methods to represent grouping, it may be convenient to remember the order of operations as GEMDAS and not PEMDAS, where G represents grouping. 6. For each of the expressions below, use the correct order of GEMDAS one step at a time, and show the

correct order of operations to obtain the given value. The first has been completed for you

a. ( ) ( )( )

− + = +

= += +=

2 23 9 5 8 3 4 83 4 6412 64 76

b. [ ]÷ + − + ===== −

24 8 6 2 7

1

c. − =•

===

25 46 2

1.75

d. • − + =====

25 4 7 6

21

e. ( )⎡ ⎤− + + =⎢ ⎥⎣ ⎦====

229 7 3 1

50

f. ( )⎡ + + ⎤ + =⎣ ⎦====

2 2 2 2 2 2

22


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7. Work with your group members to create a graphic organizer for the order of operations. 8. Trade your graphic organizer with another group so that they can test it to solve the following

exercises.

a. 26 3 3

5 6− =

+ b. 2 24 3 5+ =

c. 10 3 8 14− • = − d. ( )( )5 10 2 1 3 6 4− + − =

Page 34 AIIF

PAGE INTENTIONALLY LEFT BLANK.


AIIF

Operation Order Cutouts for Activity 2 ● 5 ●( –6 ) ●( –1 ) ● 2 ● 4 ● 10 ●( –5 )

÷( –4 ) ÷ 3 ÷ 2÷( –1 ) ÷( –8 ) ÷( –5 ) ÷10

+ 3 + ( –4) + 10 + 4 + 6 + 8 + ( –6)

–5 – ( –4) – ( –5) – 6 – 9 – ( –7) – 2

Page 36 AIIF



AIIF

Activity 3 You’ve investigated using a graphic organizer to complete operations from left to right and right to left. Another method to organize the operations could be vertical. Complete Exercises 1 and 2 using a vertical table. The input is at the top and the operations continue in order going down the table to the output. 1. Determine the outputs given the following inputs. a. Input: 8 Output: b. Input: 20 Output: 2. Determine the inputs given the following outputs. a. Output: –10 Input: b. Output: 5 Input: Using a method, such as the table, can help a person keep organized when evaluating an expression given a value of an unknown variable. For example, to evaluate the expression ( )23 4 6 x + + when x = 2, you could use the table or you could evaluate it symbolically.

( )( ) ( )2 2

2

3 2 4 6 6 4 6

10 6100 6106

+ + = + +

= += +=

3. Complete a table for the expression 4 6 122

x + − and then evaluate it for the different x values. Evaluate

it symbolically too. a. x = 3 b. x = –3 4. Write your opinion of using a table to evaluate an expression as compared to symbolically?

INPUT

6+

÷2

–10

●5

OUTPUT

INPUT: x = 2

●3

+4

Square

+6

OUTPUT: 106

INPUT:

OUTPUT:

INPUT:

OUTPUT:

Page 38 AIIF

Operation Order II Within your group make a pile of the Operation Order arrows and add the additional arrows of square and square root to your pile. Decide who will be #1, #2, #3, and #4. Your teacher will model, with a volunteer group at the front of the class, how to create and evaluate an expression with your group. Complete Exercises 5 through 6 with a similar method. Each member of your group will be responsible for one operation for each expression. 5. Have each member of your group randomly pick an arrow from the pile.

a. As a group, in the order that you chose, write the expression and fill in the table.

b. Evaluate the expression for x = 15.

c. Determine the input if the output was 30.

6. Have each member of your group randomly pick two arrows from the pile. For this exercise, you will each need to add two operations to the expression and table.

a. As a group, in the order that you chose, write the expression

and fill in the table.

b. Evaluate the expression for x = –4.

c. Determine the input if the output was 100.

INPUT: x

OUTPUT:

INPUT: x

OUTPUT:


AIIF

Additional Operation Cutouts for Activity 3

Square Square root

Square Square root

Page 40 AIIF



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Practice Exercises 1. Determine the output for the following inputs that match the diagram of operations and numbers

above. Check the outputs with your partner. a. Input: 78 Output: ________ b. Input: –10 Output: ________ 2. Determine the input for the following outputs that matches the diagram above. Check your inputs with

your partner. a. Output: 11 Input: ________ b. Output: –4 Input: ________

3. Use the expression 26 711

x − to fill in the table.

a. Evaluate the expression for an input of x = 6. b. Evaluate the expression for an input of x = –6. c. Determine the input if the output is 157. 4. Write an expression that matches the table on the right. a. Evaluate the expression for an input of x = 3. b. Determine the input if the output is 23.

Input Output –7 ●5 –3 ÷8 +2

INPUT: x

OUTPUT:

INPUT: x

+5

●3

÷2

–4

OUTPUT:

Page 42 AIIF

5. Describe how you know what order the operations should occur when determining the value of the

expression • −25 4 15

2.

6. Logan completed the following steps to determine the value of the expression ( )5 3 4+ . He made a

mistake along the way. Determine the mistake and describe why Logan may have made the mistake.

( ) ( )5 3 4 8 432

+ ==

7. Evaluate the expression 25 15

3x − for the following x value inputs.

a. x = 0 b. x = 9 8. Determine the value of x for the expression ( )25 4 7x + − for the following outputs. a. Output is 238 b. Output is 73

9. Determine the value of x for the expression −5 86x for the following outputs.

a. Output is 2 b. Output is –1/2


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Outcome Sentences I now understand that the order of operations I would conclude that adding numbers in any order I used doing and undoing operations to I was surprised that the order of operations I would conclude that determining the input when given the output I would like to find out more about Creating a diagram to match an equation is Creating a table to match the equation is How can working backwards help solve problems in mathematics?

Page 44

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Lesson 4: Solving Equations Using Order of Operations Activity 1 Today you will use doing and undoing operations with a table to solve one-variable equations. We will sometimes refer to “doing” as going forward and “undoing” as going backward.

The operations in the equation + =3 2 234

x can be represented in a table as shown.

The steps in the left column show how to start with x and work forward to 23. The steps in the right column show how to start with 23 and work backward to x.

This type of table can be used to help solve one-variable equations.

1. Study the left column and the right column. In your own words, describe how they compare?

2. Use the column on the right and work backward to undo the operations and solve for x. The equation can also be solved symbolically as shown below.

( )

3 2 23 Start with the equation.4

3 24 4 23 Multiply both sides by 4.4

3 2 923 2 2 92 2 Subtract 2 from each side.

3 90 3 90 Divide both sides by 3.3 3

30

x

x

xx

xx

x

+ =

+⎛ ⎞ =⎜ ⎟⎝ ⎠

+ =+ − = −

=

=

=

3. Compare the table method and symbolic method to solve the equation. Describe the similarities and

differences in these two methods.

Divide by 3Multiply by 3

x

Add 2

Divide by 4

Subtract 2

Multiply by 4

23

Steps to do

Step

s to

und

o

Solving One-Variable Equations Lesson 4: Solving Equations Using Order of Operations Page 45

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Solve Exercises 4 through 9 by using both a table and symbolic mathematics. Make sure to check your solutions. 4. Complete the right column in the following table that matches the equation − − = −3 15 78x and use the

table to help you solve for x.

5. Fill in the table so that it matches the equation ( )− = −3 4 5 99x and use it to solve for x.

6. Fill in the table so that it matches the equation − + = −6 50 211x and use it to solve for x.

x=

Steps to do

Step

s to

und

o

Multiply by –3

x=

Subtract 15

–78

Steps to do

Step

s to

und

o

x=

Steps to do

Step

s to

und

o

Page 46

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7. In your pair, create a table that matches the equation ⎛ ⎞− =⎜ ⎟⎝ ⎠

3 7 66x and use it to solve for the variable.

8. In your pair, create a table that matches the equation ( )+=

23 2108

4x

and use it to solve for the

variable.


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9. With your partner create a table and a matching equation. Trade the table with the other pair in your group, but don’t show them the equation. Now, create the equation that matches the table from the other pair. As a group, check your work.

A table can also be used to solve for an unknown variable in a formula. For example, the perimeter formula of a rectangle is l= +2 2P w , where P is the perimeter, l is the length, and w is the width. If we know that the perimeter is 325 inches and the width is 45 inches, we can determine the length by substituting the values into the formula, build a matching table, then solve.

= +

= + •= +

2 2325 2 2 45325 2 90

P wlll

325 2 90325 90 2 90 90

235 2235 22 2

117.5

l

ll

l

= +− = + −

=

=

=

l

Length is 117.5 inches. 10. Use this same concept to solve for the unknown radius of a circle. The formula for the area of a circle is

π= 2A r . If the area of a circle is 3,020 square centimeters, what is the approximate length of the radius? How would you check your solution?

l

l

w w

Steps to do

Step

s to

und

o Divide by 2Multiply by 2

l

Add 90 Subtract 90

325

Page 48

AIIF

There are two different methods you can use to solve equations that contain fractions. One method is to complete the opposite operations with the fractions. The second method is to first multiply both sides of the equation by the least common denominator of all the fractions then complete the opposite operations. Complete Opposite Operations First Multiply by the Least Common Denominator First

3 2 24 5

3 2 2 224 5 5 5

3 124 5

34 4 123 4 3 5

4815165

y

y

y

y

y

y

− =

− + = +

=

⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

=

=

( )

3 2 24 5

3 220 20 24 5

60 40 404 515 8 40

15 8 8 40 815 48

4815165

y

y

y

yy

y

y

y

− =

⎛ ⎞− =⎜ ⎟⎝ ⎠

− =

− =− + = +

=

=

=

11. Study the two different methods above, then describe which method you prefer to use and why. 12. Solve the following equations for the variable.

a. − =2 1 211 3

y b. + =5 3 53 4 6z

13. Solve the following equations for the variable. Use a table as needed.

a. 7.5 0.5 1.7x − = − b. 1.25 1.5 7.8x− + = −


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Activity 2 Tiles can be used to model equations with variables on both sides of the equal sign and solve the equations symbolically. 1. Solve the equation − = +2 5 3x x with tiles and draw a sketch of how you solved it using the tiles. Then,

solve the equation symbolically. 2. Solve the equation + = −3( 2) 6x x with tiles and solve it symbolically. Draw a sketch of the tiles and

describe how you solved this equation for x by manipulating the tiles and symbols.

Page 50

AIIF

3. Use tiles to solve the following problems. Draw a sketch of the tiles, then solve the problem symbolically.

a. ( ) ( )3 1 2 2x x− = + b. ( )4 2 4 5x x− + = + What if the numbers are too large in value to use algebra tiles? Use symbolic mathematics and the distributive property. 4. Solve the following problems using symbolic mathematics. a. ( )7 5 3 11x x− = − b. ( ) ( )− + + = + −12 9 6 4 12 6x x


AIIF

Practice Exercises 1. Use the table method to solve the following equations.

a. − =2 1 115

x

b. + =7 5 183

x

2. Find the mistakes in the table below for the equation −=

2 9 311y and correct the mistakes.

Steps to do

Step

s to

und

o

Steps to do

Step

s to

und

o

Divide by 11Multiply by 11

y

Add 9

Divide by 2

Subtract 9

Multiply by 2

3

Steps to do

Step

s to

und

o

Page 52

AIIF

3. The formula for the area of a trapezoid is ( )= +1 212

A b b h . If the area of a trapezoid is 672 square

inches, the height is 24 inches, and length of one of the bases is 30 inches, what is the length of the other base?

4. The Pythagorean Theorem states that for a right triangle, the sum of the square of each leg of the

triangle, sides a and b, is equal to the square of the hypotenuse, side c. + =2 2 2a b c What is the length of the unknown leg of a right triangle with a hypotenuse of 20 centimeters and one

leg with a length of 16 centimeters?

Steps to do

Step

s to

und

o

b1

24 in.

30 in.

a

b

c

a

16 cm

20 cm

Steps to do

Step

s to

und

o


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5. Solve the following equations. Use a table as needed.

a. 4 39 17x + = − b. ( )5 3 45x + = c. 13 224

y −=

d. 7 11 138

y −+ = e. ( )6 7 11 13x− + − = f. 12 5x + =

g. 2 16 65x − = h. 22 5 7x− = −

i. 5 10 3x− = − j. 4 8 2x− = − 6. Solve the following equations. Use tiles as needed. a. 3 5 13x x− = − b. 8 2 16x x− = − c. 3 8 3 10x x− = − +

d. ( )2 3 4 3x x− = − e. 4 2 5 10 3 4x x x− − = + − f. 6 12 83

x − =

g. 2 3 4x x− − = − h. 3 2 1 2 1 3x x x x+ − − = + − − i. 3 24 8x x− = + j. 4 4 23x x− = + k. − + = − +4(3 5) 9 3(3 6) 1x x l. + = −2( 3) 4x x

Page 54

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7. Solve the following equations. Round your answer to nearest tenth, if needed, or leave the answer as a fraction.

a. 0.5 1.5 1.3x x− = − b. 5.1 8.2 2 16x x− = − c. 3 6.2 3.8 10x x− = − +

d. 3 3 134 2 3

x x− = − e. 4 32 15 2x x− = + f. 6 12 8

3 5x − =

8. The following equations were solved incorrectly. Determine the mistake and solve the equation

correctly. Show your work. a. 2 5 4 7

2 2 5 7 4 2 7 712 212 22 26

x xx x x x

xx

x

− + = −− − + + = − − +

=

=

=

b. ( )2 4 3 22 4 3 2

2 2 4 2 3 2 2 26

x xx x

x x x xx

+ = −+ = −

− + + = − − +=

9. Substitute the correct values into the formula and solve for the unknown for each exercise.

a. The formula for the area of a triangle is 2bhA = where A represents the area, b represents the

base length, and h represents the height. If the height of a triangle is 21 inches and the area is 325.5 square inches, what is the base length?

21 in.

b


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b. The Pythagorean Theorem is 2 2 2a b c+ = , where a and b represent the leg lengths of a right triangle and c represents the hypotenuse length. If the one leg is 18 centimeters and the hypotenuse is 34 centimeters what is the approximate length of the other leg?

c. The volume of a cone is 213

V r hπ= , where r is the radius of the base of the cone and h is the

height of the cone. Determine the radius of a cone that has a volume of 235.5 cubic centimeters and a height of 12 centimeters. Use 3.14 for π .

d. The formula ( )1 tA P r= + represents the amount in an interest bearing account, where A is the

amount in the account, P is the original amount placed in the account, r is the annual interest rate, and t is the number of years the account has been in existence. Determine the annual interest rate for an account that began with $2,000.00 two years ago and now has a value of $2,205.00. Round to the nearest hundredth.

e. The slope of a line containing two points, ( )1 1,x y and ( )2 2,x y is 2 1

2 1

y ymx x

−=

−. Where m is the

slope, 1x is the x-value for the first point, 1y is the y-value for the first point, 2x is the x-value for the second point, and 2y is the y-value for the second point. Determine the y-value for the second point, 2y , if the slope of the line is –3, the x-value for the first point is –1, the y-value for the first point is 7, and the x-value for the second point is –6.

8. How can the order of operations be used to solve equations?

h

r

34 cm 18 cm

?

Page 56

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Outcome Sentences The table method for solving equations When using the order of operations to solve equations Tiles are useful to solve equations with variables on both sides because I need more help understanding Tiles are difficult to use when The method I prefer to use to solve equations is ________________________________________________ because

Solving One-Variable Equations Lesson 5: Solving One-Variable Inequalities Page 57

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Lesson 5: Solving One-Variable Inequalities Activity 1 1. Record the different representations for the condition you chose. 2. Write the inequality that represents w less than 3.

a. With your partner draw a number line, then each of you place a w anywhere on the number line, so that w is less than 3. You cannot choose the same place on the number line.

b. Write the placement location of the w on your number line.

c. Write your partner's placement location.

d. Whose value, or location, represents the largest number? Explain why. Now, write an inequality which represents a true statement for all three values, including the 3.

e. Write a non–integer value which is greater than your w value, but less than 3.

3. Erase the variables and then place other variables on your number line according to the conditions (or

criteria) below and write the corresponding inequalities.

a. w is greater than or equal to –2

b. x is between –3 and 2, excluding the endpoints –3 and 2

c. y is more than 0 but less than or equal to 5

d. z is greater than 5 or less than –4

Page 58

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4. Erase the variables and consider the following new criteria for the variables. w is between 0 and 1 x is between –1 and 0 y is between 4 and 5 z is between –3 and –2

a. Determine if w x< is always true, sometimes true, or never true. Give examples with numbers

to help determine the answer.

b. Determine if wy xy> is always true, sometimes true, or never true. 5. a. Write two inequality questions for the variables w, x, y, and z from Exercise 4. A sample

question could be, “Is w x< always true, sometimes true, or never true?”

b. Trade questions with your partner and answer your partner's inequality questions. 6. Discuss your inequality questions with another student pair. How were your inequality questions

similar and how were they different? 7. Below are several pairs of correctly and incorrectly graphed inequalities. For each incorrect graph

describe what is incorrect. Correctly Graphed Incorrectly Graphed a. x ≤ 5 b. x < 5 c. –1 ≤ x

0 0

0 0

0 0


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d. –1 < x e. –4 < x < 6 f. –4 ≤ x ≤ 6 8. What are the characteristics of a correctly graphed inequality? 9. Write a description for each correctly graphed inequality from Exercise 7. The first one has been

completed for you. a. x ≤ 5: All real numbers less than or equal to a positive five.

b. x < 5: c. –1 ≤ x: d. –1 < x: e. –4 < x < 6: f. –4 ≤ x ≤ 6:

0 0

0 0

0 0

Page 60

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You will often see the variable of an inequality on the left, such as the x in the inequality x ≥ 1. A written or verbal description of this inequality would generally be “x is greater than one.” There are, however, other equivalent methods to represent this inequality with symbols. There are also other methods to write or verbally describe this inequality. Below are some examples. The graph of the inequality remains the same for the different symbols, written, or verbal representations. Graph: Symbol methods: x ≥ 1, 1 ≤ x Written or verbal: x is greater than or equal to 1, 1 is less than or equal to x 10. For each written description below of an inequality, draw a graph of the inequality, then write two

different symbol inequalities and another written or verbal description.

a. w is less than or equal to positive three. Symbols:

Written or verbal:

b. y is greater than negative five. Symbols:

Written or verbal:

c. An unknown number is at least a negative ten. Symbols:

Written or verbal:

d. An unknown number is at least negative four but no more than positive eight.

Symbols:

Written or verbal:

0


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11. Write a symbolic inequality for each real-world example, then draw a graph.

a. Male adult Nile crocodiles usually mature around 10 years with an approximate length of 3 meters. One of the largest recorded adult male Nile crocodiles measured 8.6 meters.

b. An adult male Nile crocodile may easily exceed 500 pounds and many adults reach 2,200 pounds.

c. The weather report called for low temperatures ranging from 5°F to –5°F. 12. Write the symbolic inequality and a real-world example to match the graph of the inequality.

90 100

Page 62

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Activity 2

Your teacher will assign you a scenario below. You will write an inequality for the scenario then solve the inequality to answer the question.

1. The maximum safe load for each two-person chair on a ski lift is 425 pounds. The weight of one person

on a chair is 208 pounds. What additional weight can the chair carry?

2. A cell phone plan allows a maximum of 750 minutes each month. If 147 minutes

have been used this month, what range of minutes is left? 3. A family is planning a long trip across the United States that will take at least 36 hours to drive. They

drove for 14 hours on the first day. How many hours do they have left to drive? 4. Only customers 48 inches tall, or taller, can ride on most roller coasters. If Gabrielle

is only 44.5 inches tall, how much does she have to grow to be able to ride the roller coaster?

5. Create the inequality, y + 6 < 4, on the inequality tile pad using tiles. Solve the inequality with algebra

tiles. Write a description of the solution and graph the solution.

Addition Property of Inequality – Subtraction Property of Inequality –


AIIF

Use the Addition Property of Inequality and the Subtraction Property of Inequality to solve the inequalities in Exercises 6 through 8. Use tiles as needed. 6. Solve and graph the solution to the inequality 4n ≤ 3n +8. Write a descriptive solution to the inequality. 7. Solve and graph the solution to the inequality 14 ≤ n – 8. Write a descriptive solution to the inequality. 8. Solve and graph the solution to the inequality 13.5 + 7.2 + 9.8 + p > 40. Write a descriptive solution to

the inequality. The following questions are for the Inside–Outside Circles activity. 9. Describe what happened when you multiplied both sides of an inequality by a positive number. 10. Describe what happened when you multiplied both sides of an inequality by a negative number.

11. What might we do to make the inequality statement true when multiplying by a negative number? Do you think this would be true when dividing by a negative number?

Multiplication Property of Inequality: Division Property of Inequality:

Page 64

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We will now solve inequalities using the Multiplication Property of Inequality and the Division Property of Inequality. Solve and graph the solutions to following inequalities. Be careful when multiplying or dividing both sides of an inequality by a negative number. Write a descriptive solution to the inequality. 12. –6g ≤ 144

13. 75

y≥ −

−

14. 21x < –28

15. 5 158y

≥ −

16. 3 334

q−≤ −

15. 2 7 95w + ≤

16. 13k – 11 > 7k + 37 17. 7 + 3t ≤ 2(t + 3) – 2(–1 – t)


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18. The Booster Club needs to make a new school banner to display during championship tournaments. The length of the banner needs to be 15 feet. What are the possible widths, if the border of the banner can be no more than 42 feet? Write an inequality and then solve it.

19. The cheerleaders would like to purchase small pompoms to pass out at the championship game. A case

of 25 pompoms cost $20. What is the maximum number of cases that the cheerleaders can purchase if they can spend no more than $250?

20. Girl Scout Cookies™ cost $3.50 per box. If Mercedes wants to serve them at her party but only has a

budget of $20.00 for snacks, how many boxes can she purchase? 21. Your MP3 player holds 8 GB of music. Each song you put on your MP3 averages 0.004 GB. Write an

inequality that will let you know how many songs you can store on your MP3 player. How many songs at most can you store on your MP3 player?

Page 66

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Activity 3 In this activity, you will investigate compound inequalities. The word AND is a conjunction because it joins two inequalities and the solution must satisfy both inequalities simultaneously. To understand the concept of the conjunction AND, complete Exercises 1 and 2 as your teacher directs. 1. The Venn diagram on the right represents the 3 sets S,

E, and D. The set S contains the integers 1 through 15; set E contains the integers {4, 6, 10, 12}; the set D contains the integers {3, 6, 12, 15}. Shade the area in the Venn diagram which represents the numbers that are both in sets E AND D.

2. For the set A { –3, –1, 0, 1 , 3, 5} and B {–4, –3, –2, –1, 0, 1, 2, 3,

4}, draw a Venn diagram of the two sets and shade the area that contains the numbers that are common to both sets.

For Exercises 3 and 4, graph the solution to the given compound inequality. Give a written description of the solution and write the inequalities as a compound inequality. 3. a ≤ 6 AND a ≥ –2 4. –5 < x –4 AND x – 4 < 2

0

0

0

0

0

0

1

2

5 7 89

11

13 14


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Now, we will concentrate on writing compound inequalities from various application problems. For Exercises 5 through 8, write a compound inequality for the given situation then graph the inequality. 5. Starting salaries for college graduates range from $30,000, for educational services, to $56,000 for

chemical engineering. 6. According to Hooke's Law, in order to displace a spring by a certain amount of centimeters from its

"resting" length, a force F, in Newtons, needs to be applied to the spring. The equation for Hooke's Law is F = kx, where k represents the spring constant of elasticity. If the spring constant was calculated to be 1.68 and forces from 2 to 5 Newtons were applied to the spring, what will be the range of the displacements of the stretched spring? Round your answer to two decimal places.

7. The National Weather Service classifies hurricanes using the Saffir-Simpson Hurricane Scale. It is used

to inform people of potential property damage and flooding caused by the storm's surge, winds, and rain. Wind speed is the major criteria that is used to categorize hurricanes. What is the range of wind speeds for a category 3 hurricane?

8. There are several stages of sleep which affect your heart rate. During the most restful period of sleep,

your heart rate can be reduced by about 20%. During the most restless period of sleep, called REM, your heart rate can increase by about 50%. REM stands for “rapid eye movement,” and it is during REM when your dreams take place. If your normal heart rate while you’re awake is 66 beats per minute, what is the range of your heart rate while you are sleeping? Round your answer to the nearest heart beat.

Category Number

Wind Speed

1 74–95 mph 2 96–110 mph 3 111–130 mph 4 131–155mph 5 > 155 mph

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9. For sets A {-3, -1, 0} and B {1, 2, 3, 4}, draw a Venn diagram representing these sets and then shade in the portion representing A OR B. Use curly brackets to list all the items in sets A OR B.

For Exercises 10 through 12, graph the solution to the given compound inequality. Give a written description of the solution. 10. –6 > x OR x > 5 11. a ≥ 6 OR a ≤ –2 12. –2 ≥ x + 4 OR x + 4 > 9 13. Write a compound inequality and description for the graph on

the right, then write an application problem to go with your compound inequality.

0

0

0

0

0

0

0 –30 30

0

0

0


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Now, let’s write compound inequalities from various written and application type problems. For the following exercises, write a compound inequality for the given situation, find and graph its solution. Don’t forget to label the variable if necessary. 14. The product of –5 and a number is greater than 35 or less than 10.

.

15. A store is offering a $30.00 mail-in rebate on all digital

cameras costing at most $200.00 or at least $400.00.

0

0

0

0

0

0

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Practice Exercises 1. Graph the solutions to the following inequalities. Provide a written description of your solution.

a. w ≥ –5

b. x < 4

c. 80 – 24t > 32

d. 911

a− >

e. 4(y + 1) – 3(y – 5) ≥ 3(y – 1)

f. x < 5 AND x ≥ –3

g. –3 < d – 4 ≤ 1 2. How are solutions to inequalities described in written and graphed formats?


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3. Give a written and symbolic description of the graphed inequality below.

4. Different animals can hear different ranges of sound. Sound is measured in hertz which for hearing

means vibrations per second. Sounds that an animal hears are actually caused by the air pressure changing back and forth very quickly. For example, on a piano the vibration that the note “A” above “middle C” makes is actually 440 hertz. This is caused by the string inside the piano vibrating 440 times per second which makes air pressure changes of 440 times per second that your ears sense. Humans can hear sounds as low as the 20 hertz and as high as 20,000 hertz. Bats can hear in the range of 20 to 200,000 hertz

a. Write an equality that represents the range humans hear. b. Write an equality that represents range that bats hear. c. Use inequalities to represent the range of sounds bats can hear that a human cannot hear.

5. Salmon sharks thrive in water temperatures which range from

41°F to 64°F. Write a compound inequality to represent the temperatures where the salmon shark may not thrive. Graph the compound inequality.

0

0

0

0

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6. Solve the compound inequality –3h + 4 > 19 OR 7h – 3 > 18 for h. Then graph the solution. 7. Solve the compound inequality 4t – 3 < 17 OR –8 ≤ 3t –2 8. Write a compound inequality for the following graph. State the compound inequality in words.

0

0

0

0

0

0

0


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Outcome Sentences Graphing inequalities Writing a description for an inequality The difference between AND and OR for a compound inequality is I know that an AND compound inequality I know that an OR compound inequality Determining an inequality from a graph I am having trouble understanding

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Lesson 6: Solving Absolute Value Equations and Inequalities Activity 1 1. Draw a sketch of one example of two students demonstrating absolute value in front of class. 2. Record the class definition of absolute value below. 3. Determine the value of the expressions below. a. −9 b. 18 c. −1 d. −8 15 e. −7 2 f. −4 5 g. −3 6 h. −2 6 8

i. −62

j. −62

k. − •10 5 3 l. −14 182

m. − −8 5 n. − +2 5 5 o. − −8 p. − +7 4 4. Applying what you now understand about absolute value, determine what values of x would make the

following equations true. Describe how to solve these equations. a. 5x = b. 12t = c. 3812g = d. 6k = −

Solving One-Variable Equations Lesson 6: Solving Absolute Value Equations and Inequalities Page 75

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A symbolic definition of absolute value is:

if 0

if 0x x

xx x

≥⎧= ⎨− <⎩

Examples: ( )− = − − =20 20 20 because –20 < 0. =4 4 because 4 > 0. 5. Determine the value of the following expressions using the symbolic definition of absolute value. a. 3 b. −16 c. −37 d. 225 6. Solve the following equations.

a. = 3x b. = 35z

c. = 81t d. = 34

x

7. Describe how to solve the absolute value equation below by using GEMDAS. Make sure to describe

each step you used for solving it. Steps Description 4 8 16x − =

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8. Solve the following absolute value equations using opposite operations. Check your solutions with the original absolute value equation.

a. | 2x – 8 | = 6 b. | t + 5| = 3

c. | 3d – 6| – 7 = –4 d. Solve | f + 9| + 13 = 13

e. | 15v – 12| + 18 = 15 f. –2| x+ 6| = –8 9. The approximate minimum and maximum number of days of an elephant’s pregnancy can be

modeled with the equation − =630 30d . Determine the minimum and maximum numbers of days an elephant may remain pregnant.

10. Find the mistake for the solved equation, 2 6 7 11x + − = , and correct it below.

( )+ − =+ − =

− ===

2 6 7 112 6 7 11

2 1 112 12

6

xx

xxx

( )+ − = −+ − = −

− = −= −= −

2 6 7 112 6 7 11

2 1 112 10

5

xx

xxx


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Activity 2 1. In the last lesson we solved absolute value equations. Within your group, discuss what you remember

about solving absolute value equations and what the absolute value represented. Make a list below of the facts about absolute value that came up during your discussion.

2. Solve and graph the solution to the following absolute value inequalities. a. − <2 7x b. 5 10x − ≤ c. 3.5 1.5x − < d. 2 8x − < − e. 4 5x − > f. − >6 10x You can also use the symbolic definition of absolute value to solve absolute value inequalities.

if 0 if 0

x xx

x x≥⎧

= ⎨− <⎩

For example, for − <3 5x the two statements would be − <3 5x AND ( )− − <3 5x . Solving these inequalities gives –2 < x < 8. 3. Solve the following absolute value inequalities using the definition. a. − <6 10x b. 6 10x − > c. 6 10x + ≤

d. 5 7 10x − + < e. 2 6 4x − < f. 2 13 2

x − <

g. 0.7 8.5x − ≤ h. 6 6 8x − − ≥ i. 4 1 6 1x − − ≤

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Companies that manufacture products often set tolerance limits that help determine if the product passes inspection. For example, a company that produces crayons that are 10 centimeters in length might set a tolerance of 0.2 centimeters. This means every crayon created should be within 0.2 centimeters of the 10 centimeter length. This tolerance can be modeled mathematically by the absolute value inequality |x – 10| < 0.2. This represents that crayons only pass inspection if they are longer than 9.8 centimeters and shorter than 10.2 centimeters. Any crayon outside that range would be rejected by inspectors.

Sometimes companies put conditions on using their product. For example, on a can of paint you might read the directions, “Only paint when the temperature is between 50 and 90 degrees Fahrenheit.” This condition can be modeled mathematically by the absolute value inequality |x – 70| < 20. This means that if you paint with temperatures in this range the product should work properly and if you paint in a temperature that is less than or equal to 50 degrees Fahrenheit or greater than or equal to 90 degrees Fahrenheit the paint may not adhere properly or have the quality it should.

Station 1: Pencil Making This station represents a pencil manufacturing company. The company would like to make pencils that meet the conditions of the absolute value inequality |x – 18| < 0.3 for length in centimeters. Write a compound inequality that represents the range of pencils lengths that are acceptable. Write a compound inequality that represents the range of pencils lengths that do not meet the

conditions. Analyze the sample of pencils shown below. Determine which pencils pass inspection and which do

not.


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Station 2: Silicone Caulk Silicone caulk is used to seal the space between such things as walls and windows or around tubs and showers. Some silicone caulk is specifically for indoor use, some for outdoors, and some can be used for both. There are certain conditions that should occur when caulking. The surface should be clean and dry and the temperature should be between 40ºF and 100ºF. Write an absolute value inequality that represents the range of temperatures that are acceptable to apply silicone caulk. Write an absolute value inequality that represents the range of temperatures that are not acceptable to apply silicone caulk. Write a compound inequality that represents the range of temperatures that are acceptable to apply silicone caulk. Write a compound inequality that represents the range of temperatures that are not acceptable to apply silicone caulk. Station 3: Wood Product Manufacturers A wood product manufacturing company makes fiberboard panels which are wood products made from smaller wood fibers bound together into a panel or sheet. High temperature, high pressure, and glue are used to form the panels. These panels can be used for building’s structural sheathing as shown in the figure below. The company sells fiberboard panels with dimensions of 4 feet wide, 8 feet long, and 5/8 inch thick. The company lists the following tolerance specifications. Length/Width ±1/8 inch Thickness ±1/16 inch Write three different absolute value inequalities that represent the actual dimensions that can occur for length, width, and thickness. Explain why you had to change the value so that they represented the same units. Write the matching inequality for thickness.

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Station 4: Boxes for All Boxes for All is a company that manufactures boxes for packaging products, such as perfume and cologne. The company claims that each box it manufactures will be within 5% of the requested length, width, and height of the box. Jan, a perfume designer, has created a new perfume product that had great sales for the past year. She would like to update the box in which the bottle is packaged. She would like the new box to have a width of 35 millimeters, length of 45 millimeters, and a height of 60 millimeters. Write an absolute value inequality that represents the range of widths the boxes could be and fall within the condition of 5%. Write an absolute value inequality that represents the range of lengths the boxes could be and fall within the condition of 5%. Write an absolute value inequality that represents the range of heights the boxes could be and fall within the condition of 5%. Determine if the cut out for the box on the next page would pass inspection under the conditions given above.


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Cutout for Box

Top

Botto

m

Leng

th

Wid

th

Height

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PAGE INTENTIONALLY LEFT BLANK


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Practice Exercises 1. Solve the following absolute value equations. a. | 9x – 3 | = 15 b. | 5n + 2 | – 4 = 7 c. | 7z + 6 | + 12 = 9 2. Solve the absolute value inequality | x | < 5. Graph and write a description of the solution. 3. Solve and graph the absolute value inequalities. Write a description of the graphed inequality as well.

a. | y | < 4

b. | t | ≤ 6

c. | d | ≥ 1

d. –5 > | f |

e. | v | < 4

f. | g | > 3

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4. Write the absolute value inequality from the written description and graph the solution. “All real numbers less than a positive nine and greater than a negative nine and including a positive

nine and negative nine.” 5. Graph the solution to | z + 5 | < 2. 6. Graph the solution to | 3w – 4| ≥ 5. 7. Write the symbolic absolute value inequality and written description for the graphed inequality.

a.

b. 8. Graph and write the solution to the following absolute value inequalities.

a. | z | ≥ 5

0 8 –8

0


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b. | y – 4 | < 5

c. | 2t + 3| ≥ 3.

d. | 4z – 1| – 4 ≤ 3.

e. | 4d – 1| + 4 ≤ 1. 9. Write an inequality, description, and graph its solution for: “The temperature inside the freezer ranges

within 3 degrees of –18°F.” 10. Write an inequality and graph its solution for: “A cruise ship is trying to maintain a speed of 22 ± 0.6

knots while traveling to the islands.”

11. Write a compound inequality and an absolute value inequality for the following graph.

0

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Outcome Sentences When solving absolute value equations When solving absolute value inequalities Absolute value inequalities I am having a hard time distinguishing between Absolute value means What do I absolutely know?

Solving One-Variable Equations Lesson 7: Ratios, Proportions, and Percent of Change Page 87

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Lesson 7: Ratios, Proportions, and Percent of Change Activity 1 In this activity, you will work with ratios. Ratios are commonly used as an expression to compare two quantities to each other.

For each exercise below, write a ratio, in all 3 formats. 1. Bryan scores 3 points for every 2 points that Jon scores. What is the ratio of the number of points Bryan

scores compared to Jon? 2. In a cooking recipe, you will need one-and-three-quarter cups of water for each cup of rice. What is the

ratio of rice to water?

3. On a recent trip, you drove 300 miles and used 12 gallons of gas. Write a ratio comparing the miles driven to the amount of gas used. Simplify your ratio to lowest terms.

4. On the grid-side of a dry-erase board, draw a simple sailboat like the one pictured below. Make sure to

place all the vertices on the intersections of the grid lines. Draw a second sailboat on the same grid that is a different size than the original but the same shape. Use a metric ruler to measure the length of the boats drawn. Write the ratios for the length of the boat of the first sailboat drawn compared to the second sailboat drawn.

For Exercises 5 and 6, write your own ratio problem. 5. 6.

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7. For the following drawings, write a ratio comparing the area of object A to the area of object B. Write the ratio in all 3 formats.

a. b.

a. b.

d

A

B B

A


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Activity 2 Artists that sketch and paint animals use ratios to help make sure the parts of the animals are in proportion. For example, the sketch of the cat below shows that the size of the head compared to the height of the cat is 1 to 3. You may hear artist describe this as, “The head size is one-third the height” or “the height is three times the size of the head.” These ratios are important to artist in order to make animals look lifelike. For example, while the following drawing of a cat may be artistically appealing, the head size compared to the body height is no longer in a ratio that represents a real life cat. In this case the head is three-fourths the height. An artist will use the word proportional to describe an animal sketch that has a harmonious relation of parts to each other or to the whole. A mathematician will use the word proportional to state that two ratios are equal to each other. 1. Study the following drawings of dogs and determine which dogs are proportional to each other in

terms of head size compared to height. Be prepared to show your measurements and represent your findings mathematically.

a. b. c. d. e. f. g. h. i. j. k. l.

= =head size 3 unit 3height 4 unit 4

4 units 3 units

= =head size 1 unit 1height 3 unit 3

1 unit

3 units

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2. Study the dog drawings in Exercise 1 and determine which dogs are proportional in terms of head size compared to length.

Artists will often use the ratio of parts in one drawing to make a scaled version of the same drawing. Notice how in each drawing below the ratio of the head size to the height is the same even though the actual measurements are different. These ratios are a proportion because they are equal. Another way that artists use ratios and proportions is to make sure the height and length of the new drawing are the same ratio as the original drawing. If drawings are scaled in one direction more than another, the drawings would not be proportional and the ratios that represent the dimensions would not be equal. For example, the two drawings below have different ratios. 3. Measure the height and length of the following to determine which drawings are proportional. a. b. c. d. e.

= = =head size 9 mm 16 mm 1height 27 mm 48 mm 3

9 mm

27 mm

16 mm

48 mm

= =height 26 mm 13length 18 mm 9

26 mm

18 mm

88 mm

36 mm

= =height 36 mm 9length 88 mm 22

≠13 99 22


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4. An artist started to draw a scaled version of a turtle. The artist forgot to determine the actual length for the new drawing. Determine the new length and describe how you determined the new length.

5. Write a definition of proportion. 6. Solve the following proportions for the unknown variable.

a. 445 15x = b. 3 8

7 v= c.

123

6 9x =

7. A survey of the duck and geese population at a local park was conducted by a state employee of the

Department of Fish and Game. The employee counted 24 ducks to 16 geese. A few days later, the same employee counted only the ducks and ended up counting 288 ducks. The employee then estimated the number of geese using his ratio from a few days earlier. How many geese did the employee estimate were at the park?

8. A long piece of pipe was cut into two pieces, with the lengths of the pieces having a ratio of 3 to 5. If the

pipe was originally 32 meters long, what are the lengths of the two pieces of pipe?

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9. Most states have an expressway speed limit of 65 miles per hour. How many feet per second is equivalent to 65 miles per hour? Note: There are 24 hours in a day, 60 minutes in an hour, 60 seconds in a minute, and 5,280 feet in one mile.

10. A building casts a shadow of 130 feet at the same time a 60 foot flagpole casts a shadow of 5 feet. How

tall is the building? 11. Write your own proportion problem similar to Exercises 6. 12. Write your own real-world proportion problem similar to Exercises 7 through 10. 13. When biologists try to estimate the number of fish in a lake, it is not possible to count them all. The

biologists will capture a large sample size of the fish, tag them and release them back in the lake. After enough time has passed for the fish to be thoroughly mixed with the non-captured fish, the biologists then capture another sample of the fish. The biologist will count the number of recaptured fish that are tagged. Using a proportion, the biologists will then estimate how many fish are in the lake. Obtain one of the “fish” boxes from your teacher. Within your group, “capture” a small sample of fish, tag them, “mix” them back into the lake, take a second sample and determine how many fish are in your lake. Write up a detailed report on a piece of poster paper.


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Activity 3 Henrique, an online video-gamer, is trying to reach the SILVER level in order to earn the capability of invisibility for the virtual character he is playing. Friday night, before Henrique went to bed, he checked his current level. His level is displayed on the screen to the right. The gray squares represent Henrique’s current level and the unshaded squares represent how far he needs to go before reaching the SILVER level. 1. Relative to the SILVER level how might you describe the

current level of Henrique? 2. Relative to the SILVER level how might you describe the change that would need to occur for Henrique

to reach the SILVER level? 3. Relative to Henrique’s current level, how might you describe the change that would need to occur for

Henrique to reach the SILVER level? 4. Record the method to calculate the percent change as your teacher models it. After playing Saturday, Henrique checked his level points again. See the display to the right. 5. Relative to Friday’s level, how might you describe the

change that occurred to Henrique’s level as a percent? 6. Relative to the SILVER level, how might you describe the change that occurred since Friday as a

percent? 7. What percent change would need to occur for Henrique to reach the SILVER level after Saturday?

Henrique’s Current Level

Silver Level

Henrique’s Current Level

Silver Level

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Use your knowledge of percent change to determine the unknown values for the following exercises. 8. Inflation has caused the price of your favorite meal at your favorite restaurant to go up. Originally, the

cost of the meal was $14.50. The new price of the meal is $15.35. Determine the percent of change in the cost of the meal, to the nearest whole percent. State whether the percent change is an increase or a decrease.

9. A pair of jeans normally cost $55. The same pair of jeans is now 27% off. What is the

price of the jeans? 10. The National Football League’s, NFL™ , football field is 120 yards long (counting the end zones). The

Canadian Football League, CFL, is 25% longer. How long is the CFL’s football field? 11. In 2002 there were 12.1 million Federal employees. In 2006, there were 14.6 million Federal employees.

What was the percent change in Federal employees from 2002 to 2006? 12. A dining room table had a price increase of 15% to $550.00 from its original cost, due to an increase in

demand for the type of wood the furniture was constructed of. What was the original cost of the table to the nearest penny?

13. Write your own application percent of change problem.


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Practice Exercises 1. In a certain algebra class, 4 out of 5 students are planning on a career in engineering. Write this ratio in

three forms. Also, write a ratio of students who will be engineers compared to those who are not planning to be engineers.

2. For the object drawn on the left, draw a second similar object according to the ratio of 2 to 1. 3. If the current exchange rate has one U.S. dollar worth 0.68042 Euros, how many U.S. dollars can you get

for exchanging 125 Euros? Round your answer to the nearest cent. 4. For the following similar triangles, find the length of the unknown side to the nearest tenth of a

centimeter. 5. For each of the following, calculate the percent of change from the original amount and state whether

the percent of change is an increase or a decrease. Round all answers to the nearest tenth of a percent.

a. Socks: Original cost per pair is $6.00 b. Suit: Original cost per suit is $175.95 New cost per pair is $4.80 New cost per suit is $96.77

b. Clock radio: Original cost per radio is $35.00 New cost per pair is $39.00

9 cm

8 cm

5.85 cm

x

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6. Write a percent of change problem with an increase of 55%. 7. How are ratios used to model real-world phenomenon? 8. Solve the following proportions for the unknown variable. If needed, round to the nearest tenth.

a. = 812 15x b. =5.1

12.2 6.8z

c. =16 59y

d. =12 57 x

9. Solve the following proportions for the unknown variable. If needed, round to the nearest tenth.

a. − =5 0.2080

x b. −=2 23 10

x

c. −=8 3015 24

x d. − =20 0.25xx

e. − =20 0.2520

x f. −=5 68 28

x


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Outcome Sentences I know that a ratio is I know that a proportion The difference between a ratio and a fraction is The difference between a ratio and a proportion is For me, the best way to solve a proportion is to I am still having trouble with

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Vocabulary Organizer Template

Word:

Definition:

Clue (Picture or Words):

Examples:

Word:

Definition:


Examples:

Word:

Definition:


Examples:

Word:

Definition:


Examples:


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Tile Pad

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Equal Tile Pad

=

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Inequality Tile Pad

= <

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References & Resources The authors and contributors of Algebra II Foundations gratefully acknowledges the following resources: Donovan, Suzanne M.; Bransford, John D. How Students Learn Mathematics in the Classroom. Washington, DC: The

National Academies Press. 2005. Driscoll, Mark. Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10. Portsmouth, NH: Heinemann, 1999. Eves, Howard. An Introduction to the History of Mathematics (5th Edition) Philadelphia, PA: Saunders College Publishing,

1983. Harmin, Merrill. Inspiring Active Learning: A Handbook for Teachers. Alexandria, VA: Association for Supervision and

Curriculum Development, 1994. Harshbarger , Ronald J. and Reynolds, James J., Mathematical Applications for the Management, Life, and Social

Sciences Eighth Edition, Houghton Mifflin Boston, MA 2007. Hoffman, Mark S, ed. The World Almanac and Book of Facts 1992. New York, NY: World Almanac. 1992. Kagan, Spencer. Cooperative Learning. San Clemente, CA: Resources for Teachers. 1994. Karush, William. Webster’s New World Dictionary of Mathematics. New York: Simon & Schuster. 1989. McIntosh, Alistair, Barbara Reys, and Robert Reys. Number Sense: Simple Effective Number Sense Experiences. Parsippany,

New Jersey: Dale Seymour Publications. 1997. McTighe, Jay; Wiggins, Grant. Understand by Design. Alexandria, VA: Association for Supervision and Curriculum

Development. 2004. Marzano, Robert J. Building Background Knowledge for Academic Achievement. Alexandria, VA: Association for

Supervision and Curriculum Development. 2004. Marzano, Robert J.; Pickering, Debra J.; Jane E. Pollock. Classroom Instruction that Works. Alexandria, VA:

Association for Supervision and Curriculum Development. 2001. National Research Council. Adding It Up: Helping Children Learn Mathematics. Washington, DC: National Academy Press.

2001. Ogle, D.M. (1986, February). “K-W-L: A Teaching Model That Develops Active Reading of Expository Text.” The Reading

Teacher, 39(6), 564–570. The National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: The

National Council of Teachers of Mathematics. 2000. Van de Walle, Jon A. Elementary and Middle School Mathematics: Teaching Developmentally (4th Edition). New York: Addison

Wesley Longman, Inc. 2001. The authors and contributors Algebra II Foundations gratefully acknowledges the following internet resources: http://www.metalprices.com/# http://nationalzoo.si.edu/Animals/AsianElephants/factasianelephant.cfm http://hypertextbook.com/facts/1998/JuanCancel.shtml http://www.conservationinstitute.org/ocean_change/predation/salmonsharks.htm http://blogs.payscale.com/ask_dr_salary/2007/03/starting_salari.html http://www.mpaa.org/FlmRat_Ratings.asp (December 2008) http://federaljobs.net/fbijobs.htm (December 2008) http://www-pao.ksc.nasa.gov/kscpao/release/2000/103-00.htm (December 2008) www.seaworld.org http://www.dailyherald.com/story/?id=92571 http://www.infoplease.com/ipa/A0004598.html http://www.washingtonpost.com/wp-dyn/content/article/2006/10/05/AR2006100501782.html http://www.dxing.com/frequenc.htm http://www.wjhuradio.com/ http://www.wrko.com/

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http://www.census.gov http://www.ehs.washington.edu/rso/calculator/chelpdk.shtm http://en.wikipedia.org/wiki/Petroleum http://www.census.gov/ http://en.wikipedia.org/wiki/2004_Indian_Ocean_earthquake http://www.fs.fed.us/gpnf/mshnvm/ http://www.popularmechanics.com/home_journal/workshop/4224738.html http://www.fitness.gov/exerciseweight.htm http://www.economagic.com/em-cgi/data.exe/cenc25/c25q07 http://www.ndbc.noaa.gov/hurricanes/1999/floyd.shtml

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