name: unit 1 • relationships between quantities lesson 1: interpreting structure...

Unit 1 • Relationships Between QuantitiesLesson 1: Interpreting Structure in Expressions

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Assessment

CCSS IP Math I Teacher Resource© Walch Educationu1-1

Pre-AssessmentCircle the letter of the best answer.

1. How many terms are in the expression 36x3 + 27x2 – 18x – 9?

a. 3

b. 7

c. 4

d. 9

2. What are the factors in the expression 11x2 + 7x – 4?

a. 11 and x2, 7 and x

b. 11 and 7

c. There aren’t any factors in this expression.

d. x

3. What are the term(s), coefficient, and constant described by the phrase, “the cost of 4 tickets to the football game, t, and a service charge of $10”?

a. term: 4t, coefficient: 4, constant: 10

b. terms: 4t and 10, coefficient: 10, constant: 4

c. term: 14t, coefficient: 14, constant: none

d. terms: 4t and 10, coefficient: 4, constant: 10

continued


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Assessment

CCSS IP Math I Teacher Resourceu1-2

© Walch Education

4. Newer books can be downloaded for $8 each, while older books can be downloaded for $4 each. The total cost, in dollars, of 12 books is represented by the expression 8n + 4(12 – n), where n represents the number of new books downloaded. How does changing the value of n change the value of the term 4(12 – n)?

a. For values of n less than 12, the term 4(12 – n) will be negative; for values of n greater than 12, the term will be positive; for values of n equal to 12, the term will equal 0.

b. For values of n less than 12, the term 4(12 – n) will be positive; for values of n greater than 12, the term will be positive; for values of n equal to 12, the term will equal 0.

c. For values of n less than 12, the term 4(12 – n) will be positive; for values of n greater than 12, the term will be negative; for values of n equal to 12, the term will equal 0.

d. For values of n less than 12, the term 4(12 – n) will be negative; for values of n greater than 12, the term will be negative; for values of n equal to 12, the term will equal 0.

5. An amount of money deposited in a bank account earns interest according to the following expression: P(1 + r)t, where P represents the initial deposit, r represents the rate, and t represents the period of time. How does increasing the period of time change the value of P?

a. Increasing the period of time will increase the value of P.

b. Increasing the period of time does not change the value of P.

c. Increasing the period of time will decrease the value of P.

d. It is not possible to increase the period of time.

Lesson 1: Interpreting Structure in Expressions

Unit 1 • Relationships Between Quantities

Instruction


Common Core State Standards

A–SSE.1 Interpret expressions that represent a quantity in terms of its context.★

a. Interpret parts of an expression, such as terms, factors, and coefficients.

b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Essential Questions

1. How are algebraic expressions different from algebraic equations?

2. How is the order of operations applied to expressions and simple formulas at specific values?

3. How are verbal phrases translated into algebraic expressions?

WORDS TO KNOW

algebraic expression a mathematical statement that includes numbers, operations, and variables to represent a number or quantity

base the factor being multiplied together in an exponential expression; in the expression ab, a is the base

coefficient the number multiplied by a variable in an algebraic expressionconstant a quantity that does not changeexponent the number of times a factor is being multiplied together in an exponential

expression; in the expression ab, b is the exponentfactor one of two or more numbers or expressions that when multiplied produce

a given productlike terms terms that contain the same variables raised to the same powerorder of operations the order in which expressions are evaluated from left to right (grouping

symbols, evaluating exponents, completing multiplication and division, completing addition and subtraction)

term a number, a variable, or the product of a number and variable(s)variable a letter used to represent a value or unknown quantity that can change or

vary


Instruction


© Walch Education

Recommended Resources• Math-Play.com. “Algebraic Expressions Millionaire Game.”

http://walch.com/rr/CAU1L1Expressions

“Algebraic Expressions Millionaire Game” can be played alone or in two teams. For each question, players have to identify the correct mathematical expression that models a given expression.

• Quia. “Algebraic Symbolism Matching Game.”

http://walch.com/rr/CAU1L1AlgSymbolism

In this matching game, players pair each statement with its algebraic interpretation. There are 40 matches to the provided game.

http://walch.com/rr/CAU1L1Expressions

http://walch.com/rr/CAU1L1AlgSymbolism


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Lesson 1.1.1: Identifying Terms, Factors, and Coefficients

Warm-Up 1.1.1Ella purchased 2 DVDs and 3 CDs from Tyler’s Electronics at the prices listed below. After taxes, her total cost increased by $5.60.

Item Cost ($)CD c

DVD d

1. How can you write the cost of 2 DVDs as an algebraic expression?

2. How can you write the cost of 2 DVDs and 3 CDs as an algebraic expression?

3. How can you write the cost of 2 DVDs and 3 CDs, increased by $5.60 for taxes, as an algebraic expression?


Instruction


© Walch Education

Lesson 1.1.1: Identifying Terms, Factors, and CoefficientsCommon Core State Standard


a. Interpret parts of an expression, such as terms, factors, and coefficients.

Warm-Up 1.1.1 Debrief1. How can you write the cost of 2 DVDs as an algebraic expression?

The cost of 2 DVDs can be written as an expression of multiplication, 2d .

2. How can you write the cost of 2 DVDs and 3 CDs as an algebraic expression?

The cost of 2 DVDs and 3 CDs can be written as the expression 2d + 3c.

3. How can you write the cost of 2 DVDs and 3 CDs, increased by $5.60 for taxes, as an algebraic expression?

“Increased by $5.60 for taxes” means “to add 5.60.”

2d + 3c + 5.60 would represent the cost of 2 DVDs and 3 CDs increased by $5.60.

Connection to the Lesson

• In this lesson, students will be asked to identify parts of expressions given in context.

• It will be necessary for students to be able to first translate a given context into an algebraic expression prior to identifying its parts.


Instruction


Prerequisite Skills

This lesson requires the use of the following skills:

• translating verbal expressions to algebraic expressions

• evaluating expressions following the order of operations

IntroductionThoughts or feelings in language are often conveyed through expressions; however, mathematical ideas are conveyed through algebraic expressions. Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. Variables are letters used to represent values or unknown quantities that can change or vary. One example of an algebraic expression is 3x – 4. Notice the variable, x.

Key Concepts

• Expressions are made up of terms. A term is a number, a variable, or the product of a number and variable(s). An addition or subtraction sign separates each term of an expression.

• In the expression 4x2 + 3x + 7, there are 3 terms: 4x2, 3x, and 7.

• The factors of each term are the numbers or expressions that when multiplied produce a given product. In the example above, the factors of 4x2 are 4 and x2. The factors of 3x are 3 and x.

• 4 is also known as the coefficient of the term 4x2. A coefficient is the number multiplied by a variable in an algebraic expression. The coefficient of 3x is 3.

• The term 4x2 also has an exponent. Exponents indicate the number of times a factor is being multiplied by itself. In this term, 2 is the exponent and indicates that x is multiplied by itself 2 times.

• Terms that do not contain a variable are called constants because the quantity does not change. In this example, 7 is a constant.

Expression 4x2 + 3x + 7Terms 4x2 3x 7

Factors 4 and x2 3 and xCoefficients 4 3

Constants 7


Instruction


© Walch Education

• Terms with the same variable raised to the same exponent are called like terms. In the example 5x + 3x – 9, 5x and 3x are like terms. Like terms can be combined following the order of operations by evaluating grouping symbols, evaluating exponents, completing multiplication and division, and completing addition and subtraction from left to right. In this example, the sum of 5x and 3x is 8x.

Common Errors/Misconceptions

• incorrectly following the order of operations

• incorrectly identifying like terms

• incorrectly combining terms involving subtraction


Instruction


Guided Practice 1.1.1 Example 1

Identify each term, coefficient, constant, and factor of 2(3 + x) + x(1 – 4x) + 5.

1. Simplify the expression.

The expression can be simplified by following the order of operations and combining like terms.

2(3 + x) + x(1 – 4x) + 5 Distribute 2 over 3 + x.6 + 2x + x(1 – 4x) + 5 Distribute x over 1 – 4x. 6 + 2x + x – 4x2 + 5 Combine like terms: 2x and x; 6 and 5.11 + 3x – 4x2

It is common to rearrange the expression so the powers are in descending order, or go from largest to smallest power.

–4x2 + 3x + 11

2. Identify all terms.

There are three terms in the expression: –4x2, 3x, and 11.

3. Identify any factors.

The numbers or expressions that, when multiplied, produce the product –4x2 are –4 and x2. The numbers or expressions that, when multiplied, produce the product 3x are 3 and x.

4. Identify all coefficients.

The number multiplied by a variable in the term –4x2 is –4; the number multiplied by a variable in the term 3x is 3; therefore, –4 and 3 are coefficients.

5. Identify any constants.

The number that does not change in the expression is 11; therefore, 11 is a constant.


Instruction


© Walch Education

Example 2

A smartphone is on sale for 25% off its list price. The sale price of the smartphone is $149.25. What expression can be used to represent the list price of the smartphone? Identify each term, coefficient, constant, and factor of the expression described.

1. Translate the verbal expression into an algebraic expression.

Let x represent the unknown list price. Describe the situation. The list price is found by adding the discounted amount to the sale price:

sale price + discount amount

The discount amount is found by multiplying the discount percent by the unknown list price. The expression that represents the list price of the smartphone is 149.25 + 0.25x.


There are two terms described in the expression: the sale price of $149.25, and the discount of 25% off the list price, or 149.25 and 0.25x.

3. Identify the factors.

0.25x is the product of the factors 0.25 and x.


0.25 is multiplied by the variable, x; therefore, 0.25 is a coefficient.


The number that does not change in the expression is 149.25; therefore, 149.25 is a constant.


Instruction


Example 3

Helen purchased 3 books from an online bookstore and received a 20% discount. The shipping cost was $10 and was not discounted. Write an expression that can be used to represent the total amount Helen paid for 3 books plus the shipping cost. Identify each term, coefficient, constant, and factor of the expression described.

1. Translate the verbal expression into an algebraic expression.

Let x represent the unknown price. The expression used to represent the total amount Helen paid for the 3 books plus shipping is 3x – 0.20(3x) + 10.

2. Simplify the expression.

The expression can be simplified by following the order of operations and combining like terms.

3x – 0.20(3x) + 10 Multiply 0.20 and 3x.3x – 0.60x + 10 Combine like terms: 3x and –0.60x.2.4x + 10


There are two terms in the described expression: the product of 2.4 and x, and the shipping charge of $10: 2.4x and 10.

4. Identify the factors.

2.4x is the product of the factors 2.4 and x.


2.4 is multiplied by the variable, x; therefore, 2.4 is a coefficient.


The number that does not change in the expression is 10; therefore, 10 is a constant.


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© Walch Education

Problem-Based Task 1.1.1: Identifying Parts of an Expression in ContextTara and two friends had dinner at a Spanish tapas restaurant that charged $6 per tapa, or appetizer. The three of them shared several tapas. The total bill included taxes of $4.32. What are the terms, factors, and coefficients of the algebraic expression that represents the number of tapas ordered, including taxes?


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Problem-Based Task 1.1.1: Identifying Parts of an Expression in Context

Coaching a. What was the cost of each tapa without including taxes?

b. What algebraic expression can be used to represent the number of tapas ordered if each one cost $6?

c. What algebraic expression can be used to represent the cost of the tapas ordered including taxes?

d. How many terms does the expression from part c include?

e. What are the factors that make up each of the terms?

f. What are the coefficients of each term?


Instruction


© Walch Education

Problem-Based Task 1.1.1: Identifying Parts of an Expression in Context

Coaching Sample Responsesa. What was the cost of each tapa without including taxes?

Each tapa cost $6.

b. What algebraic expression can be used to represent the number of tapas ordered if each one cost $6? The number of tapas ordered is unknown. Let n represent the unknown number of tapas. The total cost of n tapas is represented by 6n.

c. What algebraic expression can be used to represent the cost of the tapas ordered including taxes? The total bill included $4.32 in taxes. The cost of the tapas is 6n. The expression that represents the cost of the tapas ordered and the taxes is 6n + 4.32.

d. How many terms does the expression from part c include? There are two terms in the expression 6n + 4.32: 6n and 4.32.

e. What are the factors that make up each of the terms? The factors are the products of the quantities of each term. The second term, 4.32, is a constant and does not have factors. The product of 6n is made up of the factors 6 and n.

f. What are the coefficients of each term? The coefficient of the term 6n is 6. The second term, 4.32, is a constant and does not have a coefficient.

Recommended Closure Activity

Select one or more of the essential questions for a class discussion or as a journal entry prompt.


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Practice 1.1.1: Identifying Terms, Factors, and CoefficientsFor problems 1 and 2, identify the terms, coefficients, constants, and factors of the given expressions.

1. 12a3 + 16a + 4

2. 21x2 + 3x – 15x2 + 9

For problems 3 and 4, translate each verbal expression to an algebraic expression. Then, identify the terms, coefficients, and constants of the given expressions.

3. half the sum of x and y decreased by one-third y

4. the product of 5 and the cube of x, increased by the difference of 6 and x3

5. Write an expression with 4 terms, containing the coefficients 3, 6, and 9.

continued


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© Walch Education

For problems 6–10, write an algebraic expression to describe each situation. Then, identify the terms, coefficients, constants, and factors.

6. Gavin agrees to buy a 6-month package deal of monthly gym passes, and in turn receives a 15% discount. Write an algebraic expression to represent the total cost of the monthly passes with the discount, if x represents the cost of each monthly pass.

7. Andre purchased 10 cans of tennis balls from an online store and received a 25% discount. Shipping cost $5.99. Write an algebraic expression to represent the total cost of the tennis balls with the shipping cost, if x represents the cost of each can.

8. Nadia and some friends went to a movie. Their total cost was $30.24, which included taxes of $2.24. Write an algebraic expression to represent the price of each movie ticket, not including taxes. Let x represent the number of Nadia’s friends that went to the movies.

9. The area of a trapezoid can be found by multiplying the height of the trapezoid by half of the sum of base

1 and base

2.

10. The surface area of a cylinder with radius r and height h is twice the product of π and the square of the radius plus twice the product of π, the radius, and the height.


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Lesson 1.1.2: Interpreting Complicated Expressions

Warm-Up 1.1.2Read the scenario below. Use words and symbols to explain your answers to the questions that follow.

At the beginning of the school year, Javier deposited $750 in an account that pays 3% of his initial deposit each year. He left the money in the bank for 5 years.

1. How much interest did Javier earn in 5 years?

2. After 5 years, what is the total amount of money that Javier has?


Instruction


© Walch Education

Lesson 1.1.2: Interpreting Complicated ExpressionsCommon Core State Standard


b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

Warm-Up 1.1.2 DebriefAt the beginning of the school year, Javier deposited $750 in an account that pays 3% of his initial deposit each year. He left the money in the bank for 5 years.

1. How much interest did Javier earn in 5 years?

The amount of interest Javier earned in 5 years is $112.50.

5(0.03 • 750) = 112.50

It is not necessary at this point for students to create and solve an equation to represent this situation, but it is important that the interest amount is calculated based on the initial deposit of $750. Look out for students who try to calculate compound interest. There are several methods to calculating interest; compound interest will be discussed later in the lesson. Focus the discussion on calculating 3% simple interest over five years.

2. After 5 years, what is the total amount of money that Javier has?

The total amount of money that Javier has after 5 years is the sum of the simple interest and the initial deposit, or $862.50.

112.50 + 750.00 = 862.50


• In this lesson, students will interpret the parts of given expressions and how changes to each part affect the expression.

• This warm-up provides an opportunity for students to begin looking at changes in expressions.


Instruction


Prerequisite Skills


• evaluating expressions using order of operations

• evaluating expressions for a given value

• identifying parts of an expression

IntroductionAlgebraic expressions, used to describe various situations, contain variables. It is important to understand how each term of an expression works and how changing the value of variables impacts the resulting quantity.

Key Concepts

• If a situation is described verbally, it is often necessary to first translate each expression into an algebraic expression. This will allow you to see mathematically how each term interacts with the other terms.

• As variables change, it is important to understand that constants will always remain the same. The change in the variable will not change the value of a given constant.

• Similarly, changing the value of a constant will not change terms containing variables.

• It is also important to follow the order of operations, as this will help guide your awareness and understanding of each term.


• incorrectly translating given verbal expressions


Instruction


© Walch Education

Guided Practice 1.1.2Example 1

A new car loses an average value of $1,800 per year for each of the first six years of ownership. When Nia bought her new car, she paid $25,000. The expression 25,000 – 1800y represents the current value of the car, where y represents the number of years since Nia bought it. What effect, if any, does the change in the number of years since Nia bought the car have on the original price of the car?

1. Refer to the expression given: 25,000 – 1800y.

The term 1800y represents the amount of value the car loses each year, y. As y increases, the product of 1800 and y also increases.

2. 25,000 represents the price of the new car.

As y increases and the product of 1800 and y increases, the original cost is not affected. 25,000 is a constant and remains unchanged.

Example 2

To calculate the perimeter of an isosceles triangle, the expression 2s + b is used, where s represents the length of the two congruent sides and b represents the length of the base. What effect, if any, does increasing the length of the congruent sides have on the expression?

1. Refer to the expression given: 2s + b.

Changing only the length of the congruent sides, s, will not impact the length of base b since b is a separate term.

2. If the value of the congruent sides, s, is increased, the product of 2s will also increase. Likewise, if the value of s is decreased, the value of 2s will also decrease.

3. If the value of s is changed, the result of the change in the terms is a doubling of the change in s while the value of b remains the same.


Instruction


Example 3

Money deposited in a bank account earns interest on the initial amount deposited as well as any interest earned as time passes. This compound interest can be described by the expression P(1 + r)n, where P represents the initial amount deposited, r represents the interest rate, and n represents the number of months that pass. How does a change in each variable affect the value of the expression?

1. Refer to the given expression: P(1 + r)n.

Notice the expression is made up of one term containing the factors P and (1 + r)n.

2. Changing the value of P does not change the value of the factor (1 + r)n, but it will change the value of the expression by a factor of P. In other words, the change in P will multiply by the result of (1 + r)n.

3. Similarly, changing r changes the base of the exponent (the number that will be multiplied by itself), but does not change the value of P. This change will affect the value of the overall expression.

4. Changing n changes the number of times (1 + r) will be multiplied by itself, but does not change the value of P. This change will affect the value of the overall expression.


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© Walch Education

Problem-Based Task 1.1.2: Searching for a Greater SavingsAustin plans to open a savings account. The amount of money in a savings account can be found by using the equation s = p • (1 + r)t, where p is the principal, or the original amount deposited into the account; r is the rate of interest; and t is the amount of time. Austin is considering two savings accounts. He will deposit $1,000.00 as the principal into either account. In Account A, the interest rate will be 0.015 per year for 5 years. In Account B, the interest rate will be 0.02 per year for 3 years. If he could, would it be wise for Austin to leave his money in the account that has less savings for an additional year? Explain your reasoning.


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Problem-Based Task 1.1.2: Searching for a Greater Savings

Coachinga. What is the total amount in Austin’s savings if he chooses Account A?

b. What is the total amount in Austin’s savings if he chooses Account B?

c. Which account has more money at the end of the term?

d. If Austin left his money in the account that has less savings for an additional year, would this change which account he might select? Explain your answer.


Instruction


© Walch Education

Problem-Based Task 1.1.2: Searching for a Greater Savings

Coaching Sample Responses

a. What is the total amount in Austin’s savings if he chooses Account A?

Use the equation s = p • (1 + r)t to determine the amount in Account A. Identify p, r, and t from the problem statement.

The principal, p, is $1,000; the interest rate, r, is 0.015; and the time, t, is 5 years.

Replace the variables in the equation with the given quantities.

s = p • (1 + r)t = (1000) • (1 + 0.015)5

Begin to evaluate the expression using the order of operations. Start by performing any operations in parentheses.

(1 + 0.015) is in parentheses. Find the sum.

(1000) • (1 + 0.015)5 = (1000) • (1.015)5

Next, evaluate any exponential expressions.

The expression (1.015)5 is an exponential expression.

(1000) • (1.015)5 ≈ (1000) • (1.08)

Next, find any products or quotients, evaluating all multiplication and division from left to right.

The final expression, (1000) • (1.08), is a product.

(1000) • (1.08) = 1080

s = (1000) • (1 + 0.015)5 ≈ 1080

If he selects Account A, after five years Austin will have approximately $1,080.00 in savings.


Instruction


b. What is the total amount in Austin’s savings if he chooses Account B?

Use the equation s = p • (1 + r)t to determine the amount in Account B. Identify p, r, and t from the problem statement.

The principal, p, is $1,000; the interest rate, r, is 0.02; and the time, t, is 3 years.

Replace the variables in the equation with the given quantities.

s = p • (1 + r)t = (1000) • (1 + 0.02)3

Begin to evaluate the expression using the order of operations. Start by performing any operations in parentheses.


(1000) • (1 + 0.02)3 = (1000) • (1.02)3



(1000) • (1.02)3 ≈ (1000) • (1.06)



(1000) • (1.06) = 1060

s = (1000) • (1 + 0.02)3 ≈ 1060

If he selects Account B, after three years Austin will have approximately $1,060.00 in savings.

c. Which account has more money at the end of the term?

Compare the results from parts a and b.

If Austin leaves the money in Account A for 5 years, he will have $1,080.00.

If Austin leaves the money in Account B for 3 years, he will have $1,060.00.

Account A has more money at the end of the term.


Instruction


© Walch Education

d. If Austin left his money in the account that has less savings for an additional year, would this change which account he might select? Explain your answer.

Re-evaluate the total amount in savings if Austin left the money in Account B, the account with the lower total savings, for one extra year.

The new value of t in the equation s = p • (1 + r)t would be 4. The values of p and r will be the same as the ones used in part b: p = 1000 and r = 0.02. Replace the values of p, r, and t in the equation s = p • (1 + r)t to find the total amount in Account B after 4 years.

s = (1000) • (1 + 0.02)4

Evaluate the equation using the order of operations. Begin by performing any operations in parentheses.


(1000) • (1 + 0.02)4 = (1000) • (1.02)4



(1000) • (1.02)4 ≈ (1000) • (1.08)



(1000) • (1.08) = 1080

s = (1000) • (1 + 0.02)4 ≈ 1080

After four years, Austin will have approximately $1,080.00 in savings in Account B. This is the same amount that would be in Account A after five years, but the money would be available one year sooner. Austin should put his money in Account B.




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Practice 1.1.2: Interpreting Complicated ExpressionsUse your understanding of terms, coefficients, factors, exponents, and the order of operations to answer each of the following questions.

1. Is the expression x5 3

2+

equal to the expression 4x? Explain your answer.

2. Is the expression 2 • 4x equal to the expression 8x? Explain your answer.

3. Is the expression (5 • 2) x equal to the expression 10 x ? Explain your answer.

4. A transfer station charges $15 for a waste disposal permit and an additional $5 for each cubic yard of garbage it disposes of. This relationship can be described using the expression 15 + 5x. What effect, if any, does changing the value of x have on the cost of the permit?

5. Absolute Cable company bills on a monthly basis. Each bill includes a $30.00 service fee plus $4.75 in taxes and $2.99 for each movie purchased. The following expression describes the cost of the cable service per month: 34.75 + 2.99m. If Absolute Cable lowers the service fee, how will the expression change?

continued


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© Walch Education

6. In order to lose weight in a healthy manner, a veterinarian suggested an overweight large-breed dog lose no more than 2 pounds per week. If the expression x – 2y represents this situation, what must be true about the value of y?

7. The product of 7, x, and y is represented by the expression 7xy. If the value of x is negative, what can be said about the value of y in order for the product to remain positive?

8. A bank account balance for an account with an initial deposit of P dollars earns interest at an annual rate of r. The amount of money in the account after n years is described using the following expression: P(1 + r)n. What effect, if any, does decreasing the value of r have on the amount of money after n years?

9. For what values of x will the result of 5 x be greater than 1?

10. A tire can hold C cubic feet of air. It loses air at a rate of r for a period of time, t. This situation can be described using the following formula: C(1 – r)t. What effect, if any, does increasing the value of r have on the value C ?

Unit 1 • Relationships Between QuantitiesLesson 2: Creating Equations and Inequalities in One Variable

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Assessment


© Walch Education

Pre-AssessmentCircle the letter of the best answer.

1. Mina bought a plane ticket to New York City and used a coupon for 15% off the ticket price. The total cost of her ticket, with the discount, was $253.30. What equation could she use to find the price of the ticket without the discount?

a. 0.15x = 253.30

b. x – 0.15(x) = 253.30

c. x = 253.30 + 0.15

d. x + 0.15(x) = 253.30

2. Lucas bought a refrigerator. His total cost of $1,331 included sales tax at the rate of 8% and an additional, untaxed delivery charge of $35. How much sales tax did he pay?

a. $72

b. $106

c. $120

d. $96

3. Your cell phone plan allows you 400 minutes to talk per month. So far this month, you have used 265 minutes and you have 7 days left on this month’s plan. Which inequality could you use to determine how many minutes at most you can use per day so that you don’t go over your monthly plan minutes?

a. 7x + 265 < 400

b. 7x + 265 > 400

c. 7x + 265 ≤ 400

d. 7x + 265 ≥ 400

4. Sydney has a $75 mall gift card. She wants to buy a sundress and a movie ticket. The movie ticket with tax costs $11.50. The sales tax on the sundress will be 4%. How much can the ticketed price of the sundress be?

a. less than or equal to $61.06

b. less than $61.06

c. less than or equal to $45.36

d. less than $45.36

5. A population of rabbits doubles every 4 months. If the population starts out with 8 rabbits, how many rabbits will there be in 1 year?

a. 128 rabbits

b. 64 rabbits

c. 32 rabbits

d. 16 rabbits

Lesson 2: Creating Equations and Inequalities in One Variable

Unit 1 • Relationships Between Quantities

Instruction


Common Core State Standards

N–Q.2 Define appropriate quantities for the purpose of descriptive modeling.★

N–Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.★

A–CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.★

Essential Questions

1. How are quantities modeled with equations and inequalities?

2. How are equations and inequalities alike and different?

3. What makes creating an exponential equation different from creating a linear equation?

WORDS TO KNOW

equation a mathematical sentence that uses an equal sign (=) to show that two quantities are equal

exponential decay an exponential equation with a base, b, that is between 0 and 1 (0 < b < 1); can be represented by the formula y = a(1 – r) t, where a is the initial value, (1 – r) is the decay rate, t is time, and y is the final value

exponential equation an equation that has a variable in the exponent; the general form is y = a • b x, where a is the initial value, b is the base, x is the time, and y

is the final output value. Another form is y abx

t= , where t is the time it takes for the base to repeat.

exponential growth an exponential equation with a base, b, greater than 1 (b > 1); can be represented by the formula y = a(1 + r) t, where a is the initial value, (1 + r) is the growth rate, t is time, and y is the final value

inequality a mathematical sentence that shows the relationship between quantities that are not equivalent

linear equation an equation that can be written in the form ax + b = c, where a, b, and c are rational numbers


Instruction


© Walch Education

quantity something that can be compared by assigning a numerical value

rate a ratio that compares different kinds of units

solution a value that makes the equation true

solution set the value or values that make a sentence or statement true

unit rate a rate per one given unit

variable a letter used to represent a value or unknown quantity that can change or vary

Recommended Resources• APlusMath. “Algebra Planet Blaster.”

http://walch.com/rr/CAU1L2LinEquations

Players solve each multi-step linear equation to find the correct planet to “blast.” Incorrect answers cause players to destroy their own ship.

• Figure This! Math Challenges for Families. “Challenge 24: Gasoline Tanks.”

http://walch.com/rr/CAU1L2Rates

This website features a description of the math involved, related occupations, a hint to get started, complete solutions, and a “Try This” section, as well as additional related problems with answers, questions to think about, and resources for further exploration. These resources could be used to lead a group discussion or as an investigation. Students could work in small groups to create a presentation related to each challenge.

• Purplemath.com. “Exponential Functions: Introduction.”

http://walch.com/rr/CAU1L2ExpEquations

This website gives an introduction of exponential equations and provides a few examples of tables of input and output values, with integers as inputs. The introduction goes into more depth about the shapes of the graphs of exponential functions and continues on to develop the concept of compound interest. It also introduces the number e.

http://walch.com/rr/CAU1L2LinEquations

http://walch.com/rr/CAU1L2Rates

http://walch.com/rr/CAU1L2ExpEquations


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Lesson 1.2.1: Creating Linear Equations in One Variable

Warm-Up 1.2.1Read the scenario and answer the questions that follow.

Andrew is practicing for a tennis tournament and needs more tennis balls. He bought 10 cans of tennis balls online and received a 25% discount. The shipping cost was $5.99. Let x represent the cost of each can.

1. Write an algebraic expression to represent the cost of the tennis balls.

2. Write an algebraic expression to represent the cost of the tennis balls with the discount.

3. Write an algebraic expression to represent the total cost of the tennis balls with the shipping cost and the discount. Simplify the expression.


Instruction


© Walch Education

Lesson 1.2.1: Creating Linear Equations in One VariableCommon Core State Standards

N–Q.2 Define appropriate quantities for the purpose of descriptive modeling.★

N–Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.★


Warm-Up 1.2.1 DebriefAndrew is practicing for a tennis tournament and needs more tennis balls. He bought 10 cans of tennis balls online and received a 25% discount. The shipping cost was $5.99. Let x represent the cost of each can.

1. Write an algebraic expression to represent the cost of the tennis balls.

Andrew purchased 10 cans of tennis balls at an unknown price, x. Therefore, the expression to represent the cost of the tennis balls is 10x.

2. Write an algebraic expression to represent the cost of the tennis balls with the discount.

First, Andrew will be charged the cost of the tennis balls (10x).

Then, 25% will be discounted or taken off the cost of the tennis balls, so –0.25(10x).

Add these amounts to arrive at the price of the tennis balls.

10x – 0.25(10x)

10x – 2.5x

7.5x

Students might have trouble remembering to convert the percentage to a decimal. Students might not have simplified.


Instruction


3. Write an algebraic expression to represent the total cost of the tennis balls with the shipping cost and the discount. Simplify the expression.

The shipping cost was $5.99. Add this to the expression found above.

10x – 0.25(10x) + 5.99

10x – 2.5x + 5.99

7.5x + 5.99

Students might try to apply the discount to the shipping cost, but the discount does not apply to shipping.

Encourage students to read the problem scenario carefully before and while writing their expressions.


• Students will extend writing expressions to writing equations in the upcoming lesson.

• The process of translating an expression is the same for translating an equation.

• The warm-up is scaffolded to help students think about breaking down the context into smaller pieces, providing an example for how to think about doing this with an equation context.

• Students will be asked to take a scenario like this a step further and solve for the unknown given a total cost.


Instruction


© Walch Education

Prerequisite Skills


• applying the order of operations

• creating ratios

• translating verbal sentences into expressions

IntroductionCreating equations from context is important since most real-world scenarios do not involve the equations being given. An equation is a mathematical sentence that uses an equal sign (=) to show that two quantities are equal. A quantity is something that can be compared by assigning a numerical value. In this lesson, contexts will be given and equations must be created from them and then used to solve the problems. Since these problems are all in context, units are essential because without them, the numbers have no meaning.

Key Concepts

• A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are rational numbers. Often, the most difficult task in turning a context into an equation is determining what the variable is and how to represent that variable.

• The variables are letters used to represent a value or unknown quantity that can change or vary. Once the equation is determined, solving for the variable is straightforward.

• The solution will be the value that makes the equation true.

• In some cases, the solution will need to be converted into different units. Multiplying by a unit rate or a ratio can do this.

• A unit rate is a rate per one given unit, and a rate is a ratio that compares different kinds of units.

• Use units that make sense, such as when reporting time; for example, if the time is less than 1 hour, report the time in minutes.

• Think about rounding and precision. The more numbers you list to the right of the decimal place, the more precise the number is.

• When using measurement in calculations, only report to the nearest decimal place of the least accurate measurement. See Guided Practice Example 5.


Instruction


Creating Equations from Context

1. Read the problem statement first.

2. Reread the scenario and make a list or a table of the known quantities.

3. Read the statement again, identifying the unknown quantity or variable.

4. Create expressions and inequalities from the known quantities and variable(s).

5. Solve the problem.

6. Interpret the solution of the equation in terms of the context of the problem and convert units when appropriate, multiplying by a unit rate.


• attempting to solve the problem without first reading/understanding the problem statement

• incorrectly setting up the equation

• misidentifying the variable

• forgetting to convert to the correct units

• setting up the ratio inversely when converting units

• reporting too many decimals—greater precision comes with more precise measuring and not with calculations


Instruction


© Walch Education


James earns $15 per hour as a teller at a bank. In one week he pays 17% of his earnings in state and federal taxes. His take-home pay for the week is $460.65. How many hours did James work?

1. Read the statement carefully.

2. Reread the scenario and make a list of the known quantities.

James earns $15 per hour.

James pays 17% of his earning in taxes.

His pay for the week is $460.65.

3. Read the statement again and look for the unknown or the variable.

The scenario asks for James’s hours for the week. The variable to solve for is hours.


James’s pay for the week was $460.65.

______ = 460.65

He earned $15 an hour. Let h represent hours.

15h

He paid 17% in taxes.

–0.17(15h)

Put this information all together.

15h – 0.17(15h) = 460.65


Instruction


5. Solve the equation.

15h – 0.17(15h) = 460.65 Multiply –0.17 and 15h.

15h – 2.55h = 460.65 Combine like terms 15h and –2.55h.

12.45h = 460.65 Divide both sides by 12.45.

h

h

12.45

12.45

460.65

12.4537 hours

=

=h = 37 hours

James worked 37 hours.

6. Convert to the appropriate units if necessary.

The scenario asked for hours and the quantity given was in terms of hours. No unit conversions are necessary.

Example 2

Brianna has saved $600 to buy a new TV. If the TV she wants costs $1,800 and she saves $20 a week, how many years will it take her to buy the TV?



The TV costs $1,800.

Brianna saved $600.

Brianna saves $20 per week.


The scenario asks for the number of years. This is tricky because the quantity is given in terms of weeks. The variable to solve for first, then, is weeks.


Instruction


© Walch Education


Brianna needs to reach $1,800.

_____ = 1800

Brianna has saved $600 so far and has to save more to reach her goal.

600 + ______ = 1800

Brianna is saving $20 a week for some unknown number of weeks to reach her goal. Let x represent the number of weeks.

600 + 20x = 1800

5. Solve the problem for the number of weeks it will take Brianna to reach her goal.

x

x

x

x

600 20 1800

600 600

20 1200

20

20

1200

2060weeks

+ =− −

=

=

=Brianna will need 60 weeks to save for her TV.

6. Convert to the appropriate units.

The problem statement asks for the number of years it will take Brianna to save for the TV. There are 52 weeks in a year.

1 year

52 weeks

60 weeks1 year

52 weeks

60 weeks1 year

52 weeks1.15 years

•

• ≈

Brianna will need approximately 1.15 years, or a little over a year, to save for her TV.


Instruction


Example 3

Suppose two brothers who live 55 miles apart decide to have lunch together. To prevent either brother from driving the entire distance, they agree to leave their homes at the same time, drive toward each other, and meet somewhere along the route. The older brother drives cautiously at an average speed of 60 miles per hour. The younger brother drives faster, at an average speed of 70 mph. How long will it take the brothers to meet each other?


2. Reread the scenario and make a table of the known quantities.

Problems involving “how fast,” “how far,” or “how long” require the distance equation, d = rt, where d is distance, r is rate of speed, and t is time.

Complete a table of the known quantities.

Rate (r) Distance (d)

Older brother 60 mph 55 miles

Younger brother 70 mph 55 mph


The scenario asks for how long, so the variable is time, t.


Instruction


© Walch Education


Step 2 showed that the distance equation is d = rt or rt = d. Together the brothers will travel a distance, d, of 55 miles.

(older brother’s rate)(t) + (younger brother’s rate)(t) = 55

The rate r of the older brother = 60 mph and the rate of the younger brother = 70 mph.

60t + 70t = 55

Expand the table from step 2 to see this another way.

Rate (r) Time (t) Distance (d)Older brother 60 mph t d = 60tYounger brother 70 mph t d = 70t

Together, they traveled 55 miles, so add the distance equations based on each brother’s rate.

60t + 70t = 55

5. Solve the problem for the time it will take for the brothers to meet each other.

60t + 70t = 55 Equation

130t = 55 Combine like terms 60t and 70t.

t

t

130

130

55

1300.42 hours

=

≈

Divide both sides by 130.

It will take the brothers 0.42 hours to meet each other.

Note: The answer was rounded to the nearest hundredth of an hour because any rounding beyond the hundredths place would not make sense. Most people wouldn’t be able to or need to process that much precision. When talking about meeting someone, it is highly unlikely that anyone would report a time that is broken down into decimals, which is why the next step will convert the units.


Instruction



Automobile speeds in the United States are typically given in miles

per hour (mph). Therefore, this unit of measurement is appropriate.

However, typically portions of an hour are reported in minutes unless

the time given is 1

2 of an hour.

Convert 0.42 hours to minutes using 60 minutes = 1 hour.

60min = 1 hr

0.42 hr •60min

1hr

0.42 hr •60min

1 hr25.2minutes=

Here again, rarely would a person report that they are meeting someone in 25.2 minutes. In this case, there is a choice of rounding to either 25 or 26 minutes. Either answer makes sense.

The two brothers will meet each other in 25 or 26 minutes.

Example 4

Think about the following scenarios. In what units should they be reported? Explain the reasoning.

a. Water filling up a swimming pool

A swimming pool, depending on the size, has between several gallons and hundreds of thousands of gallons of water.

Think about water flowing out of a faucet and picture filling up a milk jug. How long does it take? Less than a minute? The point is that gallons of water can be filled in minutes.

Report the filling of a swimming pool in terms of gallons per minute.


Instruction


© Walch Education

b. The cost of tiling a kitchen floor

Think about how big rooms are. They can be small or rather large, but typically they are measured in feet. When calculating the area, the measurement units are square feet.

Tiles cost in the dollar range.

Report the cost of tiling a kitchen floor in dollars per square foot.

c. The effect of gravity on a falling object

Think about how fast an object falls when you drop it from shoulder-height. How far is it traveling from your shoulder to the ground? It travels several feet (or meters).

How long does it take before the object hits the ground? It only takes a few seconds.

Report gravity in terms of feet or meters per second.

d. A snail traveling across the sidewalk

The context of the problem will determine the correct units. Think about how slowly a snail moves. Would a snail be able to travel at least one mile in an hour? Perhaps it makes more sense to report the distance in a smaller unit. Report the snail traveling across the sidewalk in feet per minute.

If comparing speeds of other animals to the snail’s rate, and the animals’ rates are being reported in miles per hour, then it makes sense to report the snail’s rate in miles per hour, too.

e. Painting a room

Think about how long it takes to paint a room. It takes longer than several minutes. It would probably take hours.

How is the surface area of a wall typically measured? It’s usually measured in square feet.

Report the painting of a room in square feet per hour.


Instruction


Example 5

Ernesto built a wooden car for a soap box derby. He is painting the top of the car blue and the

sides black. He already has enough black paint, but needs to buy blue paint. He needs to know the

approximate area of the top of the car to determine the size of the container of blue paint he should

buy. He measured the length to be 9 feet 111

4 inches, and the width to be

1

2 inch less than 3 feet.

What is the surface area of the top of the car? What is the most accurate area Ernesto can use to

buy his paint?



Length = 9 feet 11.25 inches

Width = 35.5 inches (3 feet = 36 inches; 36 – 1

2 inch = 35.5 inches)


The scenario asks for the surface area of the car’s top.

Work with the accuracy component after calculating the surface area.


Instruction


© Walch Education


The surface area will require some assumptions. A soap box derby car is tapered, meaning it is wider at one end than it is at another. To be sure Ernesto has enough paint, he assumes the car is rectangular with the width being measured at the widest location.

A = length × width = lw

For step 2, we listed length and width, but they are not in units that can be multiplied.

Convert the length to inches.

Length = 9 feet 11.25 inches; 9(12) + 11.25 = 119.25 inches

Width = 35.5 inches


Substitute length and width into the formula A = lw.

A = lw

A = 119.25 • 35.5 = 4233.375

This gives a numerical result for the surface area, but the problem asks for the most accurate surface area measurement that can be calculated based on Ernesto’s initial measurements. Since Ernesto only measured to the hundredths place, the answer can only be reported to the hundredths place.

The surface area of the top of Ernesto’s car is 4,233.38 square inches.


Instruction



When buying paint, the hardware store associate will ask how many square feet need to be covered. Ernesto has his answer in terms of square inches. Convert to square feet.

There are 144 square inches in a square foot.

1 ft = 144 in

4233.38 in •1 ft

144 in

4233.38 in •1 ft

144 in29.398472 ft

2 2

22

2

22

2

2=

7. Rounding must take place here again because Ernesto can only report to the hundredths place.

Ernesto’s surface area = 29.40 ft2


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© Walch Education

Problem-Based Task 1.2.1: Rafting and Hiking TripTo celebrate graduation, you and 4 of your closest friends have decided to take a 5-day white-water rafting and hiking trip. During your 5-day trip, 2 days are spent rafting. If the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day, how many miles must you raft each hour?

For the hiking portion of your trip, you and your friends carry the same amount of equipment, which works out to 35 pounds of equipment each. For extra money, you can hire an assistant, who will carry 50 pounds of equipment. Each assistant charges a flat fee of $50 and an additional $22 for each mile. The total amount you would have to pay the assistant is $512. How many miles will your group be hiking? Is it worth hiring two assistants to help you and your friends carry the equipment? Justify your answers.


naMe:


Problem-Based Task 1.2.1: Rafting and Hiking Trip

Coaching a. If the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day,

how many miles must you raft each hour?

What is the ratio of miles to days?

What is the ratio you are looking for?

What is the ratio of days to hours?

How do you convert the original ratio of miles to days into miles per hour?

b. How many miles will your group be hiking?

What is the equation of the cost of hiring an assistant?

What is the solution to this equation?

c. Is it worth hiring two assistants to help you and your friends carry the equipment?

How much weight will each of you carry without assistants?

How much weight will each of you carry with two assistants?

What is the difference in the cost per day?

Are you willing to pay more money to have someone carry your equipment?


Instruction


© Walch Education

Problem-Based Task 1.2.1: Rafting and Hiking Trip

Coaching Sample Responsesa. If the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day,

how many miles must you raft each hour?

What is the ratio of miles to days?60miles

2 days

What is the ratio you are looking for?miles

hour

What is the ratio of days to hours?1 day

8 hours

How do you convert the original ratio of miles to days into miles per hour?

Multiply the two numeric ratios together.

60miles

2 days•1 day

8 hours=60miles

16 hours= 3.75miles/hour

b. How many miles will your group be hiking?

What is the equation of the cost of hiring an assistant?

22x + 50 = 512, where x = miles

What is the solution to this equation?

x = 21 miles

c. Is it worth hiring two assistants to help you and your friends carry the equipment?

How much weight will each of you carry without assistants?

Each person will carry 35 pounds. There are 5 people.

35 × 5 = 175 pounds

How much weight will each of you carry with two assistants?


Instruction


First, determine how much less there would be to carry among the 5 of you with two assistants who each carry 50 pounds.

175 – 2(50) = 175 – 100 = 75 pounds

Now, divide 75 pounds by 5 people.

75

515 pounds=

35 – 15 = 20

If you hire assistants, each of you will carry 15 pounds, which is 20 pounds less than the original amount.

What is the difference in the cost per day?

First, determine the cost for 2 assistants if each will be paid $512.

2(512) = 1024

The cost is $1,024 for 2 assistants.

Next, determine the cost per person.

$1024

5 people$204.80/ person=

The difference is $204.80 per person or almost $205 per person.

Are you willing to pay more money to have someone carry your equipment?

Answers will vary, but should reflect and be justified by students’ calculations. Sample answer: I am willing to pay about an extra $205 to be free of 20 pounds of equipment during my 21-mile hike, so that I can enjoy my experience without being weighed down.




naMe:


© Walch Education

Practice 1.2.1: Creating Linear Equations in One VariableFor the problem below, read each scenario and give the units you would use to work with each situation.

1. What units would you use for each scenario that follows?

a. riding a bicycle

b. rainfall during a storm

c. water coming from a fire hydrant

d. watching caloric intake

For problems 2–8, read each scenario, write an equation, and then solve the problem. Remember to include the appropriate units.

2. You need to buy new tile for your kitchen. It measures 13.25 feet by 7.5 feet. What is the area of the kitchen that you calculated? What is the most accurate area you can report to your hardware store in order to purchase enough tile?

3. Zach watches TV 3 times as much as Joel. Joel watches TV 2 hours a day. How many hours a day does Zach watch TV?

4. It costs Raquel $5 in tolls to drive to work and back each day, plus she uses 3 gallons of gas. It costs her a total of $15.50 to drive to work and back each day. How much per gallon is Raquel paying for her gas? How do you know?

continued


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5. Hayden bought 4 tickets to a football game. He paid a 5% service charge for buying them from a broker. His total cost was $105.00. What was the price of each ticket, not including the service charge?

6. It cost Justin $100 to have cable TV installed in his house. Each month he pays an access fee plus tax of 7% of his monthly bill. After 6 months, Justin had paid a total of $350.38 for his access fee, taxes, and his initial installation. What is Justin’s monthly access fee not including taxes?

7. You and 3 friends divide the proceeds of a garage sale equally. The garage sale earned $412. How much money did you receive?

8. The area of Sofia’s herb garden is 1

8 the area of her vegetable garden. The area of her herb

garden is 6 square feet. What is the area of her vegetable garden?

9. Driving to your friend’s house, you travel at an average rate of 35 miles per hour. On your way home, you travel at an average rate of 40 miles per hour. If the round trip took you 45 minutes, how far is it from your house to your friend’s house?

10. Two trains heading toward each other are 400 miles apart. One train travels 15 miles per hour faster than the other train. If they arrive at the same station in 5 hours, how fast is each train traveling?


naMe:


© Walch Education

Lesson 1.2.2: Creating Linear Inequalities in One Variable

Warm-Up 1.2.2Read the scenario below, write an equation that models the situation, and then use it to answer the questions that follow.

Two people can balance on a seesaw even if they are different weights. The balance will occur when the following equation, w

1d

1 = w

2d

2, is satisfied or true. In this equation,

w1 is the weight of the first person, d

1 is the distance the first person is from the center

of the seesaw, w2 is the weight of the second person, and d

2 is the distance the second

person is from the center of the seesaw.

1. Eric and his little sister Amber enjoy playing on the seesaw at the playground. Amber weighs

65 pounds. Eric and Amber balance perfectly when Amber sits about 4 feet from the center and

Eric sits about 21

2 feet from the center. About how much does Eric weigh?

2. Their little cousin Aleah joins them and sits right next to Amber. Can Eric balance the seesaw with both Amber and Aleah on one side, if Aleah weighs about the same as Amber? If so, where should he sit? If not, why not?


Instruction


Lesson 1.2.2: Creating Linear Inequalities in One VariableCommon Core State Standard


Warm-Up 1.2.2 DebriefTwo people can balance on a seesaw even if they are different weights. The balance will occur when the following equation, w

1d

1 = w

2d

2, is satisfied or true. In this equation, w

1 is the weight of the first

person, d1 is the distance the first person is from the center of the seesaw, w

2 is the weight of the

second person, and d2 is the distance the second person is from the center of the seesaw.

1. Eric and his little sister Amber enjoy playing on the seesaw at the playground. Amber weighs

65 pounds. Eric and Amber balance perfectly when Amber sits about 4 feet from the center and

Eric sits about 21

2 feet from the center. About how much does Eric weigh?

First set up the equation. Students might struggle with which weight is represented by w1 and

which is represented by w2. If time allows, show students that as long as the associated weight

and distance stay together on the same side of the equation, it doesn’t matter which weight is w1

and which is w2.

Given: w1d

1 = w

2d

2

w1 = Amber’s weight = 65 pounds

d1 = Amber’s distance = 4 feet

w2 = Eric’s weight = x pounds

d2 = Eric’s distance = 2

1

2 feet

The unknown is Eric’s weight.

Now, make the substitutions.

(65)(4) = (x) 21

2

Simplify.

260 = 2.5x


Instruction


© Walch Education

Solve.

x = 104

Interpret the solution.

In this equation, x represented Eric’s weight. Eric weighs about 104 pounds.

2. Their little cousin Aleah joins them and sits right next to Amber. Can Eric balance the seesaw with both Amber and Aleah on one side, if Aleah weighs about the same as Amber? If so, where should he sit? If not, why not?

Set up the equation using w1d

1 = w

2d

2.

w1 = Amber and Aleah’s combined weight = 2(65) pounds

d1 = Amber and Aleah’s distance = 4 feet

w2 = Eric’s weight = 104 pounds

d2 = Eric’s distance = x feet

This time, the unknown is Eric’s distance.

Now, make the substitutions.

2(65)(4) = 104(x)

Simplify.

520 = 104x

Solve.

x = 5

Interpret the solution.

In this equation, x represented Eric’s distance from the center of the seesaw. The question asks if it’s possible for Eric to balance with Amber and Aleah. It’s possible if the seesaw is at least 5 feet long from the center, because Eric needs to sit at least 5 feet away from the center. If the seesaw is shorter than 5 feet, then he cannot balance with his sister and his cousin. If the seesaw is 5 feet or longer, then Eric can balance the seesaw.


• Students are asked to create equations much like they will be asked to create inequalities.

• Solving equations is similar to solving inequalities with a few exceptions.

• The second question’s response deals with the inequality concept “at least.”


Instruction


Prerequisite Skills


• solving simple linear equations

• comparing rational numbers

IntroductionInequalities are similar to equations in that they are mathematical sentences. They are different in that they are not equal all the time. An inequality has infinite solutions, instead of only having one solution like a linear equation. Setting up the inequalities will follow the same process as setting up the equations did. Solving them will be similar, with two exceptions, which will be described later.

Key Concepts

• The prefix in- in the word inequality means “not.” Inequalities are sentences stating that two things are not equal. Remember earlier inequalities such as 12 > 2 and 1 < 7.

• Remember that the symbols >, <, ≥, ≤, and ≠ are used with inequalities.

• Use the table below to review the meanings of the inequality symbols and the provided examples with their solution sets, or the value or values that make a sentence or statement true.

Symbol Description Example Solution set> greater than, more than x > 3 all numbers greater than 3;

does not include 3≥ greater than or equal to, at least x ≥ 3 all numbers greater than or

equal to 3; includes 3< less than x < 3 all numbers less than 3; does

not include 3≤ less than or equal to, no more than x ≤ 3 all numbers less than or equal

to 3; includes 3≠ not equal to x ≠ 3 includes all numbers except 3

• Solving a linear inequality is similar to solving a linear equation. The processes used to solve inequalities are the same processes that are used to solve equations.


Instruction


© Walch Education

• Multiplying or dividing both sides of an inequality by a negative number requires reversing the inequality symbol. Here is a number line to show the process.

• First, look at the example of the inequality 2 < 4.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10x

2 < 4

• Multiply both sides by –2 and the inequality becomes 2(–2) < 4(–2) or –4 < –8.

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10x

• Is –4 really less than –8?

• To make the statement true, you must reverse the inequality symbol: –4 > –8

–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10x

–4 > –8

Creating Inequalities from Context

1. Read the problem statement first.2. Reread the scenario and make a list or a table of the known quantities.3. Read the statement again, identifying the unknown quantity or variable.4. Create expressions and inequalities from the known quantities and variable(s).5. Solve the problem.6. Interpret the solution of the inequality in terms of the context of the problem.


• not translating the words into the correct symbols, especially with the phrases no fewer than, no more than, at least as many, and so on

• forgetting to switch the inequality symbol when multiplying or dividing by a negative

• not interpreting the solution in terms of the context of the problem—students can often mechanically solve the problem but don’t know what the solution means


Instruction



Juan has no more than $50 to spend at the mall. He wants to buy a pair of jeans and some juice. If the sales tax on the jeans is 4% and the juice with tax costs $2, what is the maximum price of jeans Juan can afford?



Sales tax is 4%.

Juice costs $2.

Juan has no more than $50.


The unknown quantity is the cost of the jeans.


The price of the jeans + the tax on the jeans + the price of the juice must be less than or equal to $50.

x + 0.04x + 2 ≤ 50


Instruction


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x + 0.04x + 2 ≤ 50 Add like terms.

1.04x + 2 ≤ 50 Subtract 2 from both sides.

1.04x ≤ 48 Divide both sides by 1.04.

x ≤ 46.153846

Normally, the answer would be rounded down to 46.15. However, when dealing with money, round up to the nearest whole cent as a retailer would.

x ≤ 46.16

6. Interpret the solution of the inequality in terms of the context of the problem.

Juan should look for jeans that are priced at or below $46.16.

Example 2

Alexis is saving to buy a laptop that costs $1,100. So far she has saved $400. She makes $12 an hour babysitting. What’s the least number of hours she needs to work in order to reach her goal?



Alexis has saved $400.

She makes $12 an hour.

She needs at least $1,100.


You need to know the least number of hours Alexis must work to make enough money. Solve for hours.


Instruction



Alexis’s saved money + her earned money must be greater than or equal to the cost of the laptop.

400 + 12h ≥ 1100


400 + 12h ≥ 1100 Subtract 400 from both sides.

12h ≥ 700 Divide both sides by 12.

h 58.33≥


In this situation, it makes sense to round up to the nearest half hour since babysitters usually get paid by the hour or half hour. Therefore, Alexis needs to work at least 58.5 hours to make enough money to save for her laptop.

Example 3

A radio station is giving away concert tickets. There are 40 tickets to start. They give away 1 pair of tickets every hour for a number of hours until they have at most 4 tickets left for a grand prize. If the contest runs from 11:00 a.m. to 1:00 p.m. each day, for how many days will the contest last?



The contest starts with 40 tickets.

The station gives away 2 tickets every hour.

The contest ends with at most 4 tickets left.


Instruction


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3. Read the statement again, identifying the unknown quantity or variable(s).

For how many days will the contest last?

This is tricky because the tickets are given away in terms of hours.

First, solve for hours.


40 tickets – 2 tickets given away each hour must be less than or equal to 4 tickets.

40 – 2h ≤ 4


40 – 2h ≤ 4 Subtract 40 from both sides.

–2h ≤ –36 Divide both sides by –2 and switch the inequality symbol.

h ≥ 18


The inequality is solved for the number of hours the contest will last. The contest will last at least 18 hours, or 18 hours or more.

The problem asks for the number of days the contest will last. If the contest lasts from 11:00 a.m. to 1:00 p.m. each day, that is 3 hours per day. Convert the units.

1 day = 3 hours

18 hours •1 day

3 hours

18 hours •1 day

3 hours6 days=

The contest will run for 6 days or more.


naMe:


Problem-Based Task 1.2.2: Free Checking AccountsThe time has come for you to open a checking account. A local bank is offering you a free checking account if you maintain a minimum balance of $200. You already have a savings account with this bank and you have $60 saved. You decide to keep saving money until you have enough to open a checking account, plus keep some money in savings. If you deposit $15 a week into the savings account, what is the minimum number of weeks it will take for you to be able to open a checking account with at least $200 and still have $25 in your savings account?


naMe:


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Problem-Based Task 1.2.2: Free Checking Accounts

Coaching a. What is the minimum amount of money you want to have to open your checking account?

b. How much are you depositing each week?

c. What inequality can be written to model the scenario?

d. What is the solution to the inequality?

e. What does the solution of the inequality represent in terms of the context of the problem?


Instruction


Problem-Based Task 1.2.2: Free Checking Accounts

Coaching Sample Responsesa. What is the minimum amount of money you want to have to open your checking account?

$225

Students at first might be inclined to write $200 because this is the minimum amount needed for a free checking account, but the problem also states that there needs to be $25 left in the savings account.

b. How much are you depositing each week?

$15/week

This is given in the problem.

c. What inequality can be written to model the scenario?

60 + 15x ≥ 225

You start with $60 and deposit $15 each week and need at least $225.

d. What is the solution to the inequality?

60 + 15x ≥ 225 Subtract 60 from both sides.

15x ≥ 165 Divide each side by 15.

x ≥ 11

e. What does the solution of the inequality represent in terms of the context of the problem?

It will take at least 11 weeks (11 weeks or more) to reach the savings goal of having $200 to open the checking account while still having $25 in the savings account.




naMe:


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Practice 1.2.2: Creating Linear Inequalities in One VariableTranslate each phrase into an algebraic inequality.

1. A tour bus can seat 55 passengers.

2. An energy-efficient lamp can only be used with light bulbs that are 60 watts or less.

Read each scenario, write an inequality to model the scenario, and then use the inequality to solve the problem.

3. Camilla is saving to purchase a new pair of bowling shoes that will cost at least $39. She has already saved $19. What is the least amount of money she needs to save for the shoes?

4. Suppose you earn $20 per hour working part time at a tax office. You want to earn at least $1,800 this month, before taxes. How many hours must you work?

5. Hiram earned a score of 83 on his semester algebra test. He needs to have a total of at least 180 points from his semester and final tests to receive an A for his grade. What score must Hiram earn on his final test to ensure his A?

6. Claire purchases DVDs from an online entertainment store. Each DVD that she orders costs $15 and shipping for her order is $10. Claire can spend no more than $100. How many DVDs can Claire purchase?

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7. The temperature outside is dropping at a rate of 1.5 degrees each hour. The weather station predicts a changeover from rain to freezing rain when the temperature reaches 41°F or below. Currently the temperature is 53°F and the time is 8:30 a.m. When do you predict the changeover to freezing rain will occur?

8. A recreation center holds a soccer game every Saturday morning for older teens. The group agreed that there should be at least 5 players on each team. One team started out with 17 players. After an hour of playing, 3 players started leaving every 5 minutes. For at least how long can the team keep playing?

9. A website has no more than $15,000 to give away for a contest. They have decided to give away $1,000 twice a day every day until they have at least $3,000 left to award as a grand prize. How many days will the contest run?

For problem 10, create your own context for the given inequality, and then solve the inequality. Be sure to express your solution in terms of the context of the problem.

10. 2x – 5 ≤ 9


naMe:


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Lesson 1.2.3: Creating Exponential Equations

Warm-Up 1.2.3Read the scenario and follow the directions.

This year, Zachary has been babysitting his young cousins after school for $70 a month. His uncle also gave him an extra bonus of $100 for his excellent work. Since school started, Zachary has earned more than $500. How many months ago did school start? Write an inequality that represents this situation. Solve it and show all your work.


Instruction


Lesson 1.2.3: Creating Exponential EquationsCommon Core State Standard


Warm-Up 1.2.3 DebriefThis year, Zachary has been babysitting his young cousins after school for $70 a month. His uncle also gave him an extra bonus of $100 for his excellent work. Since school started, Zachary has earned more than $500. How many months ago did school start? Write an inequality that represents this situation. Solve it and show all your work.

• Students will hopefully recognize this scenario as modeling a linear inequality: 70x + 100 > 500, where x is the number of months Zachary has been babysitting.

• Solving this equation for x gives x > 5 months.

• Encourage students to make meaning of the numerical answer, “School started more than 5 months ago.”


• Students will be required to create equations from context just as in the warm-up, except the equations will be exponential instead of linear.

• Students will also be required to make meaning of the solution in the upcoming lesson rather than just stopping at solving the equation for the numerical answer.


Instruction


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Prerequisite Skills


• working with exponents (raising a base to a power)

• applying the order of operations

IntroductionExponential equations are equations that have the variable in the exponent. Exponential equations are found in science, finance, sports, and many other areas of daily living. Some equations are complicated, but some are not.

Key Concepts

• The general form of an exponential equation is y = a • b x, where a is the initial value, b is the base, and x is the time. The final output value will be y.

• Since the equation has an exponent, the value increases or decreases rapidly.

• The base, b, must always be greater than 0 (b > 0).

• If the base is greater than 1 (b > 1), then the exponential equation represents exponential growth.

• If the base is between 0 and 1 (0 < b < 1), then the exponential equation represents exponential decay.

• If the time is given in units other than 1 (e.g., 1 month, 1 hour, 1 minute, 1 second), use the

equation y abx

t= , where t is the time it takes for the base to repeat.

• Another form of the exponential equation is y = a(1 ± r) t, where a is the initial value, r is the rate of growth or decay, and t is the time.

• Use y = a(1 + r)t for exponential growth (notice the plus sign). For example, if a population grows by 2% then r is 0.02, but this is less than 1 and by itself does not indicate growth.

• Substituting 0.02 for b into the formula y = a • b x requires the expression (1 + r) to arrive at the full growth rate of 102%, or 1.02.

• Use y = a(1 – r) t for exponential decay (notice the minus sign). For example, if a population decreases by 3%, then 97% is the factor being multiplied over and over again. The population from year to year is always 97% of the population from the year before (a 3% decrease). Think of this as 100% minus the rate, or in decimal form (1 – r).


Instruction


• Look for words such as double, triple, half, quarter—such words give the number of the base. For example, if an experiment begins with 1 bacterium that doubles (splits itself in two) every hour, determining how many bacteria will be present after x hours is solved with the following equation: y = (1)2 x, where 1 is the starting value, 2 is the rate, x is the number of hours, and y is the final value.

• Look for the words initial or starting to substitute in for a.

• Look for the words ended with and after—these words will be near the final value given.

• Follow the same procedure as with setting up linear equations and inequalities in one variable:

Creating Exponential Equations from Context






6. Interpret the solution of the exponential equation in terms of the context of the problem.


• multiplying a and b together before raising to the power

• misidentifying the base

• creating a linear equation instead of an exponential equation


Instruction


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A population of mice quadruples every 6 months. If a mouse nest started out with 2 mice, how many mice would there be after 2 years? Write an equation and then use it to solve the problem.

1. Read the scenario and then reread it again, this time identifying the known quantities.

The initial number of mice = 2.

The base = quadruples, so that means 4.

The amount of time = every 6 months for 2 years.


The unknown quantity is the number of mice after 2 years. Solve for the final amount of mice after 2 years.

3. Create expressions and equations from the known quantities and variable(s).

The general form of the exponential equation is y = a • b x, where y is the final value, a is the initial value, b is the base, and x is the time.

a = 2

b = 4

x = every 6 months for 2 years

Since the problem is given in months, you need to convert 2 years into 6-month time periods. How many 6-month time periods are there in 2 years?

To determine this, think about how many 6-month time periods there are in 1 year. There are 2. Multiply that by 2 for each year. Therefore, there are four 6-month time periods in 2 years.

(continued)


Instruction


Another way to determine the period is to set up ratios.

2 years •12 months

1 year24 months

24 months •1 time period

6 months4 time periods

=

=

Therefore, x = 4.

4. Substitute the values into the general form of the equation y = a • b x.

y = a • b x OR y abx

t=

y = (2) • (4)4 OR y = (2)(4)24

6

5. Follow the order of operations to solve the problem.

y = (2) • (4)4 Raise 4 to the 4th power.

y = (2) • 256 Multiply 2 and 256.

y = 512

6. Interpret the solution in terms of the context of the problem.

There will be 512 mice after 2 years if the population quadruples every 6 months.


Instruction


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Example 2

In sporting tournaments, teams are eliminated after they lose. The number of teams in the tournament then decreases by half with each round. If there are 16 teams left after 3 rounds, how many teams started out in the tournament?


The final number of teams = 16.

The reduction = 1

2.

The amount of time = 3 rounds.


The unknown quantity is the number of teams with which the tournament began. Solve for the initial or starting value, a.


The general form of the exponential equation is y = a • b x, where y is the final value, a is the initial value, b is the rate of decay or growth, and x is the time.

y = 16

b = 1

2

x = 3 rounds

4. Substitute the values into the general form of the equation y = a • b x.

y = a • b x

a16 •1

2

3

=


Instruction



a16 •1

2

3

=

Raise the base to the power of 3.

a16 •1

8= Multiply by the reciprocal.

a = 128


The tournament started with 128 teams.

Example 3

The population of a small town is increasing at a rate of 4% per year. If there are currently about 6,000 residents, about how many residents will there be in 5 years at this growth rate?


The initial number of residents = 6,000.

The growth = 4%.

The amount of time = 5 years.


The unknown quantity is the number of residents after 5 years. Solve for the final value after 5 years.


Instruction


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The general form of the exponential growth equation with a percent increase is y = a (1 + r) t, where y is the final value, a is the initial value, r is the rate of growth, and t is the amount of time.

a = 6000

r = 4% = 0.04

t = 5 years

4. Substitute the values into the general form of the equation y = a(1 + r)t.

y = a(1 + r)t

y = 6000(1 + 0.04)5


y = 6000(1 + 0.04)5 Add inside the parentheses first.

y = 6000(1.04)5 Raise the base to the power of 5.

y = 6000(1.21665) Multiply.

y ≈ 7300


If this growth rate continues for 5 years, the population will increase by more than 1,000 residents to about 7,300 people.


Instruction


Example 4

You want to reduce the size of a picture to place in a small frame. You aren’t sure what size to choose on the photocopier, so you decide to reduce the picture by 15% each time you scan it until you get it to the size you want. If the picture was 10 inches long at the start, how long is it after 3 scans?


The initial length = 10 inches.

The reduction = 15% = 0.15.

The amount of time = 3 scans.


The unknown quantity is the length of the picture after 3 scans. Solve for the final value after 3 scans.


The general form of the exponential growth equation with a percent decrease is y = a (1 – r) t , where y is the final value, a is the initial value, r is the rate of decay, and t is the amount of time.

a = 10

r = 15% = 0.15

t = 3 scans

4. Substitute the values into the general form of the equation y = a (1 – r) t.

y = a (1 – r) t

y = 10(1 – 0.15)3


Instruction


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y = 10(1 – 0.15)3 Subtract inside the parentheses first.

y = 10(0.85)3 Raise the base to the power of 3.

y = 10(0.614125) Multiply.

y ≈ 6.14


After 3 scans, the length of the picture is about 6 inches.


naMe:


Problem-Based Task 1.2.3: Population ChangeOn opposite sides of a major city two suburban towns are experiencing population changes. One town, Town A, is growing rapidly at 5% per year and has a current population of 39,000. Town B has a declining population at a rate of 2% per year. Its current population is 55,000. Economists predict that in 5 years the populations of these two towns will be about the same, but the residents of both towns are in disbelief. The economists also claim that ten years after that, Town A will double the size of Town B. Can you verify the predictions based on the data given? Do you think these predictions will come true?


naMe:


© Walch Education

Problem-Based Task 1.2.3: Population Change

Coachinga. Is Town A experiencing population growth or decay? What is the equation for Town A’s

population change after 5 years?

b. What is the solution to the equation in part a?

c. Is Town B experiencing population growth or decay? What is the equation for Town B’s population change after 5 years?

d. What is the solution to the equation in part c?

e. Are your solutions to parts b and d similar? What can you conclude about the economists’ prediction about the populations of the two towns being about the same after 5 years?

f. What will the variable, t, equal if the towns experience the same rates of growth or decline for 10 more years after that?

g. What are the equations for each town’s population change?

h. Based on your calculations, was the economists’ prediction for the town populations after 10 more years correct?

i. What factors might influence whether or not the economists’ predictions come true?


Instruction


Problem-Based Task 1.2.3: Population Change

Coaching Sample Responsesa. Is Town A experiencing population growth or decay? What is the equation for Town A’s

population change after 5 years?

Town A is experiencing growth. Fill in the information we know to find the equation:

y = a(1 + r)t

• a = 39,000 people

• r = 5% = 0.05

• t = 5 years

y = 39,000(1 + 0.05)5 is the equation.

b. What is the solution to the equation in part a?

y = a(1 + r)t

y = 39,000(1 + 0.05)5

y ≈ 49,775 people

Round up to the nearest whole person.

c. Is Town B experiencing population growth or decay? What is the equation for Town B’s population change after 5 years?

Town B is experiencing decay. Fill in the information we know to find the equation:

y = a(1 – r)t

• a = 55,000 people

• r = 2% = 0.02

• t = 5 years

y = 55,000(1 – 0.02)5 is the equation.

d. What is the solution to the equation in part c?

y = a(1 – r)t

y = 55,000(1 – 0.02)5

y ≈ 49,716 people


Instruction


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e. Are your solutions to parts b and d similar? What can you conclude about the economists’ prediction about the populations of the two towns being about the same after 5 years?

The numbers are similar, especially when considering the magnitude. If rounded to the nearest 100 people, Town A would have 49,800 and Town B would have 49,700 people. A difference in 100 people compared to tens of thousands is a small number. Based on these calculations, it would seem that the economists are correct that the populations will be about equal in 5 years.

f. What will the variable, t, equal if the towns experience the same rates of growth or decline for 10 more years after that?

The amount of time passed since the initial population count will be 5 years plus 10, which equals 15 years.

t = 5 + 10

t = 15

g. What are the equations and solutions for each town’s population change?

The equations below use the town’s current populations, with t = 15. Alternatively, students could use the value t = 10 and the population values they calculated at the 5-year mark for a.

Town A Town B

y = a(1 + r)t y = a(1 – r)t

y = 39,000(1 + 0.05)15 y = 55,000(1 – 0.02)15

y ≈ 81,079 people y ≈ 40,622 people

h. Based on your calculations, was the economists’ prediction for the town populations after 10 more years correct?

Yes, 81,079 is about twice as much as 40,622.

i. What factors might influence whether or not the economists’ predictions come true?

Many factors might contribute to the economists’ predictions coming true or not, such as the economy itself. If the economy starts to decline, the population of Town A might not continue to grow as fast. Or, a new industry might relocate to Town B, bringing jobs, and perhaps people will start moving back into that town. Or, Town A might reach its capacity before it hits 81,000 people.




naMe:


Practice 1.2.3: Creating Exponential Equations Use what you know about linear and exponential equations to complete problems 1 and 2.

1. Determine whether each scenario can be modeled by a linear or an exponential equation.

a. The price of a loaf of bread increases by $0.25 each week.

b. Each week, a loaf of bread costs twice as much as it did the week before.

2. Determine whether each scenario can be modeled by a linear or an exponential equation.

a. 10 people leave a football game every minute after the third quarter.

b. 1

4 of the people leave a football game every minute after the third quarter.

For problems 3–10, write an equation to model each scenario. Then use the equation to solve the problem.

3. A population of insects doubles every month. If there are 100 insects to start with, how many will there be after 7 months?

4. A type of bacteria in a Petri dish doubles every hour. If there were 1,073,741,824 bacteria after 24 hours, how many were there to start with?

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5. A stock loses half its value every week. If the stock was worth $125 starting out, what is it worth after 4 weeks at this rate of decline?

6. A computer depreciates (loses its value) at a rate of 1

2 the original value every 2 years. If the

computer now costs $75 after 6 years, how much did it cost when you bought it new?

7. A new car depreciates as soon as you drive it out of the parking lot. A certain car depreciates to half its original value in 4 years. If you bought a car 8 years ago and it is now worth $5,000, how much did you pay for the car originally?

8. The number of dandelions growing on your lawn triples every 3 days. If your lawn started out with 15 dandelions, how many dandelions would you have after 3 weeks?

9. The population of a town is increasing by 2% each year. The current population is 12,000. How many people will there be in 4 years?

10. The population of New York City doubles during the workday. At the end of the workday, the population decreases by half. If the population in New York City is roughly 4,000,000 people before rush hour, and the morning rush hour lasts 3 hours, what would be the population if the rush-hour growth rate continued for 12 hours instead of 3? People leave the city for home at the same rate as they come into the city. What would be the population if the decay rate continued for 12 hours at the end of the day when everyone goes home?

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