MATH MODULE 1 LESSON 1

Step 1:

Highlight key words. The phrase "twice the sum of x and 5" is shown. An arrow extends from the word twice and points to its translation of "two times." An arrow extends from the words "sum of x and 5" and points to its translation "addition—in parentheses!"

Lance and Dimetri are playing a marble game. According to their rules, 3 points are awarded for each red marble collected and 2 points are awarded for each green marble collected. The number of points per player is found using the following expression: 3r + 2g

How many points would be given for 5 red and 6 green marbles? 3(5) + 2(6) = 27 pointsHow was the expression created? You may be wondering where the expression 3r + 2g comes from. In this problem, Lance and Dimetri decide to award 3 points for each red marble and 2 points for each green marble. Because you don't know the number of red or green marbles they will end up with, variables are used to represent these unknown quantities. If you get 3 points for each red marble, the expression 3r can be used to represent the points for the red marbles. If you get 2 points for green marbles, the expression 2g can be used for the green marble score. To get the total number of points, you add the two expressions: 3r + 2g.

Samuel is mixing different-color paints for a mural he has been asked to paint in the school gymnasium. To create the color purple he wants, Samuel mixes r gallons of red paint with b gallons of blue paint. Red paint costs $25 per gallon and blue paint costs $22 per gallon. He also needs $65 to spend on paint supplies. Samuel knows the cost of his paint and supplies can be found using the expression 25r + 22b + 65.

How much does the project cost if Samuel needs 15 gallons of red paint and 12 gallons of blue paint to create the mural? Because Samuel already knows the expression to determine cost, he just substitutes the number of gallons of red for r and blue for b, then simplifies to find the total cost. 25r + 22b + 65 25(15) + 22(12) + 65 375 + 264 + 65 $704 Samuel will spend $704 painting a mural in the school gymnasium.How was this expression created? You may be wondering where the expression 25r + 22b + 65 comes from. Later in your math studies, you will learn how to create expressions or equations using a verbal statement. In this problem, the price for the red paint and the blue paint is different. The number of gallons of red paint must be multiplied by 25, whereas the number of gallons of blue paint must be multiplied by 22. Because you do not know the number of gallons of each type of paint, variables (in this case, r for red and b for blue) are needed to represent these quantities: 25r + 22b In addition to the paint, Samuel needs $65 to spend on supplies. This amount does not change, so there is no multiplication involved. This amount is represented as a constant: 65. The three values are added together to get the total: 25r + 22b + 65

Mrs. Ming earns $35.00 per hour at her job as a computer programmer. The amount she earns can be shown by the expression 35x, where x is the number of hours she works.

How much would she make if she worked 27 hours last week? 35(27) = $945.00 How was the expression created? You may be wondering where the expression 35x comes from If Mrs. Ming works 1 hour, she makes $35. If she works 2 hours, she makes 2(35), or $70. So, you multiply the number of hours worked by her rate—35x—to determine the amount of money she earns.

Role of Exponents

Same or Different?x + x = ?x + x = 2x OR x + x = x2If x = 3 then 3 + 3 = 6Let's substitute a number for x to see which is correct!What does 2x equal?2x = 2(3) = 6What does x2 equal?x2 = 32 = 9And the winner is ...x + x = 2x

Example 2

Simplify 2x − 3y + 4z − x + 2y + z. = 2x − x − 3y + 2y + 4z + z Again, notice how the subtraction signs stayed with the 3y and the x terms in this problem. = x − y + 5z

Example 1

Simplify 5x + 3y − 2x + y + 3z. Reorder the terms so like terms are next to one another. = 5x − 2x + 3y + y + 3z = 3x + 4y + 3z

Example 3

Simplify 5x − 2x2 − x + 4 + 4x2 − 10. = 5x − x − 2x2 + 4x2 + 4 − 10 = 4x + 2x2 − 6

8 less than 5 times a number Steps Examples

Step 1: Highlight Key Words Step 2: Talk It Out The phrase "less than" indicates subtraction where the order of the terms is reversed. "Five times a number" means 5 times x . So you need to subtract 8 from 5x. Step 3: Translate 5x − 8

Which operation should be done first? Which operation should be done first in the expression 12 ÷ 2 • 6: multiplication or division?Find 2x + 3y if x = 2 and y = 3

Substitute the known values into the expression, and then follow the order of operations.2x + 3y2(2) + 3(3)4 + 913

Find 4x − y − 2z if x = −2, y = 0, and z = −1

Substitute the known values into the expression, and then follow the order of operations.4x − y − 2z4(−2) − (0) − 2(−1)−8 − 0 + 2−6

Find 3x2 − 2y3 if x = 4 and y = −2

Substitute the known values into the expression. Follow the order of operations when simplifying expressions: parentheses and exponents first, then multiply or divide, and finally add or subtract. 3x2 − 2y33(4)2 − 2(−2)33(16) − 2(−8)48 + 1664

Step 2:

Talk it out. The word twice means to multiply by 2. The sum of x and 5 means to add x and 5. Because the word sum indicates the answer to the addition problem, you must put the addition in parentheses so it is done first.

Slide 3 After combining like terms, the expression 4s + 3 represents how much Alex and his family will spend to go to the movies. The cost of a student ticket is s. If s = 8, what is the solution to the expression? $28 $35 $96 $15

The answer is: $35

Slide 4 Monique has a job mowing lawns. She earns $15 for every lawn she mows and can earn extra money by pulling weeds. Last week, Monique earned an additional $25 because she pulled weeds. Which expression represents Monique's pay for the week? 25y + 15 15y - 25 15y + 25 25y - 15

The answer is: 15y + 25

Slide 2 Don's Car Rentals charges a flat rate of $35 per day plus $2 per mile. If the number of miles is represented by m, which algebraic expression represents the overall rate for the day using m? 35(2)(m) 2m − 35 35 + 2m 35m + 2

The answer is: 35 + 2m

Slide 2 The expression 3s + s + 3 represents how much Alex and his family will spend to go to the movies. Which statement explains how this expression can be simplified? 3 and s can be combined because any two terms can be divided. 3s and 3 can be combined because they are like terms. 3s and s can be combined because they are like terms. 3 and s can be combined because any two terms can be multiplied.

The answer is: 3s and s can be combined because they are like terms.

Which pair represents like terms? 10y2 and 10x2 3x2 and 2x2 3x and 2xy 2x2 and 4x

The answer is: 3x2 and 2x2

Which term is like 2x? 2y 1 3xy 5x

The answer is: 5x

Which term is not like 6y? 6x 7y 2y 10y

The answer is: 6x

Slide 5 Samuel bought a car for $8,000 three years ago. He wants to sell the car and buy another. However, its value has depreciated monthly since he bought it. If the monthly depreciation is represented by d, which algebraic expression represents how much Samuel can sell his car for today? 8,000 - 12d 8,000 + 12d 8,000 + 36d 8,000 - 36d

The answer is: 8,000 - 36d

Which pair does not represent like terms? 4x2 and x2 5z and 3z 2yz and 7yz 9x and 4xy

The answer is: 9x and 4xy

Slide 3 Julio earns a salary plus commission for his television sales job. His weekly earnings can be shown using the following expression: 200 + 5x What does the first term of the expression represent? Julio's earnings from hourly wages for the day Julio's earnings from his salary for a week Julio's earnings from his commission for a week Julio's earnings from his commission for a day

The answer is: Julio's earnings from his salary for a week

Combining Like Terms

The first step in simplifying like terms is combining like terms. To add or subtract, you must have like terms. This is not the case for multiplication or division. Any two terms can be multiplied or divided; they do not have to be like terms. Before you move on with combining like terms, it is important to understand the role of exponents.

Buying a cell phone plan Step 1: Write what you know.

The plan costs $45 plus $0.05 per text message.

Will I Ever Use This in Real Life?

You sure will. Sometimes, it's hard to see how what you're learning applies to you or the real world, especially when it's concepts like y = 2 + x or the quadratic equation. The ideas and concepts you will learn in algebra will help you in problem solving and mathematical foundations in many fields. Let's take a look at a few examples of some ways algebra is used in the real world.

Like terms

always have the same variable (the letter part) but can have different coefficients (the number part). They also always have the same power for the variable.

Factors

are numbers you multiply together to produce a product. Look at the factors that are easily seen in this expression. 2 and x2 are factors of 2x2.−3 and y are factors of −3y.−1 and z are factors of −z.

EXAMPLE

notice how there is no number in front of the variable z in this expression. It is understood that there is an invisible 1 in front of z, because 1 • z = z. Therefore, it is not necessary to write the 1 in the expression. Since the sign in front of the term belongs to that term, the coefficient of this term would be −1. The coefficients in this expression are 2, −3, and −1.

For example

the mathematical expression shown here has three unknown values because there are three variables: x, y, and z. Remember, each variable represents a different unknown quantity. Also, notice the exponent is a separate part from the variable in the term.Expressions are made up of terms separated by a plus or minus sign. Terms can contain variables, numbers, or products of variables and numbers. This expression contains four total terms: 2x2, −3y, −z, and 6. Notice how the plus or minus sign is attached to the term immediately following it.

Is the plan going to work if you send 100 text messages each month?

45 + 0.05m is the expression for the cost of the plan. Substitute 100 for m to see if the plan remains within your $50 budget. 45 + 0.05(100) = 45 + 5 = $50 even. This is the perfect plan for your needs!

Simplify: 2x + 8y + 3x + y

5x + 9y

What are the coefficients of each term in 4x − y − 2x?

4, −1, −2

Translate.

45 + 0.05m

Highlight key words

$45 plus $0.05 per text message

What are the factors of 3xy?

1, 3, 3x, 3y, x, y, xy, and 3xy

What are the common factors of 5wy and 5yz?

1, 5, y and 5y

How many terms are in the expression 3x2 + 2x + 4?

3

What are the exponents in the expression 8x3 + 5y2 + z4?

3, 2, and 4

When changing phrases from verbal expressions into algebraic expressions, simply replace the word or phrase with the mathematical symbol it represents. You may also see phrases that say "a number." In algebra, variables (letters) are used to replace numbers you don't know. So, anytime you see the phrase "a number," you can replace it with any variable. View the short presentation below to see some common examples.

Addition and subtraction appear last in line in the order of operations. In a situation where you need either one to be done first, use the special terms sum or difference. These terms ensure the operation will be placed in parentheses and moved to the front of the line!

Sum It Up

Algebraic expressions are mathematical sentences that contain numbers, variables, and operations. Like terms always have the same variable and powers for their variables, but they can have different coefficients. Combining like terms is the process of adding or subtracting like terms to simplify an expression. When simplifying expressions containing multiple operations, follow the order of operations: parentheses, exponents, multiplication and division from left to right, and addition and subtraction from left to right. If you know the value of a variable, you can substitute that value for the variable in an algebraic equation. Algebraic expressions can be created from words or phrases by replacing the word or phrase with the mathematical symbol it represents. The acronym WHAT can help you create algebraic expressions from real-world examples.Write what you know.Highlight key terms.Assign and define your variable and terms.Translate the expression.

Does 5(x + 3) equal 5x + 3 when x = 2? 5(2 + 3) 5(2) + 3 5(5) 10 + 3 25 13

No, they are not the same! Be careful with your translations because the parentheses make a BIG difference.

Coefficient:

Coefficients are numbers multiplying either a variable or some unknown quantity. Therefore, sometimes coefficients can be located outside the parentheses. To determine the coefficients, simply find the factors of the terms. Remember, if there is no number in front of a variable, it is understood as 1 times that variable and is not written. The coefficients in this expression are 4, 1, and −1.

Constant:

Constants are numbers that stand alone. They are called constants because they have fixed value. The constant of this expression is 6.

Translate "25 percent of x" into algebra0 point 2 5 x0 point 2 5 over x0 point 2 5 minus x0 point 2 5 plus x

Correct Answer: a. 0 point 2 5 xThe word "of" indicates multiplication.Write 25 percent in decimal form before multiplying. 25 percent equals 0 point 2 5. The correct answer is 0 point 2 5 x.

Translate "the product of x and y" into algebra.The quantity x times the quantity y x plus y x over yx minus y

Correct Answer: a. The quantity x times the quantity yThe word "product" indicates multiplication. "The product of x and y" is x dot y or the quantity x times the quantity y.

Translate "4 times the difference of n and 8" into algebra.4 n minus 84 times the quantity n minus 8 4 times the quantity 8 minus n 4 minus 8 n

Correct Answer: b. 4 times the quantity n minus 8The difference of n and 8 is multiplied by 4. The key word "difference" indicates subtraction. The correct answer is 4 times the quantity n minus 8.

Translate "one-third of the sum of a number and 4" into algebra.x divided by 3 plus 4The quantity x plus 4 divided by 3x plus 4 divided by 33 times x plus 4

Correct Answer: b. The quantity x plus 4 divided by 3"one-third of the sum of a number and 4" can be written as one-third times the quantity of x plus 4 or the quantity of x plus 4 divided by 3.

Translate "one-half of a number plus 7" into algebran plus 7 all over 2one-half n plus 7one-half times the quantity n plus 7 one-half plus the quantity n plus 7

Correct Answer: b. one-half n plus 7"One-half of a number" means one-half times a number, or one-half n. "Plus 7" indicates adding 7. The correct answer is one-half n plus 7

Translate "9 subtracted from x" into algebra.9 minus xx minus 99 minus 9 xx minus 9 x

Correct Answer: b. x minus 9The phrase "9 subtracted from x" indicates 9 is taken away from x. The correct answer is x minus 9.

Translate "x more than 5" into algebra.x is greater than 55 x5 plus xx minus 5

Correct Answer: c. 5 plus x The phrase "more than" indicates addition. "x more than 5" translates to 5

Translate "8 times the quotient of x and 2" into algebra8 times the quantity x plus 2 8 times the quantity x minus 2 8 times the quantity 2 x 8 times the quantity x over 2

Correct Answer: d. 8 times the quantity x over 2Multiply 8 by the quotient of x and 2. In algebra, division is often shown using a fraction. So, "the quotient of x and 2" is x over 2. The correct answer is 8 times the quantity x over 2.

Order of Operations

Order of Operations Remember to always follow the order of operations when simplifying expressions containing multiple operations. The order of operations is Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Notice, there are two groups of operations, multiplication/division and addition/subtraction, that are next to each other in the list. Keep in mind that you must do these operations that are on the same level from left to right. Let's take a look at how we solve a problem using the appropriate order of operations. (2+4)2 + 4 × 8 − 6 ÷ 2 Parentheses62 + 4 × 8 − 6 ÷ 2Exponents36 + 4 × 8 − 6 ÷ 2Multiplication/Division (left to right)36 + 32 − 3Addition/Subtraction (left to right)65

Factor:

PARentheses not only group operations together, they also mean multiplication of two factors. 4 and t + 3 are factors of the term 4(t + 3). As well, −1 and s are factors of the terms −s.

Example 2: 4(t + 3) − s Variable:

This mathematical expression contains two unknown values, or variables, identified as t and s. Term: Terms are the parts of an expression separated by a plus or minus sign. Remember, the addition and subtraction symbol could also mean positive and negative. They are grouped with the numbers behind it to form the term. There are two terms in this expression:4(t + 3) and −s. Notice, too, that the first term also contains two terms within the parentheses: t and 3.

Focus Questions

To achieve mastery of this lesson, make sure you develop responses to the following questions: What are the parts of an algebraic expression? How are algebraic expressions rewritten in different equivalent forms by simplifying them? How are algebraic expressions evaluated using substitution? How are algebraic expressions created from words or phrases? How are algebraic expressions created and interpreted in terms of the context?

Constant:

To determine the constant of an expression, find the number that stands alone. Sometimes, the constant is located in the middle of the expression. In this expression, the second factor of the term 4(t + 3) has a constant of 3.

Step 3:

Translate. Two times the quantity x plus 5, end quantity.

Algebraic Translations

Two columns are shown. The column on the left is titled "Words." The column on the right is titled "Algebra." The words "the sum of x and 2" translate to x plus 2. The words "p more than x" translate to x plus p. The words "the quotient of x and 2" translate to x divided by 2 or x over 2. The words "x decreased by 15" translate to x minus 15. The words "7 subtracted from a number" translate to x minus 7. The words "the product of a and b" translate to a times b or a b. The words "one-third of x" translate to one-third x or x over 3.

Doctors

Use algebraic formulas to calculate prescription dosages.

YOu might have to translate a real-world situation when you're calculating sale prices, interest, or depreciation. Let's try interpreting an algebraic expression about depreciation. Ellen bought a boat, and she plans to sell it in a year. However, its value depreciates, or decreases, in price or value monthly. The amount Ellen can sell the boat for in a year can be shown using the following expression: 5,000 − 12d

What does the first term of the expression represent? The purchase price of the boat.What does the second term of the expression represent? The amount the value of the boat will depreciate within one year.What does the variable represent? The amount the value of the boat will depreciate within one month.

Role of Exponents

When combining like terms, the variable part remains exactly the same. For example, 5x2 + 2x2 is 7x2. Notice that the exponent stays the same. Let's explore this idea even further by comparing 4x and x4. 4x literally means "four x's added together" or "4 times x" because multiplication is just a short way of writing the same number being added over and over again. On the other hand, x4 means x • x • x • x, which is completely different from x + x + x + x. Be sure to keep this idea at the front of your mind when you are simplifying expressions. Whether you are adding or subtracting like terms, the process is the same.

Coefficient:

When multiplying a variable and a number to write a term, the number is listed first and is called the coefficient. It is important to remember the sign in front of the term also goes with that term.If you do not see a number in front of the variable, it is understood there is an invisible 1 holding that place value. Any value multiplied by 1 is the number itself.

Larger Expressions

When translating larger phrases into algebra, the process remains the same. Just break the phrase into pieces, translating each piece as you go along. Let's start with the phrase "twice the sum of x and 5." How would you translate "twice the sum of x and 5"?

Steps When trying to create algebraic expressions from real-world examples, you can use the acronym WHAT:

Write what you know. Highlight key terms. Assign and define your variable and terms. Translate the expression.

Select X or 2 for more info on bases and exponents.

X The x is the base, or the quantity being raised to a power. 2 The 2 is the exponent and represents the number of times the base is multiplied by itself. You can read this as "x squared" or "x to the second power."

Writing Expressions with Sums and Differences

You know that the word "sum" means to add, but more specifically it means "the answer to an addition problem." The same holds true for the word "difference," which means "the answer to a subtraction problem." When you see the words "sum" or "difference" in a phrase, remember to write the addition or subtraction part in parentheses. This ensures that it is done first. Example: 5 times the sum of x and 3 Explanation: Let's start with the phrase "the sum of x and 3." You know this means x + 3. The rest of the expression says you are to multiply that part by 5. To be sure the addition part comes first, you must place the x + 3 in parentheses. Translation: 5(x + 3) Does 5x + 3 mean the same thing as 5(x + 3)? Let's see.

Slide 1 A man's age can be described as 4 more than 2 times his son's age. If his son's age is represented by the variable s, which algebraic expression represents the man's age using s? 4s + 2 2 + s + 4 2(4) + s 2s + 4

he answer is: 2s + 4

algebraic expression

mathematical sentence that contains numbers, variables, and operations, like addition or subtraction. Explore the activity to get a deeper understanding of the elements that make up algebraic expressions.

Financial analysts

use algebraic formulas to assist people in calculating loans, interest, and retirement savings.

Game designers

use algebraic formulas to calculate and code how characters move within games.

Chefs

use algebraic formulas to convert amounts of ingredients in recipes when baking and cooking.

Engineers

use algebraic formulas to design structures like bridges, roads, and buildings.

Filmmakers, animators, television producers, and photographers

use algebraic formulas to plan shots, edit film and photographs, and design sets and costumes.

What does it mean to simplify an expression?

using math processes to make a problem simpler. For example, the expression x + x + x + x can be simplified to 4x. Writing 4x is much easier than writing x + x + x + x.

Expressions

whether they are mathematical or verbal, can contain variables. A variable is a letter that holds the place for some unknown value in an expression.

What is the base in the exponential expression 3x4?

x

What are the variables in the expression 3x + 4y − z?

x, y, z

Simplify: 9p − 2p2 + 5p

−2p2 + 14p

Simplify: −5g + 2h − 3g + g − 2h

−7g