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Pre-Algebra View details

3. Algebraic Expressions

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Pre-Algebra /

Chapter 4

Algebraic Expressions

This lesson provides a comprehensive guide to understanding algebraic expressions, a cornerstone in the field of algebra. It introduces you to the basic elements like variables, which represent unknown quantities, and constants, which are fixed numbers. The lesson also explains the importance of coefficients, the numbers that multiply variables. Through real-world examples, such as calculating income or game scores, it shows how these elements are combined through mathematical operations like addition and subtraction to form algebraic expressions. The aim is to equip you with the skills to write and manipulate these expressions, making it easier to solve problems in academics and everyday life.

Student Learning Objectives:
Identify terms, variables, and coefficients Convert between algebraic expressions and verbal expressions Evaluate an algebraic expression for a given value

	16 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

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Numerical expressions deal with combinations of mathematical operations with numbers, which are known quantities. However, sometimes we need to deal with unknown quantities. This lesson will examine how to incorporate unknown quantities in expressions and how to evaluate that expression once the quantity is known.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Garage Sale to Buy a New Game

Tearrik is holding a garage sale to raise money.

His parents decided to double the money he makes and add $5 extra on top. How can this situation be written in a mathematical way without knowing how much money Tearrik will make from the garage sale?

Discussion

Variable

A variable is a symbol used to represent an unknown quantity. Often, variables represent fixed but unknown numbers. Variables are usually denoted with letters such as x. x+1=8

A variable can be used to represent a quantity that changes. Izabella's income varies depending on the number of hours she works. She receives a fixed salary of$100 per week plus$4 per hour. Izabella's weekly income could be different every week, depending on the number of hours she works. Therefore, the use of a variable is appropriate. Let x be the hours that Izabella works in a week. Her income can be written by adding the fixed $100 to the $4 per hour.

Izabella's Weekly Income 4x+100

Discussion

The Numbers Next to Variables

When working with variables, numbers are often used to complete mathematical expressions. Two common types of numbers are coefficients and constants.

Theory

Coefficient

A coefficient is a number that stands in front of a variable.

A coefficient always multiplies a variable, even if the variable is raised to some power. 5x 2x^3 A variable with a coefficient of 1 is usually written without a coefficient.

1x^2 = x^2

Discussion

Adding Numbers to a Variable

Another type of number that appear with variables is called constants.

Concept

Constant

A constant is a number with a known, fixed value. Examples of constants include regular numbers written with digits, like 15, - 23, and 11.327.

A constant is usually added to or subtracted from a variable. Consider the following example. x + 15

Here, 15 is a constant. Not every constant is written with digits. Some special constants are written with special symbols, such as the number pi, which is often written as π .

Pop Quiz

Practice with Variables and Coefficients

The applet below displays different variables multiplied by coefficients. Answer the indicated question correctly.

Discussion

Algebraic Expression

An algebraic expression is a valid combination of numbers, variables, and mathematical operations. For example, in the expression 2x+3, the variable x is being multiplied by its coefficient 2, and this product is then added to the constant 3.

Discussion

Components of an Algebraic Expression

Algebraic expressions are made by adding or subtracting smaller expressions called terms.

Concept

Term - Expression

A term is an algebraic or numeric expression that does not involve addition or subtraction. An expression contains one or more terms, separated from one another by + or - signs.

Mathematical Expression	Number of Terms	Terms
7x	1	7x
8	1	8
8x - 2(5)	2	8x and -2(5)
x^2 + y^2 +4	3	x^2, y^2, and 4
2x^2 -5x - 122	3	2x^2, -5x, and -122

Example

The Cost of Clothes

Tearrik is selling some of his old shirts and pants.

He is selling the pants for $1 more than three times the price of a shirt. If the price of a shirt is s, write an algebraic expression for the price of a pair of pants.

Hint

An algebraic expression is a combination of numbers, variables, and mathematical operations.

Solution

An algebraic expression is a combination of numbers, variables, and mathematical operations. The terms are separated by + or - signs. In verbal expressions, some words or phrases may imply certain math operations.

	Key Words and Phrases
Addition	added to, plus, sum of, more than, increased by, total of and
Subtraction	subtracted from, minus, difference of, less than, decreased by, fewer than, take away
Multiplication	multiplied by, times, product of, twice
Division	divided by, quotient of

The price of a shirt is represented by the variable s. We can identify the operations we need by examining the given information. Tearrik is selling the pants for $1 more than three times the price of a shirt. The phrase more than indicates an addition, so we will add $1 to some other quantity. 1 + The phrase three times represents a multiplication. In this case, it is 3 times the price of a shirt s. 1 + 3s There is no more information to include, so this expression represents the price of a pair of pants. 1+3s

Example

Bonus Points Calculation

Magdalena is playing a video game.

In the game, players are given bonus points for completing challenges quickly.

Help Magdalena write the information as an algebraic expression. Let the variable be t.

Hint

Identify the variable. Then, look for keywords that indicate operations.

Solution

It is important to identify the variables before writing an algebraic expression. The time it takes a player to finish a challenge can be different for different tries or for different players. Since this time can change, we will assign it the variable t. Time to Finish the Challenge: t An algebraic expression is a combination of numbers, variables, and mathematical operations. The terms are separated by + or - signs. In verbal expressions, there are words or phrases that indicate certain mathematical operations.

	Key Words and Phrases
Addition	added to, plus, sum of, more than, increased by, total of and
Subtraction	subtracted from, minus, difference of, less than, decreased by, fewer than, take away
Multiplication	multiplied by, times, product of, twice
Division	divided by, quotient of

Let's find some of these keywords. The bonus points are half the difference between400 and the time it took to finish the challenge The half indicates a division by 2 and the difference indicates a subtraction. Magdalena says to take half the difference, so let's write the difference first. This difference can be written by subtracting the time t from 400. 400 - t The division by 2 affects the result of the subtraction. In the order of operations, division operations are evaluated before subtraction. To evaluate the subtraction first, we can write it inside parentheses. 1/2(400 - t) This expression can help Magdalena determine how many bonus points she will get for completing a challenge.

Discussion

Evaluating an Algebraic Expression

Evaluating an expression consists of determining the value of an expression when the variable or variables of the expression take a specific value. This is done by substituting the given value for the variable in question into the expression and then simplifying it. Consider the following expression. (x-1)^2/2 We will evaluate this expression when x is equal to 5. There are two steps to follow.

Substitute the Given Value for x

In the expression, substitute the given value for every instance of the variable. In this case, substitute 5 for x.

(x-1)^2/2

Substitute

x= 5

( 5-1)^2/2

After the substitution, the variable disappeared and the algebraic expression turned into a numeric expression.

Simplify the Expression

Now the expression can be simplified following the order of operations.

(5-1)^2/2

SubTerms

Subtract terms

4^2/2

CalcPow

Calculate power

16/2

CalcQuot

Calculate quotient

When x=5, the given expression equals 8.

If an expression has more than one variable, each variable is replaced with its given value before all the required operations are performed.

Example

Bonus Stars on Each Level

In a video game, players can collect stars on each level to earn a bonus.

There are three special stars per level. If x is the number of levels, write an expression for the numbers of stars in the game.

There are 73 levels in the game. Use the expression from Part A to find the total number of stars in the game.

Hint

The number of stars per level indicates a multiplication.

Substitute the given value for the variable and evaluate the expression.

Solution

We can find the total number of bonus stars by adding together the number of stars on every level. Every level has 3 stars. We do not know how many levels there are, so we represent this number with the variable x. Adding the number of stars of every level is the same as multiplying 3 by the number of levels x.

3+ 3+...+ 3_x= 3x

Now we know that there are 73 levels in the game. This means that in our algebraic expression, x is 73.

x = 73 Let's evaluate our algebraic expression from Part A to find the total number of stars in the game. Substituting the value for the variable changes the algebraic expression into a numerical expression because it eliminates the variable entirely.

Substitute

x= 73

3* 73

Multiply

219

There are 219 bonus stars in the game.

Example

The Difference Between the Area and the Perimeter of a Square

Consider a square with side length s.

Write an algebraic expression for the difference between the area of the square and its perimeter.

If the side length of the square is 7, find the difference between its area and its perimeter.

Hint

The area of a square is the square of the length of a side. The perimeter of a square is the sum of the lengths of all four sides.

Evaluate the algebraic expression from Part A when s=7.

Solution

Let's start by recalling the formulas for the area and the perimeter of a square. The area of a square is given by the square of a side. The given square has a side length of s.

Area of the Square: s^2 The perimeter, on the other hand, is found by adding the lengths of all sides of a figure. Since the four sides of the square all have a length of s, the perimeter is found by multiplying s by 4. Perimeter of the Square: 4s Then, to find the difference between these two values, we subtract the perimeter from the area. s^2- 4s

The side length s of the square is 7. Then, to find the difference between the area and the perimeter, let's evaluate the expression for the difference when s= 7.

s^2-4s

Substitute

s= 7

7^2 - 4* 7

CalcPow

Calculate power

49 - 4*7

Multiply

49 - 28

SubTerms

Subtract terms

The difference between the area and the perimeter of the given square is 21.

Example

Selling Shirts and Pants

Tearrik is selling some old clothes at a garage sale.

Write an algebraic expression for the total amount of money Tearrik made from selling s shirts and p pairs of pants.

If Tearrik sold 10 shirts and 4 pairs of pants, how much did he make?

Hint

How much money Tearrik would make from each type of clothing sold? Add these amounts to find the total.

Substitute the appropriate values for the variables in the expression. Then, evaluate.

Solution

The total amount of money Tearrik made is the sum of what he got from selling shirts and pants.

Money From Shirts

Since he is selling shirts for $ 5 each and he sold s shirts, the amount of money Tearrik made from the shirts can be written as the product of 5 and s. 5* s

Money From Pants

Similarly, if p represents the number of pairs of pants sold, then the money made from selling the pants is 16p. 16* p

Total Money Made

Finally, let's combine the two expressions to find the total amount of money Tearrik made from these clothes. 5 s + 16 p

We can find the amount of money that Tearrik made from selling 10 shirts and 4 pairs of pants by evaluating the expression.

Evaluate 5 s +16 p when s = 10 and s = 4 Substitute the values for the corresponding variables and find the value of the resulting numerical expression.

5s + 16p

SubstituteII

s= 10, p= 4

5* 10 + 16 * 4

MultII

Multiply 5 by 10

50 + 16 * 4

MultII

Multiply 16 by 4

50 + 64

AddTerms

Add terms

114

Tearrik made a total of $114 from the shirts and pants sold.

Pop Quiz

Practice Evaluating Algebraic Expressions

Evaluate the given algebraic expression for the given values. a = 5 & b = 1/2 [0.8em] c = 2 & d = 14

Closure

Money Made From the Garage Sale

Tearrik's parents decided to give him money based on what he made from selling his things at a garage sale. We do not know how much money Tearrik made from the sale itself, so we can use a variable to represent ti. Consider x as the money Tearrik made at his sale. Money From the Garage Sale: x Consider what Tearrik's parents said about how much money they would give him. Phrases in this plan will indicate the necessary operations for writing the information as an algebraic expression. Double the money Tearrik made and add an additional $5 extra. The word double indicates multiplication by 2, so we will multiply x by 2. 2x Add indicates addition, so we will add $ 5 to double the money Tearrik made. 2x + 5

Now we have an expression for the total amount of money Tearrik will get from his garage sale, even though we do not know exactly how much money he made from the sales directly. Algebraic expressions are great for writing mathematical ideas easily.

Level 1

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Algebraic Expressions

Exercise 2.1

Izabella is wondering how much food she feeds her two dogs.

Every day, she feeds her little dog t cups of food and her big dog b cups of food. Help Izabella write an expression to find out how many cups of food her dogs eat in a week.

We are asked to write an algebraic expression to represent the number of cups of food that Izabella gives her dogs in a week. Let's first write the number of cups per day. We are given that each day she gives t and b cups to her dogs.

	Variable
Amount of Food Given to Small Dog	t
Amount of Food Given to Big Dog	b

The sum of these variables is the amount eaten in a day. t + b If we want to write how many cups she gives her dogs in two days, we need to add the expression to itself. t + b + t + b = 2(t + b) We multiplied the sum of t and b by 2 to find how many cups of food Izabella gives her dogs in two days. If we follow this logic, we need to multiply the sum by 7 to find how many cups of food she gives her dogs in a week. This is because each week has 7 days. Cups of Food Per Week: 7 (t + b) This expression can also be simplified by using the Distributive Property. 7t + 7b Now we have the expression to know how many cups of food Izabella feeds her dogs in a week.

LaShay evaluated an algebraic expression when x=7. In her notes, a mistake can be found.

Which of the following was LaShay's mistake? A. &LaShay substituted incorrectly. B. &LaShay did not follow the order of operations. C. &LaShay made a mistake doing an operation. D. &LaShay did not make any mistake.

We are asked to determine what mistakes, if any, LaShay made when evaluating an algebraic expression. Let's evaluate the expression ourselves to see if there are any errors. We substitute 7 for the variable x and then evaluate the resulting numerical expression.

Comparing LaShay's work with our work, we can see that substituting the value for the variable was done correctly. The mistake was that LaShay added 7 and 8 instead of multiplying 7 by 3.

We can conclude that LaShay did not follow the order of operations. She had to multiply 7 by 3 first. Therefore, the answer is B.

A local game store sells video game equipment.

Mark wants to buy a arcade sticks and c controllers. Select the correct expression that represents how much Mark has to pay.

Let's try to write the algebraic expression ourselves. We will find how much each item costs and then we will add them. We are told that each arcade stick has a price of $ 100. Mark wants to buy a arcade sticks. This means that the amount for the arcade sticks is the product of 100 by a. Cost of Arcade Sticks 100a We know that the price of the controllers is $ 75 and that Mark wants to buy c controllers. In a similar manner, we can say that the amount for the controllers is the product of 75 by c. Cost of Controllers 75c Now we have the amounts for the arcade sticks and the controllers. Things are progressing quite well. We add the expressions to write an expression for the total amount. Total Cost 100a+ 75c This is the algebraic expression that represents how much Mark has to pay. We are done! Remember that we should always be mindful when we write the products and correctly associate the variables with the appropriate coefficients.

A group of people are buying tickets to a theme park.

The group is made of a adults and c children. The total cost can be written as an expression. 50a + 26c

What is the cost of an adult ticket?

What is the cost of a child ticket?

Find the total cost for 4 adults and 9 children.

We are asked for the cost of an adult ticket. We are given an algebraic expression for the total cost of tickets for a adults and c children. 50a + 26c This expression have two terms. We can see that there is a term with the variable a. 50 a + 26c We know that a represents the number of adults. This variable has a coefficient of 50. Since the expression represents the total cost, the term 50 a alone represents the cost of tickets for the adults. Cost For the Adults 50 a This cost is the product of the number of adults a and the price for each ticket. This means that each adult ticket has a price of $ 50.

We can find the term for the total cost of tickets for the children by following the same reasoning we did in Part A. This time, we need to focus on the term with the variable c. 50a + 26 c This term is made of the variable c and the coefficient 26. Also, the term represents the cost of tickets for the adults. Cost For the Children 26 c This cost is the product of the number of children c and the entry price for each child. This means that each ticket has a price of $ 26.

We want to find the total cost of tickets for 4 adults and 9 children. This means that we need to evaluate the given algebraic expression when a= 4 and c = 9. Let's substitute these values and then evaluate the resulting numerical expression.

Our work indicates that the entry cost of 4 adults and 9 children is of $434.

Algebraic Expressions

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