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In this lesson, we will try to apply and extend your previous **understanding of arithmetic to algebraic expressions**. Moreover, we will write and evaluate numerical expressions involving whole number exponents, while at the same time writing, reading, and evaluating expressions in which letters stand for numbers.

**See the fact file below for more information on the understanding algebraic expressions or alternatively, you can download our 31-page Understanding Algebraic Expressions worksheet pack to utilise within the classroom or home environment.**

## Key Facts & Information

### ALGEBRAIC EXPRESSIONS

- Just to recall, a numerical expression is a mathematical combination of numbers, operations, and grouping symbols. It is a mathematical phrase that represents a single value. These operations include addition, subtraction, multiplication, and division.
- An algebraic expression is an expression involving variables and constants, along with algebraic operations: addition, subtraction, multiplication, and division. An example of an algebraic expression is:
- 3x + 1 and 5(x² + 3x)

- These expressions are represented with the use of unknown variables, constants, and coefficients. The combination of these three elements is called terms of expressions.
- Unlike an algebraic equation, an algebraic expression has no sides or equal sign.

### PARTS OF AN ALGEBRAIC EXPRESSION

**Variable**- A variable is a letter or symbol that represents an unknown value.
**Coefficient**- A coefficient is the number multiplied by the variable in an algebraic expression.
**Term(s)**- A term is the name given to a number, a variable, or a number and a variable combined by multiplication or division.
**Constant**- A constant is a number that cannot change its value.
- The whole expression (i.e., 5x – 3) is known to be a binomial term, as it has two unlikely terms.

### TYPES OF ALGEBRAIC EXPRESSIONS

- There are three main types of algebraic expressions, namely monomial, binomial, and polynomial expressions.
**Monomial Expression**

An algebraic expression having only one term

Examples of monomial expressions are: 8xy, 7x, 9y, 12z⁴, etc.**Binomial Expression**

An algebraic expression having two unlikely terms

Examples of binomial expressions are: 8xy + 7x, 9y + 12z⁴, etc.**Polynomial Expression**

An algebraic expression with more than one term with non-negative integral exponents of a variable

Examples of polynomial expressions are: 8xy + 7 + 9y + 12z⁴, etc.

### WRITING NUMERICAL EXPRESSIONS

- In working with algebraic expressions from verbal statements, you need to familiarize yourself with key terms representing the four operations: addition, subtraction, multiplication, division.
- Use parentheses () or brackets to help group calculations to be sure that some calculations are done in a special order.
- When you use parentheses, you are stating to “do this first”.
- Write a numerical expression given the verbal phrase below:
- The sum of eight and a number multiplied by five

- Looking at the example, you have to understand that you need to obtain the sum of eight and a number, and then multiply whatever the answer is by five.
- This should be done first – the sum of eight and a number
- Then, whatever the answer is – multiply it by five
- The operation that must be done first must be enclosed in parentheses.
- So, the algebraic expression we can get is:
- (8 + y) x 5

- Write a numerical expression given the verbal phrase below:
- The sum of eight and the product of a number and five

- Comparing it to the first example, both involve the same numbers and the same operations. Moreover, both of the two examples involve the numbers eight and five, a variable, and the addition and multiplication operations. Do they mean the same thing, though? No.
- In example 2, the operation that must be done first is to multiply a number and five, then add eight to whatever product you get.
- This should be done first – the product of a number and five
- Then, whatever the answer is – add to eight
- So, the algebraic expression we get is:
- 8 + (y x 5)

- Let’s compare the two verbal phrases.
- The sum of eight and a number multiplied by five
- (8 + y) x 5

- The sum of eight and the product of a number and five
- 8 + (y x 5)

- We can say that both verbal statements may have exactly the same numbers and may involve the same operations. However, they have different meanings. They will yield different answers when evaluated.
- Pay attention to the given phrase and group the numbers with operations that must be done first.

### ORDER OF OPERATIONS

- In an expression with more than one operation, use the rules called the Order of Operations.
- Some expressions look difficult because they include parentheses and brackets. You can think of brackets as “outside” parentheses. You evaluate inside parentheses first.
- ORDER OF OPERATIONS
- Perform all operations within the parentheses first.
- Do all multiplication and division in order from left to right.
- Do all addition and subtraction in order from left to right.

- Besides parentheses (), brackets [ ] and braces { } are other kinds of grouping symbols that are used in expressions. To evaluate an expression with different grouping symbols, perform the operation in the innermost set of grouping symbols first, then evaluate the expression from inside out.
- 2 x [(9 x 4) – (17 – 6)]
- Do the operations in the parentheses () first. Multiply, subtract, and rewrite. Do operations in the brackets [ ]. Subtract and rewrite. Multiply 2 and 25 to get 50.
- 2 x {5 + [(10 – 2) + (4 – 1)]}
- Do the operations in the parentheses first. Subtract, then rewrite. Next, do the operations in brackets [ ]. Add and rewrite. Then, do the operations in braces { }.
- Add and rewrite. Multiply 2 and 6 to get 32.

### EVALUATING ALGEBRAIC EXPRESSIONS

- To evaluate an algebraic expression, replace the variables with their values. Then, find the value of the numerical expression using the order of operations.
- a² – (b³ – 4x) if a = 7, b = 3, and x = 1

- Replace a with 7, b with 3, and x with 1.
- Evaluate 7² and 3³, then multiply 4 and 1
- Subtract

### EVALUATING LIKE TERMS

- If you have 3 bags with the same number x of books in each, you have 3x books altogether. If there are 2 more bags with x books in each, you now have 3x + 2x = 5x books.
- This can be done as the number of books in each bag is the same. The terms 3x and 2x are said to be like terms.
- Consider another example. If Aries has a trays each containing b brownies, then he has a x b brownies.
- If Jane has twice as many brownies as Aries, she has 2 x ab = 2ab brownies.
- Together, they have 2ab + ab = 3ab brownies.
**Like terms**- Two terms are called like terms if they involve exactly the same variable and each variable has the same index.
- The distributive property explains the addition and subtraction of like terms. Say for example:
- 2ab + ab = 2 x ab + 1 x ab = (2 + 1)ab = 3ab

- The terms 2a and 3b are not like terms because the variables are different. The terms 3a and 3a² are also not like terms because the indices are different.
- For the sum 8x + 3y + 7x, the terms 8x and 7x are like terms and can be added. There are no like terms for 3y, so by using the commutative property for addition, the sum is:
- 8x + 3y + 7x = 8x + 7x + 3y = 15x + 3y.

- The any-order principle for addition is used for adding like terms.
- Because of the commutative and associative property for multiplication (any-order principle for multiplication), the order of the factors in each term does not matter.
- Therefore, 5a x 3b = 15ab. It’s also the same as 15ba. Same goes with 12ab x 2b²a = 24a²b³ = 24b³a².

**Understanding Algebraic Expressions Worksheets**

This is a fantastic bundle which includes everything you need to know about the understanding algebraic expressions across 31 in-depth pages. These are** ready-to-use Understanding Algebraic Expressions worksheets that are perfect for teaching students about the understanding of arithmetic to algebraic expressions. Moreover, we will write and evaluate numerical expressions involving whole number exponents, while at the same time writing, reading, and evaluating expressions in which letters stand for numbers.
**

### Complete List Of Included Worksheets

- Lesson Plan
- Understanding Algebraic Expressions
- Two Expressions
- Put in the Jar
- Speak Algebraic Expression
- Put Into Words
- Matching Time
- Which is Which?
- Which Comes First?
- Order of Operations
- Combining Like Terms
- Test Yourself!

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