Algebraic expression

Here you will learn about algebraic expressions, including what they are, how to write them in word form, how to simplify them, and how to evaluate them.

Students will first learn about algebraic expressions as part of expressions and equations in 6th grade.

What is an algebraic expression?

An algebraic expression is an expression that includes algebraic terms.

Algebraic terms are individual letters, groups of letters, or groups of letters and numbers separated by addition or subtraction. Other operations (like multiplication, division, and exponents) happen within an algebraic term. Individual numbers are known as constant terms.

For example, this is an algebraic expression with two algebraic terms and one constant term.

Note that the term always includes the symbol or operation before it. In the example above, the constant term is -9, not just 9.

Every algebraic term has a coefficient (the number) and a variable (the letter).

For example in 5x+y-9…

• Positive 5 is the coefficient and the variable is x .
• ‘+1’ is the coefficient and the variable is y .

Even though the 1 is not written, since 1y=y, the y still has a coefficient of +1 because one positive y is being added.

Note that -9 is not a coefficient, because it is not an algebraic term. It is a constant.

In algebraic expressions, when numbers and variables are written next to each other the operation is multiplication.

For example,

4 \times a is written as 4a.

Multiplication is commutative, so 4 \times a is the same as a \times 4, but you do not write a4. When using algebraic notation for multiplication, always put the number before the variable.

When variables or numbers are being divided, they are written in fraction form.

For example,

y \div 3 is written as \, \cfrac{y}{3} \, . Note that \, \cfrac{1}{3} \times y \, can also be written as \, \cfrac{y}{3} \, .

You can convert any algebraic expression to word form or vice versa.

For example,

• m+2 can be written as “2 more than m” .
• 2a +9 can be written as “9 more than twice a number a” .
• “5 less than h” can be written as h-5 .
• “4 times the sum of 8 and t” can be written as 4 \, (8+t) .

For a step-by-step guide for writing algebraic expressions, continue to the ‘how to’ and ‘example’ sections below.

Once you can write algebraic expressions, you can start to simplify them.

Combining like terms groups similar terms together to simplify algebraic expressions. To do this you identify the like terms in an algebraic expression and combine them by adding or subtracting.

For example,

This is an expression with 4 terms.

3a and +9a are both groups of the variable a, so they are like terms. Let’s combine them.

3a is ‘3 times a’ and 9a is ‘9 times a’ which can be drawn as…

3a is 3 groups of a → +a \quad +a \quad +a

9a is 9 groups of a → +a \quad +a \quad +a \quad +a \quad +a \quad +a \quad +a \quad +a \quad +a

Counting all the a s above or adding 3a + 9a shows that combining the like terms is 12a.

+6b and -4b are both groups of the variable b, so they are also like terms. Let’s combine them.

+6b is ‘4 times b’ and -4b is ‘4 times -b’ which can be drawn as…

+6b is 6 groups of b → +b \quad +b \quad +b \quad +b \quad +b \quad +b

-4b is 4 groups of -b → -b \quad -b \quad -b \quad -b

Combining +b and -b creates a zero pair, which means together they are equal to 0.

There are four zero pairs of b s, which leaves two positive b s.

You can also think of this as 6b-4b = 2b :

Combining like terms shows that…

\begin{aligned} & 3 a+6 b+9 a-4 b \\\\ & =3 a+9 a+6 b-4 b \\\\ & =12 a+2 b \end{aligned}

This expression cannot be simplified further since 12a and the +2b are not like terms.

Step-by-step guide: Combining like terms

Continue simplifying expressions by expanding expressions when necessary.

Expanding expressions (or multiplying out) is when you use the distributive property to simplify an expression.

To do this, multiply the terms within the parentheses by the term on the outside.

Then, if needed, combine the like terms to further simplify the resulting expression.

For example,

First, look at 3(5-n) which is ‘3 times the difference of n less than 5.’

Repeated addition	Box method	Distributive property
\begin{aligned} & 3(5-n) \\ & =(5-n)+(5-n)+(5-n) \\ & =5-n+5-n+5-n \end{aligned} Combine the three groups of -\, n and the three 5s\text{:} \begin{aligned} & 5+5+5=15 \\ & -n+-\, n+-\, n=-3 n \end{aligned} So, 3(5-n)= 15-3n	Multiply 3 times 5 and 3 times - \, n\text{:} \begin{aligned} & 3 \times 5=15 \\ & 3 \times(-\, n)=-3 n \end{aligned} So, 3(5-n)= 15-3n	Distribute the 3 to the 5 and the -\, n\text{:} \begin{aligned} & 3 \times 5=15 \\ & 3 \times(- \, n)=-3 n \end{aligned} So, 3(5-n)= 15-3n

Notice that each method shows 3 \, (5-n)=15-3n.

Step-by-step guide: Simplifying expressions

Step-by-step guide: Expanding expressions

Equivalent expressions are algebraic expressions that are equal in value and can be identified using the same skills needed to simplify expressions. We use the equivalence symbol \equiv to show when two expressions are equivalent.

For example, is the statement 6x+11y\equiv11y+6x true?

The coefficients of x are the same, both positive 6x and the coefficients of y are the same, both positive 11x. This means they are equivalent expressions.

For example, is the statement 2 a-b\equiv -b+2 a true?

2a-b is the same as 2a+(-b), because subtracting b is the same as adding -b. And by the commutative property, 2a+(-b) is the same as -b+2a. This means they are equivalent expressions.

Step-by-step guide: Equivalent expressions

To evaluate the expression means to substitute the variable(s) in an algebraic expression with a value and solve.

For example,

If x=5, what is the value of 11x^2-6?

Substitute the x in the expression with the value 5 :

\begin{aligned} & 11 x^2-6 \\\\ & =11 \times 5^2-6 \\\\ & =11 \times 25-6 \\\\ & =275-6 \\\\ & =269 \end{aligned}

Step-by-step guide: Evaluate the expression

What is an algebraic expression?

Common Core State Standards

How does this relate to 6th grade math?

• Grade 6 – Expressions and Equations (6.EE.A.3)
  Apply the properties of operations to generate equivalent expressions.
  For example, apply the distributive property to the expression 3 \, (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 \, (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

How to write algebraic expressions

In order to write algebraic expressions in word form:

1. Identify the terms and operations.
2. Create a statement using the correct vocabulary.

In order to write algebraic expressions from word form:

1. Identify the terms and operations.
2. Create an expression using the correct symbols and order.

Algebraic expression examples

Example 1: expression form to word form

Write 6-r in word form.

1. Identify the terms and operations.

The terms are 6 and the term r is being subtracted.

2Create a statement using the correct vocabulary.

It helps to use what you know about subtraction. Let’s look at two examples.

If r=1, then 6-1 is “1 less than 6.”

And if r=2, then 6-2 is “2 less than 6.”

So, 6-r is written as “r less than 6.”

Example 2: expression form to word form

Write 7 \, (5+n) in word form.

Create a statement using the correct vocabulary.

It helps to use what you know about the distributive property. Let’s look at two examples.

Using the phrase “the sum of” shows that the addition happens before the multiplication in the expression, even though it is stated second.

So, 7 \, (5+n) is written as “7 times the sum of 5 and n.”

Example 3: expression form to word form

Write \, \cfrac{6 f}{11} \, in word form.

Create a statement using the correct vocabulary.

It helps to use what you know about division. Let’s look at two examples.

The phrase “the quotient” indicates that two things are being divided and the word “and” separates those two things.

So \, \cfrac{6 f}{11} \, is written as “the quotient of 6 times f and 11.”

Example 4: word form to expression

Write the expression that shows “the sum of 23 and a number.”

Example 5: word form to expression

Write the expression that shows “5 less than the quotient of c and 3.”

Example 6: word form to expression

Write the expression that shows “10 more than the product of 4 and w.”

Teaching tips for algebraic expressions

• One of the first things that is important for students to understand is exactly what makes up any given algebraic expression. Introduce algebraic expressions by connecting them to hands-on manipulatives (such as algebra tiles) or real-life situations, so students can begin to attach meaning to the symbols and their mathematical operations.

• Vocabulary is key for students to understand how to read and write algebraic expressions. And often even students who are proficient in mathematical language will read 5-r as “5 minus r” and not “r less than 5.” It is important that you give students time to grapple with these phrases and intentionally draw their attention to patterns or grammar rules that will help them understand these types of mathematical expressions.

Easy mistakes to make

• Confusing the order of operations (pemdas) when reading the word form of algebraic expressions
  Expressions will always follow the order of operations, but that does not necessarily mean that the word form will list the operations in order.
  For example,
  “the product of 5 and the sum of 2 and a number” is 5 \, (2+n) , and when solving, the addition happens before the multiplication – though it appears out of order in the word form.

• Forgetting to use parentheses
  When performing an arithmetic operation out of order, use parentheses to indicate the order. Not using parentheses changes the outcome.
  For example,
  “The double of the sum of y and 4” is 2 \, (y+4). Not including the parentheses and writing it as 2 \times y+4, leads to a different order of operations and therefore different answer.

The next lessons are

• Math equations
• Inequalities
• Math formulas
• Types of graphs
• Polynomial

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