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Algebraic expression

# 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

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…

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.’

\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

## 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.

Identify the terms and operations.

Create a statement using the correct vocabulary.

### Example 3: expression form to word form

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

Identify the terms and operations.

Create a statement using the correct vocabulary.

### Example 4: word form to expression

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

Identify the terms and operations.

Create an expression using the correct symbols and order.

### Example 5: word form to expression

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

Identify the terms and operations.

Create an expression using the correct symbols and order.

### Example 6: word form to expression

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

Identify the terms and operations.

Create an expression using the correct symbols and order.

### 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.

### Practice algebraic expression questions

1. What is the word form of 18+r?

the quotient of 18 and r

the product of 18 and r

18 minus r

r more than 18

The terms are 18 and r and they are being added.

Look at two examples.

If r=1, then 18+1 is “1 more than 18.”

And if r=2, then 18+2 is “2 more than 18.”

So, 18+r is written as “r more than 18.”

2. What is the word form of 4 \, (10-d)?

4 times 10 minus d

4 times the difference of 10 and d

4 and d less than 10

The difference of 4 and 10 and d

The terms are 10 and d and they are being subtracted then multiplied by 4.

Look at two examples using the expression.

If d = 1, then 4 \, (10-1) is “4 times the difference of 10 and 1.”

And if d=2, then 4 \, (10-2) is read “4 times the difference of 10 and 2.”

Using the phrase “the difference of” shows that the subtraction happens before the multiplication in the expression.

So, 4 \, (10-d) is written as “4 times the difference of 10 and d.”

3. What is the word form of \, \cfrac{9 h}{2} \, ?

the quotient of 9 times h and 2

the product of 9 and h and 2

9 times 2 divided by h

9 and h divided by 2

The term is 9h and it is being divided by 2.

Look at two examples.

If h = 1, then \, \cfrac{9 \times 1}{2} \, is “the quotient of 9 times 1 and 2.”

If h = 2, then \, \cfrac{9 \times 2}{2} \, is “the quotient of 9 times 2 and 2.”

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

So, \, \cfrac{9 h}{2} \, is written as “the quotient of 9 times h and 2.”

4. Which expression shows “8 less than s”?

8-s

s-8

\cfrac{8}{s}

\cfrac{s}{8}

The terms are 8 and s. “Less than” tells us the operation is subtraction.

Look at two examples,

If s = 8, then 8 less than 8” is 8-8.

And if s = 9, then 8 less than 9” is 9-8.

So, “8 less than s” is written as s-8.

5. Which expression shows “Double the sum of y and 4”?

y+y+4

y+4 \times 2

2y+4

2 \, (y+4)

The terms are y and 4 and “sum” tells us to add them. “Double” tells us to multiply the terms by 2.

Look at two examples,

If y = 1, then “Double the sum of 1 and 4” is 2 \, (1+4).

And if y = 2, then “Double the sum of 2 and 4” is 2 \, (2+4).

The parentheses show that the addition happens before the multiplication.

So, “Double the sum of y and 4” is written as 2 \, (y+4).

6. Which expression shows “6 more than the product of t and 10”?

10t+6

6t+10

6 \, (t+10)

10 \, (t+6)

The terms are t, 10 , and 6.

“The product” tells us that t and 10 are being multiplied, and “more than” tells us that 6 is being added.

Look at two examples,

If t = 1, then “6 more than the product of 1 and 10” is 10 \times 1+6.

And if t = 2, then “6 more than the product of 2 and 10” is 10 \times 2+6.

So, “6 more than the product of t and 10” is written as 10t+6.

## Algebraic expression FAQs

What do the words monomial, binomial, trinomial and polynomial mean?

These each refer to how many terms are in an expression. Monomial has one term. Binomial has two terms. Trinomial has three terms. All types are considered polynomials.

What is the difference between an algebraic expression and an algebraic equation?

An equation is a statement with an equal sign. It is like a complete mathematical sentence. An expression is a grouping of terms but does not include an equals sign. It is like a mathematical phrase.

Are there other types of algebraic expressions?

Yes, there are other rational expressions that involve coefficients including positive and negative fractions or decimals. Students also learn to work with expressions that involve square roots, and that are a part of inequalities.

Since a growing shape pattern starts with a shape and changes in a constant way from term to term, algebraic expressions can be used to write the pattern’s general rule. For example, “Start with 1 square and add 2 squares each time.” This rule can be written as the algebraic expression: 1 + 2n , where n = the pattern term number.

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