033. Algebra Terminology and Expressions

Learning Intentions

• To know the basic terminology of algebra
• identify coefficients, terms and constant terms within expressions
• write expressions from word descriptions

Pre-requisite Summary

• Understand that letters can be used to represent numbers
• Recall the operations addition, subtraction, multiplication and division
• Know that multiplication can be written without a multiplication sign in algebra, for example $3x$
• Be able to read simple mathematical symbols and brackets
• Understand that an expression is a combination of numbers, variables and operations
• Know that like terms have the same variable part
• Be able to distinguish between numbers attached to variables and numbers standing alone
• Be able to translate simple verbal phrases into arithmetic operations

Worked Examples

Worked Example 1

a) Define the terms variable, coefficient, term and constant term.
b) Identify the variable in $5x+3$.
c) State the coefficient, the terms and the constant term in $5x+3$.

Worked Example 2

Identify the coefficients, terms and constant terms in:
a) $7y-4$
b) $3a+2b+9$
c) $6m$

Worked Example 3

Write an expression for each description:
a) $4$ more than $x$
b) $3$ times $n$
c) $8$ less than $p$

Worked Example 4

Write an expression for each description:
a) the sum of $k$ and $6$
b) twice $t$ plus $5$
c) $10$ minus $q$

Worked Example 5

For the expression $4x+7-2y$:
a) list all terms
b) identify each coefficient
c) state the constant term

Worked Example 6

For each word description, write an expression and then identify its terms:
a) $5$ more than twice $r$
b) $3$ less than $4s$
c) the sum of $2p$ and $9$

Problems

Problem 1

a) Define the terms variable, coefficient, term and constant term.
b) Identify the variable in $6x+5$.
c) State the coefficient, the terms and the constant term in $6x+5$.

Problem 2

Identify the coefficients, terms and constant terms in:
a) $8y-6$
b) $2a+5b+7$
c) $9m$

Problem 3

Write an expression for each description:
a) $5$ more than $x$
b) $4$ times $n$
c) $7$ less than $p$

Problem 4

Write an expression for each description:
a) the sum of $k$ and $9$
b) twice $t$ plus $4$
c) $12$ minus $q$

Problem 5

For the expression $3x+8-4y$:
a) list all terms
b) identify each coefficient
c) state the constant term

Problem 6

For each word description, write an expression and then identify its terms:
a) $6$ more than twice $r$
b) $4$ less than $5s$
c) the sum of $3p$ and $8$

Exercises

Understanding and Fluency

1. State the meaning of each term:
   a) variable
   b) coefficient
   c) constant term

2. Identify the variable, terms and constant term in:
   a) $4x+9$
   b) $7y-2$
   c) $3m+5n+8$

3. Identify the coefficients in:
   a) $6a+1$
   b) $2p-7$
   c) $5x+3y+4$

4. List the terms and state the constant term in:
   a) $9k+6$
   b) $4r-3$
   c) $2a+7b+10$

5. Write an expression for each description:
   a) $3$ more than $x$
   b) $5$ times $n$
   c) $9$ less than $p$

6. Write an expression for each description:
   a) the sum of $t$ and $4$
   b) twice $m$ plus $7$
   c) $11$ minus $q$

7. Write an expression for each description and identify the terms:
   a) $4$ more than $3x$
   b) $2$ less than $5y$
   c) the sum of $6p$ and $1$

8. Mixed practice:
   a) In $8x+3$, state the coefficient of $x$
   b) In $5a-9$, state the constant term
   c) In $2m+4n+7$, list the terms

Reasoning

9. Explain why $5$ is called the coefficient in $5x$.

10. A student says the coefficient in $x+4$ is $0$. Explain the mistake.

11. Explain why the number $7$ in $3x+7$ is called a constant term.

12. A student writes “$4$ less than $x$” as $4-x$. Explain why this is incorrect.

Problem-solving

13. A phone plan costs a fixed $12 plus $3 for each gigabyte $g$ used. Write an expression for the total cost.

14. A taxi fare is a fixed $6 plus $2 for each kilometre $k$. Write an expression for the fare and identify the constant term.

15. A school buys $n$ notebooks at $4 each and pays a $5 delivery fee. Write an expression for the total cost.

16. A person saves $10 each week for $w$ weeks and already has $25. Write an expression for the total amount saved.

17. A rectangle has length $x$ and width $3$ less than $x$. Write an expression for the width.

18. A shop sells pens for $2 each. A customer also buys a notebook for $7. Write an expression for the total cost if $p$ pens are bought.

Potential Misunderstandings

• Students may think a variable is always a specific unknown value rather than a symbol that can represent a number
• Students may confuse a coefficient with a constant term
• Students may think a term must contain a variable, ignoring constant terms
• Students may not recognise that a single variable such as $x$ has coefficient $1$
• Students may list parts of a term separately, for example treating $3x$ as two terms instead of one
• Students may reverse word descriptions such as “$4$ less than $x$” and write $4-x$ instead of $x-4$
• Students may confuse “sum” with multiplication or subtraction
• Students may not recognise that subtraction creates a negative term in an expression

Next: 034. Substituting into Algebraic Expressions