C. Algebra · Algebra: Expressions, Equations and Coding

Variables, Terms and Writing Algebraic Expressions

C1.2 A1About 70 minutes

Learning goals

I can name the parts of an algebraic expression: variables, terms, coefficients, constants and exponents.
I can classify polynomials as monomials, binomials or trinomials and find their degree.
I can translate words, pictures and situations into algebraic expressions.
I can evaluate an expression to check that it fits the situation.

Words to know

variable
A letter or symbol that stands for a number that can change or is unknown, such as x, n or h.

algebraic expression
A combination of numbers, variables and operations, with no equals sign, such as 28 + 6h.

term
One part of an expression separated from the others by + or − signs. In 3x² − 5x + 7 the terms are 3x², −5x and 7.

coefficient
The number multiplied by the variable part of a term. In −5x the coefficient is −5. In x the coefficient is 1.

constant
A term with no variable, such as 7. Its value never changes.

polynomial
An expression made of one or more terms, where variables have whole-number exponents. A monomial has 1 term, a binomial 2 terms, a trinomial 3 terms.

degree
For a term: the sum of the exponents of its variables (4xy² has degree 1 + 2 = 3). For a polynomial: the highest degree of any of its terms.

Remember from Grade 8

In Grade 8 you used variables in pattern rules, added and subtracted monomials like 3x and −5x, added binomials with algebra tiles, and evaluated expressions by substituting numbers, including negatives and fractions. In this lesson you will use the full vocabulary of algebra and write expressions for many kinds of situations.

Let's start at the ski hill

A ski hill charges $28 for a lift ticket, plus $6 per hour to rent a snowboard.

1 hour: 28 + 6(1) = $34
2 hours: 28 + 6(2) = $40
5 hours: 28 + 6(5) = $58

Instead of a new calculation for every case, we can write one expression that works for any number of hours. Let h be the number of hours. The cost in dollars is

28 + 6h

That is the power of algebra: an expression generalizes a relationship. It describes every case at once.

The parts of an expression

Each piece of an expression has a name. Learn these words; you will use them all year.

The expression 3x² − 5x + 7 has three terms. The 3 in 3x² is a coefficient, x is the variable, 2 is the exponent, and 7 is the constant. The sign in front of a term belongs to that term, so the second term is −5x.

Name	Number of terms	Examples	Degree
monomial	1	−8, 4x, 3a²b	0, 1, 3
binomial	2	x + 5, 2y² − 7y	1, 2
trinomial	3	x² + 3x − 10	2
polynomial	1 or more	x³ − x² + 4x + 1	3

Polynomials are named by the number of terms. The degree of a term adds the exponents of all its variables: 3a²b has degree 2 + 1 = 3. A constant such as −8 has degree 0.

Turning words into expressions

Look for the operation words, then watch the order.

sum, more than, increased by, plus → +
difference, less than, decreased by, minus → −
product, times, twice (× 2), triple (× 3), of → ×
quotient, divided by, per, half of (÷ 2) → ÷
squared → exponent 2

Careful with less than: “5 less than n” means start with n and take away 5, so it is n − 5, not 5 − n.

Use brackets when a whole quantity is multiplied, divided or squared. “Twice the sum of n and 3” is 2(n + 3), but “twice n, plus 3” is 2n + 3.

Expressions from pictures

Algebra tiles let you see an expression. A large square is x² (its sides are x long), a long tile is x, and a small square is 1. Shaded tiles are positive. Unshaded tiles with a minus sign are negative.

This set of tiles shows 2x² + 3x − 4: two x²-tiles, three x-tiles and four negative unit tiles.

Worked example

Example 1: Translate phrases

Write an expression for each phrase. Use n for the number.

(a) 7 more than triple a number (b) the difference of a number and 4, squared (c) half the sum of a number and 10

(a) Triple the number is 3n. Seven more than that: 3n + 7.
(b) The difference first, in brackets: (n − 4). Then square it: (n − 4)².
(c) The sum first: (n + 10). Half of it: (n + 10) ÷ 2, also written (n + 10)/2.

(a) 3n + 7 (b) (n − 4)² (c) (n + 10)/2

A rectangle with length 2x + 1 and width x (not to scale). Its perimeter adds all four sides.

Worked example

Example 2: An expression from a diagram

Write a simplified expression for the perimeter of the rectangle above. Then find the perimeter when x = 5 cm and check it.

Perimeter = add all four sides: (2x + 1) + x + (2x + 1) + x.
Combine the x-terms: 2x + x + 2x + x = 6x. Combine the constants: 1 + 1 = 2.
Perimeter = 6x + 2.
When x = 5: 6(5) + 2 = 32 cm.
Check with the actual sides: length 2(5) + 1 = 11 cm, width 5 cm, so 11 + 5 + 11 + 5 = 32 cm. ✓

P = 6x + 2; when x = 5 cm, P = 32 cm.

Worked example

Example 3: Name the parts

For −4a²b + 3ab − 9, list the terms, their coefficients and degrees, and classify the polynomial.

Terms: −4a²b, 3ab and −9.
Coefficients: −4 and 3. The constant is −9.
Degrees: −4a²b has 2 + 1 = 3; 3ab has 1 + 1 = 2; −9 has degree 0.
Three terms make it a trinomial. Its highest term degree is 3.

A trinomial of degree 3.

Check your expression

After you write an expression for a situation, test it with a small number. If the ski expression 28 + 6h gives $34 for 1 hour, and you know 1 hour really costs $34, your expression is probably right. If it doesn't match, look again at the order and the brackets.

Hands-on activity

Make your own algebra tiles

You need: a cereal box or other thin cardboard, a ruler, a pencil, scissors (cut carefully, and ask a grown-up if the cardboard is thick), a marker.

Draw and cut 4 big squares, 5 cm by 5 cm. These are x²-tiles.
Cut 10 long tiles, 5 cm by 1.5 cm. These are x-tiles. (Their long side matches the side of an x²-tile.)
Cut 20 small squares, 1.5 cm by 1.5 cm. These are 1-tiles.
Colour or shade one side of every tile. Draw a minus sign on the other side. Shaded side up = positive; minus side up = negative.
Build these and say each expression aloud: x² + 2x + 3, 2x − 5, −x² + 4x − 1.
Ask someone at home to build a mystery expression for you to name.

Keep your tiles in an envelope. You will use them for the next several lessons.

Just for fun

Why did the binomial break up with the monomial?

It wanted a relationship with more than one term.

Your turn

Write an expression: 6 less than 4 times a number.

Write an expression: the product of 5 and the sum of a number and 2.

List the terms, coefficients and constant of 7y² − y + 12.

Classify 5x³ − 2x and give its degree.

A taxi charges $4.50 plus $1.95 per kilometre. Write an expression for the cost of a k-kilometre ride, then find the cost of a 12 km ride.

A triangle has sides x, x + 4 and 2x − 1. Write a simplified expression for its perimeter.

What is the degree of the monomial 3a²b³?

More practice: warm-up

Write an expression: a number increased by 9.

Write an expression: the quotient of a number and 8.

What is the coefficient of the term −x?

How many terms does 4x² + 3x − 2 + x³ have? What is its degree?

Evaluate 2n + 3 when n = 6.

More practice: on your own

Write an expression for 9 less than the square of a number. Why is 9 − n² wrong?

Student movie tickets cost $9 and adult tickets cost $14. Write an expression for the cost of s student and a adult tickets. Find the cost for 4 students and 2 adults.

A rectangle's length is 3 more than twice its width w. Write expressions for the length and the perimeter. Check your perimeter with w = 4.

Find the degree of each term and of the polynomial 6x²y − 3xy⁴ + 2. Classify it.

Evaluate 3n² and (3n)² when n = 2. Explain why they are different.

Challenge

A photo is x cm wide and x + 2 cm tall. It gets a 3 cm frame on every side. Write simplified expressions for the outer width, the outer height and the outer perimeter.

Write an expression for the sum of three consecutive integers, starting with n. Use it to explain why the sum of any three consecutive integers is a multiple of 3.

A phone plan costs $30 a month for up to 10 GB of data, plus $5 for each GB over 10. Write an expression for the monthly cost when g GB are used (g greater than 10). Find the cost for 14 GB and simplify your expression.

Check your understanding

Quiz

Which expression means 4 less than a number?

What is the coefficient of x in 7 − 3x?

2x² + 5 is a:

What is the degree of 4x³y²?

Sam shovels driveways. He charges $25 per driveway, plus a one-time $10 fee for the salt he buys. Which expression gives his total charge for d driveways?

A square has side length s + 2. Which expression is its perimeter?

For parents and teachers

Big idea: MTH1W expectation C1.2 asks students to create algebraic expressions that generalize relationships given in words, numbers and pictures. This lesson builds the vocabulary (term, coefficient, constant, degree, monomial, binomial, trinomial) and practises translating.

Common difficulties:

Reversing “less than”: writing 5 − n for “5 less than n”.
Missing brackets: writing 2n + 3 for “twice the sum of n and 3”.
Dropping the sign of a term: the coefficient in −5x is −5, not 5.
Mixing up 3n² and (3n)².

How to help: Make up situations from your week (a gym fee plus a cost per visit, a pizza plus a cost per topping) and ask your teen to write the expression and test it with a small number. The homemade algebra tiles will be used in later lessons, so keep them safe.