Algebra · Algebraic Expressions · Lesson 1

Writing Algebraic Expressions from Words

Market women, trotro mates and shopkeepers do algebra every day without calling it algebra: "Whatever the price of one tuber, three tubers cost three times that." In this lesson you will learn to turn word instructions like these into short algebraic expressions, using letters to stand for numbers we do not know yet.

NaCCA B7.2.2.1.1 ⏱ about 45 min

By the end you will be able to

✔Explain what a variable, a term, a coefficient and a constant are in an expression such as $5 x + 2$ .
✔Translate key words (more than, less than, times, half of, shared) into the correct operation.
✔Write algebraic expressions, with brackets where needed, for a set of instructions given in words.
✔Form expressions for real-life situations such as ages, sharing money, profit, perimeter and area.

Key words

Variable: A letter, such as $x$ or $n$ , that stands for a number we do not know or that can change.
Algebraic expression: A group of numbers, letters and operation signs with no equals sign, e.g. $5 x + 2$ .
Term: One part of an expression separated by + or −. In $5 x + 2$ the terms are $5 x$ and $2$ .
Coefficient: The number multiplying a letter. In $5 x$ the coefficient of $x$ is 5.
Constant: A term with no letter, so its value never changes, e.g. the 2 in $5 x + 2$ .

Curriculum reference

Content standard: B7.2.2.1 Simplify algebraic expressions involving the four basic operations and substituting values to evaluate algebraic expressions.

Indicator: Create simple algebraic expressions using simple logic to translate a set of instructions into an algebraic expression.

Starter

The yam seller's price list

Auntie Akua sells yam at Techiman market. The price of one tuber changes every week, so she does not print prices. Instead she tells customers, "Three tubers? Pay three times the price of one. And add GH₵2 if you want the black polythene bag." If one tuber costs GH₵x, can you write what a customer pays — without knowing x? By the end of this lesson you will write it in four symbols: $3 x + 2$ .

1. Letters that stand for numbers

In arithmetic we work with numbers we know: 3 × 8 = 24. In algebra we also work with numbers we do not know yet. We give the unknown number a name — usually a small letter such as $x$ , $n$ , $a$ or $y$ — and call it a variable. The letter is simply a box that holds a number.

When we join numbers and letters with operation signs we get an algebraic expression. An expression has no equals sign; it is like a phrase, not a full sentence. Look at the parts of the expression $5 x + 2$ :

The parts of an algebraic expression
Part	In 5x + 2	What it means
Variable	$x$	the unknown number
Term with a letter	$5 x$	5 times the number, written without the × sign
Coefficient	5	the number in front of the letter
Constant	2	a plain number that does not change

2. Key words and the operations they mean

Translating words into algebra is like translating Twi into English: you must know what the key words mean. Read the sentence slowly, find the unknown number, then look for the words that tell you what to do to it.

Words you may see	Operation	Example with a number x	Expression
more than, sum, increased by, added to, older than	+	10 more than x	$x + 10$
less than, reduced by, minus, younger than, difference	−	5 less than x	$x - 5$
times, product, multiplied by, twice (×2), thrice (×3)	×	3 times x	$3 x$
half of, shared equally among, divided by	÷	half of x	$\frac{x}{2}$

When there are two or more instructions, carry them out in the order they are given. A function machine helps you see this. Step through the animation for "2 more than 5 times a number x".

▶ Notice that the number goes through the machines in order: first × 5, then + 2. The result is 5x + 2.

3. Brackets: when an instruction acts on a whole result

Sometimes an instruction acts on the result of earlier instructions, not just on one number. Then we put the earlier result in brackets so it is treated as one quantity.

1Read: "When 8 times a certain number x is subtracted from 5, and the result is multiplied by 2."
2"8 times x" gives $8 x$ .
3" $8 x$ is subtracted from 5" means start with 5 and take away $8 x$ : $5 - 8 x$ .
4"The result is multiplied by 2": the whole of $5 - 8 x$ is doubled, so we need brackets: $2 (5 - 8 x)$ .

4. Expressions from real life

Algebra becomes powerful when we use it to describe real situations. Choose a letter for the unknown quantity, then use the story to build the expression. Here are the kinds of situations you will meet:

Situation	Unknown	Expression
Afrako is 3 years older than Maako.	Maako is m years	Afrako: $m + 3$ years
Tetteh gets GH₵35 more than Agbolosu when they share GH₵400.	Agbolosu gets GH₵a	Tetteh: GH₵ $(a + 35)$
A trader buys a basket of oranges for GH₵x and sells it for GH₵y.	cost x, selling price y	Profit: GH₵ $(y - x)$
A rectangle is t metres long and q metres wide.	t and q	Area: $tq$ m²
A rectangular garden is x m long and y m wide.	x and y	Perimeter: $2 x + 2 y$ m

The perimeter of a shape is the distance all the way round its edges. Walk round the rectangular garden in the animation and keep a running total.

▶ Walking round the rectangle adds x + y + x + y. Two lengths and two widths give 2x + 2y.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Twice a number, increased by 9

Write an expression for "9 more than twice a number n".

Show the solution

Step 1The unknown number is n.
Step 2"Twice the number" means 2 × n, written $2 n$ .
Step 3"9 more than" means add 9 to that result: $2 n + 9$ .

Answer: $2 n + 9$

Example 2

Ages

Afrako is 3 years older than Maako. If Maako is now m years old, write an expression for (a) Afrako's age now, (b) Afrako's age in 5 years' time.

Show the solution

Step 1(a) "Older than" means add: Afrako is $m + 3$ years old.
Step 2(b) In 5 years everyone is 5 years older: $m + 3 + 5 = m + 8$ years.
Step 3Check with a number: if Maako is 10, Afrako is 13 now and 18 in five years; $10 + 8 = 18$ ✓.

Answer: (a) $m + 3$ years (b) $m + 8$ years

Example 3

Sharing money

Agbolosu and Tetteh were given GH₵400.00 to share. Tetteh had GH₵35.00 more than Agbolosu. If Agbolosu's share is GH₵a, write an expression for Tetteh's share.

Show the solution

Step 1Agbolosu's share is the unknown: GH₵a.
Step 2"GH₵35 more than Agbolosu" means add 35 to a.
Step 3Tetteh's share = GH₵ $(a + 35)$ .
Step 4Notice that together they have $a + a + 35$ , which must equal 400. You will learn to solve this kind of equation later.

Answer: GH₵ $(a + 35)$

Example 4

An instruction on a whole result

Write an expression for: "When 8 times a certain number x is subtracted from 5, the result is multiplied by 2."

Show the solution

Step 18 times the number: $8 x$ .
Step 2Subtracted from 5: start with 5, so $5 - 8 x$ .
Step 3The result is multiplied by 2, so the whole result goes in brackets: $2 (5 - 8 x)$ .

Answer: $2 (5 - 8 x)$

Example 5

Area and perimeter of a plot

A rectangular plot is t metres long and q metres wide. Write expressions for (a) its area, (b) its perimeter.

Show the solution

Step 1(a) Area of a rectangle = length × width = $t \times q = tq$ square metres.
Step 2(b) Perimeter = length + width + length + width = $t + q + t + q$ .
Step 3Two t's and two q's: perimeter = $2 t + 2 q$ metres.

Answer: (a) $tq$ m² (b) $2 t + 2 q$ m

Summary

A variable is a letter that stands for a number; an expression joins numbers and letters with +, −, × and ÷ but has no equals sign.
Write $5 \times x$ as $5 x$ , $a \times b$ as $ab$ and $x \div 2$ as $\frac{x}{2}$ .
"More than" means +, "less than" means −, "times" means ×, "half of / shared" means ÷. "5 less than x" is $x - 5$ , not $5 - x$ .
Carry out instructions in the order given; use brackets when an instruction acts on a whole result, e.g. $2 (5 - 8 x)$ .
Test your expression with an easy number to check it makes sense.

The yam seller's price list

1. Letters that stand for numbers

2. Key words and the operations they mean

3. Brackets: when an instruction acts on a whole result

4. Expressions from real life

Worked examples

Twice a number, increased by 9

Ages

Sharing money

An instruction on a whole result

Area and perimeter of a plot

Summary

Practice: 15 questions