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Variables and expressions

Algebra · level 4, lesson 1 of 7

A question to start

Two friends each describe a number machine in words.

  • Ava: "Take a number, multiply it by 3, then add 2."
  • Ben: "Take a number, add 2, then multiply it by 3."

Ava writes her rule as \(3n + 2\). Ben writes his as \(3(n + 2)\). They look almost the same. Are they?

Try the number 5. Ava's machine gives \(15 + 2 = 17\). Ben's gives \(7 \times 3 = 21\). Different! A pair of brackets changed the answer by 4. In this lesson you learn to read algebra as carefully as you read words, and to write it so that it says exactly what you mean.

The idea

Variables

A variable is a letter that stands for a number. In level 3 a letter was a single hidden number, like the \(n\) in \(n \times 4 = 36\). Now we use letters in a bigger way: a letter can stand for any number, and we write rules that work for all of them at once. That is why it is called a variable: its value can vary.

If \(n\) is the number of children on a trip and each child pays 3 dollars, the total is "3 times \(n\)" dollars. When \(n\) is 10 it is 30 dollars; when \(n\) is 25 it is 75 dollars. The letter lets us say all of this in one go.

Why \(3n\) means \(3 \times n\)

In algebra we leave out the times sign between a number and a letter. \(3n\) means \(3 \times n\). There are two good reasons:

  1. The sign looks like a letter. In algebra, \(x\) is the most popular letter of all. Writing \(3 \times x\) is easy to misread as \(3xx\). Leaving out the sign avoids the confusion.
  2. It is how we already talk. "3 apples" means 3 lots of an apple. "\(3n\)" means 3 lots of \(n\). Nobody says "3 times apples".

Some other habits:

  • The number always goes first: write \(3n\), never \(n3\).
  • \(1n\) is written just \(n\), because one lot of \(n\) is \(n\).
  • Two letters side by side are multiplied too: \(ab\) means \(a \times b\).
  • A dot is sometimes used for times: \(3 \cdot n\) means \(3n\).
  • Division is usually written as a fraction: \(\dfrac{n}{4}\) means \(n \div 4\).
  • \(n^2\) ("\(n\) squared") means \(n \times n\). It is not \(2n\): when \(n = 5\), \(n^2 = 25\) but \(2n = 10\).

But between two numbers you must keep the sign: \(3 \times 4\) is 12, while "34" is thirty-four.

Expressions

An expression is a group of numbers, letters and operations, such as \(3n + 2\) or \(\dfrac{n}{4} - 1\). An expression is like a phrase in English ("three times a number, plus two"). It does not make a claim, so it is not true or false. An equation is a full sentence with an equals sign, like \(3n + 2 = 17\), and it can be true or false.

Parts of an expression have names. In \(3n + 2\):

  • the terms are the pieces joined by \(+\) or \(-\): here \(3n\) and \(2\);
  • the coefficient of \(n\) is the number multiplying it: 3;
  • a term with no letter, like \(2\), is a constant: it never changes.

A picture: cups and counters

As in level 3, a cup can stand for the unknown number \(n\), and a counter for 1. Then \(3n + 2\) is three cups and two counters, and \(3(n + 2)\) is three groups, each with a cup and two counters.

3n + 2 3(n + 2) = 3n + 6

Count the right-hand picture: three cups and six counters. So \(3(n + 2)\) is the same as \(3n + 6\), which is 4 more than \(3n + 2\), whatever \(n\) is. That is why Ava's and Ben's machines always differ by 4.

Words into algebra

Here is a dictionary for turning words into expressions. Let \(n\) be the number.

Words Expression
5 more than a number; a number increased by 5 \(n + 5\)
5 less than a number; a number decreased by 5 \(n - 5\)
a number taken away from 5 \(5 - n\)
twice a number; double a number \(2n\)
half a number \(\dfrac{n}{2}\)
a number squared \(n^2\)
3 more than twice a number \(2n + 3\)
twice the sum of a number and 3 \(2(n + 3)\)

Watch the order in "5 less than \(n\)". You start with \(n\) and take 5 away: \(n - 5\). Many people write \(5 - n\) because the 5 is said first. Test it with a number: 5 less than 12 is 7, and \(12 - 5 = 7\), but \(5 - 12\) is not 7.

Writing expressions from stories

The best way to write an expression from a story is to try it with a real number first, then replace the number with the letter. "Pencils cost 40 cents each. What do \(p\) pencils cost?" Two pencils cost \(40 \times 2 = 80\) cents; ten pencils cost \(40 \times 10 = 400\) cents. So \(p\) pencils cost \(40 \times p = 40p\) cents.

Worked examples

1. Reading algebra aloud. Say each in words: (a) \(4n - 1\); (b) \(4(n - 1)\); (c) \(\dfrac{n + 6}{2}\).

(a) "four times a number, minus one". (b) "four times (one less than the number)". (c) "a number plus six, all divided by two" (or "half of six more than a number").

2. Words to expressions. Write: (a) 7 less than three times \(x\); (b) the sum of \(x\) and 4, multiplied by 5; (c) \(y\) divided by 3, then increased by 2.

(a) \(3x - 7\). (b) \(5(x + 4)\). (c) \(\dfrac{y}{3} + 2\).

3. From a story. Mia has \(m\) stickers. Jon has 4 more than Mia. Kai has twice as many as Jon. Write Jon's and Kai's numbers.

  • Jon: \(m + 4\).
  • Kai: twice Jon's number, so \(2(m + 4)\). The brackets matter: Kai doubles all of Jon's stickers, not just the \(m\). Try \(m = 10\): Jon has 14, Kai has 28, and \(2(10 + 4) = 28\). ✓ (Without brackets, \(2m + 4 = 24\), wrong.)

4. Matchstick squares. A row of squares is made from matchsticks. One square uses 4 sticks; each extra square shares a side with the one before, so it needs 3 more. Write the number of sticks for \(s\) squares.

  • 1 square: 4; 2 squares: 7; 3 squares: 10.
  • Think of it as 1 starting stick, plus 3 for each square: \(3s + 1\). Check: \(3 \times 3 + 1 = 10\). ✓

5. Naming parts. In \(7a - 2b + 9\), name the terms, the coefficients and the constant.

Terms: \(7a\), \(-2b\) and \(9\). The coefficient of \(a\) is 7, of \(b\) is \(-2\) (the minus sign belongs to the term). The constant is 9.

Try it

  1. Write with algebra: (a) 8 more than \(k\); (b) \(k\) multiplied by 8; (c) 8 less than \(k\); (d) \(k\) less than 8.
  2. Write in words: (a) \(2t + 5\); (b) \(2(t + 5)\); (c) \(t^2 - 1\).
  3. A ticket costs 12 dollars. Write the cost of \(t\) tickets, and of \(t\) tickets plus a 5 dollar booking fee.
  4. Leo is \(L\) years old. Write: (a) his age in 6 years; (b) his age 3 years ago; (c) the age of his mother, who is three times as old as he is.
  5. Is \(3n + 2\) an expression or an equation? What about \(3n + 2 = 11\)?
  6. Explain the difference between \(n^2\) and \(2n\), and find a number where they are equal.
  7. A row of triangles is made of matchsticks: 1 triangle needs 3 sticks, 2 need 5, 3 need 7. Write the number of sticks for \(t\) triangles.
  8. Show that \(5(n + 1)\) and \(5n + 1\) are different by trying a number. How much bigger is the first, always?
Answers
  1. (a) \(k + 8\). (b) \(8k\). (c) \(k - 8\). (d) \(8 - k\).
  2. (a) "twice \(t\), plus 5". (b) "twice the sum of \(t\) and 5". (c) "\(t\) squared, minus 1".
  3. \(12t\) dollars; with the fee, \(12t + 5\) dollars.
  4. (a) \(L + 6\). (b) \(L - 3\). (c) \(3L\).
  5. \(3n + 2\) is an expression (no equals sign, no claim). \(3n + 2 = 11\) is an equation; it is true only when \(n = 3\).
  6. \(n^2 = n \times n\) and \(2n = n + n\). They are equal when \(n = 2\) (\(4 = 4\)), and also when \(n = 0\).
  7. Each new triangle adds 2 sticks, starting from 1 extra: \(2t + 1\). Check: \(t = 3\) gives 7. ✓
  8. With \(n = 2\): \(5(2 + 1) = 15\) but \(5 \times 2 + 1 = 11\). The first is \(5n + 5\), so it is always 4 more than \(5n + 1\).

Common mistake

Dropping the brackets. "Twice the sum of \(x\) and 3" is \(2(x + 3)\), not \(2x + 3\). If the words say to do something to a whole group, the group needs brackets.

Writing the subtraction backwards. "6 less than \(y\)" is \(y - 6\). Check with a number: 6 less than 10 is 4, and \(10 - 6 = 4\).

Where it leads