🔢 Algebra Machine: Letters Stand for Numbers

There is a number inside this box, but we don’t know which one. In maths we give a secret number like this a letter: n. Today we will learn to work with letters just like numbers.

We don’t know the number in the box, but we can write 3 more than the box: one box and three balls. That is n + 3. If we open the box and find 5, then 5 + 3 makes 8.

2 times the box means two boxes: n + n. We write that in short as 2 × n. 3 times the box is three boxes: 3 × n.

Careful, the order of the words matters! 3 more than 2 times n: two boxes and three balls, so 2 × n + 3. 2 times (3 more than n): a group of a box and three balls, twice, so 2 × (n + 3). They are different expressions. “Times” means multiply, “more than” means add.

Now a number machine. We drop a number in. The machine first multiplies by 2, then adds 3. Let’s put in 4: 4 times 2 is 8. Then 8 plus 3 is 11. The machine’s rule is 2 × n + 3.

Let’s try other numbers. If we put 1 in for n, 5 comes out. 2 gives 7, 5 gives 13 and 10 gives 23. This is called finding the value of the expression: write the number in place of n and work it out step by step.

Let’s name the parts of 2 × n + 3. The letter n is the variable. The 2 in front of n is the coefficient: it tells us how many n there are. The 3 on its own is the constant term. This expression has two terms: 2 × n and 3.

Now let’s build squares from sticks. One square needs 4 sticks. A chain of two squares needs 7, three squares need 10, four squares need 13. Every new square adds only 3 sticks.

Let’s find the rule. The stick at the far left stands alone: + 1. Every square brings three sticks. For n squares that is 3 × n sticks. So a chain of n squares has 3 × n + 1 sticks. Now, however many squares there are, we can find the sticks without counting!

Summary: We give an unknown number a letter. 3 more than n is n + 3, and 2 times n is 2 × n. The coefficient is the number in front of the letter, and the constant term is the number on its own. We put a number in for n to find the value of the expression. The rule of a shape pattern is an algebraic expression too. Now try it in the Algebra Machine!