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Y13 · Letters and Expressions
KO

Middle school maths, year 1 · unit 3

Letters and Expressions

Surprised that an x has suddenly turned up in maths? x is a pouch you cannot see inside. We just do not know yet how many marbles are in it; it is simply a number. We start by putting marbles into the pouch.

Two students looking at pouches and marbles on a desk

x is a pouch you cannot see inside

A board to arrange

When we do not know how many marbles are in the pouch, we call that number x. Drag marbles from the pile into the pouch.

The times sign is left out

Joining board

Just as x × 3 is written 3x, there are agreements for writing expressions briefly. Drag from the dot on a left card to the dot on its partner on the right.

Pouches with pouches, marbles with marbles

A board to arrange

However long an expression is, gathering like with like shortens it. Drag each group to the right tray.

A bracket is one set

A board to arrange

2(x + 3) means two sets of x + 3. Drag set trays into place and see how many pouches and marbles there are altogether.

Try it yourself

Use what you found on the boards. The numbers change every time and the easy questions come first. Get three right and the lamp lights.

What to take away from this unit

  1. A number we do not know yet, or one that can change, is written as a letter. Putting a number in place of the letter is substitution, and the result is the value of the expression. When x = 2, x + 3 is worth 5.
  2. The times sign is left out. Numbers go before letters (x × 3 = 3x), a 1 is not written (1 × x = x), the same letter multiplied by itself becomes a power (x × x = x²), and division becomes a fraction. 3x is not "thirty-something"; it is 3 × x.
  3. The pieces split off by + and − are terms, the number in front of a letter is its coefficient, and a term that is only a number is a constant. An expression like 2x + 3, where the letter is multiplied in just once, is a linear expression.
  4. Terms with the same letter are like terms. Only like terms can be gathered. 3x + 2 + 2x + 1 = 5x + 3. The 5x and the 3 cannot be combined any further.
  5. The number in front of a bracket multiplies every term inside it. This is the distributive law. 2(x + 3) = 2x + 6.

See where this unit sits on the maths family tree