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Algebra · Formulae & rearranging

Chapter 1 · 3

The idea

Number machines and function machines

A number machine is a list of instructions carried out in order. How to run one forwards, how to run it backwards by undoing the last box first, why the order of the boxes matters, how to write a machine as an expression with the brackets in the right place, how to find the number a machine leaves unchanged or a box that is missing, and how to join two machines, undo one in algebra and solve with them.

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Algebra · Formulae & rearranging

Number machines and function machines

A number machine is a list of instructions carried out in order. How to run one forwards, how to run it backwards by undoing the last box first, why the order of the boxes matters, how to write a machine as an expression with the brackets in the right place, how to find the number a machine leaves unchanged or a box that is missing, and how to join two machines, undo one in algebra and solve with them.

Why it works

Follow the arrows, one box at a time

"Here is a number machine." Those words open a question on many papers. A machine is a list of instructions, carried out in order from left to right:

Input→×4→− 9→Output\text{Input} \to \boxed{\times 4} \to \boxed{-\,9} \to \text{Output}

Put 55 in. The first box multiplies by 44, giving 2020; the second subtracts 99, giving the output, 1111. Write the number on each arrow as it goes:

5→×420→−9115 \xrightarrow{\times 4} 20 \xrightarrow{-9} 11

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