Concept 6 of 6
Machine Rules: Two Steps, and the Order Matters
Today's big idea
Write the middle number down, because the second step acts on it, not on the input.
Double then subtract 3: 5 → 10 → 7. Swap the order and 5 gives 4 instead.
How it works
👀 What's going on?
A number machine is a rule in a box. A number goes in one end, the machine changes it by a fixed rule, and a new number comes out the other end. A simple machine does one step, like 'add 5'. This machine does two steps, one after the other, and the order they happen in changes the answer.
💡 Here's the trick
Look at the machine above. Its rule has two steps: double, then subtract 3. Put 5 in the top. Step 1 doubles it, so 5 becomes 10. Write that 10 down; it is the middle number. Step 2 subtracts 3 from the middle number, so 10 becomes 7. So 5 goes in and 7 comes out. The order is everything: if the machine subtracted 3 first and then doubled, 5 would give 2, then 4, a completely different answer.
🪄 How to do it
- 1.Read the machine's two steps in order, top to bottom, in the picture above.
- 2.Do the first step to your number and write down that middle number.
- 3.Do the second step to the middle number, never to the number you started with.
✍️ Try it
One machine adds 4 and then doubles. What comes out when 6 goes in? Another doubles and then adds 4. Send the same 6 through it.
Adding 4 to 6 gives 10, and doubling that gives 20. Doubling 6 gives 12, and adding 4 gives 16. Same two steps, same starting number, two different answers.
⚠️ Watch out
You might think you can do the easier step first to save time. You cannot. The machine sets the order, and the answer depends on it, just like the order-of-operations rule (BODMAS).
🌟 Why this matters
A recipe, a set of directions and a discount applied after a delivery charge all care about order. Doing the right steps in the wrong order gives the wrong result.
Exam tip: Write the middle number down rather than holding it in your head.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A machine multiplies by 3 then subtracts 2. What comes out when 5 goes in?”
- 1
First step only
5 times 3 is 15. Write that middle number down.
- 2
Second step on the middle number
15 minus 2, never 5 minus 2.
- 3
Why the order matters
Subtracting first would give 3, then 9, a different answer entirely.
- 4
Answer ✓
13
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Do the first step, write the middle number, then the second.
TWO-STEP MACHINE · ROUND 1 OF 3
Step 1 · what is the middle number?
Quick check
✅ VERIFICATION
A machine multiplies by 3 then subtracts 2. What comes out when 5 goes in?
⚡ Finish the activity above first to answer.
Keep going!