Lesson Guide
Explanation
A function machine takes a number in, does one job to it, and sends a different number out. This one multiplies by 3: feed it 5 and out comes 15. Feed it 2 and out comes 6. The machine always does the same job, which is what makes it useful — once you know the job, you can work out any answer without being told.
Your Turn
If this machine is fed 7, what comes out? And what would you feed it to get 24? Check: 21 comes out, and you would feed it 8.
Watch Out
The machine does the same job every time. If a table of numbers does not follow one job, you have the wrong rule.
Explanation
Every operation has an opposite that undoes it. Adding 6 is undone by subtracting 6. Multiplying by 4 is undone by dividing by 4. Mathematicians call the opposite the inverse. Test it: start at 5, add 6 to get 11, then subtract 6 and you are back at 5 exactly. You already met this idea in fact families — 3 times 4 is 12, so 12 divided by 4 is 3. Same idea, bigger job.
Your Turn
Name the inverse of each: subtract 9, divide by 5, add 20. Check: add 9, multiply by 5, subtract 20.
Watch Out
The inverse of times 4 is divide by 4, not times a quarter of something else, and definitely not minus 4.
Explanation
Now the useful bit. Suppose you know 24 came out of the times-3 machine, but nobody told you what went in. Run the machine backwards: instead of multiplying by 3, divide by 3. 24 divided by 3 is 8, so 8 went in. Always check it forwards afterwards — 8 times 3 really is 24, so you are right. That forward check takes two seconds and catches almost every mistake.
Your Turn
A machine subtracts 7. Out comes 15. What went in? Check: undo with add 7, so 22 went in. Forward check: 22 minus 7 = 15.
Watch Out
Going backwards means using the INVERSE. Doing the same operation again is the trap: 24 times 3 is 72, not the answer.
Explanation
With two machines the order matters. The number went through times 2 first, then plus 5. To get back you undo them in the opposite order: undo the plus 5 first, then undo the times 2. So 17 minus 5 is 12, and 12 divided by 2 is 6. Think about putting on socks then shoes — to undo it you take the shoes off first. Last on, first off. Then check forwards: 6 times 2 is 12, plus 5 is 17.
Your Turn
A number is multiplied by 4 then has 3 subtracted, giving 25. What was it? Check: add 3 to get 28, divide by 4 to get 7. Forward check: 7 times 4 is 28, minus 3 is 25.
Watch Out
Undoing in the SAME order as the machines is the classic error. Reverse the order, and reverse each operation.
Explanation
Test questions almost never say the words function machine. They say: I think of a number, double it, add 7, and get 31 — what was my number? That is the same thing with the diagram rubbed out, so your first move is to draw the diagram back in. Write the chain of machines, then undo it backwards. Students who draw it get these right; students who try to hold it in their head get muddled. Draw it every time, even when it feels easy.
Your Turn
I think of a number, add 4, then multiply by 3, and get 27. Draw the chain and undo it. Check: 27 divided by 3 is 9, minus 4 is 5. Forward check: 5 + 4 = 9, times 3 = 27.
Watch Out
Do not start calculating until you have written the chain down. The order of the machines is the whole question.
📌 Key Points — this lesson in a nutshell
- A function machine does the same job to every number you feed it.
- The inverse undoes an operation: + and − undo each other, × and ÷ undo each other.
- To find the start from the answer, run the machines backwards using inverses.
- With two machines, undo the last one first — shoes off before socks.
- Always check forwards. It takes two seconds and catches nearly every slip.
Step 1: Introduction
Step 1 of 5
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