At a glance. One lesson of about 60 minutes. Students run one- and two-step machines forwards, write the rule as an expression, and run the machine backwards by undoing steps in the reverse order.
Calculator: no calculator. Nothing here needs one. The numbers are small, and the structure of forwards and backwards is the point.
What students will learn
Know
- Input, rule and output.
- The inverse of add is subtract, and the inverse of multiply is divide.
- A two-step machine can be written as an expression.
Understand
- The order of the steps changes the machine.
- Running a machine backwards undoes the steps in the reverse order.
- Backwards from an output gives the input, and a check runs it forwards again.
Do
- Find the output of a one- or two-step machine for a given input.
- Write the rule of a machine as an expression.
- Find the input from the output by undoing the steps.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: running machines forwards and backwards and identifying inverse operations (questions 1 to 6).
- Criterion B. Investigating patterns: the first figure. Ask students to find inputs for which two machines agree and to explain what the swapped order does to the rule.
- Criterion C. Communicating: writing a machine rule in symbols and explaining the order of undoing (question 9).
- Criterion D. Applying mathematics in real-life contexts: the Dhaka data kiosk, run forwards and then backwards from 220 taka (the hook and ‘Back to the kiosk’), a price per metre plus a fixed charge, and a measurement rule (questions 4 and 5).
The one thing to do with the figure
Ask for the output before moving the input. “Multiply by 4 then add 3. What comes out for 5?” Then press the swap button and ask what will change before they see it.
On the backwards figure, cover the bottom row first. Ask students to say the two undoing steps aloud, in order, before the row appears, then pick an output that gives a non-whole input to discuss.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure, running backwards. |
| 10 min | Worked example; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose B, return to the order of the steps. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 32; 6a + 3; 6. |
|---|---|
| Worked example | Input 9 gives output 50, and output 50 comes from input 9. |
| Faded example | 16. 15 − 7 = 8, then 8 × 2. |
| 1. Running a machine forwards | A, 14. B does the steps in the wrong order; C adds where the machine subtracts; D adds 3 instead of multiplying. |
| 2. Two-step machine, forwards | 37. 7 × 5 + 2. |
| 3. Running backwards | 6. (19 + 5) ÷ 4 = 6. |
| 4. A machine in a context | 3900. 3 × 800 + 1500. |
| 5. Two-step rule | 20. 48 ÷ 3 + 4. |
| 6. Inverse machine | B. A repeats the machine; C has the correct inverses in the wrong order; D has the right order with the wrong operations. |
| 7. Same output | 4. Both give 17. |
| 8. Finding a missing step | 7. 24 + 7 = 31. |
| 9. Order of undoing | B. A has the inverses in the wrong order; C uses add where subtract is needed; D multiplies where it should divide. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Did the steps in the wrong order, or added where the machine subtracted. | Have students write each intermediate value on the machine diagram. |
| 45 or 14 in question 2 | Added before multiplying, or added every number to the input. | Draw two boxes in order and write the first step in the first box. |
| 3.5 or 9.75 in question 3 | Used the wrong inverse, or undid the steps in the same order as they were done. | Ask what the last step was, and what undoes it. Do that first. |
| 2300 or 6900 in question 4 | Added 800 and 1500, or charged the work for every metre. | Ask which charge is paid for each metre and which is paid once. |
| 16 or 52 in question 5 | Stopped after one step, or skipped the divide. | Number the steps in the question: step 1, step 2. |
| A, C or D in question 6 | Repeated the machine, reversed the inverses, or used wrong operations. | Write the original steps, then the inverse of each, then reverse the list. |
| 17 or 12 in question 7 | Gave the output rather than the input, or stopped at the difference. | Make a three-column table of input, P and Q. |
| 25 or 124 in question 8 | Ignored the multiplication, or multiplied the output. | Work forwards from the input: 6, then 24, then 31. |
Questions worth asking
- Why must the steps be undone in reverse order? Think of putting on socks then shoes: to undo, shoes come off first. Listen for “the last one done is the first one undone”.
- Is a machine with + 3 then × 4 the same as × 4 then + 3? No. 5 gives 32 and 23. Ask for the two rules written in symbols.
- Can every output be produced by the machine? Not always: with + 3 at the end, no output below 3 is possible for non-negative inputs. In the second figure the output 2 has no input of 0 or more.
- How do you check a backwards answer? Run it forwards again. If you get the output you started with, the input is right.
Differentiation
A way in for anyone stuck
- Draw the machine as boxes on paper and write the number in each gap. Use arrows for the steps.
- Start with one-step machines before two-step ones, and let students write the inverse of each step in a table.
- Let them run backwards with a number line: add and subtract by hops, multiply and divide by repeated jumps.
Stretch
- Make a two-step machine whose output is always 7 more than its input. How many different ones can you find? (Add 3 then add 4; add 6 then add 1; and so on.)
- Two machines in a row: × 3 then + 2, followed by × 2 then − 4. Find one machine that does the same job. (6x + 4 − 4 = 6x: just × 6.)
Students learning in English as an additional language
The words input, output and machine are new. Link them to a physical object: a coin goes in, a ticket comes out. Inverse is a hard word; use “the opposite step” as well.
| Word | Plain meaning | Example |
|---|---|---|
| function machine | a rule that turns each input into one output | multiply by 4, then add 3 |
| input | the number that goes in | 5 in the machine multiply by 4 then add 3 |
| output | the number that comes out | 23 |
| rule | what the machine does, written as an expression | 4x + 3 |
| inverse | the step that undoes another | subtract 3 undoes add 3 |
| order | which step comes first | multiply then add is not add then multiply |
Sentence starters for explaining aloud or in writing:
- The machine multiplies by … and then …, so the output for … is …
- To go backwards, I undo … first because …
- I checked by running … forwards and got …
Two things not to say
“Just do the opposite of everything.” The opposites are needed, and so is the reverse order.
“Guess the input until it fits.” Guessing finds small cases. Undoing finds any case.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ForwardsA machine multiplies by 3 then adds 5. What is the output for 8?
29. - BackwardsA machine divides by 2 then adds 10. The output is 30. What was the input?
40. 30 − 10 = 20, then 20 × 2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA baker uses 150 g of flour per person plus 50 g for dusting the table. How much flour is needed for 8 people?
1250 g. 150 × 8 + 50. - Earlier work, like termsSimplify 6a + 3 + 2a − 1.
8a + 2.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningA two-step machine of the form multiply by k, then add b, turns 3 into 14 and 5 into 22. Find k and b.
k = 4, b = 2. The step of 2 in the input gives a step of 8 in the output. - Earlier work, position rulesThe position rule of a sequence is 5n − 2. Which term is 48?
The 10th term. 48 + 2 = 50, and 50 ÷ 5 = 10.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys or Enter) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw the machine on the board with a number written in each gap.