Did you know? ‘Add five, then double’ and ‘double, then add five’ are different number machines. With an input of three, the first gives sixteen and the second eleven. The arithmetic is simple; understanding which operation happens first is what prepares students for Secondary 1 algebra.
For families seeking a Secondary 1 G2 Mathematics tutor in Bukit Timah, number machines provide a clear way to form one-variable equations, undo operations in reverse order and check each answer by running the original rule forwards. This bridge from Primary arithmetic to Secondary algebra makes algebraic symbols less mysterious.
The approach develops linear equation and algebraic expression skills in the MOE G2 Mathematics syllabus. The unchanged eduKateSG Secondary 1 small-group tutorial reference describes weekly 90-minute tuition for up to three students near Sixth Avenue MRT. The worked machines here are instructional illustrations rather than a separate national exam chapter.
Number machines in one minute
- Label an input x and the stated output y.
- Draw the operation boxes in the order described.
- Use brackets when an operation applies to an entire earlier result.
- To find a missing input, undo the final operation first.
- Undo + with −, − with +, × with ÷, and ÷ with ×, where the divisor is nonzero.
- Preserve equality by applying valid inverse operations to both sides.
- Check the candidate input by running every original step forwards.
- Read the physical context before deciding if negative or fractional inputs are allowed.
Two machines, two different inverse routes
Suppose the rule is ‘add five, then multiply by two’ and the output is 26. The equation is 2(x+5)=26. Reverse the last step first: 26÷2=13; then 13−5=8. The input is eight. Check forwards: 8+5=13, and double thirteen gives twenty-six.
Change the rule to ‘multiply by two, then add five’. The equation becomes 2x+5=26. Now undo addition first, giving 21, then divide by two, giving x=10.5. Both solutions are correct for their own machines. The difference comes from the written rule, not a calculator.
31 first-principles lessons for reversing equations
1. A number machine is a rule, not a mysterious device
A number machine takes an input, applies one or more stated operations in order, and gives an output. In algebra, the input can be called x and the output y. Both are numbers, but they play different roles.
Worked illustration: Input 4, add 3, then multiply by 2: output is (4+3)×2=14.
Teaching check: Write each operation in a box and connect the boxes with arrows before translating them to symbols.
2. The first action changes the input before the second
Operations in a sequential machine are carried out in the order on the arrows. Swapping the boxes usually changes the result. The full algebraic expression needs brackets when the first action is addition or subtraction.
Worked illustration: Starting with 4, add 3 then double gives 14; double then add 3 gives 11.
Teaching check: Compare both results so the importance of order becomes visible.
3. Brackets show which part is multiplied
When the rule says add five, then triple, the triple applies to the new whole amount. The correct rule is 3(x+5). The unbracketed 3x+5 instead triples only the original x.
Worked illustration: For x=2, 3(x+5)=21 while 3x+5=11.
Teaching check: Ask the child to draw three complete copies of x+5.
4. The output can be written as an equation
An expression gives a general output rule; an equation with a known output allows the input to be found. This is the connection to solving linear equations in one variable.
Worked illustration: For a machine add 5 then double, y=2(x+5). If y=26, solve 2(x+5)=26.
Teaching check: Check that the given output is on the equality’s correct side.
5. Undo the last action first
A backwards journey through a number machine reverses the order of the boxes. The final operation is undone first because it was applied last. This is like reversing a route through a sequence of changes.
Worked illustration: If add 5 then double produced 26, halve 26 to get 13, then subtract 5 to obtain input 8.
Teaching check: Perform the operations forwards on 8 to confirm 26.
6. Addition is undone by subtraction
Adding a fixed amount changes a number’s position on the number line. Subtracting the same amount returns to the original position, provided no other operation intervenes.
Worked illustration: If x+7=19, then x=19−7=12.
Teaching check: Read both the forward and reverse movement aloud.
7. Subtraction is undone by addition
When a fixed amount is subtracted from an input, restoring that same amount recovers the input. Keep the order of subtraction clear, since a−x and x−a mean different things.
Worked illustration: If x−4=15, x=19. But 4−x=15 gives x=−11.
Teaching check: Never treat every minus sign as though it could be moved without considering order.
8. Multiplication by nonzero a is undone by division by a
A nonzero multiplier can be reversed by dividing the product by the same multiplier. The restriction nonzero matters because division by zero is undefined.
Worked illustration: If 6x=42, divide by six and find x=7.
Teaching check: Multiply the answer by six to reconstruct forty-two.
9. Division by nonzero a is undone by multiplication by a
If the machine divides an input by five, multiplying the output by five returns the original value. The nonzero condition applies to the divisor.
Worked illustration: If x/5=9, then x=45.
Teaching check: Check 45/5=9.
10. A negative multiplier is still reversible
Multiplication by a negative nonzero number can be undone by division by that same negative value. Predicting the sign helps avoid an incorrect positive input.
Worked illustration: If −3x=24, then x=−8.
Teaching check: Calculate −3×(−8)=24 to verify the sign.
11. The inverse of adding a negative is not adding a positive again
Adding a negative value is equivalent to subtraction. To reverse it, add the opposite signed quantity. Brackets help prevent students from dropping signs.
Worked illustration: If x+(−4)=9, then x=13 because 13−4=9.
Teaching check: Place the signed number on a number line.
12. The expression x/3+4 differs from (x+4)/3
The first divides before adding four, while the second adds four before dividing. Their inverses are therefore different. Parentheses are statements about operation order.
Worked illustration: If x=6, x/3+4=6, but (x+4)/3=10/3.
Teaching check: Ask which operation occurs inside each bracket and which occurs last.
13. Two-step equations can be solved by balance
Undoing a machine is a useful picture, but formal equation work must preserve equality. Doing the same valid operation to both sides guarantees this.
Worked illustration: For 4x+3=23, subtract three from each side: 4x=20; divide by four: x=5.
Teaching check: Use one true equality per line.
14. The unknown might be inside a bracket
An equation a(x+b)=c means the bracket is a single intermediate quantity. Divide c by a, then undo the internal addition.
Worked illustration: For 5(x+2)=35, x+2=7, so x=5.
Teaching check: Check 5(5+2)=35.
15. A positive output can come from a negative input
The sign of a machine’s output depends on the full sequence, not simply on the input sign. A negative input may become positive after a large addition.
Worked illustration: For x=−3, add eight then double gives (−3+8)×2=10.
Teaching check: Do not reject a negative input unless the story requires a nonnegative count.
16. A zero output can still determine an input
Zero is a valid result of many two-step rules. If the multiplier is nonzero, it can still be inverted to find a unique input.
Worked illustration: If 3(x−4)=0, dividing by three gives x−4=0 and x=4.
Teaching check: Check the bracket truly becomes zero.
17. An algebraic solution is not necessarily a whole-object count
A pure number machine can accept fractional or negative inputs. But an equation for numbers of pupils or boxes may require a nonnegative integer, so interpretation must follow the context.
Worked illustration: A machine input x=2.5 is a valid number, but it cannot ordinarily represent two and a half distinct pupils.
Teaching check: Identify what x represents in the story.
18. A changed output can reveal a constant rate
If a rule has the form y=3x+5, each one-unit increase in x increases y by three, although the fixed five remains unchanged. This links number machines to algebraic pattern reading.
Worked illustration: At x=2, y=11; at x=3, y=14. The increase is three.
Teaching check: Label the coefficient three as the repeated change and five as the offset.
19. Different-looking formulas can describe the same rule
Expanding brackets may give an algebraically equivalent expression. A student should not interpret a new written form as a new machine.
Worked illustration: 3(x+4) equals 3x+12 for every x.
Teaching check: Try substitution at x=0 and x=5, then explain distribution.
20. One matching input does not prove equivalent machines
Two different rules may happen to produce the same output at one special input. Matching one test is useful but not a complete proof of algebraic identity.
Worked illustration: Rules 2x+4 and 3x both produce twelve when x=4, but at x=5 they produce fourteen and fifteen.
Teaching check: Look for an actual algebraic reason that forms are equivalent.
21. A word description should specify each action unambiguously
The phrase ‘double a number and add five’ usually means 2x+5, while ‘double the sum of a number and five’ means 2(x+5). Reading carefully precedes calculation.
Worked illustration: At x=3, the two rules produce eleven and sixteen.
Teaching check: Underline ‘the sum’ and draw operation boxes.
22. An incorrect output can identify the first wrong arrow
If a learner’s machine calculation fails, compare the output after each operation with the stated rule. The first wrong intermediate value is the most efficient teaching target.
Worked illustration: For input 5, subtract two then multiply by four: intermediate three, output twelve. If a learner shows twenty, the subtraction was skipped.
Teaching check: Repair the intermediate value and retest with input seven.
23. A two-step process can contain rational results
A school problem may lead to a fractional input when the output is not a neat multiple of the stated coefficient. Linear-equation reasoning still works, though an integer-only story may rule the solution out.
Worked illustration: If 2x+3=8, then x=5/2. That is valid for a pure number machine.
Teaching check: Check 2(5/2)+3=8 exactly.
24. Inverse operations should be recorded in the proper order
A common error is to undo the first action first. For add five then multiply by three, dividing by three must happen before subtracting five.
Worked illustration: An output 39 from 3(x+5) gives x=8: 39÷3=13, then 13−5=8.
Teaching check: Draw reverse arrows, not merely a memorised rearrangement.
25. The opposite of a number and its reciprocal play different roles
The additive inverse of a is −a, while the multiplicative inverse of nonzero a is 1/a. Reversing addition uses the opposite; reversing multiplication uses the reciprocal.
Worked illustration: Undo +5 with −5, but undo ×5 with ÷5.
Teaching check: Do not apply reciprocals to addition questions.
26. Divide by the entire coefficient, including its sign
If a machine’s final action multiplies by −4, dividing the output by positive four is not the inverse. The negative sign is part of the multiplier.
Worked illustration: For −4(x+1)=12, first divide by −4 to get x+1=−3, so x=−4.
Teaching check: Check −4(−4+1)=12.
27. The simplest reverse calculation is often mental
Understanding inverse operations makes small equations solvable without elaborate symbol manipulation, but a written line is still useful when the task demands clear working.
Worked illustration: For 2(x−7)=10, halve to five then add seven, obtaining x=12.
Teaching check: Express the reasoning in one sentence.
28. A missing input can be found from a table
If a table gives outputs for several inputs under one rule, the reverse operation can locate which input produced a target output. The rule must be stated or reliably inferred from the situation.
Worked illustration: Under y=4x+1, output 29 corresponds to input seven.
Teaching check: Substitute x=7 and compare with the table.
29. Two machines may be chained without losing the meaning
More operations can be added by extending the arrow diagram. A three-step machine is still reversed from the last box to the first, but students should master two steps first.
Worked illustration: Add two, multiply by three, subtract four: y=3(x+2)−4. Output seventeen gives x=5.
Teaching check: Reverse +4, ÷3, −2 and then check forwards.
30. An equation can have no valid contextual input even when solvable algebraically
A model x represents an item count. If the equation returns negative two, the algebra is consistent but the physical count may not be. The story must be checked.
Worked illustration: A fictional box count x satisfying 3x+5=−1 gives x=−2, not a feasible count of boxes.
Teaching check: Distinguish the real-number solution from the allowed context.
31. A coherent solution ends with a forward test
After reversing the machine, enter the candidate input into every forward operation in the stated order. This is an independent validation of the answer.
Worked illustration: If y=5(x−2) and y=30, x=8. Forward: 8−2=6, times five is thirty.
Teaching check: If the output differs, return to the earliest incorrect inverse step.
64 worked number-machine questions
Cover each answer before working. Decide whether the machine adds then multiplies, multiplies then adds, subtracts then multiplies, or divides after adding. Inverse steps should be shown in the opposite order from the forward description.
Machine clinic 1: Machine: add 3, then multiply by 2. Output 16. Find input.
Answer: x=5. Inverse reasoning: Reverse: 16÷2=8, then subtract 3 to get 5.
Forward check: Forward check 2(5+3)=16. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 2: Machine: add 4, then multiply by 3. Output 33. Find input.
Answer: x=7. Inverse reasoning: Reverse: 33÷3=11, then subtract 4 to get 7.
Forward check: Forward check 3(7+4)=33. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 3: Machine: add 5, then multiply by 4. Output 44. Find input.
Answer: x=6. Inverse reasoning: Reverse: 44÷4=11, then subtract 5 to get 6.
Forward check: Forward check 4(6+5)=44. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 4: Machine: add 2, then multiply by 5. Output 50. Find input.
Answer: x=8. Inverse reasoning: Reverse: 50÷5=10, then subtract 2 to get 8.
Forward check: Forward check 5(8+2)=50. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 5: Machine: add 1, then multiply by 6. Output 60. Find input.
Answer: x=9. Inverse reasoning: Reverse: 60÷6=10, then subtract 1 to get 9.
Forward check: Forward check 6(9+1)=60. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 6: Machine: add 3, then multiply by 7. Output 35. Find input.
Answer: x=2. Inverse reasoning: Reverse: 35÷7=5, then subtract 3 to get 2.
Forward check: Forward check 7(2+3)=35. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 7: Machine: add 5, then multiply by 8. Output 72. Find input.
Answer: x=4. Inverse reasoning: Reverse: 72÷8=9, then subtract 5 to get 4.
Forward check: Forward check 8(4+5)=72. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 8: Machine: add 2, then multiply by 9. Output 108. Find input.
Answer: x=10. Inverse reasoning: Reverse: 108÷9=12, then subtract 2 to get 10.
Forward check: Forward check 9(10+2)=108. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 9: Machine: add 7, then multiply by 4. Output 20. Find input.
Answer: x=-2. Inverse reasoning: Reverse: 20÷4=5, then subtract 7 to get -2.
Forward check: Forward check 4(-2+7)=20. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 10: Machine: add 8, then multiply by 3. Output 9. Find input.
Answer: x=-5. Inverse reasoning: Reverse: 9÷3=3, then subtract 8 to get -5.
Forward check: Forward check 3(-5+8)=9. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 11: Machine: add 9, then multiply by 5. Output 45. Find input.
Answer: x=0. Inverse reasoning: Reverse: 45÷5=9, then subtract 9 to get 0.
Forward check: Forward check 5(0+9)=45. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 12: Machine: add 4, then multiply by 6. Output 18. Find input.
Answer: x=-1. Inverse reasoning: Reverse: 18÷6=3, then subtract 4 to get -1.
Forward check: Forward check 6(-1+4)=18. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 13: Machine: add 11, then multiply by 2. Output 48. Find input.
Answer: x=13. Inverse reasoning: Reverse: 48÷2=24, then subtract 11 to get 13.
Forward check: Forward check 2(13+11)=48. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 14: Machine: add 3, then multiply by 10. Output -10. Find input.
Answer: x=-4. Inverse reasoning: Reverse: -10÷10=-1, then subtract 3 to get -4.
Forward check: Forward check 10(-4+3)=-10. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 15: Machine: add 1, then multiply by 12. Output 96. Find input.
Answer: x=7. Inverse reasoning: Reverse: 96÷12=8, then subtract 1 to get 7.
Forward check: Forward check 12(7+1)=96. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 16: Machine: multiply by 2, then add 3. Output 19. Find input.
Answer: x=8. Inverse reasoning: Subtract 3: 16, then divide by 2 to get 8.
Forward check: Forward check 2×(8)+3=19. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 17: Machine: multiply by 3, then add 5. Output 17. Find input.
Answer: x=4. Inverse reasoning: Subtract 5: 12, then divide by 3 to get 4.
Forward check: Forward check 3×(4)+5=17. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 18: Machine: multiply by 4, then add 7. Output 31. Find input.
Answer: x=6. Inverse reasoning: Subtract 7: 24, then divide by 4 to get 6.
Forward check: Forward check 4×(6)+7=31. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 19: Machine: multiply by 5, then add 2. Output -13. Find input.
Answer: x=-3. Inverse reasoning: Subtract 2: -15, then divide by 5 to get -3.
Forward check: Forward check 5×(-3)+2=-13. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 20: Machine: multiply by 6, then add 11. Output 23. Find input.
Answer: x=2. Inverse reasoning: Subtract 11: 12, then divide by 6 to get 2.
Forward check: Forward check 6×(2)+11=23. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 21: Machine: multiply by 7, then add 8. Output 71. Find input.
Answer: x=9. Inverse reasoning: Subtract 8: 63, then divide by 7 to get 9.
Forward check: Forward check 7×(9)+8=71. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 22: Machine: multiply by 8, then add 1. Output -15. Find input.
Answer: x=-2. Inverse reasoning: Subtract 1: -16, then divide by 8 to get -2.
Forward check: Forward check 8×(-2)+1=-15. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 23: Machine: multiply by 9, then add 4. Output 4. Find input.
Answer: x=0. Inverse reasoning: Subtract 4: 0, then divide by 9 to get 0.
Forward check: Forward check 9×(0)+4=4. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 24: Machine: multiply by 3, then add 12. Output -3. Find input.
Answer: x=-5. Inverse reasoning: Subtract 12: -15, then divide by 3 to get -5.
Forward check: Forward check 3×(-5)+12=-3. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 25: Machine: multiply by 5, then add 10. Output 45. Find input.
Answer: x=7. Inverse reasoning: Subtract 10: 35, then divide by 5 to get 7.
Forward check: Forward check 5×(7)+10=45. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 26: Machine: multiply by 2, then add 9. Output 39. Find input.
Answer: x=15. Inverse reasoning: Subtract 9: 30, then divide by 2 to get 15.
Forward check: Forward check 2×(15)+9=39. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 27: Machine: multiply by 4, then add 13. Output 57. Find input.
Answer: x=11. Inverse reasoning: Subtract 13: 44, then divide by 4 to get 11.
Forward check: Forward check 4×(11)+13=57. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 28: Machine: multiply by 6, then add 2. Output 80. Find input.
Answer: x=13. Inverse reasoning: Subtract 2: 78, then divide by 6 to get 13.
Forward check: Forward check 6×(13)+2=80. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 29: Machine: multiply by 10, then add 3. Output 123. Find input.
Answer: x=12. Inverse reasoning: Subtract 3: 120, then divide by 10 to get 12.
Forward check: Forward check 10×(12)+3=123. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 30: Machine: multiply by 12, then add 5. Output -43. Find input.
Answer: x=-4. Inverse reasoning: Subtract 5: -48, then divide by 12 to get -4.
Forward check: Forward check 12×(-4)+5=-43. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 31: Machine: subtract 3, then multiply by 2. Output 12. Find input.
Answer: x=9. Inverse reasoning: Divide 12 by 2 to get 6, then add 3 to get 9.
Forward check: Check 2(9−3)=12. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 32: Machine: subtract 5, then multiply by 3. Output 9. Find input.
Answer: x=8. Inverse reasoning: Divide 9 by 3 to get 3, then add 5 to get 8.
Forward check: Check 3(8−5)=9. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 33: Machine: subtract 2, then multiply by 4. Output 16. Find input.
Answer: x=6. Inverse reasoning: Divide 16 by 4 to get 4, then add 2 to get 6.
Forward check: Check 4(6−2)=16. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 34: Machine: subtract 7, then multiply by 5. Output 20. Find input.
Answer: x=11. Inverse reasoning: Divide 20 by 5 to get 4, then add 7 to get 11.
Forward check: Check 5(11−7)=20. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 35: Machine: subtract 1, then multiply by 6. Output -18. Find input.
Answer: x=-2. Inverse reasoning: Divide -18 by 6 to get -3, then add 1 to get -2.
Forward check: Check 6(-2−1)=-18. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 36: Machine: subtract 3, then multiply by 7. Output -21. Find input.
Answer: x=0. Inverse reasoning: Divide -21 by 7 to get -3, then add 3 to get 0.
Forward check: Check 7(0−3)=-21. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 37: Machine: subtract 2, then multiply by 8. Output 16. Find input.
Answer: x=4. Inverse reasoning: Divide 16 by 8 to get 2, then add 2 to get 4.
Forward check: Check 8(4−2)=16. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 38: Machine: subtract 8, then multiply by 9. Output 63. Find input.
Answer: x=15. Inverse reasoning: Divide 63 by 9 to get 7, then add 8 to get 15.
Forward check: Check 9(15−8)=63. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 39: Machine: subtract 6, then multiply by 3. Output 18. Find input.
Answer: x=12. Inverse reasoning: Divide 18 by 3 to get 6, then add 6 to get 12.
Forward check: Check 3(12−6)=18. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 40: Machine: subtract 5, then multiply by 4. Output -32. Find input.
Answer: x=-3. Inverse reasoning: Divide -32 by 4 to get -8, then add 5 to get -3.
Forward check: Check 4(-3−5)=-32. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 41: Machine: add 4, divide by 2. Output 3. Find input.
Answer: x=2. Inverse reasoning: Reverse: multiply 3 by 2 to obtain 6, then subtract 4 to get 2.
Forward check: Forward check (2+4)/2=3. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 42: Machine: add 6, divide by 3. Output 5. Find input.
Answer: x=9. Inverse reasoning: Reverse: multiply 5 by 3 to obtain 15, then subtract 6 to get 9.
Forward check: Forward check (9+6)/3=5. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 43: Machine: add 5, divide by 4. Output 2. Find input.
Answer: x=3. Inverse reasoning: Reverse: multiply 2 by 4 to obtain 8, then subtract 5 to get 3.
Forward check: Forward check (3+5)/4=2. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 44: Machine: add 10, divide by 5. Output 7. Find input.
Answer: x=25. Inverse reasoning: Reverse: multiply 7 by 5 to obtain 35, then subtract 10 to get 25.
Forward check: Forward check (25+10)/5=7. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 45: Machine: add 2, divide by 6. Output -2. Find input.
Answer: x=-14. Inverse reasoning: Reverse: multiply -2 by 6 to obtain -12, then subtract 2 to get -14.
Forward check: Forward check (-14+2)/6=-2. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 46: Machine: add 7, divide by 7. Output 0. Find input.
Answer: x=-7. Inverse reasoning: Reverse: multiply 0 by 7 to obtain 0, then subtract 7 to get -7.
Forward check: Forward check (-7+7)/7=0. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 47: Machine: add 4, divide by 8. Output 9. Find input.
Answer: x=68. Inverse reasoning: Reverse: multiply 9 by 8 to obtain 72, then subtract 4 to get 68.
Forward check: Forward check (68+4)/8=9. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 48: Machine: add 9, divide by 9. Output -3. Find input.
Answer: x=-36. Inverse reasoning: Reverse: multiply -3 by 9 to obtain -27, then subtract 9 to get -36.
Forward check: Forward check (-36+9)/9=-3. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 49: Machine: add 12, divide by 3. Output 6. Find input.
Answer: x=6. Inverse reasoning: Reverse: multiply 6 by 3 to obtain 18, then subtract 12 to get 6.
Forward check: Forward check (6+12)/3=6. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 50: Machine: add 8, divide by 4. Output 10. Find input.
Answer: x=32. Inverse reasoning: Reverse: multiply 10 by 4 to obtain 40, then subtract 8 to get 32.
Forward check: Forward check (32+8)/4=10. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 51: Input x, add 5 then double, result 26. Find x.
Answer: 8. Inverse reasoning: Divide 26 by two to get thirteen, then subtract five.
Forward check: 2(8+5)=26. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 52: Input x, double then add five, result 26. Find x.
Answer: 10.5. Inverse reasoning: Subtract five to get 21, then halve.
Forward check: 2×10.5+5=26. Not the same machine as add five then double. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 53: Input x, triple then subtract four, result 17. Find x.
Answer: 7. Inverse reasoning: Add four then divide by three.
Forward check: 3×7−4=17. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 54: Input x, subtract four then triple, result 18. Find x.
Answer: 10. Inverse reasoning: Divide eighteen by three, then add four.
Forward check: 3(10−4)=18. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 55: Rule y=−4(x+1); output 12. Input?
Answer: −4. Inverse reasoning: Divide by −4 to get x+1=−3; subtract one.
Forward check: −4(−4+1)=12. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 56: Rule y=2x+3; output eight. Input?
Answer: 5/2. Inverse reasoning: Subtract three and divide by two.
Forward check: Two times 5/2 plus three equals eight. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 57: Rule y=3x+5; input two then three. How much does output rise?
Answer: 3. Inverse reasoning: Outputs eleven and fourteen differ by three.
Forward check: Constant coefficient is three per input step. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 58: Rule 3(x+4) compared with 3x+12: equivalent?
Answer: Yes. Inverse reasoning: Distribution gives 3x+12.
Forward check: Both produce the same output for every x. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 59: Rule 2x+4 versus 3x at x=4: same output?
Answer: Yes, both twelve. Inverse reasoning: But the rules are not equivalent for all x.
Forward check: At x=5, outputs fourteen and fifteen. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 60: A machine output 0 after 3(x−4). Input?
Answer: 4. Inverse reasoning: Divide zero by three, then add four.
Forward check: Three times zero is zero. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 61: A machine adds eight then doubles, input −3. Output?
Answer: 10. Inverse reasoning: −3+8=5 then ×2=10.
Forward check: Negative input can produce positive output. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 62: A three-step machine adds two, triples then subtracts four; output seventeen. Input?
Answer: 5. Inverse reasoning: Undo −4 by adding four; divide by three; subtract two.
Forward check: Forward 3(5+2)−4=17. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 63: Does a rule y=3x+5 work as an item count if y=−1?
Answer: Not for a nonnegative item-count input. Inverse reasoning: Algebra gives x=−2, but item counts cannot be negative.
Forward check: State the domain restriction. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Machine clinic 64: Rule x/3+4 and rule (x+4)/3 at x=6: outputs?
Answer: 6 and 10/3. Inverse reasoning: In the first, divide first; in the second, add first.
Forward check: Brackets determine operations. Changing just one operation or the output makes an excellent new test of whether the input can be reconstructed independently.
Three activities that connect boxes to algebra
Make an operation-arrow card
Write ‘input → add 4 → multiply by 3 → output’ on a card. Choose an input such as two and calculate 18. Reverse the card: start with 18, divide by three, then subtract four, recovering two. Replace the order of the two boxes and compare the answer.
Give an unfamiliar input or output once the model is removed. Require the child to name the inverse step rather than simply writing the result.
Use a balance picture
Draw two equally heavy sides of a balance and write 3x+5=20. Remove five from both sides, leaving 3x=15. Share each side into three equal parts to find x=5. Explain why an equality operation must affect both sides.
Give an unfamiliar input or output once the model is removed. Require the child to name the inverse step rather than simply writing the result.
Compare a bracket with a missing bracket
Place 2(x+3) and 2x+3 on cards. For x=4, they produce fourteen and eleven. Tell a short story matching each rule, then choose a second input to show they are not equivalent.
Give an unfamiliar input or output once the model is removed. Require the child to name the inverse step rather than simply writing the result.
Four-week repair plan for one-variable equations
Week 1: learn the inverse pairs
Use addition/subtraction and multiplication/division with positive integers, including clear number-line and equal-group pictures.
Keep practice proportional to school work, and use the school’s actual topic sequence. A successful unfamiliar check is more useful than pages of copied solutions.
Week 2: two actions in different orders
Contrast add-then-multiply against multiply-then-add. Build arrow diagrams, brackets and inverse steps; verify every result forwards.
Keep practice proportional to school work, and use the school’s actual topic sequence. A successful unfamiliar check is more useful than pages of copied solutions.
Week 3: signed values and bracket equations
Introduce negative multipliers, negative inputs and simple equations such as −3(x−2)=12. Keep every sign visible on the reverse arrow path.
Keep practice proportional to school work, and use the school’s actual topic sequence. A successful unfamiliar check is more useful than pages of copied solutions.
Week 4: contextual models without hints
Mix number machine, rate-plus-fixed-fee and simple worded equations. Ask the student to define x, model, reverse and check independently. Repeat a corrected type after several days.
Keep practice proportional to school work, and use the school’s actual topic sequence. A successful unfamiliar check is more useful than pages of copied solutions.
How a three-student tutorial diagnoses the exact error
One pupil may reverse the operations in the original order, another may lose brackets, and a third may divide by the wrong sign. These are different causes despite similar wrong answers. Listening to the learner’s first inverse step allows the tutor to target the problem rather than repeat all of algebra.
The eduKateSG Secondary 1 reference describes 90-minute weekly classes for three learners near Sixth Avenue MRT. Parents should verify current class availability and individual suitability directly; no score is guaranteed.
Frequently asked questions
Why use a number machine for Secondary 1 G2 Mathematics?
It helps visualise how a one-variable equation is built and why inverse operations must be carried out in reverse order.
What does ‘undo the last step first’ mean?
The output was produced by a sequence of steps; reverse the final action, then the preceding action.
How do I reverse ‘add five then triple’?
Divide the output by three, then subtract five.
How do I reverse ‘triple then add five’?
Subtract five first, then divide by three. The order differs from the previous machine.
Why do brackets matter?
They show whether the multiplier acts on the input alone or on an already modified quantity.
Can negative numbers be machine inputs?
Yes in a pure number task. Contextual quantities such as numbers of pupils may have restrictions.
Can the input be a fraction?
Yes. A linear equation can have a fractional solution even when classroom demonstrations often use whole numbers.
Do I need to move terms across the equals sign?
No. You can preserve equality by doing the same inverse operation on both sides, which is more reliable than an unexplained ‘move and change sign’ rule.
How do I check the input?
Use the solved input in the forward machine and verify that it reproduces the stated output.
Is this in the G2 Sec 1 syllabus?
The syllabus includes linear equations in one variable and their formulation for solving problems. A number machine is a teaching model for those skills.
How should a tutor help if a pupil gets stuck?
Ask for the last action and its inverse first. Then help the learner work backwards one box at a time and check forwards independently.
Can I use a calculator?
It can check arithmetic, but the operations and order must be correctly modelled before keystrokes can help.
Official syllabus and connected G2 learning
The MOE G2 Mathematics syllabus provides the one-variable equation objectives. The unchanged eduKateSG tutorial reference describes the teaching model.
- Forming linear equations from word problems
- Simplifying expressions with brackets
- Algebraic equivalence and checking substitutions
- Consecutive integer number puzzles
- Number patterns and nth-term rules
The idea that makes algebra approachable is simple: every action has a meaning, and its inverse reverses that action. Follow the arrows in the correct order and the unknown becomes a number you can justify and check.