Work Backwards to Recover a Start Value
Given the end state of a chain of operations (or a calculation done wrongly), invert each step in reverse order to recover the original number or position. It begins with undoing simple add/subtract or skip-counts, then reversing a wrong product, undoing geometric moves, and unwinding multi-step decimal and fraction operations. Inversion is the universal 'the result was...' strategy.
grade 2–6 GeometryMeasurement & dataBase-ten numbersFractionsOperations Work BackwardsConvert to Algebra
Progression (15)
g4 1. Skip-count backward to the start Base-ten numbers
g3 2. Recover the unknown, then recompute Operations
g2 3. Undo more/less to find original length Measurement & data
g3 4. Reverse the operations to find the start Operations
g3 5. Recover the original number from a wrong product Operations
g4 6. Reverse the moves to recover original Geometry
g4 7. Undo a wrong fraction operation to recover the start Fractions
g5 8. Reverse decimal operations to recover the start Base-ten numbers
g5 9. Undo a wrong decimal operation to recover the start Base-ten numbers
g5 10. Reverse the operation order to find the unknown Operations
g6 11. Symbolize the unknown, then build an expression ExpressionsOperations
g5 12. Reverse the steps to recover the original fraction Fractions
g5 13. Recover the original number from a wrong calculation Fractions
g6 14. Both sides of an equation can be multiplied or divided by the same number Expressions
g6 15. Name the unknown with a symbol and build an equation ExpressionsGeometry
Bridges to competition (7)
Reasoning problems that extend this idea toward competition:
Work backwards from the result Logic
Fraction word problems via backward reasoning Other
Work backwards in fraction word problems Other
Measure a target in fewest pourings Optimization
Measure a target with unmarked jugs Logic
Find the unknown by working backwards Logic
Recover the starting number Number arrangement