Track a Quantity Through Changes
A measured quantity — container weight, water level, bar length, work done — changes as things are added, removed, or accumulate at a net rate; the unknown is recovered by following the running total. Learners handle before/after weight, net fill-with-leak, bouncing-ball fractions, submerged bars, and combined work rates. The thread is conservation: every change is accounted for. In middle school the same ledger runs on grams of salt carried from cup to cup and on whole-job fractions, with a letter standing for the amount that is not yet known.
grade 3–8 Measurement & dataBase-ten numbersFractionsOperationsRatiosExpressions Draw a DiagramWork Backwards
Progression (17)
g3 1. Bouncing ball rises a fraction of its fall FractionsOperations
g3 2. Find container weight after adding or removing Measurement & data
g3 3. Net fill rate is inflow minus outflow Measurement & dataOperations
g3 4. Add poured water to find total capacity Measurement & data
g4 5. Add per-day work fractions to find combined output Fractions
g4 6. Use submerged and exposed parts to find bar length Fractions
g5 7. Use weight differences to find the empty container Base-ten numbersMeasurement & data
g5 8. Find empty container weight by subtracting fractions Fractions
g6 9. Use what was poured out to weigh the container Number systemExpressionsRatios
g6 10. Moving some across leaves the total unchanged ExpressionsRatios
g7 11. Track grams of salt through two pourings RatiosExpressions
g7 12. Compare two drain-outs to isolate the inflow ExpressionsRatios
g7 13. Subtract the full-time pump's share first ExpressionsRatios
g7 14. Salt balance turns a mixture into proportion RatiosExpressions
g7 15. Swapping mixtures shrinks the concentration gap RatiosExpressions
g8 16. Track the salt through each pour ExpressionsRatios
g8 17. Subtract team rates to isolate one worker ExpressionsNumber system
Bridges to competition (2)
Reasoning problems that extend this idea toward competition:
Set up the correct add/subtract equation Logic
Broken clock with constant drift Pattern & rule