6.8 Variation and applications

Ideas in this lesson

Inverse variation

For inverse variation, the product of input and output is constant. Use a known pair to determine that constant.

Find a variation constant

Substitute the given quantities into the variation model and solve for its constant.

Combined variation

Translate directly varying quantities into numerator factors and inversely varying quantities into denominator factors.

Rate and work applications

A work rate is the reciprocal of completion time. Add simultaneous rates and use realistic positive times.

Classify direct, inverse or neither from equations, including misleading rearrangements and additive offsets

classify direct, inverse or neither from equations, including misleading rearrangements and additive offsets.

Classify exact/approximate data, state an explicit estimation method for the constant and assess the model fit before predicting

classify exact/approximate data, state an explicit estimation method for the constant and assess the model fit before predicting.

Construct joint variation equations from words/data, including products of more than two quantities and powers

construct joint variation equations from words/data, including products of more than two quantities and powers.

Translate combined verbal variation with powers and roots into an equation and solve for any requested variable

translate combined verbal variation with powers and roots into an equation and solve for any requested variable.

Compare multiplicative changes, scientific-notation scales and chained variation relations; justify the factors algebraically

compare multiplicative changes, scientific-notation scales and chained variation relations; justify the factors algebraically.

Create an inverse-variation model from a measurement/experimental setting, build a table, predict and validate with appropriate rounding and limitations

create an inverse-variation model from a measurement/experimental setting, build a table, predict and validate with appropriate rounding and limitations.

Build average-cost and changing-success-fraction models, solve for a target and interpret discrete counts and asymptotes

build average-cost and changing-success-fraction models, solve for a target and interpret discrete counts and asymptotes.

Build mixture/concentration and weighted-percentage equations, solve and generalize symbolically with meaningful bounds

build mixture/concentration and weighted-percentage equations, solve and generalize symbolically with meaningful bounds.

Build equal-time speed/distance, opposite-current/wind and round-trip equations, solve all candidates and reject impossible speeds/times

build equal-time speed/distance, opposite-current/wind and round-trip equations, solve all candidates and reject impossible speeds/times.

Construct geometric proportion equations, including a golden-ratio-style problem with an exact irrational solution and positive-domain check

construct geometric proportion equations, including a golden-ratio-style problem with an exact irrational solution and positive-domain check.

Select, construct, compare and critique rational models across verbal, tabular, graphical and symbolic representations, including units, feasible domains, approximate results and limits of extrapolation

select, construct, compare and critique rational models across verbal, tabular, graphical and symbolic representations, including units, feasible domains, approximate results and limits of extrapolation. Include open-ended direct-plus-inverse rules satisfying a value constraint.