Direct and inverse proportional relationships · Foundation
導入済み| English | 意味 |
|---|---|
| constant of proportionality/ˈkɒnstənt ɒv prəˌpɔːʃəˈnælɪti/ | 比例定数 |
Buying twice as much fabric at a fixed price doubles the bill. Sharing one fixed job among twice as many equally productive workers halves its duration.
- Buying twice as much fabric at a fixed price doubles the bill. Sharing one fixed job among twice as many equally productive workers halves its duration.
- This lesson studies constant of proportionality 比例常数: The fixed multiplier in a stated proportional relationship.
Choose the mathematical structure
- Direct proportion y=kx preserves y/x for nonzero x and gives a straight line through the origin. Inverse proportion y=k/x preserves xy and gives a reciprocal curve, with x=0 excluded. State the fixed assumptions: a start-up charge breaks direct proportion, while changing productivity can break inverse proportion. Higher constructs the equation from a known pair; Foundation interprets and uses a given equation.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
For the inverse model t=24/w, what happens when the number of workers doubles?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Inverse proportion keeps the product constant: wt=24. Doubling w to 8 forces t=3 - half the previous 6 hours. The model assumes equally productive workers on one fixed job.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For a given fabric cost C=3L, lengths 2,4,6 m cost 6,12,18 yuan: C/L=3. A 5-yuan fixed fee instead gives C=5+3L, which is linear but not directly proportional. For a given fixed-job model t=24/w, workers 2,4,6 need 12,6,4 hours, and wt=24 worker-hours. Doubling workers halves time. The inverse model assumes equally productive workers on one fixed job.
Direct and inverse proportional relationships
Direct proportion y=kx preserves y/x for nonzero x and gives a straight line through the origin
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Use C=3L to find C for L=6.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Substitute L=6: C=18.
Test a tempting shortcut
- An increasing graph need not be direct proportion. An inverse relationship is proportional to 1/x, not to -x. Keep a product constant for inverse proportion and a quotient constant for direct proportion.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every straight line represents direct proportion. This claim is false. Explain which definition or assumption it violates.
Use t=24/w to find t for w=4.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Substitute w=4: t=24/4=6 hours.
Every straight line represents direct proportion.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
An increasing graph need not be direct proportion. An inverse relationship is proportional to 1/x, not to -x. Keep a product constant for inverse proportion and a quotient constant for direct proportion.
Interpret a new situation
- AQA R10/R13/R14 covers numerical, algebraic and graphical direct/inverse proportion. Interpret the units of k and the gradient. In Foundation use the supplied equations; Higher also constructs and interprets them from the context.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the constant product wt for this inverse model.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Multiply: w×(24/w)=24 for w≠0.
Match each relationship to its equation form.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Direct proportion keeps the quotient fixed; inverse proportion keeps the product fixed.
Use this in your course
- 8300 · Foundation · 3.3. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The fixed multiplier in a stated proportional relationship. Choose the relationship, show the method, check its assumptions and interpret the result.