Identities, equivalence and algebraic arguments · Foundation
| English | Español |
|---|---|
| equivalent expression/ɪˈkwɪvələnt ekˈspreʃn/ | equivalent expression |
Does one successful test prove a claim?
- Two students test x=1 and get equal results. Does that prove their expressions agree for every x?
- This lesson studies equivalent expression 等价表达式: An expression with the same value as another for every allowed input.
Choose the mathematical structure
- An equation may hold only for some values, while an identity holds for every allowed input. Establish equivalent expressions by valid expansion or factorisation. A few matching inputs are checks rather than a general argument.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines equivalent expression?
An expression with the same value as another for every allowed input.
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.
Expand 3(x+2)-x=3x+6-x=2x+6. This chain of valid steps shows the expressions agree for every x. However, x²=x holds only when x=0 or x=1; at x=2, 4≠2. Evaluating (2n+1)² at n=3 gives 49, but this one calculation does not establish an all-integers claim.
Identities, equivalence and algebraic arguments
An equation may hold only for some values, while an identity holds for every allowed input
Compare the model with the worked case and explain one change.
Find the coefficient of x in 3(x+2)-x.
Expand then collect: 3x-x=2x.
Test a tempting shortcut
- Start from an expression or the assumptions, not from the conclusion as though it were already true. An example can disprove an all-values claim, but one confirming example cannot prove it.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Testing two expressions at one input proves that they are identical. This claim is false. Explain which definition or assumption it violates.
Find x²-x at x=2.
4-2=2, a nonzero counterexample.
Testing two expressions at one input proves that they are identical.
Start from an expression or the assumptions, not from the conclusion as though it were already true. An example can disprove an all-values claim, but one confirming example cannot prove it.
Interpret a new situation
- AQA A3/A6 Foundation distinguishes expression, equation and identity and argues equivalence using algebra. General parity and divisibility proofs belong to the Higher variant.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find (2n+1)² at n=3.
(6+1)²=49.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Foundation · 3.2. 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.
An expression with the same value as another for every allowed input. Choose the relationship, show the method, check its assumptions and interpret the result.