Identities, equivalence and algebraic arguments · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| equivalent expression/ɪˈkwɪvələnt ekˈspreʃn/ | 等价表达式 | děng jià biǎo dá shì |
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 be true only at certain values. An identity is true at every allowed value. Establish equivalence by valid expansion or factorisation; testing a few inputs is only a check. Higher proofs use a general integer or algebraic variable and a conclusion tied to its definition.
- 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.
Expanding 3(x+2)-x gives 3x+6-x=2x+6, proving equivalence for every x. But x²=x holds only for x=0 or 1. A counterexample x=2 rejects an all-values claim. For Higher, an odd integer is 2n+1; its square is 4n²+4n+1=2(2n²+2n)+1, so it is odd for every integer n.
Identities, equivalence and algebraic arguments
An equation may be true only at certain values
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 A6 Foundation distinguishes equation/identity and argues equivalence. Higher extends this to algebraic proofs. State integer restrictions when using parity or consecutive integers.
- 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 · Higher · 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.