g10-13-04 · Additional mathematics
Does cancellation repair a hole?
Are equal values on a shared domain the same as equal domains?
1 MIN · PREDICT ON PAPER
Compare (x−a)(x+c)/[(x−a)(x−b)] with (x+c)/(x−b), especially at x=a.
Discuss or write first. No typed answer is required to continue.
Original excludes 1,-1; the written simplified formula excludes -1.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Cancellation preserves values only at original allowed inputs. The simplified formula may extend the domain, but the original hole remains excluded. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: a≠b. Original domain excludes a,b; the written simplified formula still excludes b.
Mathematical justification: The factor x−a is nonzero on the original domain, so division by it is legal there. At x=a the original denominator vanishes regardless of the finite simplified value.
1 MIN · EXIT CHECK
Is (x²−1)/(x−1) the same function as x+1 on R?
Check after answering
No; the original excludes 1, although the values agree elsewhere.
Teacher guidance and extension
Before: Domains, cancellation and function equality
Ask: Ask pupils to write both domains before comparing the plotted curves.
Watch for: A removable discontinuity is not permission to redefine the original function silently.
Challenge: Choose c=−b and classify what remains at both excluded inputs.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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