g10-05-07 · Core topics
Swap two function machines
Does doubling then adding 3 give the same result as adding 3 then doubling?
1 MIN · PREDICT ON PAPER
For input 2, predict both intermediate values and final outputs before sending the token.
Discuss or write first. No typed answer is required to continue.
Input 2; order: f(u) = 2u → g(u) = u + 3.
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
The orders give 2x+3 and 2(x+3)=2x+6, differing by 3. With square roots, order can also change which inputs are allowed.
Conditions: Every intermediate result must belong to the next machine’s domain. Stop when a stage is undefined.
Mathematical justification: Algebra gives (2x+6)−(2x+3)=3 for every real x. For the root pair, the domains are x≥1 and x≥0, so they cannot be equal functions.
1 MIN · EXIT CHECK
At x=0, which is defined: √(x−1) or √x−1?
Check after answering
Only √x−1, whose value is −1.
Teacher guidance and extension
Before: Read function inputs, outputs and the stated domain; write your first prediction on paper.
Ask: Read the pipeline left to right; require the intermediate value as well as the final answer.
Watch for: The same two rules always commute.
Challenge: Compare √(x−1) and √x−1 at x=0 and 1; explain why one equal sample does not prove equality.
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
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore