088i - Tables from Function Rules

Lesson 88 — Generating Tables of Values from Function Rules

Strand: Algebra | Descriptor: AC9M7A05 | Duration: 45 minutes

Learning Intentions

• To generate a table of values from a given rule, including with negative inputs.
• To understand a rule as a machine pairing every input with exactly one output.

Success Criteria

I can:

1. Substitute a sequence of inputs (including negatives) into a rule accurately.
2. Build a table of pairs from a rule.
3. Read a table backwards: find the input that gave an output.
4. Decide whether a given pair “belongs” to a rule.

Warmup

(6 minutes — substitution retrieval from Lesson 24, mini whiteboards)

For , find when:

5. Which gives ?

Answers: ; ; ; ; .

Note the negatives: they behave exactly as Lesson 24 drilled — brackets around every substituted value.

Activities

Activity 1 — Explicit Instruction: the Function Machine and Its Table (12 min)

The machine picture: input output , i.e. .

I do — build the table left to right, narrating each substitution:

Three observations to draw from the table:

1. The constant step: climbs by each time climbs by — the coefficient is the step (Lesson 87’s bridge, now with negatives).
2. The entry is the constant term, — the machine’s output for “nothing in”.
3. Every input has exactly one output — but an output can point back to only one input for these rules (contrast: for , two inputs share most outputs — preview only).

We do — including a decreasing rule. :

The step is : falls as grows. Where does hit zero? At — read from the table, verified by .

Activity 2 — Table Production Line (14 min)

You do — build each table for to :

Then answer, using your tables:

7. For rule 3: which gives ?
8. For rule 4: when is negative?
9. For rule 5: which entries are not whole numbers, and why?
10. Does the pair belong to rule 6? Show how you decide.

(Answers: tables as computed; 7. — beyond the table: extend or solve ; 8. from onwards; 9. odd give halves — the rule happily accepts any input; 10. ✓ yes.)

Emphasise Q7’s move: the table has edges; the rule does not. When the answer lies off the table, solve the equation — tables and equations are two interfaces to one rule (Lesson 31–36’s skills now serve A05).

Activity 3 — Inquiry: Guess My Rule (13 min)

Pairs. One writes a secret linear rule; the other requests inputs and receives outputs, aiming to name the rule in as few queries as possible.

Round 1: free play, three rules each.

Round 2 — the strategy question:

Prompt	Purpose
Which single input reveals the most?	exposes the constant term outright.
And the second-best query?	: the output minus the constant is the coefficient.
So how many queries suffice for any linear rule?	Two — and determine then in .
Could one query ever suffice?	No — one point fits infinitely many lines. (Two points, one line: this is Lesson 89’s punchline, previewed numerically.)
What if you may not use ?	Any two inputs work: the step between outputs, divided by the step between inputs, is the coefficient.

Round 3: replay with the two-query strategy; keep score of queries used.

Closing note: the class has just discovered that two points determine a linear rule — tomorrow, on the Cartesian plane, this becomes “two points determine a line”.

Checks for Understanding

(5 minutes — exit ticket)

1. Build the table for , to .
2. From your table: the step size, and the entry.
3. Which gives ?
4. Does belong to ?
5. Reasoning. A classmate found a secret rule with the queries and . Name the rule and explain each step.

Answers: 1. ; 2. step ; entry ; 3. ; 4. ✓ yes; 5. Constant (the output); coefficient ; rule (or ).

Common Misconceptions

Misconception	How to pre-empt it
Sign slips at negative inputs, e.g. .	Brackets around every substituted value — the Lesson 24 rule, re-enforced.
Filling the -row by pattern-continuation without substituting.	Legitimate after two genuine substitutions confirm the step; require the anchors.
Believing the table is the rule.	Q7’s off-table value; the rule generates any entry on demand.
Reading the step from non-adjacent columns without dividing.	If steps by , the -step is twice the coefficient — trap it explicitly in Activity 2’s rule 5.
Thinking decreasing rules are “not proper rules”.	and are first-class citizens from the start.
One query “solving” guess-my-rule.	The inquiry’s infinitely-many-lines discussion.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). For : which input gives an output of ?

Answer

.

E2 (AMC Junior style). A table for a linear rule shows and . Find the rule.

Answer

Step: ; then : .

E3 (Challenge). For which rule do the outputs at read the same forwards and backwards (a palindromic triple)? Find all linear rules with this property.

Answer

Outputs are , , ; palindromic needs , so : exactly the constant rules .

E4 (Challenge). Two rules and share one table entry. Find it, and explain what “sharing an entry” will mean on tomorrow’s graphs.

Answer

, : the shared pair is . On the plane, the two lines cross at that point.

Homework

1. Build tables for to : (a) (b) (c) (d) .
2. For rule (c): (a) the step size (b) the entry (c) which gives (d) does belong?
3. A linear rule’s table shows and . Name the rule and give its value at .
4. A linear rule’s table shows and . Find the rule.
5. The rule : (a) build the table to (b) where does hit zero? (c) for which is negative?
6. Reasoning. In a table for listing only even (), the -step is . Explain why this does not contradict “the coefficient is the step”.
7. Reasoning. Explain why two table entries determine a linear rule, but one cannot.
8. Challenge. A rule pairs with the remainder when is divided by (for whole ). Tabulate to . Is the rule linear? Does every input still get exactly one output?

Answers: Q2 — (a) (b) (c) (d) ✓ yes. Q3 — ; . Q4 — step ; : . Q5 — (b) (c) . Q6 — the coefficient is the step per unit of ; stepping by doubles the visible step. Q7 — one entry fits infinitely many lines (any slope through one point); a second entry fixes the slope, and slope plus point fix everything. Q8 — table ; not linear (no constant step), yet still a function — one output each. Rules need not be linear to be rules.