090i - Consolidation Patterns and the Plane

Lesson 90 — Problem Solving and Consolidation: Patterns and the Cartesian Plane

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

Learning Intentions

• To move fluently between pattern, table, rule and graph — the four faces of one relationship.
• To choose the representation best suited to a question.

Success Criteria

I can:

1. Start from any face (picture, table, rule, graph) and produce the other three.
2. Use the most convenient face to answer a given question.
3. Connect coefficient/constant to step/start and steepness/crossing.
4. Solve unfamiliar problems by switching representations deliberately.

Warmup

(6 minutes — the four faces, whole class)

Board shows the same relationship four ways: a matchstick pattern (), its table, the rule , and its plotted line.

1. Point to where the "" lives in each face.
2. Point to where the "" lives in each face.
3. Which face answers fastest: “value at ?” “when does it pass ?” “is the growth steady?”

Answers: 1. The new matches per shape; the constant table step; the coefficient; the steepness; 2. The starting extra match; the entry (or fitting at ); the constant; the -axis crossing; 3. Rule (); rule/equation (); table or graph at a glance.

The lesson’s motto: four faces, one relationship — switch to whichever face makes the question easy.

Activities

Activity 1 — Representation Relay (16 min)

Pairs. Each task enters at a different face and demands the rest.

Task A — enter at the picture. A pattern of H-shapes uses matches.

1. Table to ; rule with structural justification; plot the first five points.
2. Which shape uses matches?

Task B — enter at the table.

3. Rule; plot and draw the line; describe a situation this could model.

Task C — enter at the rule. .

4. Table to ; plot; read the -axis crossing from the graph and confirm by equation.

Task D — enter at the graph. A printed line passes through and .

5. Read off steepness and crossing; name the rule; produce the table for to ; invent a pattern-picture the rule could describe.

Socratic prompts while circulating:

Prompt	Purpose
Which face did you use for that answer — and was it the easiest one?	Makes switching a conscious choice.
For B: what kind of situation falls steadily to zero?	E.g. water draining, credit running out — falling rules deserve stories too.
For D: your picture adds per step but the rule says…	Steepness ✓ — picture must match the coefficient.

Answers: A — rule ; : . B — ; e.g. L draining at L/min. C — table ; crossing at , from . D — steepness , crossing : ; table ; e.g. tables-and-chairs with per table plus ends.

Activity 2 — Mixed Problems (14 min)

Pairs. Free choice of representation; answers must name the face used.

Problem 1. A phone repairer charges by the rule . (a) Make the table to and plot. (b) What job length costs 1654025$ mean on the graph and in money?

Problem 2. Two patterns race: and . (a) Tabulate both to . (b) When does A overtake B? Answer from the table, then the equation, then say what the graphs would show.

Problem 3. The points came from a linear experiment, but one was recorded wrongly. Which one, and what should it be?

Problem 4. A staircase pattern has values . Show it is not linear, plot the points, and describe how the graph announces the non-linearity.

Answers: 1. (b) hours (c) 40y$256n + 2 > 4n + 14n = 7n = 638(6, 38)(4, 15)+3(4, 14)y = 3x + 23, 5, 7, 9$ grow: the plotted points curve upward — a bending point-trail is the graph’s way of saying “not linear”.

Activity 3 — Inquiry: Design the Question (9 min)

Pairs, then swap — the block’s closing construction task.

Build a “four faces” puzzle for another pair: choose a linear rule, then present it as one face only (picture, table, graph, or two points), with three questions whose easiest answers live on different faces.

Solve your own puzzle first; note beside each question which face you expect solvers to use.

Quality bar (displayed): the rule fits on the grid; the three questions genuinely favour different faces; one question must require going beyond the given data (extrapolate or solve).

Debrief prompt after the swap: did solvers use the faces you predicted? Mismatches are the interesting part — some brains reach for tables, others for graphs, and both arrive.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A pattern has table values from . Rule?
2. For : the graph’s steepness and -crossing?
3. The line through falling per step: rule, and its -axis crossing?
4. Values : linear or not? Evidence?
5. Reasoning. Give one question best answered by a table, and one best answered by a rule, about the same relationship.

Answers: 1. ; 2. ; ; 3. ; ; 4. Not linear — differences ; 5. E.g. table: “list the first five values / spot the step”; rule: “value at / which gives “.

Common Misconceptions

Misconception	How to pre-empt it
Staying loyal to one representation for every question.	The relay forces entry at all four faces; answers must name the face used.
Table-pattern extended blindly past a data error.	Problem 3 rewards checking each step, not assuming.
Reading a curving point-trail with a ruler anyway.	Problem 4: linearity is checked, not presumed.
Confusing -indexed patterns (start ) with -tables (often ).	Name the indexing each time; the constant lives at / either way.
Graph answers given without scale-reading care.	Craft checklist carried over from L89.
Believing the four faces are four topics.	The warmup’s “point to the ” ritual — one relationship throughout.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A pattern’s rule is . Give its table step, its th value, and the position holding .

Answer

Step ; ; .

E2 (AMC Junior style). Lines and are drawn. Find the crossing point and state which line is steeper.

Answer

, : ; the first ().

E3 (Challenge). A linear pattern’s th value is and its th is . Find the rule, the first value, and the position of the value .

Answer

Step ; : rule ; first value ; .

E4 (Challenge). The three lines , and are drawn. Show that all three pass through one point.

Answer

; check the third: ✓. All meet at — three rules, one shared pair.

Homework

1. For the pattern : (a) rule (b) th value (c) position of (d) plot the first five points and describe the trail.
2. For : (a) table to (b) plot and draw (c) both axis crossings (d) a one-sentence situation it could model.
3. The line through and : (a) rule (b) table to (c) which gives ?
4. Patterns and : (a) tabulate to (b) when does Q pass P? (c) what would the crossing look like graphically?
5. One of these points is off the line through the others: . Identify, correct, and name the line.
6. Reasoning. “The rule, the table and the graph never disagree.” Explain why, referring to how each is produced.
7. Challenge. Design a four-faces puzzle (as in class) on the rule of your choice, present it entering at the graph, and provide the full answer key.

Answers: Q1 — (a) (b) (c) (d) straight, climbing per step. Q2 — (c) and . Q3 — (a) steepness : (c) . Q4 — (b) : from (equal at : both ) (c) two lines crossing at , Q steeper. Q5 — ; should be ; line . Q6 — the table is computed from the rule, and the graph plots the table; each face is derived from the same rule, so any disagreement means a computational error, not a mathematical one.