087i - Tables from Growing Patterns

Lesson 87 — Generating Tables of Values from Growing Patterns

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

Learning Intentions

• To generate a table of values from a visually growing pattern.
• To find and express the rule connecting position to value.

Success Criteria

I can:

1. Count systematically and tabulate a growing pattern.
2. Distinguish the recursive view (add each time) from the positional rule (from ).
3. Write the rule as a formula, using the structure of the picture.
4. Use the rule to predict far-off terms and to find a position from a value.

Warmup

(6 minutes — the L-shape, individual then pairs)

Draw the sequence: an L of squares, then , then (-th L: a vertical arm of and horizontal arm of , sharing a corner).

1. Draw the next shape.
2. Complete: | Shape | 1 | 2 | 3 | 4 | 5 | — | Squares | 3 | 5 | 7 | ? | ? |
3. How many squares in shape ? How did you get it without drawing?

Answers: 2. , ; 3. — most will say “keep adding ”; some will spot . Both are honoured — and today distinguishes them.

Activities

Activity 1 — Explicit Instruction: Two Ways to See a Pattern (14 min)

Recursive view: each shape adds squares to the last. Fine for the next shape; painful for shape .

Positional view: shape has squares — computed directly from , no history needed.

Where the rule lives: in the picture. For the L-shape, colour the two arms: a vertical arm of squares and a horizontal arm of squares that starts beside the corner square, so nothing is double-counted: total . Model the colouring so the algebra names the picture (Lesson 23’s matchstick idea, revisited).

The constant-difference bridge (Lesson 26 revisited): a pattern adding each step has a rule of the form ; the value at pins down . Here gives .

I do — a second pattern. Square tables seat ; tables joined in a row seat

Tables	1	2	3	4
Seats	4	6	8	10

Structure: seats per table (top and bottom) plus ends: . Check at : ✓

We do: Matchstick triangles in a strip: matches.

1. Tabulate to .
2. Rule from the structure? (First triangle , each new one adds : .)
3. Matches for triangles? (61.)
4. How many triangles from matches? (Solve : .)

Activity 2 — Pattern Circuit (14 min)

Pairs. For each pattern: table to , rule with a structural justification, value at , and one reverse question.

Pattern A — Fence panels. Each panel needs rails; adjoining panels share a post, and posts are: posts for panels. Tabulate posts against panels. (Rule .)

Pattern B — Window crosses. A cross of squares grows by one square on each of its four arms: (Rule .)

Pattern C — Staircases. Steps built from squares: , then , then , then (-th staircase adds a column of ). (Not linear! The differences grow: — rule , met in Lesson 26 E4. Students tabulate and describe; the formula is offered, not demanded.)

Pattern D — Tiles around a pond. A square pond of side is bordered by a single ring of unit tiles: (Rule ; structure: four sides of plus four corners.)

Socratic scaffolding for Pattern C:

Prompt	Purpose
Tabulate. What is different from every other pattern today?	The difference isn’t constant: .
So can the rule be “something plus something”?	No — constant difference is exactly what linear rules produce.
Describe the growth recursively instead.	Each staircase adds a column of height .
Sum view: staircase is .	Connects to the handshake/sum formula.
Check the offered formula at .	✓
Looking back	Tables diagnose linearity before you hunt for a rule — check differences first.

Answers (value at ; reverse example): A — posts; panels need posts. B — ; . C — ; which staircase uses squares? . D — ; .

Activity 3 — Inquiry: Same Table, Different Pictures (6 min)

Pairs.

Two students’ patterns both give the table

Priya’s picture: rows of dots with extra at the start. Marco’s picture: a square of dots gaining an L of each step.

1. Write each student’s rule as their picture suggests it.
2. Show the rules are the same expression.
3. What does this say about pictures and rules?

Answers: 1. Priya: ; Marco: ; 2. ✓ (Lesson 23 E3’s move); 3. Different structural seeings give different-looking but equivalent rules — simplification reveals the agreement.

Checks for Understanding

(5 minutes — exit ticket)

1. A pattern gives Tabulate to and state the rule.
2. Use your rule: value at ?
3. Which position has the value ?
4. A pattern’s values are . Show it is not linear.
5. Reasoning. Why does a constant difference of force a rule of the form ?

Answers: 1. at ; rule ; 2. ; 3. ; 4. Differences — not constant; 5. Each step up in adds exactly , which is precisely what multiplying by produces; adjusts the starting value.

Common Misconceptions

Misconception	How to pre-empt it
Rule difference, e.g. "" offered as the whole rule.	Recursive vs positional named explicitly; shape exposes the gap.
Reading ‘s as the previous value rather than the position.	The table’s two labelled rows; substitution checks.
Assuming every pattern is linear.	Pattern C’s staircases break the assumption inside the circuit.
Pinning from when the table starts at .	Use the value; extend to only as a check.
Double-counting shared elements (corner squares, shared posts).	The arm-colouring of the L and the fence’s shared posts both rehearse it.
Believing different-looking rules must disagree.	The inquiry: simplify before judging.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A pattern’s rule is . Which position gives ?

Answer

.

E2 (AMC Junior style). Hexagonal tables in a row seat Find the rule and the seats at tables.

Answer

; at : seats.

E3 (Challenge). The pond border rule is . A border uses exactly tiles. Find the pond, or prove none exists.

Answer

— a pond ✓. (Contrast: a border of tiles is impossible, since is always a multiple of , and is not.)

E4 (Challenge). Staircase numbers: Show that adding two consecutive staircase numbers always gives a perfect square, and say which square.

Answer

. Two consecutive staircases interlock into an square — draw it.

Homework

1. For each sequence, tabulate to , give the rule, and find the value at : (a) (b) (c)
2. Chairs around joined trapezoid tables seat (a) Rule? (b) Seats at tables? (c) Tables needed for people?
3. A tile pattern uses tiles then adds each step. (a) Rule. (b) Which step uses tiles? (c) Can a step use exactly tiles? Justify.
4. The pattern : show it is not linear, and describe its growth.
5. Draw two different pictures whose pattern is , and explain how each picture shows the rule.
6. Reasoning. A student says the rule for is “add “. Explain what is right, what is missing, and supply the full rule.
7. Challenge. The border of an rectangular pond uses tiles. Verify for a pond by drawing, and find all ponds (with ) whose border uses exactly tiles.

Answers: Q1 — (a) ; (b) ; (c) ; (negative — discuss whether the pattern can continue physically). Q2 — (a) (b) (c) . Q3 — (a) (b) (c) — no; values are always one more than a multiple of . Q4 — differences : growing, so not linear; the differences themselves grow by (values are ). Q6 — “add ” describes the step (recursive) but not where to start; the positional rule is . Q7 — : ✓ by drawing; : ponds .