LESSON PLAN

Patterns with Words, Numbers, Symbols, and Variables

Apothem Team

Grade 5 · Algebra & Patterning

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Show the pattern 3, 7, 11, 15. Rule in words: start at 3, add 4. Can I write this shorter? Term n = 4n - 1. Check: n=1: 4-1=3. n=2: 8-1=7. n=3: 12-1=11. Yes! Now: what is term 25? 4x25-1=99. Immediate, no listing required.

Explore

Pattern rule challenge: each group receives a table of values with 5 rows. They must: (1) identify the constant difference, (2) find the variable rule (term = ?n + ?), (3) verify with 3 term numbers, (4) find term 15, (5) find which term has value 100 (set up and solve the equation).

Formalize

Solving for the term number: the rule is term n = 5n + 2. Which term equals 47? 5n + 2 = 47. Subtract 2 from both sides: 5n = 45. Divide by 5: n = 9. Verify: 5x9+2=47. Yes. This is a two-step equation at Grade 5, accessible through the pattern context.

Patterns with Words, Numbers, Symbols, and Variables

Connect to real-world: a cedar tree grows 15 cm per year. It is currently 45 cm tall. The rule: height after n years = 15n + 45. When will it reach 150 cm? 15n+45=150. 15n=105. n=7 years. This connects the algebraic pattern rule to a natural growth model.

Practice

Students find rules for 4 tables of values, use each rule to find term 10, and solve for the term number that gives a specified value. Exit ticket: what is term 12 in the pattern with rule 4n + 5?

Exit ticket

Students find rules for 4 tables of values, use each rule to find term 10, and solve for the term number that gives a specified value. Exit ticket: what is term 12 in the pattern with rule 4n + 5?

TIP Always verify a rule by substituting at least two term numbers. If both check out, the rule is likely correct. If one fails, the rule needs adjustment.

WORKED EXAMPLES

Example 1 — From pattern to variable: the toothpick squares, once and for all

The pattern: a row of joined squares built from toothpicks. 1 square: 4 picks. 2 squares: 7. 3 squares: 10.

Step 1: Chart and find the growth: +3 per new square (each new square borrows a wall from its neighbour).

Step 2: Hunt the direct rule and TIE IT TO THE PICTURE: every square costs 3 picks (top, bottom, right) after an initial left wall: picks = 3 × squares + 1.

Step 3: Introduce the variable as the chart's shorthand: let n be the number of squares; then picks = 3n + 1. The letter isn't new math — it's the phrase "any number of squares" compressed.

Step 4: Cash it: n = 25 → 76 picks. Reverse it: 100 picks → 3n + 1 = 100 → n = 33 squares.

Step 5: The moment to savour: three DIFFERENT student rules — 3n + 1, 4 + 3(n−1), 4n − (n−1) — all simplify to the same thing, because each narrates the same picture differently (first square then additions; all squares minus shared walls). Equivalent expressions exist because explanations differ. That's algebra's origin story.

Example 2 — Reading a rule table with two operations: y = 4x − 2

Step 1: Fill the table from the rule: x = 1 → 4(1) − 2 = 2; x = 2 → 6; x = 3 → 10; x = 5 → 18; x = 10 → 38.

Step 2: Notice the down-the-table growth: +4 each time x steps by 1 — the ×4 in the rule IS the growth rate; the −2 sets where the pattern starts. Two operations, two jobs.

Step 3: Reverse: which x gives y = 30? Undo in reverse order: 30 + 2 = 32, then 32 ÷ 4 = 8. Check: 4(8) − 2 = 30 ✓.

Step 4: The discrimination task: match tables to rules among y = 4x − 2, y = 2x + 4, y = 3x. Strategy that experts use: check the GROWTH first (eliminates by the coefficient), then one point (fixes the adjustment). Two checks, done — no full-table testing.

Exit: invent a two-operation rule whose table contains (3, 13). (Many answers: y = 4x + 1, y = 3x + 4, y = 5x − 2 … the multiplicity is the lesson.)

Example 3 — The tournament handshake pattern: when growth itself grows

The question: at a tournament, every team plays every other team once. 2 teams: 1 game. 3 teams: 3 games. 4 teams: 6 games. How many games for 10 teams?

Step 1: Chart it and inspect the growth: 1, 3, 6, 10, 15 … differences are +2, +3, +4, +5 — the growth GROWS. Not a steady +k pattern; our linear tools don't fit. Name the species: these are the triangular numbers.

Step 2: Reason it directly: each of the 10 teams plays 9 others → 10 × 9 = 90 — but that counts every game twice (A-vs-B and B-vs-A are one game). Halve: 45 games.

Step 3: Check against the chart's march: 1, 3, 6, 10, 15, 21, 28, 36, 45 ✓ (nine steps).

Step 4: The takeaway pair: (1) not every pattern is linear — check whether the DIFFERENCES are constant before reaching for ×n rules; (2) a clever counting argument (count-double-then-halve) can leapfrog a chart entirely.

Extension for the hungry: n teams → n × (n−1) ÷ 2 games. Where else does this formula live? (Handshakes, chess round-robins, toasts at a table…)

MATERIALS

•Table of values sheets

•Pattern cards

•Hundred chart

•Algebraic pattern-rule recording sheets

WATCH FOR

!Students may confuse the constant difference with the constant term in the rule. The constant difference is the coefficient of n; the constant term is found by solving for when n=1.

!Students may not verify their rules. Two checks are the minimum: if both pass, the rule is almost certainly correct.