Increasing and Decreasing Patterns with Tables and Charts
Warm-up
Fish stock data: a lake had 8,400 fish. Each year, 1,200 are removed. Show the table: year 0=8400, year 1=7200, year 2=6000... What is the rule? (Subtract 1,200.) In what year will there be no fish? (Year 7: 8400 - 7x1200 = 8400 - 8400 = 0.)
Explore
Pattern investigation: each group receives a real-world data scenario (fish stocks, tree growth, population change). They build a table of values, identify the rule, extend the table 3 more terms, and answer the question: when does the value first exceed/fall below a given target?
Formalize
Multiply the connection to multiplication: the pattern 7, 14, 21, 28 is both an additive pattern (add 7) and the 7-times table (7x1, 7x2, 7x3, 7x4). Term 10 = 7x10 = 70. This double representation shows why multiplication tables are worth knowing: they let us jump to any term without listing all previous ones.
Increasing and Decreasing Patterns with Tables and Charts
Graph the fish stock data on a number line (or simple line graph): year on the horizontal axis, fish count on the vertical. The pattern is visible as a straight line going down. This is a linear relationship, and Grade 4 students are experiencing it before the word linear exists in their vocabulary.
Practice
Students build tables for 3 real-world scenarios, identify rules, predict terms 10 and 20, and write one sentence about what the prediction means in context. Exit ticket: what is the 15th term in a pattern starting at 3, adding 7 each step?
Exit ticket
Students build tables for 3 real-world scenarios, identify rules, predict terms 10 and 20, and write one sentence about what the prediction means in context. Exit ticket: what is the 15th term in a pattern starting at 3, adding 7 each step?
The pattern: necklace 1 uses 5 beads; each next necklace uses 3 more beads than the last.
Step 1: Build the T-chart: necklace 1 → 5, 2 → 8, 3 → 11, 4 → 14. Confirm the down-the-chart rule: +3.
Step 2: Hunt the across-the-chart rule: try "×3 then adjust": 1×3=3, need 5 → +2. Test on all rows: 2×3+2=8 ✓, 4×3+2=14 ✓. Direct rule: beads = 3 × necklace-number + 2.
Step 3: Predict necklace 10 without the chart: 3×10+2 = 32 beads.
Step 4: The reverse question, which is where reasoning shows: "which necklace uses exactly 26 beads?" Undo the rule: 26 − 2 = 24; 24 ÷ 3 = 8. Necklace 8. Verify with the rule forward: 3×8+2 = 26 ✓.
Say the meta-lesson: the +3 rule explains the GROWTH; the 3n+2 rule answers QUESTIONS. Charts are where you discover the second by staring at the first.
The data: the copier tray starts Monday with 500 sheets; the class uses 65 sheets per day.
Step 1: Chart it: day 0 → 500, day 1 → 435, day 2 → 370, day 3 → 305, day 4 → 240 …
Step 2: The question with consequences: "on which day does the tray die?" Continue: day 5 → 175, day 6 → 110, day 7 → 45, day 8 → −20?? Paper can't go negative — the tray empties DURING day 8's printing.
Step 3: Answer as a decision, not a number: refill by the START of day 8 (or during day 7 to be safe).
Step 4: The faster route for older-thinking students: how many full days does 500 support? 500 ÷ 65 = 7 remainder 45 — seven full days, then 45 sheets of grace. Division answers what the chart marched to.
The modelling habit: decreasing patterns in the world end at zero, and "when do we hit zero?" is usually the entire point of tracking them.
The race: Pattern A starts at 40 and grows +5 per step. Pattern B starts at 2 and DOUBLES per step. Who's ahead at step 6? Does B ever pass A?
Step 1: Chart both side by side. A: 40, 45, 50, 55, 60, 65, 70. B: 2, 4, 8, 16, 32, 64, 128.
Step 2: Read the race: A leads for five steps (steady, comfortable). At step 5 it's 65 vs 64 — a photo finish. Step 6: B blows past, 128 to 70, and the gap only explodes from there.
Step 3: Name the two growth personalities: ADDING patterns climb stairs; DOUBLING patterns ride rockets — slow to leave the pad, unbeatable after. No head start saves a stair-climber from a rocket.
Step 4: Where students will meet this again: savings vs compound interest, population growth, viral videos — Grade 4 meets exponential-vs-linear with cubes and a chart.
Exit: "Pattern C starts at 1,000 and adds 10. Roughly which step does B (2, doubling) catch it?" Chart until caught — it takes only about 10 steps. Rockets.