Three growing patterns arrive at your board: a row of tables gaining chairs, a staircase gaining steps, and a paper strip whose pieces double with every cut. Step 1, step 2, step 3 of each — and then the questions start.
The task
For each pattern, answer three escalating questions. What comes next? — easy, draw it. What comes 100th? — drawing is now a trap; you need a rule. What comes th? — say the rule two ways: as a recipe that builds each step from the one before it, and as a formula that jumps straight to any step from nowhere. Then the real question: one of your three patterns is fundamentally unlike the other two. Which one, and what exactly is the difference? Finish by inventing a fourth pattern that is unlike all three, and hand it to a neighbouring group.
Facilitation notes — for the teacher
Choose the three so that one grows by a constant difference, one by a constant difference in disguise (the staircase — its totals are quadratic, a lovely rabbit hole to permit briefly), and one by a constant ratio. The recursive rule always arrives first — “you just add four” — and the jump-to- formula is the productive struggle; the bridge is asking how many times the adding happened by step 100. Groups rarely name the doubling pattern’s difference correctly at first: push the table of values until someone divides neighbouring terms instead of subtracting. The invented fourth pattern flushes out Fibonacci-style rules — welcome them; they are tomorrow’s example of a recipe with two ingredients.
What mathematics tends to surface
The two ways of saying a rule — build-from-previous and jump-to-any — are the recursive and explicit representations of a sequence, and groups discover that each is good at what the other is bad at. The constant-difference versus constant-ratio split separates arithmetic from geometric sequences before either word is spoken. Step numbers are natural numbers, so the graphs come out as dots, not curves — discreteness made visible. Sequences and Their Rules names all of it, and Visual Patterns keeps the instinct warm all semester.
Where it leads
Adding up the steps instead of naming them is the next move — Series does it without adding — and a savings account is a geometric pattern counted in dollars, which is where this unit is headed.
The answer is not on this page
The three patterns and their rules are handed out in class, not printed here. The fourth pattern is yours to invent.
Curriculum connection
C1.2
determine and describe (e.g., in words; using flow charts) a recursive procedure for generating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, …), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)
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C1.4
represent a sequence algebraically using a recursion formula, function notation, or the formula for the nth term [e.g., represent 2, 4, 8, 16, 32, 64, … as ; , as , or as , or represent , , , , , , … as ; , as , or as , where is a natural number], and describe the information that can be obtained by inspecting each representation (e.g., function notation or the formula for the nth term may show the type of function; a recursion formula shows the relationship between terms) Sample problem: Represent the sequence 0, 3, 8, 15, 24, 35, … using a recursion formula, function notation, and the formula for the nth term. Explain why this sequence can be described as a discrete quadratic function. Explore how to identify a sequence as a discrete quadratic function by inspecting the recursion formula.
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C2.2
determine the formula for the general term of an arithmetic sequence [i.e., ] or geometric sequence (i.e., ), through investigation using a variety of tools (e.g., linking cubes, algebra tiles, diagrams, calculators) and strategies (e.g., patterning; connecting the steps in a numerical example to the steps in the algebraic development), and apply the formula to calculate any term in a sequence
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