Grade 4: From Counting On to a Rule
Ms. Tremblay's Grade 4 class is working on a growing pattern built from tiles: position 1 uses 5 tiles, position 2 uses 8, position 3 uses 11. Almost everyone extends correctly to position 4. Asked how many tiles are at position 10, most students count on seven times and several lose track. Asked which position uses 47 tiles, only three students begin, all by counting backwards one step at a time. One student, Priya, writes '3n + 2' and cannot explain what n means. The Grade 4 program expectation is to identify, describe, and extend growing patterns; algebraic expressions are formalised in later grades.
1. Priya's expression is correct but unexplained. Compare the costs of praising it, ignoring it, and making it public, and state what you would do.
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- Praising it privately is safe but signals to Priya and to the class that the symbolic form is the summit. It rewards the answer-getting move the unit is trying to move beyond.
- Ignoring it wastes the strongest artifact in the room and quietly tells Priya that her thinking is off-syllabus.
- Making it public unguarded produces imitation: several students will write '3n + 2' on the next task without being able to generate it for a new pattern, which is worse than counting, because it hides the gap.
What I would do: make it public under a condition - Priya must show where the 3 and the 2 live in the tiles. Three tiles are added per position; two tiles sit in the fixed part. The class then tests the expression at positions 1, 3, and 4 against the tile models. The target for everyone stays 'describe the rule in words'; the expression is exhibited as one accurate way to record the description, credited to Priya but owned by the structure.
2. Design one question and one scaffold that move the class from counting on to a rule they can use, staying within Grade 4.
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Question: 'If you already know position 3 uses 11 tiles, how could you get position 10 without counting every step? Show it in two steps or fewer.'
This is answerable from the visual structure rather than from algebra: from position 3, seven more steps of 3 gives 11 + 21 = 32; from position 1, nine steps of 3 gives 5 + 27 = 32. Both routes are legitimate Grade 4 reasoning, and both are far more efficient than counting on seven times.
Scaffold: a three-column table - position, tiles, and 'tiles added since position 1' - plus the sentence frame 'It starts with ___ tiles, and each new position adds ___ , so for position ___ I need ___ .'
The table makes the constant difference visible and the frame forces the two-part structure (fixed part, growing part) into words. For the 47-tile question, students use the same frame backwards: 'From 47, take away the 2 fixed tiles, then share 45 into groups of 3.' That is division Grade 4 students already have, applied to a structure they built themselves.
3. A parent emails: 'Why not just teach them the formula 3n + 2 and save everyone the counting?' Draft your reply, addressing the reasoning behind the sequencing.
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I would thank the parent for raising it, agree that 3n + 2 is where the class is heading, and be specific about the sequence.
'The formula is the destination, not the starting line. Right now the goal is for your child to look at an unfamiliar pattern and work out its rule - that is the Grade 4 expectation, and it is what later grades build on when letters start standing for numbers. If I teach 3n + 2 first, many students can substitute into it but cannot build a rule for a pattern they have not seen, and cannot say what n means. Priya found the expression this week and could not explain it, which is exactly the risk.
Here is something you can try at home: ask your child to show you where the 2 comes from in the tiles and where the 3 comes from. If they can, they own the formula. If they cannot yet, they are doing the work the formula depends on.'
I would also mention the expression will be on the board this week, credited to a classmate, so the parent sees the formula is not being withheld - it is being earned.