Describing Rules & Patterns
Describe a pattern as a suggested rule, test it against new cases, and explain why matching examples do not prove it always works.
What a learner can do afterwards
- Describe a suggested rule in their own words and identify the entries or cases it covers.
- Use a rule to predict a result, compare it with a new checked result, and identify any mismatch.
- Distinguish matching examples from a reason covering every allowed case, including a check of every member of a complete small group.
1 · Read
You have practised spotting patterns. Describe what repeats or changes as a suggested rule. Compare neighbouring entries in order. In 12, 11, 10, each move subtracts 1. Name the action and its size. A short list can fit more than one rule.
A rule gives a prediction. Mia sees 5, 10, 15, 20 and predicts 25 by adding 5. To test it, she reveals a covered entry from the same list and compares. If it shows 25, that case fits. Writing 25 herself follows her rule but does not check the covered entry.
To use 'double each time', add the last number to itself. Double 6 is 6 + 6 = 12. Adding 0 is different: it adds nothing, so 6 + 0 = 6. That same reason explains why adding 0 leaves any counting number unchanged.
Check every move covered by a rule. In 10, 12, 14, 17, two jumps add 2, but the last adds 3. One mismatch shows that 'always add 2' does not fit this list. For a claim about classroom pencils, check other classroom pencils, not pens or pencils already checked.
For a repeating pattern, name the whole repeating part and its order. In red, blue, blue, red, blue, blue, the part is red, blue, blue. 'Use colours' leaves out the order. 'Repeat red, blue, blue' tells someone how to continue.
Jumps can follow a changing rule. Start at 1, add 2, then make each later jump 1 larger. This gives 1, 3, 6, 10. The jumps are 2, 3, 4. The next jump is 5, so the next entry is 15. Use the stated rule when predicting.
A counterexample is a case that breaks a claim. 'One dress' breaks the claim that every word ending in s names more than one thing. Several matches cannot prove a claim about endless cases. Proof needs a reason covering them all. For a complete small group, checking every member can settle a claim about that group.
Two rules can fit the same short list. 'Keep adding 1' and 'add 1 twice, then add 2' both fit 1, 2, 3. They predict 4 and 5 next. Ask for the intended rule or check another entry. Matching the shown entries does not make the next predictions equal.
Describe a suggested rule, predict with it and check new cases. Matching examples support it. An 'always' claim needs a reason covering every allowed case.
Original lesson, questions and flashcards about describing rules, comparing predictions and distinguishing examples from proof, written for this dataset.
Original extra questions and readings written for this dataset. External records informed activity design; their text and illustrations are not reproduced.
More to read
Make a prediction before looking
A game has numbered cards in a row. You can see 8, 12 and 16. The next card is face down.
You suggest adding 4 each time and predict 20. Record that prediction before turning over the card. Then compare the revealed number with 20.
A match supports your suggestion for this new case. A different number shows that your suggestion does not fit the whole revealed row.
Making your own card with 20 on it would follow your rule. It would not check the hidden card. Keep these two actions separate: making a prediction and checking it against a result you did not choose.
Which group did you check?
You inspect every cup on one tray. Each cup has a handle. You have checked the claim that all cups currently on that tray have handles.
That does not check every cup in the building. The claim changed when the named group grew.
Before checking a claim, say which cases it covers. If it covers one small group, can you inspect the whole group? If you checked only some members, describe those checks without claiming that the rest must match.
For a claim about every possible case, a handful of matching examples is not enough. You need a reason that covers all the allowed cases.
Write instructions someone can follow
Arrange paper cards like this: white, orange, orange, white, orange, orange.
Ask someone to continue the row using only your spoken rule. Saying 'use white and orange' leaves several choices open. Saying 'repeat one white card followed by two orange cards' gives the repeating part and its order.
Check their row against your instructions. If it differs from yours, reread the instructions together. Did you leave out a colour, a count or the order?
A useful rule gives enough detail for someone else to follow it. A pattern's name alone may not supply those details.
A short list can fit different rules
A list begins 20, 21, 22. One instruction says to keep adding 1. Under that instruction, the next entry is 23.
Another instruction says to add 1 twice, then add 5. That also produces the three entries shown, but it predicts 27 next.
The instructions agree so far and disagree later. Neither can be selected from the first three entries alone.
If a task supplies an instruction, follow it. If someone shows only a few entries, explain the rule you suggest. You can ask for the intended instruction or compare your prediction with another checked entry.
2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.