Activities for Week 6
Activity W 6.1
Discovering Patterns Through Arrangements
Students may be encouraged to perform these activities.
They may observe the patterns and explain about it.
Objective:
To help students recognize and analyze patterns using real-life variables such as height, weight, or other attributes, and connect them to numbers through reasoning and logic.
Procedure
1. Look at this picture. You can see that some children are standing in a line in a park. Each one is saying a number.
2. Ask the students to tell what these numbers might mean.
3. Does it have something to do with their heights?
Students should discuss and try to find out as to how it could be related to their heights.
4. The children then re-arrange themselves again and each of them says a number based on the new arrangement.
Students may be motivated to think and try to answer the following questions with their reasoning.
Q1: Can the children re-arrange themselves so that the children standing at the ends say ‘2’?
ANSWER:
Yes, if there are 5 children, and “2” represents the number of children shorter than that person, then placing a middle-height child (3rd tallest) at both ends would give both ends the value “2”.
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Example: Heights in order → 1, 2, 3, 4, 5
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Arrangement: [3, 1, 5, 2, 4] → First and last are "3rd tallest", so 2 kids shorter than them.
Q2: Can we arrange the children in a line so that all say only 0’s?
ANSWER:
Yes, if “0” means there are 0 children shorter than me (i.e., I am the shortest), then all children would need to be the same height.
But since it’s said they’re all of different heights:
No, unless 0 represents something else like distance from a reference point, which isn’t the case here.
Q3: Can two children standing next to each other say the same number?
ANSWER:
Yes, if the numbers represent how many children are shorter than them, or relative position, it’s possible that two children of different heights can still have the same number if others in the group are taller or shorter equally.
Q4: There are 5 children in a group, all of different heights. Can they stand such that four of them say ‘1’ and the last one says ‘0’? Why or why not?
ANSWER:
No, this is not possible if the numbers represent unique rankings or counts of shorter children.
Only one child can be the shortest (say ‘0’), and only two children can have exactly one shorter child below them. So four saying ‘1’ cannot happen logically.
For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?
Q5: For this group of 5 children, is the sequence 1, 1, 1, 1, 1 possible?
ANSWER:
No, again, this is not possible if the numbers represent unique rankings. You can’t have 5 children each with exactly one person shorter than them—it violates basic order.
Q6: Is the sequence 0, 1, 2, 1, 0 possible? Why or why not?
ANSWER:
Yes, this is possible if the numbers represent distance from the tallest child in the center.
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This pattern is symmetric: the middle child is tallest, others decrease outwardly.
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Think of it as:
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Child 1: shortest → 0
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Child 2: a bit taller → 1
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Child 3: tallest → 2
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Child 4: a bit taller than 1 → 1
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Child 5: shortest → 0
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How would you re-arrange the five children, so that the maximum number of children say ‘2’?
Q7: How would you re-arrange the five children so that the maximum number of children say ‘2’?
ANSWER:
If the number "2" represents number of children shorter than me, then the middle child in height (3rd tallest) would have two people shorter than them.
To maximize the number of children saying '2', we need as many children as possible to have exactly two children shorter than them.
Best possible:
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Heights (in increasing order): A, B, C, D, E
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Arrangement: [C, B, D, A, E]
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Those with value "2" = C, B, D (middle 3 children)
Maximum of 3 children can say ‘2’. More than that is not possible with unique heights.
Extension
Based on their weights or some other features, students may draw diagrams associating them with the numbers.
They may present them to other students and ask them to guess it
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