Third graders are looking for patterns and rules in sets of whole numbers so they can predict what numbers will come next in the set. Which of the following shows an error in student reasoning?
1, 4, 16, 25, 36, ... The first number is 1, and the next number is found by doubling the previous number and adding 1.
In this sequence, the reasoning is flawed because the pattern is not consistent. While the first number is indeed 1, the subsequent numbers do not follow the proposed rule of doubling the previous number and adding 1; rather, they follow a different pattern based on perfect squares.
This choice correctly identifies a consistent pattern where each number increases by 12. The reasoning is accurate, as the sequence can be described as an arithmetic progression with a common difference of 12.
Here lies the error; the numbers do not follow the given rule. Instead, they represent perfect squares (1^2, 2^2, 4^2, 5^2, 6^2), which means the reasoning for determining the next number is incorrect, as the method described does not produce the correct subsequent terms.
This sequence accurately describes the Fibonacci sequence, where each number is indeed the sum of the two preceding ones. The reasoning is valid, and the pattern is maintained throughout the sequence.
This choice correctly identifies a consistent pattern where each number decreases by 2. The reasoning is sound, as the sequence represents an arithmetic progression with a common difference of -2.
Among the choices presented, only option B contains an error in reasoning regarding the pattern of the sequence. While A, C, and D correctly describe their respective patterns, B fails to accurately represent how the numbers progress based on their mathematical properties. Understanding these patterns is essential for third graders as they learn to identify rules in sequences and develop their predictive skills in mathematics.
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