The Tenth Number and the Even Claim
Mr. Bell writes a pattern on the board: 3, 7, 11, 15, ... He asks, 'What is the 10th number in this pattern?' Ana says 39. She explains, 'I added 4 each time and counted them out: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39.' Ben says 43. He explains, 'You multiply 4 by 10, which is 40, and add the starting 3, so 43.' Mr. Bell writes both answers on the board and does not say who is right. Then he asks two more questions: 'Is 50 in this pattern?' and 'How could you check your answer without writing the whole list?' Several students say 50 must be in the pattern because the pattern goes on forever and 50 is a counting number. Mr. Bell asks the class to test that claim with the first four numbers before anyone changes an answer.
1. Who is correct, Ana or Ben? Name Ben's mistake precisely and show how to tell without listing all ten numbers.
Read the worked answer
Ana is correct: 39.
Ben's number rule is almost right, but his jump count is off by one. The first number is 3, so getting to the 10th number takes nine jumps of 4, not ten:
- 3 + 4 × 9 = 3 + 36 = 39
Quick checks that do not require a full list:
- Ben's answer, 43, is what you get for the 11th number (3 + 4 × 10). So his answer is in the pattern, just one place too far along.
- Test his rule on the 2nd number: 4 × 2 + 3 = 11, but the 2nd number is 7. A rule that fails on an early term cannot be trusted later.
The error is the classic off-by-one: counting the starting number as if it were a jump.
2. Is 50 in this pattern? Support your answer using the first four numbers, without writing the list.
Read the worked answer
No, 50 is not in the pattern.
Two independent ways to see it:
- Odd/even reasoning. The first four numbers are 3, 7, 11, 15, all odd. Adding 4 to an odd number always gives an odd number (odd + even = odd). So every number in the pattern is odd, and 50 is even.
- Subtract-the-start reasoning. Every number in the pattern is 3 plus some number of 4s. 50 − 3 = 47, and 47 is not a multiple of 4 (4 × 11 = 44, 4 × 12 = 48). So 50 would need 11 and three-quarters of a jump, which is not allowed.
What the students who said 'it goes on forever' missed: a pattern can go on forever and still skip most numbers. 'Forever' tells you the list does not stop; it does not tell you which numbers are in it.
3. Write one sentence Mr. Bell could say to Ben that helps him find the error himself, without giving the answer.
Read the worked answer
A good sentence points Ben at a testable step rather than the result:
'Try your rule on the 2nd number and the 3rd number. Does 4 × 2 + 3 give you the number that is actually second?'
Why this works:
- It uses Ben's own rule instead of replacing it.
- It picks small cases where he can check by hand, so he discovers the off-by-one himself.
- It does not say 'you multiplied by 10 instead of 9,' which would let him fix this problem without fixing the thinking.
A weaker response would be 'count again' - that gives no target and no reason to test the rule.