Sixth graders will need to be able to revise a generalization to make it a conjecture. Let’s say we have the generalization, “All multiples of 5 are also multiples of 10.” We would begin by using the steps in lesson tip 2 to determine if the generalization is a conjecture. In doing this, we would find that some multiples of 5 are not multiples of 10 (i.e. 5, 15, 25). However, we would also notice that some multiples of 5 are multiples of 10. This is where mathematical reasoning comes in. We know that a conjecture needs to be a generalization that is believed to be true all of the time. So, this generalization will need to be revised.
1. Think about what is true all of the time when looking at your data.
5: 5, 10, 15, 20, 15, 30, 35, 40, 45, 50, 55, 60…
10: 10, 20, 30, 40, 50, 60
The evidence shows that all multiples of 5 end in a 5 or a zero. It shows that all multiples of 10 end in a zero. So, we have disproved the statement that, “All multiples of 5 are also multiples of 10.”
2. Find a way to rephrase the statement so that it is true.
“All multiples of 10 are also multiples of 5.”
This is a conjecture. As we look at the data for the multiples of 10, we see these numbers also in the multiples of 5.