Can you give me an example on how to find two fractions with a sum that is between the two given numbers? [ex:] #20. 0 and 1/2.
I like this question because it is completely unlike how I was taught how to learn and do math in middle school. It made me think about how much more in-depth this kind of math question is. Here is my reply.
... mostly, these kinds of questions in the books are thinking exercises and not calculation exercises.
This first lesson doesn’t focus on exact methods of solving the problem. It is focusing the kids on thinking about the benchmark fractions that they know “by heart.” It is asking them to reason things out without needing to come up with a specific answer (estimation skills). A typical way of thinking about it might go like this [imagine a student thinking it to themselves]:
I know what 1/2 is. I know that 1/2 is the same as 1/4 + 1/4. So since 1/4 + 1/4 is too big for the problem (because they equal 1/2 exactly), I need to use a fraction that is smaller than 1/4. Any fraction smaller than 1/4. I'll try half of a quarter: 1/8. So 1/8 + 1/4 has to be less than 1/2 but it will still be more than 0.
No need for an exact answer to the problem, the students know from their benchmark values that the sum must be less than 1/2 but more than 0, so 1/8 + 1/4 meets those criteria.
This kind of open ended question provides the kids practice with algebraic reasoning. It sets the students up to be much more independent mathematical thinkers so that they can reason about their math class instead of just memorizing steps.Of course, the chosen fraction could have been anything smaller than 1/4. But I thought this was a great example to illustrate what these "thinking" questions are trying to get the students to do and how they might approach it.
