What should you look for?

Each question is shown below, with some suggestions for dealing with the demonstrations students offer.

1. Write these fractions in order, smallest to largest.

This question is an example of the usefulness of manipulatives—if the student writes the answer incorrectly before doing the proof, the manipulatives will show him what the correct answer is, and he will change his written answer to line up with the proof before you get there. You can come along and sign off on the demonstration without having to mark anything wrong, and can then engage in a conversation about how counter-intuitive the structure is, and have some conversation about the meaning, i.e., that the bottom number tells how many pieces the whole thing is divided into, and if you cut more pieces, each piece will have to be smaller.

2. Circle the smaller fraction.

Ask, "How can you tell by the way the fractions are written which one is smaller?" Look for an answer like "A bigger bottom number means the whole is cut into more pieces, so each piece is smaller.

3. Circle the smaller fraction.

Ask, "How can you tell by the way the fractions are written which one is smaller?" An answer like "When the bottom numbers are the same, you know the pieces are the same size, so you can just look at the top number to see how many pieces there are," means that the student is beginning to see the necessity of a common denominator when comparing fractions.

4. Write a fraction that equals one whole.

Ask, "How can you tell by the way the fraction is written that it is equal to 1?" You are asking the student to articulate the fact that a fraction with the same top and bottom number always equals 1. (Except 0.) The manipulatives give the student a chance to find this out for himself, rather than hearing it from you.

5. Write a fraction that equals one half.

Ask, "How do you know your answer is right?" and expect a variety of answers on the theme of "The top number is half of the bottom number."