 showing. Just as I did before, I instruct students to fold the paper (1/4) in half again. I instruct students to open their paper. Again, I ask questions very similar to the ones above.
How many parts are now in the whole? (Eight, which is the denominator) How many of these eight equal parts are shaded? (Four, which is the numerator) What fraction is shaded? (4/8) Again, I ask what the relationship between the fractions is and I record the fractions on the board for students to see.
Next, I instruct my students to fold their paper up again so they can only see 1/8. I repeat the same process again, having students fold the paper in half and make observations about what is happening to the shaded part (numerator) and the whole piece of paper (denominator.)
Then I repeat the above procedures again, but have students start with another piece of paper and fold it into thirds. I direct students to color in two thirds. I then go through a series of questions about what students notice. I guide students to make connections between patterns they see, which ultimately leads students to discover how to make equivalent fractions by multiplying the numerator and denominator by the same number, which represents a whole or 1.
In this photo you can see a student working on thirds as well as see the paper on the right with the fold marks from coloring 1/2 and making equivalent fractions.
You can see in this student video below, a student who is really trying to make sense of equivalent fractions noticing the pattern that appears between numerators and denominators. This student is explaining how he knows that 1/3 is equivalent to 4/12. I had written on the board 1/3 = 2/6 = 4/12. You can hear in this very short clip, (and see this student's productive struggle) that he notices the pattern of multiplying both numerator and denominator by 2 each time.
You will hear me ask this student how he/she could tell that 1/3 is equivalent to 4/12. Notice the wait time I give this student. While it's not immense, it is definitely a strategy I think about often. Allowing students time to think is critical. I can be guilty of jumping in to quickly to give students answers or hints. Notice in this video a redirected question and then wait time to allow the student to think.