This week, I am fortunate to have another school visit to a private school in Boston, Massachusetts. We are working on a 4-week lesson study, where we work together with the school teachers on lesson planning, lesson observation and lesson evaluation. The school has recently adopted Singapore Math and teachers are new to Singapore Strategies. It was a very enriching experience for us and the teachers, as we talked about the classroom challenges and what specific strategies to use for classes.
One important area we worked on is Fractions – a topic that many students have difficulty in. A crucial concept for Fractions is Equivalent Fractions. It is tempting to dive straight into mathematical procedures, “telling” the students to multiply the numerator and denominator by the same number to find equivalent fractions. However, doing so will deprive students of an excellent opportunity to derive their own “rules” through induction.
In our lesson study, we started the lesson with paper folding, where students had to fold a long strip of paper into halves, then into quarter, then into eighths, and finally into sixteenths. Students are told to transfer their findings to bar models. For induction, students compare denominators and numerators of pairs of equivalent fractions, and are asked the following questions:
- What do you notice?
- Is there a pattern?
- Is there a rule?
Many students are quick to induce their own “rules” for equivalent fractions; some are faulty while some stood the test. Students know that they need to reason logically and be prepared to defend their “rules” and convince the rest that it works! It was an engaging and meaningful discussion as we bring up important vocabulary and concepts in fractions such as numerator, denominator, same whole, equal value, same value on the number line!
Here is a draft from one of the students.
Exploring Equivalent FractionsAs an extension, students were asked to find the midpoint of 3/8 and 4/8, e.g. a value/point that is midway between 3/8 and 4/8. This proves to be very challenging to many students! Some mentioned 3½/8 and some mentioned 3.5/8. While these seem logical, these are not conventions that many would use so we need to find alternatives. It didn’t take long to someone to realize that we need to change 3/8 to 6/16 and 4/8 to 8/16, and with that, finding the fraction midway between 3/8 and 4/8 is straightforward! It is 7/16!
If you’re looking for additional resources, an excellent reference is Teaching to Mastery Mathematics: Teaching of Fractions
by Dr Douglas Edge and Dr Yeap Ban Har. It explains fractions in a very conceptual way and is easy to follow.
Are you teaching equivalent fractions in your classroom? What challenges do you face? Share them in the comments below!
