Consumed by Fractions!

The students in my classroom are already seated in groupings that are not consistent. This provides for the varying needs of students to work with several or only one other student during shared activities. The setting is ideal for this lesson because you NEED different groupings. In my class of 27, for example, I have 3 groups of 4, 3 groups of 3 and 3 groups of 2. At the groups of 4, I place 6 M&Ms, 5 M&Ms and 3 M&Ms. At the groups of 3 I place 5 M&Ms, 4 M&Ms, and 2 M&Ms. At the groups of 2 I place 5 M&Ms, 4 M&Ms, and 3 M&Ms. Now, before the M&Ms are consumed, I ask if the M&Ms have been distributed fairly. This elicits lively discussion as I argue that since 3 groups all have the same number of people, it must be even; or that since 2 groups have the same number of M&Ms, it must be fair. Although discussion begins in less mathematical terms of each person in a group getting more or less M&Ms than each person in another group, it consistently migrates to fractions. Students benefit from the visual that the M&Ms are the numerators and the group members are the denominators. Eventually I give every student an equal number of M&Ms to enjoy as a reward for good discussion. This exercise focuses on non-equal fractions. If photo resources were available, I would have students make large cards for the number of M&Ms and the number of team members at their group. Then we'd photograph, or video, the number beside the M&Ms, the number beside the members grouped together, and then show the M&M-number becoming the numerator and the member-number becoming the denominator. Then we could post the images in the classroom, on the class website, or both. Next I move to strips of card stock (any size will do, but larger are better visually, since students work in groups)and rearrange students to be in generally even teams. Although all the strips are equal length, they have lines on them that basically divide the strips into segments - 3 segments, 4 segments, 5 segments, 6 segments, and 10 segments. Each team gets one set of 5 bars as visuals. Students are to determine, if each strip were owned by a different person, all the different ways each person could trade a portion of their strip with anyone else such that the trades were always even. This could also be done with food "strips" if you want to keep the food theme going. Or students can imagine that the strips are anything "trade-able" - they have great imaginations! The focus of this segment is equivalent fractions. For example, half (5/10) of the 10-segment strip can be traded for half (3/6) of the 6-segment strip or half (2/4) of the 4-segment strip, but there is NO way to trade segments of the 10-segment strip for segments from the 3-segment strip. The visual aids facilitate the students' understanding of equivalent fractions. I conclude the lesson by giving every student a small index card or sticky-note with a fraction on it. Their objective is to find everyone else in the room that has a fraction that is equivalent to theirs. I generally try to design the activity to have 4 equivalent fractions in each set.