Less than one, equal to one, greater than one

Courtesy of Dan Meyer’s 3-Act Math resource, students began today’s class in detective mode. A brief, 30 second video clip captured their attention as two CSI investigators pulled out a sawed-off limb. The portion of the video where the percent of the mass of the lower leg to the body mass was bleeped out. Students were up and on their way asking questions about the scenario. I love questions from 6th graders and we spent some time wondering how the leg portion could have shown up on the screen before returning to the math.

I asked for an estimate that students thought would be too high and another they thought to be too low (sometimes I inadvertently let this step slip by but it is oh so revealing). The range began at 1% to 500% and after a bit of discussion was narrowed down to a range of 10-50%.

A group’s thoughts in process… We’ve been discussing ratios and are moving to percents so I was happy that groups began looking for comparisons and writing them as ratios. Today’s surprise came as I rotated through groups. A group wound up with a ratio of \frac{42}{147} and wanted to turn in into a percent but was not sure how. (A benefit of spending large amounts of time on a single problem is that opportunities for quick mini-lessons to groups of students always pops up.)

I asked the students if they believe the fraction to be less than one, equal to one or greater than one. Jumping that hurdle seems to help students position their thoughts and gives me a good idea of what they are thinking. Two of the three members in this group believed the fraction to be greater than one. Their reasoning was that 147 is greater than 100 so the fraction must be greater than one. They were convinced. Pulling a value out of a hat, I asked if \frac{13}{14} would be a smaller or larger amount. The two students indicated that the 147 would be larger though neither had a explanation as to why. Soon, they were busy breaking equal sized rectangles into portions to represent the two fractions. The student in charge of 147 pieces quickly became frustrated at having to make so many small pieces and the group had a good discussion about part to whole relationships.

Looking back, I wonder what was the foundation of the students’ original thoughts and at the moment lean towards a developing concept of the relationships between fractions, decimals and percents. They appeared to understand that a percent greater than 100% is greater than one whole. Did they transfer the thought to fractions and think that the value must be bigger than 100%? I’m working on some follow-up activities to bring out these relationships but additional ideas would be appreciated.

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