1. 9third Grade James Test

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This is one of the first lessons I teach on division. It can also be used as a prelude to fractions. It works for either!

To begin, I ask students to silently think of at least 2 things they know about division and write them down. If they say they don't know anything, I prompt them to even write down something as simple as, 'I know I will learn it in 3rd grade,' or 'I know what the symbol looks like.'

Next, students share with a buddy and then 4 students share what their partner said with the entire class. I challenge students to be prepared to reflect on how their thinking has changed or evolved by the end of the lesson.

Then the fun part. I bring out a fruit that is NOT in a quantity conducive to equal sharing. For example, this year I brought 7 bananas. I have 30 students. (I love having a class number that works so well as a divisor or factor!) The students can very clearly see that 7 is not enough for each child to have one, and many of them quickly add that cutting them in half also won't be enough, but they still suggest that as the next step.

At this point, of course, several children will say they don't like bananas or whatever the fruit is and the teacher tells them to play along, for math's sake.

I follow their suggestions from there on. If they suggest uneven pieces (say cutting one of the halves in half but leaving the other intact), I do what they suggest but then guide them to assess if these are fair shares.

I follow the students' lead, so the results are different every time I teach this lesson. But there's always a way to use their misconceptions to build on how I guide the discussion. This year, when the bananas were cut into 1/2, 1/4, 1/4, we then had 21 uneven pieces. A child did see that if we cut the remaining 1/2 into 2 equal pieces, all the pieces (quarters) would be the same size.

This still did not solve our problem, as we have 30 students, two of whom were grudgingly pretending they ate bananas. Little did they know apples were arriving on the scene later, to alleviate their banana disappointment.

9third Grade James Test

A student suggested I split two of the fourths (they were not using that term) in 1/2 again, so for two of the bananas I had 1/8 + 1/8 + 1/4 + 1/4 + 1/4. I made sure the children realized that the 1/8 were not equal pieces, and then let them eat their mini snack anyways.