Students who are working to become efficient with fractions must learn to seek out “friendly” numbers — a shifting target depending on the problem’s denominators. Once they recognize multiplication/division relationships, students can exploit the properties of multiplication to simplify computation. The following string encourages them to do just that.
Our fifth grade team was trying to encourage students to use a visual model to represent their thinking when they multiplied fractions. So many students were so fast — multiplying the numerators, then multiplying the denominators — but had no context and demonstrated very little number sense. Did their answer, the product, make any sense? What would happen to the size of the first fraction as it was multiplied by the second fraction? Was the product bigger or smaller than the fractions? Should it be? We saw students who were simply carrying out some steps without thinking about what multiplying fractions really means.
A 5th grade teaching team I work with recently raised the issue of how to model the problem 1/2 x 3/10 on an array. They saw their students use a variety of models and one teacher got “bothered” by how some of the models felt “imprecise” or “not to scale.” We had a conversation together about this issue. To prepare myself for the conversation I did a little sketching of possible models that kids might use. What do you think?
Fractions as operators,
What is ½ of $1.00?
What is 1/4 of $1.00?
What is 2/4 of $1.00?
What is 1/8 of $1.00?
What is 3/8 of $1.00?
What is 1/4 of $2.00?
Note: I modeled this on an open (double ) number line.
This is intended as an early fraction string, best done with kids as they are beginning to think about fractions, and already have exposure to the open number line.