Dividing by a fraction inquiry
Mathematical inquiry processes: Extend a pattern; find connections; generalise; reason about structure. Conceptual field of inquiry: Division of whole numbers by fractions; division and multiplication.
The prompt was designed to expose and correct a common misconception among secondary school students. When shown the prompt, they will often claim that fifty divided by a half is 25 and 20 divided by a fifth is four. Therefore, they reason, the two sides are unequal and the equation is false.
The inquiry normally starts in one of two ways. Either there will universal agreement that the prompt is not true, or one student or a few will argue that it is. (If the majority can explain why it is true, then the prompt is not appropriate for the class because it will hold no intrigue.)
In the first case, the teacher should assert the equation is true and ask each student to think why that might be before students pair up to discuss and then share their thinking. Often, at this point, students will speculate that the reason is related to 50 x 2 = 20 x 5.
The teacher, using a diagram if appropriate, should link the speculation to the fact that there are two halves in one whole, four halves in two, six halves in three and so on. Following the pattern, we have 100 halves in 50. Similarly, there are five fifths in one whole, 10 fifths in two, and 100 fifths in 20.
In the second case, the teacher will draw on the existing knowledge of one or a few students to co-construct a convincing argument.
Once the class accepts the prompt is true, the teacher might structure the inquiry by following the first line of inquiry below. Alternatively, if the class has experience in carrying out inquiries, the teacher might offer a selection of regulatory cards:
Students select a card and explain how they will continue the inquiry. They might justify their selection in one of the following ways:
Find more examples: "I will extend the the chain of equations that equal 100."
Practise a procedure: "I would like questions with which to develop fluency in dividing by a fraction."
Change the prompt: "I want to find another chain by starting with a different number or by making the equations equal to a different number (not 100)."
Explain a pattern: "I would like to explain why the prompt is true."
Lines of inquiry
1. Extending the prompt
Is it possible to find more equations of the same type (i.e. with unit fractions) that equal 100? Can we construct a chain of the equations? How do we know when the chain is complete? What is the connection with the factors of 100? Is it possible to extend the chain to the left and continue to use unit fractions? (N.B. We could extend the chain to the right using a half, a quarter, an eighth and so on, but the teacher might decide to restrict the inquiry to whole numbers in order to emphasise the connection to factors.)
2. Starting with a different number
What would happen if we were to start the chain with a different whole number? Would the chain be longer or shorter than the one we made from the prompt? What number could we start with to make the chain longer? Is it possible to make a chain of equal length?
3. Using multiples
What would happen if we were to construct a chain of equations in which the whole numbers are consecutive multiples? Would there be a connection between the fractions? Could we generalise for any chain of the same type?
4. Using a different rule
What would happen if the whole numbers were to follow a different rule? What if they doubled each time or the difference between them increased by 10 (for example, 20, 30, 50, 80)? Is there a connection between the fractions? Could we generalise for any chain of the same type?
5. Creating problems
Is it possible to find the missing fractions in these equations? Is it possible to create an equation of the same type that has no solution?