Mixed numbers inquiry

The prompt

Mathematical inquiry processes: Verify; analyse structure; find more examples; generalise and prove. Conceptual field of inquiry: Subtraction and division of mixed numbers by integers.

This prompt comes from division = subtraction? on Don Steward's website. Students notice the properties of the prompt: the mixed and whole numbers are the same on each side of the equation, but the operation is different. Finding another example of the same type, however, can prove challenging for students. By focusing on the mathematical structure of the equation, the teacher can support the search process.

minuend – subtrahend = difference

dividend ÷ divisor = quotient

The quotient equals the difference, which means, in the case of the prompt, the subtrahend is three times greater than the quotient. In general, if you divide the subtrahend by one less than the divisor, you get the quotient.

Students have asked in classroom inquiry:

  • Is the equation true?

  • How do you subtract and/or divide mixed numbers by whole numbers?

  • Are there other examples that work the same?

  • Does it work with addition and multiplication?

In the next phase of the inquiry, students might verify the equation is true before using its structure to search for more examples.

Algebraic generalisation

The inquiry might conclude with an attempt to form an algebraic generalisation of these types of equation.The teacher could lead a class discussion to manipulate the algebraic forms with the aim of showing that the two sides simplify to the same expression.