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Invented multiplication strategy

Task Statement

Thomas and his class have been working on invented strategies for 27 x 18 his teacher has him put his work on the board for a class discussion:


  1. What is Thomas demonstrating that he knows?  Under what operations would his strategy be applicable?
  2. Create an area model to justify and explain Thomas’ strategy, in order to further develop his reasoning.  How does the symbolic calculation of this situation map to the area model?
  3. How might we use Thomas’ strategy of compensation to solve this problem in a simpler way?


Task of Teaching

Interpreting and explaining student work

IM Commentary

Many teachers might dismiss Thomas’ strategy because it arrives at the wrong answer.  However, it brings in some complex mathematics in terms of the distributive property.  The distributive property is essential in developing number sense with students and becomes important in higher mathematics.  Using the area model allows students to visualize the distributive property that can be useful in algebra.  The area model also highlights a common mistake that many people make when trying to figure out what needs to be subtracted in this situation.  Most subtract either too much or too little and the “6” becomes problematic, seeing this connection to the distributive property can be a powerful lesson in representations and linking to student strategies.

Teachers need to be able to interpret student work and determine the mathematics behind their thinking so that they can build on their understanding.  This problem illustrates a situation where the students thinking was partially correct and could be modified to produce an effective strategy.