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Background For Teachers:
In this lesson, students will organize information about running an ice cream production business into matrices. They will then explore ways to combine the data in the matrices to create new matrices that contain relevant information about their business. Students will share strategies for multiplying and organizing their data and recognize the need to standardize the process of multiplying matrices.
The task is set in the context of organizing data and combining information for making batches of ice cream so that students can use the context to determine which factors need to be multiplied, which terms need to be added, and how data might be arranged to keep track of the operations in an organized way. Since the row-by-column procedure for multiplying matrices is a mathematical convention, students may initially chose to organize the data in a different way. The need for a standard, agreed upon procedure will be motivated by the move to technology where matrix multiplication is performed according to established rules. Inventing their own procedures for multiplying matrices will help students make sense of the conventional row-by-column procedure.
To highlight the context of the task, it is recommended that students name each of the matrices using labels such as the "ingredient by recipe" matrix or the "recipe by day" matrix. They will need to be familiar with the convention of describing matrices first by rows, then by columns. Hence, the name "ingredient by recipe" matrix would suggest that the rows contain information about the ingredients used to make the ice cream, and the columns represent the amount of each ingredient used in different ice cream recipes. Using the conventions of matrix multiplication, the result of multiplying an "ingredient by recipe" matrix by a "recipe by day" matrix will be an "ingredient by day" matrix. Teachers will need to introduce the "row-by-column" naming structure to students, but students should sort out how these naming stuctures help them keep track of the operations to be performed and the meaning of the data in the product matrix.
For additional examples of developing matrix multiplication within a context, see the unit Meadows or Malls from the Interactive Mathematics Program published by Key Curriculum Press.
Student Prior Knowledge:
This series of lessons would typically follow lessons on adding matrices within a real world context.
Intended Learning Outcomes: