Mathematical goals

This lesson unit is intended to help you assess whether students recognize relationships of direct proportion and how well they solve problems that involve proportional reasoning. In particular, it is intended to help you identify those students who:

• Use inappropriate additive strategies in scaling problems, which have a multiplicative structure.
• Rely on piecemeal and inefficient strategies such as doubling, halving, and decomposition and have not developed a single multiplier strategy for solving proportionality problems.
• See multiplication as making numbers bigger and division as making numbers smaller.

Introduction

This lesson unit is structured in the following way:

• Before the lesson, students work individually on a task designed to reveal their current levels of understanding and difficulties. You review their work, writing questions to help them to improve.
• During the lesson, students first work in pairs or threes on the same task. Then working in the same small groups, they analyze work produced by other students on the task.
• In a whole-class discussion, students compare and evaluate the methods they have seen and used.
• Finally, students review their initial, individual response, using their learning to complete a new similar task.

Materials required

• Each individual student will need a mini-whiteboard, pen, and eraser, a calculator and a copy of the tasks A Sense of Scale and A Sense of Scale (revisited).
• Each small group of students will need a large sheet of poster paper and a copy of each of the Sample Responses to Discuss.

Time needed

15 minutes before the lesson, a 60-minute lesson, and 15 minutes in a follow-up lesson. These timings are approximate. Exact timings will depend on the needs of your students.