# Generalizing Patterns: Table Tiles

## Mathematical goals

This lesson unit is intended to help you assess how well students are able to identify linear and quadratic relationships in a realistic context: the number of tiles of different types needed for a range of square tabletops. In particular, this unit aims to identify and help students who have difficulties:

- Choosing an appropriate, systematic way to collect and organize data.
- Examining the data and looking for patterns; finding invariance and covariance in the numbers of different types of tile.
- Generalizing using numerical, geometrical or algebraic structure.
- Describing and explaining findings clearly and effectively.

## Introduction

The unit is structured in the following way:

- Before the lesson, students attempt the task individually. You then review their work and formulate questions for students to answer in order for them to improve their work.
- At the start of the lesson, students work individually to answer your questions.
- Next, they work collaboratively, in small groups, to produce a better joint solution than those they produced alone. In the same small groups, students critique examples of other students’ work.
- In a whole-class discussion, students explain and compare the approaches seen and used.
- Finally, students reflect on their work and their learning.

## Materials required

- Each student will need the
*Table Tiles*task, a copy of the*Grid Paper*and a copy of the*How Did You**Work?*questionnaire. - Each small group of students will need a copy of the
*Grid Paper*and the*Sample Responses to Discuss*. - There is a projector resource to help you with the whole-class discussions.

Time needed:

15 minutes before the lesson, a 1-hour lesson, and 10 minutes in a follow-up lesson. All timings are approximate. Exact timings will depend on the needs of the class.

## Lesson Type

## Mathematical Practices

This lesson involves a range of *mathematical practices* from the standards, with emphasis on:

- MP1: Make sense of problems and persevere in solving them
- MP2: Reason abstractly and quantitatively
- MP3: Construct viable arguments and critique the reasoning of others
- MP4: Model with mathematics
- MP5: Use appropriate tools strategically
- MP6: Attend to precision
- MP7: Look for and make use of structure
- MP8: Look for and express regularity in repeated reasoning

## Mathematical Content Standards

This lesson asks students to select and apply mathematical content from across the grades, including the *content standards:*