Lesson 1 Checkerboard Borders Develop Understanding

Jump Start

The pattern of tiles shown here is made with a $5 \times 5$ square of white tiles surrounded by a border with alternating colored squares. Show how you would group the tiles to quickly count the number of colored squares in the pattern.

Learning Focus

Use variables to describe ways of seeing a pattern.

When I look at a pattern, how can I describe what’s changing and what’s staying the same?

Can seeing groups help in understanding diagrams?

How do variables express the way we see patterns?

Technology guidance for today’s lesson:

Basic Calculations — Add and Subtract: Casio ClassPad Casio fx-9750GIII
Basic Calculations — Add, Subtract, Multiply, and Divide: Casio ClassPad Casio fx-9750GIII
Basic Calculations — Exponents: Casio ClassPad Casio fx-9750GIII
Basic Calculations — Square Root: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

In preparation for going back to school, the school administration plans to replace the tile in the cafeteria. They would like to have a checkerboard pattern of tiles two rows wide to surround the tables and serving carts.

Below is an example of the border that the administration is thinking of using to surround a $6 \times 6$ square of white tiles.

1.

Find the number of colored tiles in the checkerboard border using a strategy that either you or one of your peers used in the Jump Start. Show your calculations.

2.

The contractor who was hired to lay the tile in the cafeteria is trying to generalize a way to calculate the number of colored tiles needed for a checkerboard border surrounding a square of tiles with any dimensions. To represent this general situation, the contractor started sketching the square below. Since he wants the diagram to work for any number of tiles in the middle square, he just called the side length $N$ for the number of tiles on that side.

Use the same counting strategy that you used for the $5 \times 5$ and the $6 \times 6$ square to find an expression for the number of colored border tiles needed for any $N \times N$ square center.

Ready for More?

Find an expression to calculate the number of colored tiles in the two-row checkerboard border for any rectangle. Be prepared to share your strategy and justify your work. Create models to assist you in your work.

Takeaways

Strategies to help us see patterns in diagrams:

Vocabulary

Lesson Summary

In this lesson, we found ways to see patterns in diagrams that help us to be efficient in counting. We modeled these patterns using variables to show how the pattern would work for any number of squares.

Retrieval

Evaluate each expression using the given value.

1.

$10 - 3 g$ , $g = 5$

2.

Given: $\frac{3 x}{4} - 12$ , $x = 8$

3.

Complete the table of values.

$y = 3 x - 5$

$x$	$y$
$0$
$1$
$2$
$3$

4.

Graph the points from the table on the graph.

$x$	$y$
$1$	$16$
$3$	$13$
$5$	$10$
$7$	$7$