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# The Rectangle Game

## Overview and Objective

In this lesson, students explore a game to build the background knowledge needed to classify and define the terms factor, multiple, prime, and composite. The game introduces the idea as to why 1 is neither prime or composite. The game is played by taking turns selecting a number on a hundreds chart, and then collecting points. A point is scored for each unique rectangular array that they can create with that number of square tiles.

This activity can be used with younger students who can identify numbers up to 100, and who have the dexterity to use a mouse. This activity can also be used with high school and upper level mathematics as it explores number systems such as highly composite, abundant numbers, superior numbers and their intersections.

## Warm-Up

Invite students to explore how many unique rectangles they can make with 100 square tiles. Share this Polypad with students so they can explore this question. This video shows how to “push and pull” the merged squares to find rectangles and how to duplicate and recolor rectangles so that students can see all of their findings. Click here to learn more about using Number Tiles on Polypad.

Be sure to include a 1 x 100 and a 100 x 1 as different rectangles. This will set the groundwork for why 1 is not prime or composite. Also, 100 is a perfect square, so remind students that squares are a type of rectangle, and should be included.

There are a total of 9 unique rectangles: 1x100, 2x50, 4x25, 5x20, 10x10, 20x5, 25x4, 50x2, and 100x1.

## Main Activity

Play the Rectangle Game. Make a copy of this gameboard and then share a copy with students.

Each player clicks on a number on the gameboard to open a Polypad with that number of square tiles. They find as many unique rectangles as possible (note: 1 x 3 is different than 3 x 1). Then score a point for each unique rectangle.

Example:

3 has two unique rectangles (3x1 and 1x3), so this would score 2 points.

4 has three unique rectangles (4x1, 2x2, 1x4), so this would score 3 points.

Once points are scored, players can block that number using a colored chip.  After four rounds, the player with the most points is the winner. Students can play additional games as time allow.

## Closure

Gather as a class to discuss the following.

Worst Gameboard Number: 1. It only has 1 rectangle (1x1).  Because it only has 1 factor, it is not prime, but it is also not composite.

Bad Gameboard Numbers: 2, 3, 7, 11, and other prime numbers because you can only make 2 rectangles.

Good Gameboard Numbers: 4, 6, 8, 9, 10, 12 and other composite numbers because you can make multiple rectangles.

Best Gameboard Numbers: 60, 72 and 96 because you can make the most (12 rectangles) with these 3 numbers.  These numbers have the most factors of any number under 100.

Amount of different rectangles: factors

Flipped rectangles: (example: 3 x 2 and 2x 3) use the same factors and are an example of the commutative property of multiplication.

Square rectangles: (example: 2 x 2) these are square numbers because the two sides use the same factor.

• *Smallest Gameboard number with the most rectangles: 60 because it is a highly composite number. Highly composite numbers is a positive integer with more divisors than any smaller positive integer has.
• *Upper level mathematics number theory terms: Highly composite number, smooth number, abundant number, superior number

## Support and Extension

To help support students, ask them the following: Sarah picked 100, because it was the biggest number. She scored 9 points because she found 9 different rectangles (one of which was a square). Zarah found another number on the gameboard that would give her more points than Sarah. What number could it be?

For an extension question, ask students the following: What if the gameboard extended to 1,000.  What are the best gameboard numbers?

## Polypad and Gameboard For This Lesson

To assign these to your classes in Mathigon, save a copy to your Mathigon account. Click here to learn how to share Polypads with students and how to view their work.