Download Tricky rectangles

Tricky rectangles
Purpose:
The purpose of this multi-level task is to engage students in an investigation of
Achievement Objectives:
NA5-7: Form and solve linear and simple quadratic equations.
Description of mathematics:
The background knowledge presumed for this task is outlined in the diagram below:
The task can be presented with graded expectations to provide appropriate challenge for individual
learning needs.
Activity:
Task: The sides of a rectangle, in metres, are each a whole number, less than 10.
The area of the rectangle is the same value (in m2) as the perimeter (in m).
Is this possible?
The arithmetic approach
The student forms algebraic equations as a description of the steps taken in calculations. They
calculate with numbers first, enabling them to focus on the steps they took as they generalise
with algebra.
Prompts from the teacher could be:
1. Consider the case that this rectangle has all four sides of equal length (ie, is a square).
2. Try all the possibilities for a solution using numbers.
3. Show how you found your successful solution using x for the length of a side of the
square.
The procedural algebraic approach
The student carries out directed calculations that will lead them to form and use a quadratic
equation to solve a problem.
Prompts from the teacher could be:
1. Sketch a variety of possibilities. Label the sides x, and/or a multiple of x.
2. For each of these rectangles, form equations to solve for x.
The conceptual algebraic approach
The student carries out an exhaustive algebraic investigation where they to form and use algebraic
equations, including quadratics, to solve a problem.
Further exploration of the task can be encouraged, by suggesting:
Use algebra or otherwise to find all the possibilities for a rectangle to satisfy these conditions.