Lesson 10 Rectangle functions

Rectangular functions

´╗┐Continuous variables Resources
The focus is on the continuous as opposed to whole-number, or discrete value of the variables length, width, area and perimeter. Both graphical and algebraic representation are explored. Starting from the idea that there are many rectangles with the same area, pupils look for a pattern linking the sides of rectangles with the same area with the value of that area.They address the ideas of continuity and infinity in number values. This activity follows on from TM9: Framed tiles, and combines reasoning strands in shape and space, and in algebra and graphs.
Aims Curriculum links
Exploring the contrast between area as a multiplicative relationship and perimeter as an additive relationship.
  • Aspects of continuity,
  • Infinite and infinitesimal lengths.
    Area as a function
    Pupils find a number of rectangles with area 12.They are shown how to arrange them from a common origin so that the corners of the rectangles opposite the origin are aligned on a smooth curve. Pupils then construct their own curves and investigate the other possible rectangles with area 12 and 24. Discussion is focused on the inverse proportionality, the continuous nature of the relationship and what happens when the height or the width approaches zero. ln many cases classes may engage in these advanced mathematical ideas to make this episode a full hour lesson.
    Perimeter as a function
    Pupils work out the width and length for a number of' whole-number' side rectangles with perimeters equal to 12 and to 24.This information is plotted on a graph. As in the area case, pupils investigate the continuous nature of the relationship and what happens as one dimension approaches zero. Discussion highlights the similarities and differences between the area function and the perimeter function, and why this might be so.
    Exploring area and perimeter problems
    Higher ability pupils, or pupils working in mixed-ability groups, can use these problems to move from visually supported work to direct mental operations and calculations involving inverse functional relationships, which is equivalent to the change of subject in algebraic Formulae. Some pupils may also be ready to handle quadratic proportionality through the change of units of measure of area.


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