Lesson 10 Rectangle functions

# Episode 1

Reasoning Resources: Worksheet 1, 1/2 cm squared paper, OHTs and pens (optional), Calculators
Whole class preparation
The two variables, length and width, are normally seen as features of the rectangle itself, indicating the shorter and longer sides.

Some early ideas on fractional values are possible.

Alternatively, as height and width (or 'across'), they are features of orientation on the page, indicating the vertical and horizontal dimensions. The context here calls for this type of label, with terms to be agreed by the class. Focus on the general pattern rather than accuracy of drawing. You may wish to present this as a continuation of the story in TM9. When Hassan and his apprentice were fitting decorative tiles they sometimes had to cover different rectangular shapes with the same number of tiles, i.e. the same area. The apprentice thought that there were only six different ways to form a rectangle with an area of 12. How many rectangles do you think can be drawn with an area of 12?
Take initial ideas and record them before a short session of individual and paired work to identify some of the rectangles. Pupils will recognise the equivalence of the rectangles in different orientations. But in the context of decoration suggest that they are different. Some pupils may suggest breaking tiles in halves or more parts. Tell them that anything imaginable is possible.

Class feedback allows a more extensive list of rectangles to be shown in sketches, recorded as numbers and orientation to be addressed. Any half values offered should be used. Use some but not all values of height and width for rectangles with an area of 12 to draw the rectangles using a common origin point for one corner, as shown in the diagram. Draw this in front of the class, without being accurate and with the approximate scale. Show how the corners of each rectangle opposite the origin are aligned on a smooth curve. Ideas should emerge on the use of a proper scale or axes as well as clarification of the two dimensions. Give out Worksheet 1 and 1/2 cm squared paper. Pair and group work
The inverse proportional relationship.

=Fixing one of the three variables (height, width and area) as a parameter allows the relationship to be represented graphically. Here fixing the area to 12 units, gives the equation 12 = h x w. Plotted with width on the horizontal and height on the vertical axes the graph effectively becomes that for w = 12/h, a hyperbola in the first quadrant. Similarly for area = 24.
Working in pairs the pupils plot for themselves the rectangles and then the curve for area equal to 12 on a scaled graph (with both axes up to 25). Pupils may find it easier to draw curves on OHT overlays. They then use the curve to find rectangles with non-whole-number dimensions through visual estimation and checking with a calculator. They then consider where the curve for rectangles of area 24 will lie on the grid. To demonstrate continuity convincingly it will be necessary to look at rectangles with sides that are not whole numbers, using calculators. Some pupils may also begin to see the inverse nature of the relationship by selecting a side length and dividing the fixed area by it to find the other side length. Allowing time for pupils, exploration at this stage will help reinforce the idea of continuity, which although implicit in drawing the curve is not by this act proven. Pairs should be asked to prepare to talk about what they did and what the curves mean.
Whole class sharing and discussion
The continuity of relationships; describing them in their own words.

Intuitively approaching infinitely large and infinitely small values in a visual context.
Choose groups to report their ideas to the class (the less advanced groups first). Encourage them to talk about the meaning and usefulness of the curves. Discuss what happens to one dimension if you increase the other while the area remains unchanged. Typical answers may be: 'As the width gets bigger the length gets smaller'; 'if you double the length then the width is halved'; 'There are hundreds of rectangles of area equal to 12'; 'The length times the height stays the same'.
Discuss doubling the length of a rectangle which starts with a unit height. Ask pupils what happens to the height as the length is repeatedly doubled if the area stays the same.
Whole class reflection
If the class has engaged well with the advanced mathematical ideas of continuity (infinite number of values between any discrete values for the width and height), infinitely small or large values, and what happens when the height (or width) approaches zero, then they could talk about these discoveries and describe them in their own terms.