1. Developing the concepts and formulae for perimeter and area

2. Measurement involves finding and reporting the amount of an attribute that is present in an object or event.
Length (width, depth, height, distance, etc.)
Mass
Capacity
Time
Angle
Length involving perimeter
Area
Volume

3. Perimeter is not a new measurement concept.
It is the length concept.
A formulae can be used to find the perimeter of some shapes.

4. Child's language
boundary edge
short long

5. These are sometimes called non-standard units.
Materials language
links on the border
bricks around the edge
This step must stress counting.
These are sometimes called non-standard units.

6. Mathematical language
for rectangles –
double the numbers that represent the length and the width and add; or
add the numbers that represent the length and the width and then double
length
width

7. Perimeter of a rectangle is
Symbol language
Perimeter of a rectangle is
2 L + 2 w = P or
2 (L + w ) = P
For general formulae, mathematicians use the raised dot rather than an ‘x,’ to avoid the confusion - does ‘x’ mean multiply or is it an unknown?
As well, it helps to have a symbol for the operation rather than using the expression 2L + 2W = P.

8. What is the concept of area? What is used to measure the amount of area?

9. Units of Measure Approach
Language Approach
Units of Measure Approach
Child’s language
Direct comparison – no units of measure

10. How could we compare directly? What are the words the students use?
Child's language - area
How could we compare directly?
What are the words the students use?
Area is an amount of covering of a surface (surface automatically implies it is 2-dimensional).

11. Units of Measure Approach Units of Measure Approach
Language Approach
Units of Measure Approach
Child’s language
Direct comparison – no units of measure
Language Approach
Units of Measure Approach
Material’s language
Use non-standard units with materials

12. Materials language - area
What could we use in this stage?
This step must stress counting.
These are sometimes called non-standard units.

13. Units of Measure Approach
Language Approach
Units of Measure Approach
Child’s language
Direct comparison – no units of measure
Language Approach
Units of Measure Approach
Material’s language
Use non-standard units with materials
Language Approach
Units of Measure Approach
Mathematical language
Use standard units with materials

14. Mathematical language - area
What squares do we use?

15. Trace around your closed hand. How many squares does it cover?
What is the length and width of your closed hand?
How does that help make an estimate?

17. Units of Measure Approach
Language Approach
Units of Measure Approach
Child’s language
Direct comparison – no units of measure
Language Approach
Units of Measure Approach
Material’s language
Use non-standard units with materials
Language Approach
Units of Measure Approach
Mathematical language
Use standard units with materials
Language Approach
Units of Measure Approach
Symbolic language
Use standard units with tools

18. There is no tool that is used to ‘read off’ the area of a region.
Symbol language
cm2 m2 km2 ha
There is no tool that is used to ‘read off’ the area of a region.

19. The two aspects associated with measurement – concepts and skills are made clear with area:
The concept involves the idea of covering with square units.
The skill (in this case a formulae) uses at least one operation to help count the squares.

20. How can you work out the number of squares without seeing them?

21. The skills associated with area
How could we quickly count the number of squares that cover the interior of a rectangle?
length
width

22. Mathematical and symbolic language
Area of the interior of a rectangle
The number of rows (one cm wide) – the width - by the number of squares (one cm long) in each row – the length.
A = L w

23. Anne has 80 metres of fencing
Anne has 80 metres of fencing. What are the possible dimensions of rectangular shaped grazing areas she could build for pet goat?
Which rectangle will give the greatest area?

24. Anne purchases a second goat and now needs to double the area
Anne purchases a second goat and now needs to double the area. How much extra fencing does she need?

25. Ben has 100 square pavers (each paver is 50cm by 50cm)
Ben has 100 square pavers (each paver is 50cm by 50cm). What are the dimensions of the rectangle that will require the least perimeter for edging?

26. A square is a rectangle with special features so the rule is the same.

27. A parallelogram can easily be transformed into a rectangle so the rule is the same.

28. Draw a rectangle at the top of a sheet of grid paper
Draw a rectangle at the top of a sheet of grid paper. It height should be 4 cm and the length of its base should be 6 cm.
Then draw a parallelogram with the same dimensions. What do you notice about the area?
Draw parallelograms that look different but have the same area (24 cm2).
What are some of the different possibilities?

29. Mathematical and symbolic language
Area of the interior of a parallelogram
The number of rows - now described as the height - by the number of squares (one cm long) in each row – now described as
the length of the base
A = b h

30. A rhombus is a special parallelogram so the rule is the same.

31. Two copies of a trapezium forms a parallelogram.

32. Mathematical and symbolic language
Area of the interior of a trapezium is developed from the parallelogram.
How is this rule changed for a trapezium.
A = b h

33. quadrilaterals
trapeziums
parallelograms
rectangles
rhombuses

34. simple quadrilaterals
The development of this structure involves many carefully planned activities and typifies what the curriculum is all about.
What is the structure for quadrilaterals? This is the mathematically preferred classification. (inclusive)
simple quadrilaterals
kite
trapezium
other
parallelogram
other
rhombus
other
rectangle
square
oblong

35. Mathematical and symbolic language
Area of the interior of a triangle
Start with a parallelogram with the same dimensions as the triangle.
Find the area of the parallelogram (which is too much).
Cut the area of the parallelogram into 2 equal parts to find the area of the triangle.
A = b h ÷ 2

36. Once a rule has been established to find the area of the interior of a parallelogram, it can be used to find the area of the interior of a triangle.

37. Mathematical and symbolic language
Area of the interior of a triangle
Note: The Australian curriculum says to develop the rule for a triangle from a rectangle.
This works only for right triangles – very limited!

38. What about circles?

40. There is a constant relationship between the diameter and the circumference of a circle.
This relationship is close to …
C  d = 

41. If you know 12  4 = 3 C  d =  This is the same: 12 = 3 4 C =  d
are the same
This is the same:
12 = 3 4
C =  d

42. The rule to find the area of the interior of a parallelogram can be used to find the area of the interior of a circle.

43. What were the steps we used to find the area of a parallelogram?