Spaces 3 Sat, 26th Feb 2011

9:30 - 11:00 Areas and perimeters 11:30 - 13:00 Circles and ∏ 14:00 - 15:30 Volumes and Surface Areas

Activity In your books, create a Venn-diagram to show the relationships among parallelograms, rhombuses, rectangles and squares.

Activity In your books, create a Venn-diagram to show the relationships among parallelograms, rhombuses, rectangles, and squares. Parallelograms Rhombuses Rectangles Squares

Definitions A square is a quadrilateral with all sides equal and all angles equal. A rhombus is a parallelogram with all sides equal. A parallelogram is a quadrilateral whose opposite sides are parallel and consequently equal in length. A rectangle is a parallelogram with an interior angle of 90°.

9:30 - 11:00 Areas and perimeters to revisit areas and perimeters of 2D shapes to address some of the misconceptions pupils have related to these concepts

What is the perimeter of this shape? To find the perimeter of a shape we add together the length of all the sides. What is the perimeter of this shape? Starting point 1 cm 3 Perimeter = 3 + 3 + 2 + 1 + 1 + 2 = 12 cm 2 3 1 1 Ask pupils if they know how many dimensions measurements of perimeter have. Establish that they only have one dimension, length, even though the measurement is used for two-dimensional shapes. Tell pupils that when finding the perimeter of a shape with many sides it is a good idea to mark a starting point and then work from there adding up the lengths of all the sides. 2

Which shape has a different perimeter from the first shape? Equal perimeters Which shape has a different perimeter from the first shape? A B C B Ask pupils to decide which shape has a different perimeter from the other three and to explain how they can tell that the other three shapes have the same perimeter. For the first set all of the shapes have a perimeter of 16 units except shape B. For the third set all of the shapes have a perimeter of 14 units except shape A. A A B C

Area of a rectangle The area of a shape is a measure of how much surface the shape takes up. Area is measured in square units. For example, we can use mm2, cm2, m2 or km2. The 2 tells us that there are two dimensions involved, length and width. We can find the area of a rectangle by multiplying the length and the width of the rectangle together. length, l width, w Area of a rectangle = length × width = lw This formula should be revision from key stage 2 work.

Area of a triangle The area of any triangle can be found using the formula: Area of a triangle = × base × perpendicular height 1 2 base perpendicular height Ask pupils to learn this formula. Can you prove this formula? Or using letter symbols, Area of a triangle = bh 1 2 GSP interactive

Area of a right-angled (or right angle) triangle Calculate the area of this right-angled triangle. To work out the area of this triangle we only need the length of the base and the height. Why? 8 cm 6 cm 10 cm Talk through this example. Make sure that pupils are able to identify which length is the base and which length is the height. The lengths of the sides may be modified to make the arithmetic more difficult. Area = 1 2 × base × height = × 8 × 6 1 2 = 24 cm2

Activity 2 (in pairs) Calculate the area of an equilateral triangle of side 8cm. Calculate the area of an isosceles triangle whose sides are of 8cm, 8cm and 6cm. For b) what difference does the choice of ‘base of a triangle’ make? GSP GSP

Area formulae of 2-D shapes You should know the following formulae: b h Area of a triangle = bh 1 2 b h Area of a parallelogram = bh a h b Use this slide to summarize or review key formulae. Area of a trapezium = (a + b)h 1 2 Can you derive this formula yourself?

Using units in formulae Remember, when using formulae we must make sure that all values are written in the same units. For example, find the area of this trapezium. 76 cm Let’s write all the lengths in cm. 518 mm = 51.8 cm 518 mm 1.24 m = 124 cm 1.24 m Stress that when substituting different lengths into a formula the units must be the same. Link: S7 Measures – converting units. Area of the trapezium = ½(76 + 124) × 51.8 Don’t forget to put the units at the end. = ½ × 200 × 51.8 = 5180 cm2

Activity 3: Doubling statements (SS5-7 first four cards) Resources: A large sheet (poster) Glue stick Card statements In groups of three or four, divide the sheet into three columns and head the columns Always true, Sometimes true, Never true. Extension: what if you treble the lengths?

How would you guide the learners to tackle this activity? Understand the problem Try some examples (look at a special case) Make conjectures Try to disprove or justify the conjectures Step 1. Understand the problem Explain that the learners have to decide whether the statement is always true, sometimes true, or never true. What does this mean? Well, we need to decide whether the statement is true or not for all possible squares and rectangles that have the same area. What does the word ‘area’ mean? How do we calculate it? What does the word ‘perimeter’ mean? How do we calculate it? Step 2. Try some examples In this way, help learners to generate examples and then to offer conjectures. Numbers as in measurements, etc Step 3. Make conjectures Learners may begin to notice...Thus the statement appears to be always true. Step 4. Try to disprove or justify the conjectures Can we see why the statement must be true? This statement might be too difficult for learners to prove algebraically, but they may be able to test particular cases in an organised way.

Misconceptions in geometry Areas and perimeters Language confusion Forgetting which formula refers to which idea for a particular shape Thinking that the perimeter and area are in some way interrelated Cognitive Conflict needed!

Always true, Sometimes true, Never true The Standards Unit Box, Malcom Swan http://nationalstrategies.standards.dcsf.gov.uk/node/366857 Always true, Sometimes true, Never true Pupils commonly confuse the ideas of perimeter and area. This seems to be more than the mere language xconfusion as to which word refers to each idea and it is more than forgetting which formula refers to which idea for a particular shape. There is a deeper problem to do with understanding the subtle relationship between the two ideas which require an appreciation that changing one does not necessarily change the other. The misconception is that somehow the perimeter and area are in some way interrelated, that an increase in one is thought to lead to an increase in the other. Equal perimeters implies equal areas. Ths is a persistent misconception and will not necessarily be eliminated by making reference to a single counter-example or through a single lesson task investigating the perimeter of rectangles of constant are. It requires frequent encounters through a variety of examples which draw out the conflict between faulty intuition and geometrical results. Step 1. Understand the problem Explain that the learners have to decide whether the statement is always true, sometimes true, or never true. What does this mean? Well, we need to decide whether the statement is true or not for all possible squares and rectangles that have the same area. What does the word ‘area’ mean? How do we calculate it? What does the word ‘perimeter’ mean? How do we calculate it? Step 2. Try some examples Give me some dimensions for a square. (6 cm × 6 cm) What is the area of that square? (36 cm2) What is its perimeter? (24 cm) Now give me some dimensions for a rectangle with the same area as the square. (12 cm × 3 cm) What is the perimeter of the rectangle? (30 cm) So is the statement true for this example? (Yes) Can you give me a different rectangle with the same area? (9 cm × 4 cm) What is the perimeter of the rectangle? (26 cm) In this way, help learners to generate examples and then to offer conjectures

These activities help pupils to: Understand concepts of lengths (perimeters) and area in more depth; Develop reasoning through considering areas of plane compound shapes; Construct their own examples and counter-examples to help justify or refute conjectures.