Module 9  Area & Volume 
Area & Volume  Teacher Notes
Solvemymaths has produced pages like this for lots of topics
NRICH Tasks
NRICH: Cut
Nets 
NRICH: An
Unusual Shape NRICH: Areas of Parallelograms NRICH: Isosceles Triangles NRICH: Trapezium Four NRICH: Cuboids NRICH: Sending a Parcel 
NRICH: Efficient
Cutting NRICH: Semi Circles NRICH: Partly Circles NRICH: Triangles and Petals NRICH: Cola Can NRICH: Funnel 
Extras
Video: Salaries 
This topic provides an opportunity for exploration and investigation and the traditional approach of telling the pupils the formulae and asking them to complete 20 questions should be avoided. Pupils should be challenged to think for themselves at every opportunity, I will give a few specific examples but the approaches can be easily adapted. Areas of Rectangles Source: Pupils could be presented with the following images and the teacher should invite (pairs of) students to make a statement (observation) or pose a question about the prompt, providing the class with stems (examples below) if appropriate.
This might lead to the class posing or answering some of the following questions in its first response to the prompt: Image § What is different and the same about the rectangles? § How many rectangles are possible with the same area? § Which has the longest perimeter? ... the shortest? § Is there a rectangle with an area equal to the length of its perimeter?

Area Puzzle Draw lines along the grid edges to divide the grid into rectangles, so that each rectangle contains exactly one numbered square and its area is equal to the number written in that square – Click here
Area Game
Roll Two Dice and draw a rectangle with the dimensions defined by the two numbers or an area equal to the product of the dice. Largest total area wins?

Area of Compound Rectangles
Fencing of Areas Investigation  Click here
FlashMaths: Area of 2D Shapes – Click here
Volume and Surface Area of Cuboid
This GeoGebra file allows you to adjust the dimensions of a cuboid, with dimensions as a multiple of 10 or integer or 1dp. It also allows you to colour pairs of sides to help when finding the surface area  Click here


Area v Perimeter
Coordinates Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc. Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points. 

Triangles and Quadrilaterals
Easy investigation showing that two identical triangles always form a rectangle  Click here Interactive Version  Click here
By Cutting and Rearranging  Click here
This activity involves presenting the pupils with a selection of 9 shapes and challenging them to cut each shape into pieces and then rearranging to make a rectangle and hence find the area. By looking at the resulting calculation it is possible to derive the formula for each of the shapes
Which Triangle has the larger area? If you draw in a vertical you end up with two 3,4,5 triangles in each shape so the areas are equal!

Largest Triangle on a Square Grid
This NRICH Task starts with a 6x6 grid and possible solution by exhaustion but then leads onto the generalised solution for any size grid – Click here
The areas of a rectangle, a triangle and a circle are equal  Source While this seems a rather simple prompt, it can lead to a number of difficult questions for students.
How do you work out the area of an obtuseangled triangle? There are a number of related prompts that can be used on their own or in conjunction with the main prompt. All of the following have led to successful inquiries: (1) The areas of three different types of quadrilaterals are equal. (2) The perimeters of a rectangle, a triangle and a circle are equal. (3) The volume of a cuboid, a triangular prism and a cylinder are equal

Area v Perimeter


Areas of Trapeziums Isometric shape areas  Click here
Students can be asked to draw several isosceles trapeziums
on an isometric grid and try to find relationships between
the three variables and the number of (unit) triangles in
the shape, this task is made more interesting by an
involvement of three variables
Many students identified that a + b = c and were able to offer some explanation for this Many rules can be identified including the following
But it can also be noted that the Area = (a + c)*b which leads onto the formula for a trapezium, it is interesting to try to relate the versions of the rules for the area to each other

Areas and Perimeter of Circles Drawing Freehand Circles – Click here Introducing Finding the Circumference Get an old tire, paint it and roll it to demonstrate circumference. Builds understanding for those bike exam questions too! Finding the perimeter of a circle is often introduced by measuring the perimeter using string and then finding that the Circumference is approximately 3 times the Diameter, an alternative to this which I prefer is to cut a circle into sectors as shown
Then the results are recorded into a table and the connection can be deduced.
Calculating Pi by Weighing – Video Clip

What are the measurements? Rather than asking pupils to answer 20 questions present them with this diagram and ask them to find a Trapezium with an area of 40cm^{2}.

Great Visual Proof of the Area of a Trapezium  Click here

Area and Volume Mazes 
Obstacle Course – What Maths is required Volume by Paper
USING THESE ITEMS ONLY make enough approximate measurements to
calculate an estimated volume of the object. Comparing Areas Draw a range of shapes without measurements onto A3 paper and pose the challenge to order the shapes according to their area, pupils could be asked to estimate the order first and then calculate subsequently. The shapes can all be of the same type or include a variety of shapes.

Taken from http://brilliant.org/ 
C ≈ 3r + 3r = 6r as we don't need the short sides and A ≈ 3 x r x r = 3r^{2}.
Alternatively show this gif looking at the perimeter
of a circle 
Click here Value of Pi digit by digit as the game Higher or Lower  Click here 
Surface Area of a Sphere
Walk into the classroom with a large fluorescent ball
under your arm and you've instantly got the attention of the
class.
This is a very engaging activity. As pupils can't talk unless they're holding the ball, you'll see them wildly waving their arms about, silently begging their friends to pass them the ball so they can share their ideas!

Volume I would encourage you to approach this by using a real life problem or scenario. One activity I have used is Packaging Salad This task looks at the redesign of some packaging for Salad
Surface Areas Volumes
Act 2  Measurements for calculating Volume  Click
here

Coke Cans This lesson idea initially explores the optimum way of packaging 12 coke cans and is adapted from Mr Piccini's Idea  Click here Question 1: As a customer which box would you choose to buy? Question 2: If you were Coca Cola which box would you choose to make?
Possible initial questions that need to be considered
Extension Ideas

Areas of Coins This provides lots of challenge and you could ask them to construct the shapes as well, highlighting the fact that many coins (the three below for example) have a constant diameter so they work in vending machines. Numberphile Video
Wrapping Presents Dr. Sara Santos is a popular mathematician and speaker on mathematics. She has worked out a method for wrapping boxes (rectangular prisms) as efficiently as possible – Click here Roofs Garage Roofs or the roofs of buildings provides an interesting starting point:


What questions (and answers) can you ask about this image? 
