Module 9 Area & Volume
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Area & Volume - Teacher Notes

Solvemymaths has produced pages like this for lots of topics

NRICH Tasks

NRICH: Cut Nets
NRICH: A Puzzling Cube
NRICH: Torn Shapes
NRICH: Numerically Equal
NRICH: Fence It
NRICH: Warmsnug Double Glazing
 

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
Area Worksheets: Click here

 

 

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?

 

Source

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 Co-ordinates

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 obtuse-angled triangle?
How do you work out the area of a circle?
Can the radius of the circle be a whole number if its area is a whole number?
Can the dimensions of the three shapes all be whole numbers or must there be decimals?

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 
it can be hard (but good) for some students to grasp the notion of measuring 'area' with unit triangles rather than unit squares (using only sides that follow the isometric grid lines)


the three sides can be labelled so that algebraic rules can be recorded easily.

 

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 40cm2

 

Great Visual Proof of the Area of a Trapezium - Click here

Area and Volume Mazes

   Blog

Obstacle Course – What Maths is required

Video Clip
 

Volume by Paper


You have been given an object, pair of compasses, a pencil and this sheet of A4 paper, which has dimensions 210mm by 296mm.

 

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 = 3r2.

Alternatively show this gif looking at the perimeter of a circle - Click here
and this one looking at the area of a circle pause and discuss -
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.

Students stand in a circle and are told they're only allowed to speak if they're holding the ball. Ask "how could we work out the surface area of this ball?" and throw it to a random student. Students throw the ball around the circle as they share their thoughts and suggest ideas. At some point a teacher prompt may be necessary, along the lines of, "why did I bring a tennis ball? Why not a football or netball or golf ball?". In my experience this prompt may lead some students to 'eureka' moments as they start thinking about the markings on the surface of the tennis ball. Eventually someone will spot that the surface is made up of 4 circles wrapped around each other, and they can then deduce that the formula for the surface area is 4πr2. If they need convincing, show them this 
slide.



 

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



Act 1 - What is the question? - 
Click here
Act 1b - Some additional information - 
Click here

Surface Areas

Act 2 - Measurements for calculating Surface Area - 
Click here
Act 3 - Calculation of the two Surface Areas - 
Click here

Volumes

Act 2 - Measurements for calculating Volume - Click here
Act 3 - Practically comparing the two volumes - are they equal? - 
Click here
Act 3 - Calculation of the two Volumes - 
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
How many cans?
Estimate Dimensions?
Packing in the fridge
Size of pallet: 1m by 1.2m
How many stacked boxes would be the same as your height?
Volume wasted in the box
Value for money (can versus bottle)
Surface Area
Packing on lorries?
Base not printed so cheaper to manufacture
Do we consider flaps?

Possible Progression Route

1.       Estimate Can size and hence sizes of box (real size 11.5 by 6.3cm)
2.       Round can size to 12 by 6 for lower group
3.       Size of boxes?
4.       Sketch net and write measurements on and then discuss Scale
5.       Lower ability draw rectangles for each face on A4 squared paper (or  
          photocopy squares onto card) and then sellotape them together or lay them
          out on large sugar paper.
6.       Draw nets to scale (1:2) will fit on A3 easily
7.       Surface area?
8.       Fit onto pallet?
9.       What about slim cans
10.    Cost of packaging (Note base cheaper)
11.     Extra 100% or 50% free recalculate
 

Extension Ideas

Boxes that are not cuboids: PowerPoint  - 
Click here
Design a Container which holds 300ml - 
Click here
Using Similarity with Drinks Cans - Click here

 

 

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?