Comparing Areas
MAG 4.3.5
DRAFT-This is a work in progress. MAG Writing Project Year 4 2013
Australian Curriculum YR 4
ACMMG087 Compare the areas of regular and
irregular shapes by informal means
Key Ideas
Compare the areas of regular and irregular shapes by informal
means
Measure the areas of common 2-D shapes using a square centimetre grid overlay e.g. measure the area of a regular
hexagon
Compare how different placements of a grid make measuring
area easier or harder
Develop strategies for counting partial units in the total area of
the shape
Measure the areas of irregular shapes using a squarecentimetre grid overlay
Explain why two students may obtain different measurements
of the area of the same irregular shape
Resources
● FISH problem solving kit
● A5 blank paper
● Grid paper (cm squares)
Introductory Activity Process
1. Hand out to each student an A5 piece of paper.
2. Ask the students, How can we measure how big the
piece of paper is without using a standard tool such as a
ruler? How can we cover it? Record and discuss
suggestions.
3. Ask student what kinds of items could we use as a
comparison e.g. how many pattern blocks would we
need?
How many MAB tens blocks? How many MAB ones
blocks?
4. Have students investigate the above mentioned and discuss
findings. What was the problem? Each way of measuring resulted in a
different answer - inconsistent.
5. Ask; is there a better way of measuring? Guide discussion into using
square units of measure.
Activity Process -Describing Polygons
1. Ask students to identify and name the shape of the A5 piece of
paper
2. Ask students to explain what kind of shape it is. Encourage the use
of labels e.g. polygon, plane shape etc.
3. Ask the class to work in collaborative groups and open a new
popplet page (iPad) or work in student journals to create a rectangle,
which is labelled with the following
● what kind of plane shape is it?
● how many sides does it have?
● how many angles does it have?
● are all the sides equal?
● are all the angles equal?
4. Ask students to create a square
● with four equal sides
● with four equal angles
5. Ask students to look, discuss and share what are the differences
between these two shapes.
6. Introduce the labels regular and irregular. Explain that the prefix re
means ‘again and again’ and that the word equal means the same length of sides and degree of angles-these shapes are regular. The
same sides and angles repeated.
7. Shapes where the sides and angles are not the same are called
irregular - explain the prefix ir means ‘not’ or the ‘opposite of’ in this
case irregular.
8. Many artists use these kinds of shapes in their artwork.
Activity Process - Identifying Regular and Irregular
Polygons
1. Organise cooperative groups of four
2. Use different resource sheets from
http://commoncoresheets.com/Shapes.php
and ask students to label shapes as regular-the same or
irregular-not the same. Remind learners that they need to focus
on the attributes of the length of sides and the degree of angles,
when sorting shapes
Activity Process - Cover Up
1. Use masking tape to create regular and irregular large shapes
on the classroom floor.
2. Encourage students to estimate as a guess be. Ask again after
the shape has been a third or half completed to assist in
developing a sense of the size of the unit. The skill of estimating
is a used as often as exact measurement in real life.
3. Cover the shape using informal (non-standard) units and
count the number used. Ask students to focus on the process of
repeatedly using a familiar unit as a measuring device e.g. their
hands, a sheet of paper.
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Ask students to use a circle as a unit and investigate why this
presents coverage problems? Discuss the gaps that are left.
Circles do not tessellate in the plane like squares and
rectangles. Also discuss the idea of arrays that can be easily
counted by multiplication.
5. Encourage students to think about the appropriateness of
units of measure when compared to the overall size of the area
to be measured. Ask them to demonstrate an inappropriate
informal unit and an appropriate unit.
Activity Process - Chalking Up Monsters
1. Have the students draw a monster on 2cm graph paper. The
picture must contain a minimum of eight half squares.
2. Using chalk the teacher creates a drawing of an irregular
shape in the playground. Students estimate how many monster
shapes will be needed to cover the shape and the apply the cut
out shapes of monsters to
make a determination
Part B- Working with Square Centimetres
1. Invite students who drew unusual or different shapes in Part
A to come forward and describe how they counted the number
of whole squares and then part squares inside their paper. They
then counted how many squares the shapes took up. After this
they tore a piece out of some of their shapes to make a puzzle
for the other groups to do.
Part C - Working with Square Centimetres
1. Give each student a piece of grid paper (or use their grid
maths books).
2. Challenge them to draw a quadrilateral and a triangle that
cover the same amount of square centimetres.
3. Students compare answers and strategies.
Digital Learning
SCOOTLE L383: Finding the area of compound shapes
.
Extensions and Variations
Part A - Working with Square Centimetres
1. Give each student an A4 piece of 1cm square grid. Ask them
to fold it in half.
2. Get them to draw a polygon using a ruler and the grid lines.
3. Ask each student to count the number of squares that covers
the inside of the shape and record the number on the sheet.
4. Have students share their shapes and areas.
5. On the other side, have students draw another polygon using
a ruler that does not follow the grid lines.
6. Ask them to count the number of squares that covers the
inside of the shape and record the number on the sheet.
7. Discuss any challenges or difficulties faced in calculating the
size of the inside - half squares, quarter squares etc.
Find the area of compound shapes based on rectangles on a
grid. Explore how the formula works for finding a rectangle’s
area. First, estimate the area of a compound shape based on
rectangles on a grid. Second, work out the correct formula for
finding area placing rows and columns of squares inside two
rectangles. Then, compare the actual area of the original shape
with your first estimate. Practise applying the formula directly
to a range of compound shapes based on rectangles.
IXL - Area (Year 4 Maths Practice)
Reviewing square units.
http://au.ixl.com/math/year-4/area
Context for Learning
Real life experiences:
FENCE IT
This challenging problem is for able students to work
systematically while applying their knowledge of areas of
rectangles. It offers opportunities for higher level mathematical
thinking.
(Student Page: http://nrich.maths.org/2663)
(Teacher Page: http://nrich.maths.org/2663/note)
Assessment Year 4 Achievement Standard
By the end of Year 4, students choose appropriate strategies for
calculations involving multiplication and division. They recognise
common equivalent fractions in familiar contexts and make
connections between fraction and decimal notations up to two
decimal places. Students solve simple purchasing problems.
They identify unknown quantities in number sentences. They
describe number patterns resulting from multiplication.
Students compare areas of regular and irregular shapes using
informal units. They solve problems involving time duration.
They interpret information contained in maps. Students identify
dependent and independent events. They describe different
methods for data collection and representation, and evaluate
their effectiveness.
Students use the properties of odd and even numbers. They
recall multiplication facts to 10 x 10 and related division facts.
Students locate familiar fractions on a number line. They
continue number sequences involving multiples of single digit
numbers. Students use scaled instruments to measure
temperatures, lengths, shapes and objects. They convert
between units of time. Students create symmetrical shapes and
patterns. They classify angles in relation to a right angle.
Students list the probabilities of everyday events. They construct
data displays from given or collected data
Assessment Tasks
Wallpaper (irregular shapes)
http://nrich.maths.org/4964
Arrange these pieces of wallpaper in order of size. Put the
smallest first. Wallpaper is designed to help children begin to
understand the meaning of area. Can you explain how you did
it?
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Torn Shapes (regular based shapes)
http://nrich.maths.org/4963
Jason's class cut out rectangles and some shapes which were
two rectangles joined together from one centimetre squared
Background
Torn Shapes is a challenge that encourages children to adopt a
different technique for finding area rather than simply counting
squares.
Area relates to the measurement of 2D space in the same way
that volume and capacity relate to the measurement of 3-D
space.
Students should appreciate that measuring area with a squarecentimetre grid overlay is more difficult when the shape to be
measured is not rectangular. This leads to an appreciation of
the usefulness of the various formulas for calculating areas that
are developed in later stage
Natural shapes are often irregular and incongruent with
flowing and rhythmical lines.
The artist Matisse in later life painted with scissors. He loved to
use his scissors to make various shapes overlapping to create a
collage. He rarely threw any scraps away. The purpose of these
pictures, he always asserted, was to give pleasure. For Matisse,
painting was the rhythmic arrangement of line and colour on a
flat plane. He created the technique of striking contrasts,
unmixed hues, and flat planes of colour
The Baltimore Museum of Art has a Matisse for Kids website
which can be used to familiarise learners with the kinds of
shapes that inspired his work.
http://www.artbma.org/flash/F_conekids.swf
PROJECT:
Create a Matisse style organic shape composition
Materials: Glue stick, Construction paper, Scissors
Step1: glue one large block of colour to your white sheet
Step2: glue a second block of colour to your white sheet
Step3: begin to cut organic shapes in all colours and glue them
to your paper
Step4: create a composition from your random shapes
Step5: pull the piece together by adding smaller pieces in a
pattern
Ask student to explain how they have used regular or irregular
shapes in their work.
This topic provides an opportunity to explore the notion of
shapes as organic or inorganic as compared to regular and
irregular and thirdly natural and man-made.
Word Wall:
http://www.artsconnected.org/toolkit/encyc_shapegeorganic.h
tml
The Artist’s Toolbox site provides examples of the way in which
artist use both in their work.
Links to other MAGs
Area, space, array, formal, informal, count, shapes
polygon, plane shapes, square units
An important concept to check for understanding is that
wherever the ends of a continuous line meet a shape is formed.
This shape can be organic or geometric.
Definition of Terms
Natural shapes can be organic based on living things. E.g.
plants, animals. They can be inorganic based on non-living
things E.g. stones, water, clouds
Manufactured (man-made) shapes are often congruent, based
on squares, circles and triangles and seen as regular shapes.
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