Puzzling over perimeter, area and volume
Band of development:
Later adolescence
Curriculum organiser:
Mathematics
Year level(s):
Proposed duration:
Essential Learning
Achievements
9
6 weeks
17. The student chooses and uses measures.
18. The student recognises and represents patterns and relationships.
ACKNOWLEDGEMENT:
Thank you to the Literacy and Numeracy Team for developing and sharing this unit.
This is a sample unit of work. Teachers need to consider its usefulness within the context of their own students’ needs
and school’s curriculum plan and adapt it accordingly.
Posted:
November 2007
PUZZLING OVER PERIMETER, AREA AND VOLUME
BAND OF DEVELOPMENT:
Later adolescence
CURRICULUM ORGANISER:
Mathematics
YEAR LEVEL:
9
PROPOSED DURATION:
6 weeks
UNIT DESCRIPTION
OUTCOMES:
Students explore perimeter, area and volume using stimulus materials they
have created.
Students will:
PRIOR KNOWLEDGE
Concepts of basic shapes; including beginning notion of perimeter and
area.
Students should also be able to identify different shapes according to their
properties.











design Puzzle Box to meet specifications
find the perimeter, area, volume of basic and compound shapes
construct nets associated with a variety of shapes
estimate size and space
explore relationships between perimeter, area volume
use and understand a decision tree
analyse applications of area and volume
analyse own thinking and application critically
appraise puzzle box
make own puzzle box
build their own decision tree
2
ATTITUDES AND VALUES:
In this unit, students have opportunities to:
 develop creativity in using different representational forms and tools, including ICT, to represent thinking
 develop a positive attitude towards measurement and exploring its applications in mathematical and everyday situations
 develop confidence, perseverance, creativity and flexibility in solving problems using measurement
 take responsibility for measuring accurately when the situation requires it (e.g. safety)
ESSENTIONAL LEARNING ACHIEVEMENTS & ESSENTIAL CONTENT
(ACT Curriculum Framework)
Later adolescence (note: italics indicate aspects of the essential content explicitly covered in this unit of work)
17. The student chooses and uses and measures
1.
measurement error, including recording measurements as values that lie within a given interval of measurement error, judgments about acceptable or
reasonable error in a measurement context, strategies to minimise error and estimation of error rates to provide confidence in measurement results,
and risks of compounding error by repetition and calculation
6.
choose and use instruments, technologies, strategies and formulas to estimate, measure and calculate measures of attributes, including mass, duration,
temperature, angle and simple derived measures such as rates
7.
work routinely with International System (SI) and other units with respect to both everyday and technical measurement contexts, including derived
measures, choose units appropriate to the order of magnitude involved and estimate values that lie between marked graduations on scales of
measuring instruments
18. The student recognises and represents patterns and relationships
1.
a broad range of 2D shapes (eg. quadrilaterals, polygons, ellipses), composite shapes and 3D objects, including those with curved surfaces (e.g.
cylinders, cones, packages and containers), with respect to properties involving line, length, angle and surface
10.
draw by hand representations of common 2D shapes and 3D objects (and their cross-sections) with attention to their geometric properties and scale
LINKS TO OTHER ELAs
This unit is also linked to The student understands and applies number, The student understands and applies scientific knowledge and The student designs,
makes and appraises using technology through applications of measurement.
3
KEY
WORK SAMPLE TO BE COLLECTED
OCCUPATIONAL HEALTH AND SAFETY CONSIDERATION
ABOUT THIS UNIT
This unit has been developed and trialled over the last three years. It has grown from a need to ensure students develop deep conceptual
understandings surrounding the concepts of perimeter, area and volume.
This unit:

provides the opportunity for students to have ownership of the stimulus materials, resulting in higher engagement by the students

is easily differentiated and currently all Year 9 students complete this unit

forms the basis for the entire assessment of Perimeter, Area and Volume; that is – assessment is ongoing.
Assessment:

The assessment is continuous, work samples are collected throughout the unit.
Teaching points:

Record any extra support that is required for students to succeed. Provides scaffolds for student learning as required.

Explicit teaching, investigation, consolidation and assessment are ongoing throughout the entire unit.
The following table identifies what is essential, what can be added and what can be changed.
4
Summary of Process
Intro
TOPIC
ACTIVITY
Exploring Shapes –
Barrier Game
Defining properties of
shapes using a Data Set
Puzzle Box
Decision Trees
Twenty Questions
1
How does Twenty
Questions Work?
Examining a Decision
Tree
Where can decision trees
be used?
Selecting questions for a
decision tree
DOMAIN1
Experiencing the known
Conceptualising by naming
PURPOSE


Experiencing the new
Conceptualising with theory

Experiencing the new

Analysing functionality

Conceptualising with theory


introduces a tool for discussion of effective
questioning
classification of objects Decision trees as useful
organisers.
makes connections between maths and other KLAs
particularly Science.
demonstrating purpose and functionality
effective questioning techniques – links to definition
of attributes
Applying appropriately
Experiencing the new
Design layout
Conceptualising with theory
Mark and Cut and Sand
Puzzle box
Applying appropriately
ELA18 LA 1
ELA18 LA 1
An effective decision Tree Analysing critically
Build your own decision
tree
Introducing the Puzzle
Box
introduces language specific to the topic of 2D and
3D shapes
formulates the language into a defined set of
attributes for each shape
ELA
ELA18 LA 12


by creating their own puzzle box to use throughout
the unit, students level of engagement is increased
highlights the importance of accuracy in
measurement
ELA17 LA 1
ELA17 LA 6
ELA17 LA 7
ELA18 LA 1
FROM LEARNING BY DESIGN FRAMEWORK,COMMON GROUND PUBLISHING, 2007, COMMON GROUND, ILLINOIS, VIEWED 14 NOVEMBER 2007, WWW.COMMONGROUNDGROUP.COM
5
Perimeter
Defining Perimeter
Estimation of different
perimeters
Defining Wire-Perimeter
Using a Placemat
Language focus
Conceptualising with theory
Estimation
Conceptualising by naming
Language
Finding the Perimeter of
Objects
Applying appropriately
Explicit teaching and Consolidation
Find the Perimeter of the
Puzzle Box Piece
Applying appropriately
Assessment
Post-it notes on the board
How many post it notes
on the board?
Defining Area and
surface area
Area
Conceptualising by naming
Experiencing the new
ELA17 LA 6
ELA17 LA 6
ELA17 LA 7
ELA18 LA 1
ELA18 LA 10
ELA17 LA 6
ELA17 LA 7
ELA18 LA 1
ELA18 LA 10
ELA18 LA 1
Concept of area
Conceptualising with theory
Conceptualising by naming
Language
Arrays
Experiencing the new


How do Arrays help us?
Analysing functionality
Links to application of area outside the classroom
Making Shapes with
Paper
Experiencing the new
How does the rectangle
relate?
Conceptualising with theory
Finding the Area of
Objects
Applying appropriately
Finding the area of the
puzzle box
Applying appropriately
estimation, mental computation and multiplication.
highlight strategies explicitly and provide
opportunity for consolidation
ELA17 LA 6
area as a measurement of space
area of shapes as linked to rectangles
ELA17 LA 6
students creating their own understanding, no formal
ELA17 LA 7
rules are give, students investigate and discover
them
ELA17 LA 6
Consolidation
ELA17 LA 7
ELA18 LA 10
ELA17 LA 6
Assessment
ELA17 LA 7
ELA18 LA 10



6
Volume
Stacking Furniture
Experiencing the new
Volume as a measurement of 3D space
Defining Volume by a
Gallery Walk
Conceptualising by naming
Language
Stacking the shape
Experiencing the new
Non – uniform volume
Conceptualising by naming
How would I find the
volume of these shapes?
Finding the Volume of
Objects
Puzzle Box Volume


volume as a measurement of 3D space
volume of prisms as a repeated layering of area
Conceptualising with theory
Consolidation
Applying appropriately
Applying appropriately
Assessment
ELA18 LA 1
ELA17 LA 6
ELA17 LA 7
ELA18 LA 1
ELA17 LA 6
ELA17 LA 7
ELA18 LA 1
ELA18 LA 10
ELA17 LA 6
ELA17 LA 7
ELA18 LA 1
ELA18 LA 10
7
Teaching and Learning Experiences / Activities
Exploring Shapes – Barrier Game
One student in a pair is given a basic shape, (square, triangle- different types of triangles can be
used for more advanced classes, rhombus, parallelogram etc.). This student describes the properties
associated with different shapes so that the other person can accurately replicate the shape
needed. Restrictions are imposed that the students are to imagine the ‘drawer’ has very limited
capacity for language. So…for example ‘draw a square’, Would be answered with ‘What does
that look like?’. ‘draw four lines’ etc The teacher pushes the students to be as specific possible.
Emphasising the unique elements of each shape.
Example
Student
Draw a square
It has four lines
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
Student 2
What’s a square?
Draws…
Make sure the lines all touch
All lines have to be the same length
8
Teaching and Learning Experiences / Activities
Essential Content
They need to make an enclosed shape….. so you
could colour in the middle say…
All the sides need to meet at 90degrees
What’s 90 degrees?
Right angle
What’s a right angle?
Like the top part of a door
Ok….
Continue with other basic shapes, rectangle, triangle, parallelogram, rhombus.
Defining properties of shapes using a Data Set
In pairs students define a list of properties that different shapes will have and build a data set to
represent.
Example
Shape
Has four sides
Equal Length
Square
*
*
Oblong
*
9
Teaching and Learning Experiences / Activities
Twenty Questions
Teacher explains the rules of ‘twenty questions’ and ‘celebrity heads’
The rules:
A student thinks of a celebrity or item and their partner has only twenty questions to identify
correctly the celebrity or item
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
Students play twenty questions as a class and keep track of the amount and type of questions
asked before they reach and answer.
Introduce and play the computer game of Twenty Questions. Can be accessed at 20Q the –net on
the internet, 2007, Radica Games, USA, viewed 14 November 2007 http://www.20q.net
How does Twenty Questions Work?
Think Pair Share, Square Pair Share
Students are asked the question how does Twenty questions work?
They think for 5 minutes in a quiet spot. Find a partner and discuss their answer. They then
construct their own answer and write in their book. Students then form into groups of four and
select the best answer to represent their groups to the class.
10
Teaching and Learning Experiences / Activities
Decision Trees
Teacher presents an assortment of decision trees to the class.
Essential Content
10. draw by hand representations of common
2D shapes and 3D objects (and their crosssections) with attention to their geometric
properties and scale
Where can decision trees be used?
Class Round Robin Tournament
In table groups students brainstorm as many ideas as to where decision trees could be used.
Students stand up and each group has to offer an answer if someone from the groups speaks out of
turn the group is out, if an answer has been repeated the group is out, if they have exhausted
answers they are out. When they are ‘out’ students sit down.
Selecting questions for a decision tree
Using their data set students write the questions needed for a decision based on identifying shapes
An effective decision Tree
Students select what questions are needed to construct the most effective decision tree
Build your own decision tree
Individually students make their own decision tree for identifying shapes.
(collect for work sample)
11
Teaching and Learning Experiences / Activities
Puzzle Box
Hand out the Puzzle Box Rich task, allow students to read and ask questions. Talk about what an
excellent job on the assessment task would look like. What is important? What is not so important?
Collect the responses from the students for construction of a rubric to be used by the students on
what great work looks like.
Design layout
Students need to design their own puzzle box according to the given criteria. Creativity is
encouraged. Point out though the difficulty of cutting out on the band saw pieces that are too
small.
Once a design is reached that meets the criteria, students draw their designs onto the template
provided on the rich task. (see attachment A)
Mark and Cut and Sand Puzzle box
Students having already experienced work in the woodwork rooms are
familiar with safety. But the safety precautions should be reminded.
Students mark out on the wood, cut using the band saw and sand their
puzzle boxes.
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
10. draw by hand representations of common
2D shapes and 3D objects (and their crosssections) with attention to their geometric
properties and scale
ELA 17
1. measurement error, including recording
measurements as values that lie within a given
interval of measurement error, judgments about
acceptable or reasonable error in a
measurement context, strategies to minimise
error and estimation of error rates to provide
confidence in measurement results, and risks of
compounding error by repetition and
calculation
6. choose and use instruments, technologies,
strategies and formulas to estimate, measure
and calculate measures of attributes, including
mass, duration, temperature, angle and simple
derived measures such as rates
7. work routinely with International System (SI)
and other units with respect to both everyday
and technical measurement contexts, including
derived measures, choose units appropriate to
the order of magnitude involved and estimate
values that lie between marked graduations
on scales of measuring instruments
12
Teaching and Learning Experiences / Activities
Perimeter
Defining Perimeter
Think Pair Share
Using prior knowledge students construct an individual definition then share with a partner and then
construct a group definition.
Estimation of different perimeters
Students complete estimation sheet and collate as a class
Thing
Measurement
Desk
White Board
Room
School
Suburb
ACT
Australian Coastline
(example only)
Discuss different units of measurement for different sizes
What is the attribute we are measuring? What is best to measure this in? How else could we
measure it? What else could we measure it in?
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
10. draw by hand representations of common
2D shapes and 3D objects (and their crosssections) with attention to their geometric
properties and scale
ELA 17
6. choose and use instruments, technologies,
strategies and formulas to estimate, measure
and calculate measures of attributes, including
mass, duration, temperature, angle and simple
derived measures such as rates
7. work routinely with International System (SI)
and other units with respect to both everyday
and technical measurement contexts, including
derived measures, choose units appropriate to
the order of magnitude involved and estimate
Defining Wire-Perimeter Using a Placemat
values that lie between marked graduations
Students come up with a definition for the perimeter of a 3-dimensional object. Each writing in their on scales of measuring instruments
own part of the placement, combining strategies and definitions to form a group solution.
Finding the Perimeter of Objects
Provide opportunity for consolidation of finding perimeters, both 2d, and 3d, estimating, exact.
Find the Perimeter of the Puzzle Box Piece
Students find the surface area of the appropriate puzzle box piece.
13
Teaching and Learning Experiences / Activities
Area
Post-it notes on the board
Cover the board with post-it notes of the same size.
How many post it notes on the board?
In pairs students devise a way of calculating the amount of post-notes needed to cover the board.
Pairs form Square and share strategies and vote on the best one.
Individually they reflect and write their answer in their book with a diagram to explain.
Picture
Words
Defining Area and surface area
Individually students write down their own definition for what Area and Surface Area is.
Individuals share their results with three others, combing to form a group of four and the small
group comes up with a suggested definition. These get written on A4 paper and stuck up on the
walls. A class definition is then decided upon, through discussion, amalgamation of student
responses.
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
10. draw by hand representations of common
2D shapes and 3D objects (and their crosssections) with attention to their geometric
properties and scale
ELA 17
6. choose and use instruments, technologies,
strategies and formulas to estimate, measure
and calculate measures of attributes, including
mass, duration, temperature, angle and simple
derived measures such as rates
7. work routinely with International System (SI)
and other units with respect to both everyday
and technical measurement contexts, including
derived measures, choose units appropriate to
the order of magnitude involved and estimate
values that lie between marked graduations
on scales of measuring instruments
14
Teaching and Learning Experiences / Activities
Arrays
What is an array structure? Students create their own 20 x 10 array using anything they like.
(for example, stickers, drawings, grid paper)
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
How do Arrays help us?
How does the concept of arrays help us in maths? In general? Do they have specific purpose? Link
to use in multiplication. (Follow with activities for mental computation strengthening and
multiplicative thinking)
Making Shapes with Paper
Start with a making a rectangle. Students measure and draw accurately 12, 12cm x 8cm
rectangles on coloured paper.
Then construct other shapes, sticking to dimensions close to these….
Square, 8x8
Right angle triangle, construct 2 or 3 triangles from the rectangle. Measure and cut out.
Other type of triangle, construct from the rectangle, measure and cut out.
Parallelogram, construct from the 12x8 rectangle
Rhombus, construct from the 12x8 rectangle
Using a whole rectangle as a guide, students show and display in their books how they constructed
the shapes from the rectangle.
15
Teaching and Learning Experiences / Activities
How does the rectangle relate?
Using a whole rectangle as a guide, students show and display in their books how they constructed
the shapes from the rectangle. Knowing that areas of rectangles relate to arrays, students explore
how areas of all these other shapes are constructed from knowledge of the area of a rectangle.
Finding the Area of Objects
Provide opportunity for consolidation of finding areas and surface areas, both 2d, and 3d,
estimating, exact.
Finding the area of the puzzle box
Students find the surface area of the appropriate puzzle box piece.
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
curved surfaces (e.g. cylinders, cones,
packages and containers), with respect to
properties involving line, length, angle and
surface
10. draw by hand representations of common
2D shapes and 3D objects (and their crosssections) with attention to their geometric
properties and scale
ELA 17
6. choose and use instruments, technologies,
strategies and formulas to estimate, measure
and calculate measures of attributes, including
mass, duration, temperature, angle and simple
derived measures such as rates
7. work routinely with International System (SI)
and other units with respect to both everyday
and technical measurement contexts, including
derived measures, choose units appropriate to
the order of magnitude involved and estimate
values that lie between marked graduations
on scales of measuring instruments
16
Teaching and Learning Experiences / Activities
Essential Content
ELA 18
1. a broad range of 2D shapes (eg.
Quadrilaterals, polygons, ellipses), composite
shapes and 3D objects, including those with
Stacking Furniture
curved surfaces (e.g. cylinders, cones,
Students are to fill the room with desks.
packages and containers), with respect to
properties involving line, length, angle and
surface
Defining Volume by a Gallery Walk
10. draw by hand representations of common
Students at their table groups come up with a definition for what volume is, writing it up on an A4
2D shapes and 3D objects (and their crosspiece of paper and leaving it on the table. The groups then do a gallery walk and look around at sections) with attention to their geometric
the other definitions, returning to their table and discuss how to refine, change or improve their own. properties and scale
Class definition to be reached.
Volume
Stacking the shape
Build a range of prisms using various items, units used should be consistent in size, eg blocks, milk
crates… Also introduce 3-dimensional drawings of prisms, how to create
Non – uniform volume
Units of measuring volume. Standard units, and non-standard units.
How would I find the volume of these shapes
By recalling their building with objects, students determine how volume of prisms are constructed by
repeated layers of the same sized area pieces.
ELA 17
6. choose and use instruments, technologies,
strategies and formulas to estimate, measure
and calculate measures of attributes, including
mass, duration, temperature, angle and simple
derived measures such as rates
7. work routinely with International System (SI)
and other units with respect to both everyday
and technical measurement contexts, including
derived measures, choose units appropriate to
the order of magnitude involved and estimate
values that lie between marked graduations
on scales of measuring instruments
Finding the Volume of Objects
Provide opportunity for consolidation of finding volumes 3d, estimating, exact.
Puzzle Box Volume
Students calculate the volume of their puzzle box piece.
17
Teacher Reflection
In this unit how has the teaching and learning demonstrated:
that every student can learn?
the maximising of student learning?
sustained opportunities for students to learn?
depth of student understanding and expertise?
equitable and inclusive opportunities for learning?
ethical practice?
content, assessment and pedagogy that is coherent and aligned?
a dynamic an responsive approach?
18
Student Reflection
Something I have really enjoyed in this unit is…
What I have learned from this unit is….
Something I was very good at was…
What would you like to change or do better at?
19
PUZZLE BOX
The task is in three parts.
PART ONE:
To create a puzzle box that meets certain criteria.
(Clarify, Choose, Use, Communicate)




The box needs to be 10 x 14cm
Use no more than 9 straight lines to break up the shape
At least one shape has more than 4 sides
You must have…. Rectangle, Square, Right Angle Triangle, One other type of triangle, Parallelogram or Rhombus.
One this page of your submission, draw a scaled drawing of the top view of the box.
20
PUZZLE BOX
PART TWO:
Use the following guidelines and submit the estimates for the Rectangular Prism, Square Prism, One of the
Triangular Pieces (make sure you state which one), the Parralelogram Prism or Rhomboid (whichever you used)
and your more than four sided shape.
(Choose and Communicate)
SHAPE NAME
(sketch of the solid here)
I estimate the perimeter of the shape to be ____________
(Explain how you estimated this)
I estimate the surface area of the shape to be ______________
(Explain how you estimated this)
I estimate the volume of the shape to be ______________
(Explain how you estimated this)
21
Submit this for marking, you will then get PART THREE.
22
PUZZLE BOX
PART THREE:
You need to complete the following four pages.
Make sure you explain your thinking in each step.
Consider the inquiry model to complete the sections in part three.
CLARIFY

Make clarifying statements. eg I have three shapes that are classified as rectangles, but one is a square.
CHOOSE

Choose a method to use to draw the figures or make your calculations.

Identify what tools and technologies can make this job easier.
USE
 Use the tools and methods you have identified.
 Draw the drawings and make the calculations.
 Be sure to communicate well and make clear all steps you have made in this process.
INTERPRET
 Look at your answers, are the results appropriate. Answer the questions.
 Would there be a more practical way of doing this?
 Give examples of where these skills could be used in everyday life.
COMMUNICATE
 Make and present your puzzle box.
23
PERIMETER OF THE Rectangular or Square Prism.
(pick one of these to make actual calculations on)
My estimate for the perimeter of the ______________________________
(insert name of shape that your doing here)
was ________________________.
Calculate the perimeter of the shape here. Make sure you explain what and how you have done it.
24
SURFACE AREA OF one of the Triangular Prisms.
(pick one of these to make actual calculations on)
My estimate for the surface area of the ____________________________
(insert name of shape that your doing here)
was ________________________.
Calculate the surface area of the shape here. Make sure you explain what and how you have done it.
25
VOLUME OF either the Parallelogram or Rhombus Prisms.
(pick one of these to make actual calculations on)
My estimate for the volume of the ______________________________
(insert name of shape that your doing here)
was ________________________.
Calculate the volume of the shape here. Make sure you explain what and how you have done it.
26