Download Measurement and Geometry – 2D 58G

Measurement and Geometry – 2D 58G
Identify diagonals on convex two-dimensional shapes, recognising the endpoints as the vertices of the shape
Draw all the diagonals of convex two-dimensional shapes, comparing the diagonals on different shapes
Identify which of the special quadrilaterals (parallelograms (squares, rectangles and kites), rhombuses and trapeziums)
have diagonals that are equal in length
Identify whether any of the diagonals are also lines (axes) of symmetry of the shape
Resources: protractors, rulers, pencil, paper
EXPLICIT LEARNING - TRIANGLES
Focuses
children’s
thoughts on the
concept,
exposing current
understanding
and any
misconceptions
Reviews
constructing
regular
quadrilaterals –
squares - with a
protractor and
ruler
What could we do?
What language could we use to explain and ask questions?
Children think about, talk and listen to a friend about, then have the
opportunity to share what they already know.
Children construct a regular
quadrilateral – a square (See
Measurement and
Geometry, 2D - Level 49G),
for example,
4 cm
4 cm
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►
Today brings an investigation about diagonals in twodimensional shapes.
►
What do you know about diagonals in two-dimensional
shapes?
►
Talk about diagonals in two-dimensional shapes with a
friend.
►
Is anyone ready to share what they are thinking about
diagonals in two-dimensional shapes?
►
Let’s construct a regular quadrilateral, a square with a
ruler and a protractor.
4 cm
4 cm
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Display the square without the angles and side
measurement labelled, for example,
Introduces
diagonals
Record, for example, diagonal
Record the diagonals on the square, for example,
Record, for example, diagonal = straight line
Record, for example, diagonal = straight line that joins 2 vertices.
Record ‘opposite’ between the words ‘2 and vertices’, for example,
diagonal = straight line that joins 2 opposite vertices.
Record, for example, the end point of diagonals are the vertices of
the shape.
Introduces the
number of
diagonals on a
square
Introduces
length of the
diagonals
Display the square with the diagonals recorded,
for example,
Measure the length of the diagonals, then record, for example, equal
length.
Introduces
Fold the square on the diagonals then record for example, axes of
diagonals as lines symmetry.
of symmetry
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►
So we’ve constructed a regular quadrilateral, a square.
►
And we know that this quadrilateral has sides and
vertices.
►
Does it also have diagonals?
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I’ve recorded diagonals on this square.
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How would you describe a diagonal?
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Is a diagonal a straight line?
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Does a diagonal join 2 vertices?
►
Are the vertices adjacent or opposite one another?
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Are the vertices opposite?
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Where are the end points of the diagonals?
►
Are the end points the vertices of the shape?
►
Let’s investigate the diagonals on this square.
►
How many diagonals on this square?
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Are there 2 diagonals?
►
How could we describe the diagonals?
►
Are the diagonals equal length? Let’s measure them.
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Yes, the diagonals are equal in length!
►
Are the diagonals, lines of symmetry? Let’s fold the
square on the diagonals to check.
►
We’ve been calling these ‘lines’ of symmetry.
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Introduces axes
of symmetry
Record for example, axis of symmetry.
Record, for example, axis – singular
►
The mathematical term is ‘axis’ of symmetry.
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An axis is a central point, so an axis of symmetry is a
central line.
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Just like vertex is singular and vertices are plural, axis is
singular, but the plural is not axises! The plural is axes.
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Are the diagonal axes of symmetry?
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The diagonals are axes of symmetry!
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At what angle do the diagonals cross?
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Could we measure the angles where the diagonals cross?
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At what angle do the diagonals cross?
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Do the diagonals cross at right angles?
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Yes the diagonals do cross at right angles!
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So a square has 4 equal 90 degree vertices and 4 equal
sides, 2 equal length diagonals that are axes of symmetry
that cross at right angles.
Record, for example, axes - plural
Introduces
measuring the
angle at which
the diagonals
cross
Use a protractor to measure the angles where the diagonals cross, for
example,
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Introduces
recording
diagonals of a
square in a table
Record, for example,
Shape
Square
Vertices
4 equal
►
Sides
4 equal
Diagonals
length
2 equal
Diagonals
axes of
symmetry?
Diagonals
cross at
right
angles?
yes
yes
Reviews
constructing
irregular
Children construct an irregular quadrilateral – a rectangle that is not a
quadrilaterals –
square, (See Measurement and Geometry, 2D - Level 49G) for
rectangles that
example,
are not squares 4 cm
with a protractor
6 cm
and ruler
4 cm
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6 cm
►
Could we record this in a table?
Let’s construct an irregular quadrilateral – a rectangle that
is not a square - using a protractor and a ruler!
►
Is this a regular quadrilateral?
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How could we check?
►
Could we measure the length of the sides?
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Are the sides equal or unequal?
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Are opposite sides equal?
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Could we measure the size of the angles?
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Are the angles equal or unequal?
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Because the sides are unequal while the angles are
equal, is this an irregular quadrilateral?
►
Because opposite sides are equal, and all angles are right
angles, is this irregular quadrilateral, a rectangle?
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Reviews
diagonals
Display the irregular quadrilateral, a rectangle,
without the angle sizes and side lengths labelled, for
example,
►
So we know that this quadrilateral has both sides and
vertices.
►
Does it also have diagonals?
►
How could we draw diagonals on this rectangle?
Record, for example, diagonal
►
How would you describe a diagonal?
Record the diagonals on the rectangle, for example,
►
Is a diagonal a straight line?
►
Does a diagonal join 2 vertices?
Record, for example, diagonal = straight line
►
Are the 2 vertices adjacent or opposite one another?
Record, for example, diagonal = straight line that joins 2 vertices.
►
Are the vertices opposite?
Record ‘opposite’ between the words ‘2 and vertices’, for example,
diagonal = straight line that joins 2 opposite vertices.
►
Where are the end points of the diagonals?
►
Are the end points the vertices of the shape?
Reviews the
number of
diagonals on a
rectangle that is
not square
Record, for example, the end point of diagonals are the vertices of
the shape.
►
How many diagonals on this rectangle?
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Are there 2 diagonals?
Reviews length
of the diagonals
Measure the length of the diagonals, then record, for example, equal
length.
►
How could we describe the diagonals?
►
Are the diagonals equal length? Let’s measure them.
►
Yes, the diagonals are equal in length!
►
Are the diagonals axes of symmetry? Let’s fold the
square on the diagonals to check.
►
No, the diagonals are not axes of symmetry!
Fold the rectangle on the diagonals then record for example, not axes
Reviews
diagonals axes of of symmetry.
symmetry
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Introduces
measuring the
angle at which
the diagonals
cross
Use a protractor to measure the angles where the diagonals cross, for
example,
Record, for example,
Introduces
recording
Shape
Vertices
Sides
diagonals of a
rectangle that is
not a square, in a
Square
4 equal
4 equal
table
Rectangle
Reviews
differentiating
the investigation
for children as
they
demonstrate
understanding
4 equal
4, 2
opposite
pairs equal
Diagonals
length
Diagonals
axes of
symmetry?
Diagonals
cross at
right
angles?
2 equal
yes
yes
2 equal
no
no
►
Could we measure the angles where the diagonals cross?
►
At what angle do the diagonals cross?
►
Do the diagonals cross at right angles?
►
No, the diagonals do not cross at right angles.
►
So a rectangle that is not a square, has 4 sides with
opposite sides equal, 4 equal angles, 2 equal length
diagonals that are axes of symmetry that do not cross at
right angles.
►
Could we record this in a table?
Allow children time now to engage in guided and independent
investigation (at the end of this teaching plan) of describing
diagonals in quadrilaterals.
A child who has not demonstrated understanding of measuring and
constructing angles will continue to investigate this before
progressing to describing diagonals in quadrilaterals.
A child could be sitting next to a child who is investigating at a
different level. They will explain their current levels of understanding
to one another as they investigate. This is a research-based way to
accelerate learning for children at all levels.
Children move to Guided and Independent Investigation now to investigate the concept at increasing levels of understanding over many learning sessions
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GUIDED INVESTIGATION
INDEPENDENT INVESTIGATION
REFLECTION
Resources: protractors, rulers, pencil, paper
What could we do?
Children:
1. construct
quadrilaterals using
a protractor and
ruler, identifying
the properties of
the shape as guided
by the teacher
2. identify and
describe the
diagonals
What language could we use to ask
questions and explain?
What could we do?
 How could we use the protractor
and ruler to construct a
quadrilateral?
Children:
1. sit in pairs
2. construct triangles and
 What are the properties of the
quadrilaterals using a
quadrilateral?
protractor and ruler,
identifying the
 How many diagonals?
properties of the shape,
 Are the diagonals equal in length?
that are neither too
 Do the diagonals cross at right
easy nor too
angles?
challenging
 Are the diagonals axes of
3. identify and describe
symmetry?
the diagonals
What language could we use to explain?
 I constructed a quadrilateral using a
protractor and ruler, by making the
sum of the angles 360 degrees.
 The properties of my quadrilateral are
4 sides and 4 angles, … equal sides, …
equal angles, … pairs of parallel
sides, … axes of symmetry.
 I drew the diagonals.
 There are … diagonals.
 The diagonals are / are not equal
length.
 The diagonals do / do not cross at
right angles.
 The diagonals are / are not axes of
symmetry.
What questions could
children discuss and record a
response to?
How can we construct a
quadrilateral with a
protractor and ruler?
What is a diagonal?
How can we draw diagonals
on quadrilaterals?
Which quadrilaterals have
diagonals that are equal in
length?
Which quadrilaterals have
diagonals that are axes of
symmetry?
Which quadrilaterals have
diagonals that cross at right
angles?
Children may be investigating concepts at a level that varies from other children. In one class, there may be children investigating the concept at Level 1 while
another child is investigating the concept at Level 4, Level 12 or even higher.
Regardless of the child's current grade, children need to investigate concepts at the level of their current understanding. This means that a child in a given
grade, who has current understanding at Level 5, will investigate at Level 6, then Level 7 etc.
If this makes you worried that they are investigating at a level much lower than their grade level, consider this: If the child is made to try to investigate at a
higher level than their current level of understanding, they will be building on an unstable knowledge base with gaps, and will continue to use inefficient strategies
often based on misconceptions, guaranteeing that their level of understanding will be the same at the end of the year as it was at the beginning of the year. If the
child is allowed to investigate the concept at their current level of understanding, they will correct misconceptions, fill gaps in their understanding and build a firm
knowledge base, as they move through the levels, investigating at a higher level by the end of the year.
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OTHER LEARNING ACTIVITIES
MEASUREMENT AND GEOMETRY – 2D 58G IDENTIFY DIAGONALS ON CONVEX TWO-DIMENSIONAL SHAPES, RECOGNISING THE
ENDPOINTS AS THE VERTICES OF THE SHAPE, DRAW ALL THE DIAGONALS OF CONVEX TWO-DIMENSIONAL SHAPES, COMPARING
THE DIAGONALS ON DIFFERENT SHAPES, IDENTIFY WHICH OF THE SPECIAL QUADRILATERALS (PARALLELOGRAMS (SQUARES,
RECTANGLES AND KITES), RHOMBUSES AND TRAPEZIUMS) HAVE DIAGONALS THAT ARE EQUAL IN LENGTH, IDENTIFY WHETHER
ANY OF THE DIAGONALS ARE ALSO LINES (AXES) OF SYMMETRY OF THE SHAPE: OTHER LEARNING ACTIVITIES
These learning activities allow children to investigate and explain the concept in various situations. ‘Doing’ mathematics is not
enough and is not a good indicator of understanding. As Einstein said, ‘If you can’t explain it simply, you don’t understand it’!
Investigation takes time as children develop both the capacity and meta-language to explain mathematical concepts at their current
level of understanding. Differentiate learning for children working at all levels of the concept, including those requiring extension,
and allow children to differentiate their own learning, by varying the range and size of numbers investigated.

In pairs, children construct special quadrilaterals, including regular quadrilaterals – squares, irregular quadrilaterals –
rectangles that are not squares, rhombuses that are not squares, kites that are not rhombuses or squares, trapeziums (See
Measurement and Geometry, 2D - Level 49G). They describe the side and angle properties and draw in diagonals, recording
their data in a table, for example,
Shape
Vertices
Sides
Diagonals length
Diagonals axes
of symmetry?
Diagonals cross
at right angles?
Square
4 equal
4 equal
2 equal
yes
yes
Irregular Rectangle
4 equal
4, 2 opposite pairs
equal
2 equal
no
no
Irregular Rhombus
4, 2 opposite pairs
equal
4,
1 opposite pair equal, 1
opposite pair unequal
4 equal
2 unequal
yes
yes
4,
1 adjacent pair equal, 1
opposite pair unequal
2, unequal
yes
yes
4, 2 adjacent pairs
equal
4, 3 adjacent equal, 1
twice as long
2, equal
no
no
4, zero equal
4, zero equal
2, unequal
no
no
Irregular Kite that is not a
rhombus
Isosceles Trapezium
Non-isosceles Trapezium

Children review the data in the table, identifying that quadrilaterals with
►
2 equal length diagonals also have 3 or 4 sides and angles equal
►
diagonals that are axes of symmetry also have 2 or 4 adjacent sides equal
►
diagonals that cross at right angles also have 2 or 4 adjacent sides equal
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
Children construct different quadrilaterals using a protractor and a
ruler. They cut the quadrilateral into 4 parts with an angle is each
part. They rotate the parts to place the angles together, identifying
that the 4 angles create a rotation, which is 360 degrees, for
example,

In pairs, children each construct a quadrilateral from a description of its angles, for example 4 equal vertices, 4 equal sides, 2
equal diagonals, diagonals cross at right angles and diagonals are equal length

They share and classify their quadrilateral. Children could also investigate concave quadrilaterals, for example,
Children draw the diagonals, identifying that in a concave quadrilateral, a diagonal will lie outside the
shape, for example,

In small groups, children have 8 metres of string tied into a loop, and an extra length of string with which to construct and
compare diagonals. Children determine a strategy to construct specified quadrilaterals using only the loop of string. They
justify their shape by describing its side and angle properties. Children use the extra string to construct and compare
diagonals of their quadrilateral.

In pairs children have 2 craft sticks of the same length. They cross sticks at right angles (perpendicular) to create diagonals.
They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based on the
diagonals.

In pairs children have 2 craft sticks of different lengths. They cross sticks at right angles (perpendicular) to create diagonals.
They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based on the
diagonals.

In pairs children have 2 craft sticks of the same length. They cross sticks at non-right angles (non-perpendicular) to create
diagonals. They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based
on the diagonals.

In pairs children have 2 craft sticks of different lengths. They cross sticks at non-right angles (non-perpendicular) to create
diagonals. They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based
on the diagonals.
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
In pairs, children make quadrilaterals with and without diagonals using chenille sticks in straws. They test the rigidity of
quadrilaterals without diagonals and with diagonals by pushing with equal force on all 4 vertices at the same time.

EXTENSION: In pairs children could investigate the relationship between
the number of vertices and the number of diagonals. For example, a
quadrilateral has 2 diagonals and 4 vertices.
Each vertex is the meeting point of 1 diagonal and 2 sides, for example,
So each vertex has 1 diagonal, but that would mean counting each diagonal twice – once at each end. In a quadrilateral, every
vertex (4) has a diagonal to every vertex except itself (1) and the 2 adjacent vertices (2). So in a quadrilateral, 4 vertices have
diagonals to 4 minus 1 minus 2 vertices. 4 minus 1 minus 2 = 1. So in a quadrilateral each vertex has 1 diagonal. Because each
end of a diagonal ends in a vertex, that means we have counted every diagonal twice – once at each end. So we need to halve
the number of diagonals = 2. (The formula in Year 7 is Number of vertices – 3 divided by 2, which means the number of
vertices, minus the 3 vertices that each vertex does not have diagonal to - itself and the 2 adjacent vertices -, divided by 2
because we counted each diagonal at both ends - in both vertices)

In pairs children investigate whether triangles have diagonals.

EXTENSION: In pairs children could investigate diagonals in other two-dimensional shapes, pentagons, hexagons, octagons.
They could investigate the relationship that they found between the number of vertices and the number of diagonals in
quadrilaterals, in pentagons, hexagons, octagons.
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PROBLEM SOLVING
MEASUREMENT AND GEOMETRY – 2D 49G MEASURE LENGTHS OF SIDES AND SIZES OF ANGLES TO IDENTIFY THE SIDE AND ANGLE
PROPERTIES OF TRIANGLES AND QUADRILATERALS, CONSTRUCT AND CLASSIFY TWO-DIMENSIONAL SHAPES, INCLUDING
TRIANGLES AND QUADRILATERALS, FROM A DESCRIPTION OF THEIR SIDE AND ANGLE PROPERTIES, ENLARGE TWO-DIMENSIONAL
SHAPES, COMPARING SIDE PROPORTIONS AND ANGLES AND IDENTIFYING ONLY THE AREA HAS CHANGED: PROBLEM SOLVING
Problems allow children to investigate concepts in various situations. Any problem worth solving takes time and effort – that’s why
they’re called problems! Problems are designed to develop and use higher order thinking. Allowing children to grapple with
problems, providing minimal support by asking strategic questions, is key. Differentiating problems allows children to solve simpler
problems, before solving more complex problems on a concept. Problems may not always be solved the first time they are
presented. Returning to a problem after further learning, develops both resilience and increased confidence as children take the
necessary time and input the necessary effort. As Einstein said, ‘It’s not that I’m so smart – I just stay with problems longer’. The
problem solving steps may be followed to solve problems.

Mark constructed a shape that had 4 equal vertices, 4 equal sides, 2 equal diagonals, diagonals cross at right angles and
diagonals are equal length. What could the shape have looked like?

Which one of these shapes has diagonals that cross at right-angles?

Which one of these shapes has diagonals that are axes of symmetry?

Which 2 of these shapes has diagonals that are unequal lengths?
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Problem Solving Steps (back to Problems)
1. Read
2. Understand
3. Choose a strategy
Read the part that is
asking you to find out.
Read the information you
need to find it out.
Think about what you
could do to work it out.
4. Work it out
5. Check
6. Share
Use your strategy to work
it out.
Read the part that asked
you to find out.
Share and compare your
strategy and answer with a
friend’s strategy and
answer.
Did you find it out?
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