Download KS3 Shape 5 Constructions and loci 53.77KB



3D Shapes

Names of 3D solids
Prisms
Volume
Surface Area
Plans, elevations and nets
Isometric Drawing
Units of measure
Length, area and volume
Capacity / Mass
Metric and imperial measure
Conversion / conversion graphs
Real-life graphs
Speed, distance time

Parts of
Circumference
Area
Circle geometry
Equation of a circle


Right-angled triangles
Finding missing side lengths
Angles and sine, cosine and
tangent ratios
KS3 Shape, Space and
Measure
Constructions
Triangles
Similar & Congruent Shapes
Bisectors
Loci
Angles

Angles on a straight line
Angles at a point
Parallel lines and transversals alternating / corresponding /opposite angles
Supplementary angles
Polygons: interior & exterior angles
Extension work: Bearings
Lines & Angles
Line Segments
Vertical / Horizontal
Perpendicular / Parallel
Types of angle
Estimating measuring and drawing
Direction of turn
Compass directions
Describing angles (90o= ¼ of turn)
Pythagoras and
Trigonometry
Perimeter and Area
Dimensions (including volume)
Units of measure (including volume)
Counting squares
Intrinsic and Extrinsic information
Rectangles, triangles and compound shapes
Using Formulae (Rectangle, Triangle,
Trapezium, Parallelogram)

Circles

Properties of 2D Shapes
Names of polygons up to 10 sides
Special Quadrilaterals and Triangles
Geometric Properties
Tessellation
Transformations

Basic congruent & similar shapes
Coordinate geometry
Reflection (include lines of symmetry)
Rotation (include rotational symmetry)
Translation (including vector notation)
Enlargement
All topics can be covered by the end
of year 8.
Shape 5: Constructions and Loci

Constructions and Loci
Must
Should
Understand that a construction
requires the use of correct geometrical
equipment including a ruler, pair of
compasses and angle measurer or
protractor
Construct a triangle accurately given; Construct a triangle accurately
The length of two sides and the angle given the lengths of all three sides
formed at the intersection of those two using a pair of compasses and ruler
sides or;
The length of one side and the two
angles at the end of that side
Understand the term, ‘congruent’ and Know that congruent shapes are
identify congruent shapes
formed after transformations
involving rotations, reflections and
enlargements
Understand the term, ‘similar’ and
Know that ‘similar’ shapes have
identify similar shapes
corresponding angles of the same
size and lengths in the same ratio
Know the terms, ‘perpendicular’ and
‘bisect’
Construct the perpendicular bisector
of a line segment
Construct the bisector of an angle
Know that a rhombus can be
constructed by constructing the
perpendicular of a line segment
Understand the terms, ‘locus’ and
‘loci’
Construct the locus of all points
equidistant from a point or a line
segment
Construct an angle of 60o
Could
Construct tessellations using
appropriate congruent shapes
(such as regular hexagons)
Construct scale factor
enlargement of given shapes (such
as triangles)
Understand that the area of a
similar shape will be equivalent to
n2 × the area of the original shape,
where the original shape has been
enlarged by scale factor, n
Construct the perpendicular to a
line which intersects the line
segment at a given point
Know that an equilateral triangle
can be constructed by constructing
an angle of 60o
Construct the locus of all points
equidistant from two given points
Key Words: construct, sketch, ruler, protractor, angle measurer, pair of compasses, vertex, vertices, side, line
segment, congruent, similar, enlarge, scale factor, perpendicular, bisect / bisector, locus, loci, equidistant
Starters:
Sorting activity with shapes that have been constructed and shapes that have been sketched or drawn
Identify congruent shapes
Identify similar shapes
Practise measuring and drawing line segments and angles
What shape is this? How do we know? eg recapping properties of shapes (such as rhombus or kite)
Activities:
Construct congruent / similar shapes
Loci work: eg goat problems as in 10 ticks worksheets
Construct a diagram involving bearings / scale drawings
Shape 5: Constructions and Loci
Plenaries:
Learning framework questions:
- What is the difference between a construction and a sketch or drawing?
- Following an enlargement, how many times does the original shape fit inside the enlargement? Is this
what you expected?
- What does ‘bisect’ actually mean?
- What equipment would you expect to use for a construction?
Resources:
Pair of compasses / angle measurer / protractor / ruler
10 ticks worksheets
Possible Homeworks:
Investigate the golden ratio and shapes / patterns which follow the ratio
By re-examining the properties of shapes, construct common triangles, quadrilaterals and other polygons
Try to construct a 45 degree / 30 degree / 15 degree angle
Investigate the pattern for the number of parts that are formed by bisecting a line segment, then bisecting each
bisected part and so on.
Describe the locus for an everyday moving object. Does it follow a predictable pattern or is it random /
chaotic?
Teaching Methods/Points:
CONSTRUCTION
Students should know that most constructions require the use of a pencil, a ruler, a pair of compasses and an
angle measurer or protractor. A sketch or drawing will still require the use of a pencil and ruler in Maths, but
the important difference between a sketch and a construction is the accuracy of the lengths and angles in the
diagram. Construction lines should remain clear, the geometric equivalent of ‘showing your workings /
process’.
The topic of construction, in particular, provides an opportunity to remind students of some important 2D
shapes and of their important properties. For example;
A rhombus has diagonals which perpendicularly bisect each other.
A kite has just one diagonal which perpendicularly bisects the other.
An equilateral triangle has three angles of 60o.
It is also important to ensure students are familiar and conversant with the following terms;
Perpendicular lines are two lines that meet each other at right angles (90 degrees)
Bisect means to divide or cut something into 2 equal parts, so a bisector is a line which cuts another line
into 2 equal parts.
Construct means use appropriate geometric equipment to make an accurate drawing.
Locus (plural loci) is the pathway of a moving object
CONSTRUCTING A TRIANGLE
The simplest and most familiar task in terms of using prior skills is that of constructing a triangle requiring
precise measurement of length and measurement of angle. This is covered as part of ‘Shape 1’ as a means to
consolidate skills drawing line segments and angles. In order to develop construction skills, tasks can be
structured as follows;
Shape 5: Constructions and Loci
1) To construct a triangle given the length of line segment, AB, the size of angle BAC and the length of AC;
Required: Ruler, angle measurer or protractor
Ensure students are required to measure and draw line segments in both centimetres (eg 3.4 centimetres) and
millimetres (eg 34 millimetres) and stretch them to make judgements on even more precise measurements
such as (34.5 millimetres). The accuracy of the angle is important and this provides an opportunity to
practise using angle measurers and protractors.
2) To construct a triangle given the length of line segment, AB and the size of angles BAC and ABC;
Required: Ruler, angle measurer or protractor
Students should be encouraged to draw line segments AC and BC faintly at first (and beyond the length that
they are likely to need in order to ensure the line segments intersect) once line segment AB has been
accurately drawn and the angles have been measured.
3) To construct a triangle given the length of three line segments, AB, BC, and AC.
Required: Ruler, pair of compasses
This construction relies on the understanding that the intersection of the arcs drawn from centre points A and
B respectively (once line segment AB has been drawn accurately), with radii equal to the lengths BC and AC
respectively, provides the location of point C. They must avoid the temptation to simply estimate the
location of point C by using a ruler and a trial and error approach!
Construction of triangle ABC such that AB = 4.3 cm,
AC = 3.9cm and BC = 2.3 cm
C
3.9 cm
2.3 cm
A
1. Draw line segment AB, length 4.3 cm
2. Draw arc with centre A, radius 3.9 cm
3. Draw arc with centre B, radius 2.3 cm
B
4.3 cm
OTHER CONSTRUCTIONS
Constructing a perpendicular bisector of a line
Draw line segment AB.
Set the radius of the pair of compasses to be more than half of length AB.
Put the point on A and draw an arc above and below the line.
Put the point on B and draw an arc above and below the line.
Make sure the two arcs cross each other.
Label the points where the two arcs cross, C and D then draw line segment CD.
CD is the perpendicular bisector of AB.
D
A
B
C
Shape 5: Constructions and Loci
Note: Shape ADBC is a
rhombus.
Constructing a perpendicular line, MN, which meets line segment, AB at a specific point, P
Draw line segment AB.
Identify point P which is along the length of line segment AB.
Measure and mark an equal distance away from point P in either direction along line segment AB.
Ensure the radius of the pair of compasses is sufficient to allow for the arcs to intersect.
Put the point on the marks (placed an equal distance away from point P) and draw arcs above and below the
line.
Make sure the two arcs cross each other.
Label the points where the two arcs cross, M and N then draw line segment MN.
MN is a line perpendicular to AB which meets AB at a specific point, P.
M
A
P
B
Note: Shape AMBN is a
kite.
N
Constructing a bisector of an angle
Draw angle ABC.
Measure and mark an equal distance along line segments BA and BC from corner point B.
Set the radius of the pair of compasses to ensure that the arcs will intersect.
Draw arcs from the marked points (an equal distance away from point B) so that the arcs intersect.
Draw a line which passes through B and D, the point of intersection of the two arcs.
BD is the bisector of angle ABC.
A
D
B
C
Constructing a 60 degree angle
Draw line segment AB.
Set the radius of the pair of compasses to exactly the same length as the length of line segment AB.
Draw an arc from point A.
Draw an arc from point B, such that it intersects the first arc at point C
C
Draw line segment AB.
Note: Shape ABC is an
equilateral triangle
A
Shape 5: Constructions and Loci
60o
B
SIMILARITY AND CONGRUENCE
Congruent shapes have exactly the same size and shape (as explained in Shape 3: Transformations).
Students must understand that shapes that have been translated, reflected and rotated are always congruent.
Similar shapes have corresponding angles of the same size, and corresponding lengths in the same ratio.
Any scale factor enlargement of a shape creates a similar shape.
For example;
scale factor 2 enlargement
Here are examples of other shapes that are similar (with corresponding angles of the same size and
corresponding sides in the same ratio.)
These examples are also given in the notes for enlargement (Shape 4: Transformations)
eg
A
E
D
B
C
F
Shapes A and B are similar because the corresponding angles are the same size and all of the lengths of
Shape B are exactly twice the size of the lengths of Shape A. The ratio of the lengths is 1 : 2
Shapes C and D are similar because the corresponding angles are the same size and all of the lengths of
Shape D are three times the size of the lengths of Shape C. The ratio of the lengths is 1 : 3
Shapes E and F are similar because the corresponding angles are the same size and all of the lengths of
Shape F are 2.5 times the size of the lengths of Shape E. The ratio of the lengths is 1 : 2.5
[this can be
written as 2 : 5]
Every length is twice as long. It is useful to investigate the resulting increase in area, leading to the
conclusion that for a scale factor enlargement of n, the area will be n2 times as big. Similarly, applying the
same investigative approach to volume will lead to the conclusions that a scale factor enlargement of n for a
solid will lead to a volume which is n3 times as big.
Incorporate activities which require students to identify similar shapes (by considering the ratio of their
lengths.
While work on constructions is not directly linked to ‘congruent’ and ‘similar’ shapes (which are actually
dealt with as part of the topic of ‘Transformations’), this is an opportunity to consolidate student
understanding and extend the nature of the tasks. For example, given a triangle ABC, such that AB = 3cm,
BC = 4 cm and angle ABC is 60 degrees, construct the enlargement of triangle ABC by scale factor 2.
Shape 5: Constructions and Loci
LOCI
A locus (or loci in the plural) is defined as the path of a moving point. Students must understand the term;
equidistant meaning “equal distance away”
You have to be able to draw a locus given a description of the rule that the locus follows, and you have to be
able to describe the rule that a locus follows.
Locus of points a fixed distance from point A
A
Locus of points a fixed distance from line AB
A
B
B
Construction for locus of points equidistant
from points A and B
A
B
Construction of locus of points equidistant
from AB and AC
A
C
The topic of loci provides an opportunity to apply construction skills to a range of real-life problems. While
the above examples demonstrate specific skills that students should learn when drawing a locus, and
illustrate the type of descriptions that students need to give in words given a diagram of a locus, the topic can
be used to deal with other real-life problems typified by a goat in a field. These problems generally require
students to use their geometric equipment to construct specific scenarios, such as a goat tethered at a
particular point with a rope of a certain length. These problems are important in developing geometric
reasoning skills!
Shape 5: Constructions and Loci
Constructions: Help Sheet
Construct means use appropriate geometric equipment to make an accurate drawing.
Perpendicular lines are two lines that meet each other at right angles (90 degrees)
Bisect means to divide or cut something into 2 equal parts
Constructing a triangle given the length of three line segments, AB, BC, and AC.
Construction of triangle ABC such that AB = 4.3 cm,
AC = 3.9cm and BC = 2.3 cm
C
3.9 cm
4. Draw line segment AB, length 4.3 cm
5. Draw arc with centre A, radius 3.9 cm
6. Draw arc with centre B, radius 2.3 cm
2.3 cm
A
B
4.3 cm
Draw line segment AB.
Set the radius of the pair of compasses to be more than half
of length AB.
Put the point on A and draw an arc above and below the line,
then do the same at B so that the arcs intersect each other
Label the points where the two arcs cross, C and D then draw
line segment CD.
CD is the perpendicular bisector of AB.
Constructing a perpendicular bisector of a line
D
A
Note: Shape
ADBC is a
rhombus.
B
C
Constructing a perpendicular line, MN, which meets line segment, AB at a specific point, P
M
A
P
Note: Shape
AMBN is a kite.
B
N
Draw line segment AB and locate point P on the line segment.
Measure and mark an equal distance away from point P in either
direction along line segment AB.
Put the point on the marks (placed an equal distance away from
point P) and draw arcs above and below the line.
Make sure the two arcs cross each other.
Label the points where the two arcs cross, M and N.
MN is a line perpendicular to AB which meets AB at point, P.
Constructing a bisector of an angle
A
Draw angle ABC.
Measure and mark an equal distance along line segments BA
and BC from corner point B.
Draw arcs from the marked points (an equal distance away
from point B) so that the arcs intersect. Label the point of
intersection, D.
BD is the bisector of angle ABC.
D
B
C
Constructing a 60 degree angle
C
Draw line segment AB.
Make sure the radius of the pair of compasses is equal to the
length of line segment AB.
Draw arcs using both points A and B as centres so that the
arcs intersect. Label the point of intersection, C.
Angle CAB is 60o.
Note: Shape ABC
is an equilateral
triangle
A
Shape 5: Constructions and Loci
60o
B
Loci: Help Sheet
A locus (or loci in the plural) is the pathway of a moving point.
Equidistant meaning “equal distance away”
Examples of common loci:
Locus of points a fixed distance from point A
A
Locus of points a fixed distance from line AB
A
B
B
Construction for locus of points equidistant
from points A and B
A
B
Construction of locus of points equidistant
from AB and AC
A
C
LOCI AND GRAPHS (equation of a circle)
10
Using a coordinate grid, you may be asked to
construct the locus of a circle given an ‘equation’,
or describe the locus of a circle by writing the
equation.
The equation of a circle is given by the equation
x2 + y2 = r2
So, for example; x2 + y2 = 25 is a circle with centre
on the origin, r2 = 25, so radius is 5. The locus
is shown here
Shape 5: Constructions and Loci
5
y
5
x
-10
-5
5
-5
5
-5
-5
-10
10