Download Ideas beyond Number SO SOLID Activity worksheets

Ideas beyond Number
Activity worksheets
Activity sheet 1
Regular polygons and tesselation
Which regular polygons tessellate? Square tiling is all around us, but are there any others?
Questions
1. What is it about the angles and side lengths of these square tiles that guarantees the bottom left tile will
fit exactly into the space?
2. What is it about regular octagons that prevents them tiling on their own, but does allow them to
tessellate with another kind of regular polygon? (Which kind?)
3. Make sketches (or accurate drawings, if you prefer) to show how particular combinations of regular
polygons tessellate.
4. Can you find a tessellation that involves regular decagons, either on their own (No! Why not?) or in
combination with something else?
5. Describe each of your tilings using this system.
SO SOLID Activity worksheets
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Activity sheet 2
Constructing polygons
This part of the unit is concerned with regular polygons. In particular, we will be looking at how we can use our
knowledge of the symmetrical properties of these polygons to construct them, using the traditional instruments
of Euclidean geometry: a pair of compasses and a straight edge only – NO PROTRACTORS! In order to tackle
most of these, we will need to be able to easily and accurately find the mid-point of a given line segment;
construct a perpendicular to a given line at a given point, and to bisect an angle.
Discuss, experiment, investigate:
♦
What property of a rhombus (or a kite) can we use to locate the midpoint of a given line segment?
♦
What property of a rhombus (or a kite) can we use to bisect an angle?
♦
What angles can we construct?
♦
Can we use angle bisection to find any other useful angles?
♦
What regular polygons can be constructed, then, using just these construction techniques?
♦
Is it possible to construct a regular pentagon?
♦
Are there any polygons that can’t be constructed, do you think?
♦
What are the angle properties of the polygons we can construct?
♦
What are the areas of the polygons we can construct, if the side length is one unit? Can we use a
spreadsheet to generate these calculations efficiently? What is the radius of the circumcircle? The
inscribed circle (this has every side of the polygon as a tangent to the circle)?
SO SOLID Activity worksheets
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Activity sheet 3
Pentagons
° at the centre of its circumscribing circle. We cannot
A regular pentagon has sides which subtend an angle of
use multiples of
° and angle bisection alone to create this angle.
Tasks
Consider this sequence of bisections:
A. Construct
° and
°. Bisect the angle between them to find
B. Bisect the angle between
° and
° to find
C. Bisect the angle between
° and
° to find
D. Bisect the angle between
° and
° to find
°.
°. Close, but no cigar.
°.
°. Agonising, isn’t it? Only
of a degree out and, to
all intents and purposes this might do if we were just looking for something which looks okay but, hey,
we’re mathematicians!
E. What would you do now? How close can you get? Use a spreadsheet or a calculator, if that would help.
How many operations does it take before you think you are close enough?
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Now consider the rather beautiful triangle below:
°
°
Question
1. What type of triangle is it? How do you know? What does this triangle have to do with our quest to
construct a regular pentagon?
Task
Now bisect angle , and let the bisector intersect
at
°
°
SO SOLID Activity worksheets
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.
Question
2. What do you notice: about triangle
? About triangle
There is a powerful relationship between the triangles
?
and
which we can use to help us work out the
relationship between lengths of sides. If we could construct the distance we need, rather than worrying about
bisecting angles all the time, we might be in luck.
Let’s imagine that our original triangle,
, has side
= 2 units.
It would be helpful if we could know (and construct!) the length corresponding to
. Let’s call that distance .
3. What other distances on our diagram are now equal to ? Can we express any other distances in terms
of ?
4. The ratio of base to equal side in our similar triangles must be the same, so
=
Which side should replace the question mark in the equation above?
5. Replace the names of the distances in this relationship with their lengths (where known) or the
expressions in terms of .
6. How can this help you to find the exact value of ? (For a hint, see the end of the worksheet.) Can you
show that
is equal to
− ?
7. Now how can we construct that distance, when all we know for sure is that we have a line (
2? Well, we can find a distance of 1 unit by bisecting
? Think about it. If we can find
SO SOLID Activity worksheets
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) of length
. Can we use distances of 1 and 2 units to find
, can we find a distance of
− ? I think so!
Task
A. Now to the pentagon! You may well have already formulated your own ideas about how to use what
we’ve done to construct your regular pentagon. If so, great! Go ahead and do it. If not, take a look at the
following.
is a radius of length 2.
Construct
, a radius, perpendicular to
Construct point , the midpoint of
and
.
. How long are
,
?
Bisect angle
, to find
on
.
Now construct
, perpendicular to
.
is now one side of your pentagon. Set your compasses
to
as radius and step around the circle to find the other
vertices.
Question
8. How is this linked to what we looked at earlier? Where is the “
SO SOLID Activity worksheets
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− ” in this diagram?
Is there another way?
Tasks
What about:
B. Start with a line
of length 2, in the middle of a clean sheet of paper.
C. Construct the perpendicular to
D. Construct the midpoint of
E. Join
at
to find
and mark a distance 2 along it, at .
, a distance of 1 from .
.
F. With centre
and radius
, draw an arc to intersect
at . What is the length of
?
G. Construct the perpendicular bisector of
.
H. Now fix your compasses to the radius
from your original line. With centre , draw an arc to intersect
with the perpendicular bisector of
°,
°,
I.
is now the
J.
If we can construct a
at .
° triangle that we were considering.
° angle, we can construct our regular pentagon!
K. Finish the pentagon.
Questions
9. In what ways are these two methods linked?
10. Are there more efficient methods? Did you find one of your own?
Hint for finding the value of :
Form the equation
−
=
. Rearrange it to form a quadratic equation:
Expand it, tidy it up. Then solve it exactly. Do not use a calculator.
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−
=
Activity sheet 4
Regular Polyhedra
When we were considering whether particular regular polygons tessellate or not, we will have seen that three
regular pentagons, arranged as shown below, leave a small gap:
Questions
1. What is the angle between the two free sides of the pentagons at the gap?
2. Now imagine joining the tiles with a flexible hinge, along the common sides and then bringing together
the two free sides which form this gap. What do you see? If you’re having difficulty visualising it, why not
make a model?
(You don’t have to keep constructing these objects. You may be able to get your hands on some ready made
tiles (ATM Mats, for example) or you may want to use a computer drawing package to do this for you. The
picture above was created using nothing more sophisticated than Microsoft Word’s Drawing Tools. I drew one
regular pentagon – Hint: Choose Basic Shapes, Pentagon and then Drag and draw it while holding Shift down to
keep the shape regular – and then simply copied and pasted it twice more. Very quick, very easy.)
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This 3D vertex (could we call it “
”?) might possibly be repeated to give a closed, convex regular
polyhedron. Then again, it might not. Let’s find out. What we are going to be looking at here is whether
particular 3D vertices repeat to give us regular polyhedra.
One regular polyhedron with which we are all familiar is the cube. All its vertices are
– that is, three
squares meeting at one point.
Questions
3. What would the angle of the “gap” be, if we had begun with three squares lying flat, as in our pentagon
example, above?
4. Why can’t we have a polyhedron with
5. Is there a polyhedron with
vertices? Or
vertices?
vertices? What others can you find? Do you know the names for the
polyhedra you find?
One very important idea here is that the vertices should all be identical.
The vertex
(triangle, square, triangle, square) produces a particularly beautiful polyhedron called a
cuboctahedron. The “missing” angle, if these were to be laid flat would be:
−
Now if we were to make a vertex
+
+
+
=
°
, the same “gap” would appear in our attempted tiling.
Questions
6. Does the vertex repeat to give the same polyhedron? (This may well be worth trying. Make enough tiles
to build one. Start with a single vertex: triangle, triangle, square, square. Now add tiles to it, so that each
vertex is the same. What happens?)
SO SOLID Activity worksheets
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If we insist on all the vertices being identical, all the faces being regular polygons, and any polyhedra being
convex, how many different polyhedra can be made? (This is not a trivial question – although the answer has
been known for a couple of thousand years, it took some powerful minds to get to it.)
Of particular interest amongst this group of polyhedra are those that only use one kind of polygon – the socalled “Platonic Solids” (the cube is one of these!).
(Why is it impossible for there to be a Platonic solid with octagonal faces?)
All of the other polyhedra use at least two types of face. The amazingly-named rhombicosidodecahedron (!)
uses three different polygons. Its vertices are of the form
.
Task
Join forces with someone and make it.
(Can you predict how many of each face will be needed? How many vertices there will be? How many edges
will the finished polyhedron have? I do mean predict, not guess! It might be helpful to begin by looking at much
simpler models.)
Question
7. Are there any relationships between the type of face, the number of faces, the number of vertices and
the number of edges which are always true? Here’s one to get you started.
Cube:
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Vertex description:
Number of faces:
6
Number of vertices:
8
Number of edges:
12