Shape and Space Problems Year 9
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Examples Page Ref |
Problem
Title |
Objectives
Ref |
Description |
Key Words |
169, 183, 193 |
Paper
Folding |
Similarity,
Gradients |
A
square is simply folded over and then opened out - what does the crease look like ? |
Similar,
Slope, Gradient |
185, 188, 209 |
Overlapping
Figures |
Angles, Symmetry,
Similar triangles |
How much overlap
between 2 congruent regular polygons ? |
Congruent,
Corresponding, Interior angle, Isometric |
185 |
P
Where You Like |
Properties
of Triangles |
Prove that the sum of distances from any point in an equilateral triangle to the sides is constant |
Proof,
Triangle |
185, 187 |
L-Shape |
Area, Proof |
How to divide any
2-rectangle 'composite' in half? |
Symmetry,
Proof, Bisect |
123, 185, 247 |
Overlapping
Squares 2 |
2-D Shape, Proof,
Trigonometry,
Surds |
Rotate 2 squares
- how much of the lower one can you see? Simple to state, scope to
explore... |
Congruent,
Symmetry, Pythagoras, Surds, Proof |
187,
199 |
A
Conic Journey |
Pythagoras,
2-D and 3-D Shape |
Find
the shortest route across a rather special cone. |
Pythagoras,
Cross-section, Chord |
91, 127, 189 |
The
Magic of Pythagoras |
Pythagoras |
This Problem is a variation on the Well in the
Courtyard problem... and has a lovely, surprising answer as unwanted
terms 'cancel out' ! |
Pythagoras,
Subject of the formula |
127,
189 |
Truncated
Square |
Pythagoras |
Another
'not-enough-information-surely?' problem that comes out nicely when
Pythagoras is liberally applied ... |
Pythagoras,
Subject of the formula |
189 |
Double
Squares |
Pythagoras |
A
surprisingly simple trick to writing numbers as the sums of squares - a
must-see ! |
Pythagoras |
189, 219 |
Quickest
Route |
Co-ordinate
distances, Pythagoras |
Jeeves
needs to escape across a swimming pool - but what's his quickest escape
route ? |
Pythagoras |
191, 193, 217 |
Triangle
Dissections |
Congruence,
Similarity, Enlargement |
A
series of puzzles concerned with dissecting equilateral triangles into
smaller parts |
Congruent,
Similar |
191 |
The
Radius |
Congruence,
Similarity |
A
simple application of Similar triangles |
Congruent,
Similar |
193,61, 81,
215 |
Pendants |
Similarity,
Enlargement |
'Area Factors'
under enlargement give quick answers to this problem |
Similar,
Enlargement, Scale factor |
193, 215, 234 |
Sculpt
Big |
Ratio,
Enlargement, Similarity, Cylinders |
A
sculptor chooses between similar large and small designs, in order to
maximize profits... |
Scale
factor, Volume, Enlarge, Proportional |
193,
235, 237 |
Shortest
Half |
2-D
Shape,
Dynamic Geometry |
What
is the most efficient way to divide an equilateral triangle into two equal
areas ? |
Arcs,
Sectors, Scale Factor |
200, 207 |
TetraCubes |
3D
Shape |
Identify
shapes, then use to build mini
Soma Cubes |
Plan,
View, Symmetry |
201 |
Sliced
Cube |
3-D
Shape, Pythagoras |
Visualize,
then prove, a result about a bisected cube. |
Cross-Section,
Plane, Properties |
161, 165, 203 |
Graphic
Convergence |
Co-ordinates,
Mappings, Combination of Transformations |
A
pair of mappings for x and y lead to a convergent sequence of points,
which can also be viewed as a combination of transformations - quite
pretty. |
Map,
Invariant |
213 |
Square
in a Triangle |
Transformations,
Symmetries |
How
to squeeze the largest possible square inside any triangle? |
Enlargement |
215, 221 |
Forensic
Triangles |
Constructions,
Dynamic Geometry |
Reconstruct the
original triangle from the sides' mid-points |
Medians,
Similar, Parallel |
219 |
Points
In Between |
Co-ordinates |
Finding
mid-points and points of trisection, using 'weighted average'
co-ordinates, and hence finding the 'centroid' |
Co-ordinates,
Line, Mid-point, Average, Graphs |
221 |
Eggs |
Constructions,
Compasses |
First
take
an egg... |
Compasses, Perpendicular, Arc, Tangent |
221 |
Triangle
in a Square |
Constructions,
Dynamic Geometry |
How
to squeeze the largest possible equilateral triangle inside a square? |
Compasses,
Rotation |
223, 245 |
Height
of the Tower |
Similar Triangles,
Scale Drawing |
There's a tower,
see, across this river, and what you've got to do is... |
Similarity,
Elevation |
133, 227 |
Overlapping
Squares 1 |
Loci, Graphs, 2-D shape |
This problem offers an
element of surprise in that the locus of possible solutions isn�t the
straight line that pupils may well expect |
Region,
Proof, Locus |
81, 91, 233 |
Average
Speed |
Speed, Ratio
+ Proportionality |
A nice
introduction to ratio methods for combining average speeds over 2 sections
of a journey |
Speed,
Average |
91,
233 |
A
Walk In The Bush |
Fractions,
Measurements |
An
'average speed' problem that comes out very sweetly - involving some up
and down hills |
Speed,
Average |
235 |
How
Deep Is The Well |
Circles |
Fairly
straightforward circumference calculations |
Pi,
Circumference |
237 |
Pie
Free Circles |
Circles,
Area |
A
section of a circle turns out to have an area independent of Pi |
Pi,
Arc, Pythagoras |
237 |
Snake
Eyes |
Circles,
Pythagoras |
A circular area
that turns out not to involve Pi ! |
Pi,
Hypotenuse, Pythagoras |
237 |
Target
Practice |
Circles |
The
middle ring of a circular target has a simple area... |
Pi,
Proportion |
237 |
Loo
Roll Emergency |
Circles |
When
the loo roll looks half-size, how much is really left? |
Pi,
Circle, Area |
35, 237 |
Wiggly
Paths |
Circles |
Surprisingly
pretty result about the area of winding pathways...(cf P.35 Garden Path) |
Pi,
Radius, Arc |
237 |
Square
Peg, Round Hole |
Circles |
Which
fits better-a square peg in a round hole, or a round peg in a...? |
Pi |
155,
237 |
Pick
A Shape |
Generate
sequence, Find nth term |
Pick's
Theorem for Areas on a dotty grid |
Formula,
Generate Tn, Proof |
|