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#### Something in Common Rich Tasks

This page contains a truly wonderful set of activities created and kindly shared by John Burke. They allow consolidation of key skills, prevent students from copying each other (as all the questions are different), make marking and assessing easy for the teacher (as all the answers are the same!), and provide a lovely extra challenge for students as they try to figure out exactly what is going on! In short, I think they are amazing, and thank you so much to John for sharing them with us all.

Please note: the "Earliest Year Group" is just a guideline. I have often been able to use a "Year 8 Task" with my Year 11s, for example.

#### Contents

#### The Background*keyboard_arrow_up*
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This set of resources came into being from my desire, as a teacher, to get pupils to think for themselves and not copy work from their neighbours. I wish I could say that I did all the research and came up with this scheme but, in fact, it was serendipitous. I put together a task for some sixth formers (SIC_6 "Annulus Volume of Revolution") where they all had different input parameters but all had the same answer, using a result I remembered from my sixth form days. It went very well so I developed more resources for lower down the school. As the pupils caught on there was the risk that they would just sit back and await the answer so one ruse I use is to give each worksheet a letter and say that I have a table of answers that I look up. Then I introduced some activities where the answers are different but can easily be computed from the input parameters (if you know how) to avoid this issue. A good example of this is SIC_11 "Triangle xy Area" where the teacher only has to multiply two integers together to know the answer but the pupils don't know that - yet. I started off labelling my resources "Different Problem, Same Answer!" but that didn't really fit in the end so I ended up with "Something In Common".

Not all of the resources are mine. Like all teachers I take inspiration from wherever I find it and so have included, for example, some RISPs by Jonny Griffiths and tasks from the nRich website and other places and I give due credit for a task where I remember where it came from.

Each resource has a suggested earliest year group. This is solely for use by the Mathematics department at my school so that we can avoid the situation whereby a teacher has planned to use a resource and then find that some pupils say they have already completed the task in a previous year. Please use a resource wherever you see fit. Indeed, the earliest year group can be a bit misleading because although the scheme of work indicates that the knowledge to perform the task is imparted to that year group the maturity of their problem solving capabilities may not be up to it and so would need a lot of scaffolding anyway. So, in a way, you can ignore the earliest year group information! Just review the activity and use it where you see fit.

Because the worksheets are all different I have placed them in pdf format with 1-up, 2-up or 4-up printing already prepared. All you need do is print the appropriate number of pages for the size of your class. Every accompanying Powerpoint has a worked solution and the original worksheets can be found at the back of it (so you can create separate ones for visually impaired pupils, for example, or change wording). Some activities have additional resources like Geogebra files for demonstration. One even has a Scratch program!

I use the resources in different ways. Sometimes I use them
to introduce a topic (circle theorems and sine rule, for example),
sometimes to consolidate the learning of a topic and sometimes to
introduce a wow factor on a Friday. But in every case I
encourage/challenge them to find out how I did it and quite often
they are up for the challenge, which I find very satisfying, and I
hope they will tell their friends.

I enjoy creating these resources and as of October 2016 there are
50 in the set. They do take time but I and my colleagues at
school reuse them a lot and I hope you will too. As a result
I would be grateful for any feedback, whether it is about mistakes
or suggestions for improvement or indeed new ideas for a resource.

John Burke

(The Manchester Grammar School)

[email protected]

#### Earliest Year Group: Year 7*keyboard_arrow_up*
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###### 12. Quad Tessellate

Get pupils to draw a quadrilateral that they think will not
tessellate. Then reproduce it using the Geogebra file and
show that it does. Then show a picture proof.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

None | Quad Tessellate Proof | Quad
Tessellate (Geogebra) Quad Tessellate Animation (Geogebra) |

###### 15. Magic Fractions

nRich. Using their own numbers (following a rule) the
pupils form fractions and carry out operations on them. All
answers are the same.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

None | Magic Fractions | None |

#### Earliest Year Group: Year 8*keyboard_arrow_up*
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###### 1. Think of a Number

Simple game that finishes on 1, whatever your starting
number. Leads to creating expressions and proof.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

None | Think
of a Number |
None |

###### 3. Parallelogram in Quadrilateral - Grid

Bisect the sides of a quadrilateral to get a parallelogram.
All quadrilaterals are different but all parallelograms are
identical. The grid allows easy bisection.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Parallelogram in Quadrilateral Grid | Parallelogram in Quadrilateral Grid | Parallelogram in Quadrilateral Animation (Geogebra) |

###### 31. Straight Line

A very simple (but illuminating) task where pupils follow simple
instructions and generate the equation of a straight line from
consideration of areas of triangles. Each equation is
different but easy for the teacher to ascertain the answer by
visual inspection.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Straight Line | Straight Line | Straight Line (Geogebra) |

###### 39. Letter Wheel

Pupils have to work out how much the last wheel of five connected
wheels turns when told how much the first wheel turns.
Sixteen different answers but teacher knows each answer from
information coded in the worksheet. Can introduce modulo
arithmetic.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Letter Wheel | Letter Wheel | Letter Wheel (.sb2) |

###### 43. Pick a Polygon

Pupils discover Pick's Theorem. Each worksheet has
different polygons but polygons allocated to a given letter have
the same area. They are invited to draw more polygons and
discover the formula for the area.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Pick a Polygon | Pick a Polygon | Proof of Pick |

###### 47. How Many Sides?

On the face of it this looks like a very tricky problem but in
fact it is quite straightforward, only requiring knowledge of
external angles and the sum of angles in a triangle! Teacher
can easily infer the answer and there is an interesting result at
the end.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

How Many Sides? | How Many Sides? | None |

#### Earliest Year Group: Year 9*keyboard_arrow_up*
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###### 4. Parallelogram in Quadrilateral - Compass

Bisect the sides of a quadrilateral to get a parallelogram.
All quadrilaterals are different but all parallelograms are
identical. A compass will be needed to bisect the sides.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Parallelogram in Quadrilateral Compass | Parallelogram in Quadrilateral Compass | Parallelogram in Quadrilateral Compass Animation |

###### 7. Trapezium and Diagonals

Work out the area of two triangles given the area of two others
which all fit in a trapezium. Each problem is different but
all the answers are the same.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Trapezium and Diagonals | Trapezium and Diagonals | None |

###### 35. Strange Guys

Classic problem involving masts and guy ropes solved using the
application of similar triangles. The activity brings out
the fact that the distance between the masts is irrelevant to the
question posed. All the answers are the same (90 m).

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Strange Guys | Strange Guys | None |

###### 41. Cross Bear!

Three bears start scoffing porridge at different rates and then
start scoffing each other's. Pupils have to work out the
individual rates. Teacher can find the answer encoded in the
worksheet by multiplying a number by 10.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Cross Bear! | Cross
Bear! |
None |

###### 42. The Obedient Ray

A light ray bounces between two mirrors and retraces its
steps. The angle (to be determined) between the mirrors is
the same but the height of each ray is different. Simple
geometry yields the answer but pupils are invited to look at a
more general case.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

The Obedient Ray | The Obedient Ray | None |

###### 45. Piscatorial Percentages

A counter-intuitive result involving percentages. It's a
nice little problem to solve algebraically but there is also a
very simple way to do it which makes it even more worth
doing. I couldn't resist using some Escher artwork.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Piscatorial Percentages | Piscatorial
Percentages |
None |

#### Earliest Year Group: Year 10*keyboard_arrow_up*
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###### 16. Angle at Centre, Angle on Arc

Discovery of two circle theorems. Pupils mark out angles on
a chord and cut them out and compare.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Angle at Centre, Angle on Arc | Angle at Centre, Angle on Arc | Angle
at Centre, Angle on Arc Resources Angle at Centre, Angle on Arc (Geogebra) |

###### 17. Multi-Simultaneous

Pupils are presented with a number of simultaneous equations to
solve. How quickly will they spot that they only need to
select one pair? How many will select the easiest pair
rather than just the first two in the list?.......

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Multi Simultaneous | Multi Simultaneous | Multi Simultaneous (Excel) |

###### 18. Polygon in Annulus

Pupils need to work out the area of a couple of rings with
different dimensions - but the same area.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Polygon in Annulus | Polygon in Annulus | Polygon in Annulus (Geogebra) |

###### 21. Leonardo of Pisa

Pupils are given the nth term of a sequence and are asked to find
specific (non consecutive) terms in the sequence. It is the
Fibonacci sequence. The task involves surds and
indices. Then they are asked to prove that the ratio of
consecutive terms tends to the golden ratio.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Leonardo of Pisa | Leonardo of Pisa | Leonardo of Pisa (Excel) |

###### 25. Crossed Lines

This tests the pupils' knowledge of straight line graphs and
y=mx+c and the significance of gradients of perpendicular lines.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Crossed Lines | Crossed Lines | Crossed Lines (Excel) |

###### 36. Quadthagoras!

Problem involving forming and solving a quadratic equation.
Not very hard and won't take long. Can the pupils work out
how all the answers are the same?

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Quadthagoras | Quadthagoras | None |

###### 44. Sea Shell Product

Pupils discover an intriguing result (that will be beyond them to
prove) - the product of the lengths of the chords equals the
number of equally spaced dots on the circle. Quite simple
trigonometry and lengths of sides of triangles.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Sea
Shell Product |
Sea Shell Product | None |

###### 46. Squaring the Rectangle

Pupils are asked to construct a square with area equal to a given
rectangle, using construction methods only. Gives practice
in constructions and application of the intersecting chords
theorem. Leads to determining the square root of a number by
construction.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Squaring
the Rectangle |
Squaring the Rectangle | None |

###### 48. Three Circles and a Tangent

Pupils seem to be fazed when they first look at this problem, so
it's worth doing. It only requires Pythagoras' Theorem (like
most Sangaku problems). There is a neat answer to the
general case.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Three
Circles and a Tangent |
Three Circles and a Tangent | None |

#### Earliest Year Group: Year 11*keyboard_arrow_up*
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###### 2. Sine Rule Discovery

Measure the sides and angles and calculate the ratios. All
ratios in the class are the same, yet all of the triangles are
different.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Sine Rule Discovery | Sine Rule Discovery | None |

###### 8. Four Crescents

Four crescents are drawn around a rectangle. All rectangles
are different and the area of the crescents is equal to the
rectangle. Surd (area = 1) and non-surd (area = 900)
versions available.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Four Crescents | Four Crescents | Four
Crescents 900 Four Crescents (Geogebra) |

###### 11. Triangle xy Area

Work out the area of a triangle with an inscribed circle.
Answers are all different but easily calculated from the given
parameters.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Triangle xy area | Triangle xy area | None |

###### 34. L-centre

Hands-on activity to find the centre of mass of an L-shaped plane
figure by construction only. But there is something special
about them all - each centre of mass is at the inner elbow of the
L. Can the (Year 13) pupils work out why this is?

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

L-centre | L-centre | L-centre
1 (Geogebra) L-centre 2 (Geogebra) |

###### 38. Awkward Triangle

Given the side lengths of a scalene triangle (in surd form) can
the pupils work out its area? Tests use of the Cosine Rule
and multiplication and simplification of surds. For
interest, Heron's formula is also used to solve the problem..

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Awkward Triangle | Awkward Triangle | Awkward
Triangle (Geogebra) Awkward Triangle (Excel) |

###### 49. What's the Sine Rule Ratio?

Pupils are asked to use their knowledge of sine rule, cosine rule
and even a circle theorem or two to determine angles and lengths
in a triangle.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

What's the Sine Rule Ratio? | What's the Sine Rule Ratio? | None |

#### Core 1*keyboard_arrow_up*
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###### 5. A Circle Property

RISP 9. Pupils choose certain values and draw their own
circle. All circles go through the origin.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

A Circle Property | A Circle Property | A
Circle Property (Geogebra) |

###### 9. Brackets Out, Brackets In

RISP 3. Pupils have to simplify some algebra, that they
have generated. All have the factor (x+1)

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Brackets Out, Brackets In | Brackets Out, Brackets In | Brackets Out, Brackets In (Excel) |

###### 19. Cubic Tangent Circle

Pupils need to work out the equation of a cubic and draw a
tangent at a specific place. All tangents should go through
the first root.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Cubic Tangent Circle | Cubic Tangent Circle | None |

###### 20. Hyper Parabola

Pupils need to find out where a parabola and a hyperbola just
touch.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Hyper Parabola | Hyper Parabola | Hyper
Parabola (Geogebra) |

###### 28. Survey Plot

Imagine you needed to remove one metre depth of topsoil from a
new building plot. What volume does this represent?
How many wagons will you need, etc? By approximating this
area to a polygon (where you know the coordinates of the vertices)
makes the task very easy (even easier with a spreadsheet).

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Survey Plot | Survey Plot | Survey
Plot (Excel) Survey Plot (Geogebra) |

###### 32. How Close

A coordinate geometry task. Pupils have the coordinates of
two points on a line. How close is this line to a third
point? All pairs of coordinates are different but lie on
three different lines - but the closest distance is the same for
all.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

How Close | How Close | How
Close (Geogebra) How Close (Excel) |

###### 37. Find f(2)

A slightly cryptic question based on a quadratic function.
Not too difficult but a little different from the
norm. As an extension you could ask the pupils to come
up with their own parameters that give the same solution.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Find
f(2) |
Find
f(2) |
Find
f(2) (Geogebra) |

#### Core 2*keyboard_arrow_up*
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###### 13. A Geometric Progression Problem

From clues given about a geometric progression work out the
possible solutions - all common ratios are the same and first
terms are five times each other.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

A Geometric Progression Problem | A Geometric Progression Problem | A Geometric Progress Problem (Excel) |

###### 23. A Fractional Sequence

An inductive formula is given with each pupil having different
starting values. They have to find the 11th and 12th
terms. The sequences loop (but they don't know that) so they
don't need to go that far. Then, of course, they have to
prove it always loops!

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

A Fractional Sequence | A Fractional Sequence | A Fractional Sequence (Excel) |

###### 24. Area between Parabolas

An integration activity around the area bounded by two
parabolas. Some pupils are confused if the area
straddles the x-axis. This activity addresses this.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Area between Parabolas | Area between Parabolas | Area between Parabolas (Geogebra) |

###### 30. Parabola in Parallelogram

Given certain facts about a parabola the pupils have to determine
its equation. Then they need to integrate the function to
work out ratios of areas. There is a neat solution that
makes it very easy to do - which you can show them at the
end. A further extension is to challenge them to do it
without integration.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Parabola in Parallelogram | Parabola in Parallelogram | Parabola
in Parallelogram (Excel) Parabolas in Parallelogram - no integration (Geogebra) |

###### 40. Quadratic Segments

Pupils have to work out the area of three segments formed by
chords of equal horizontal widths drawn on a parabola.
Interestingly, this area is invariant for a given width.
They discover this and then hopefully prove it.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Quadratic Segments | Quadratic
Segments |
Quadratic Segments (Geogebra) |

#### Core 3*keyboard_arrow_up*
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###### 6. Annulus Volume of Revolution

Pupils are given different segments of circles and asked to
calculate their volumes of revolution. They all have the
same volume.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Annulus Volume of Revolution | Annulus Volume of Revolution | Annulus
Volume of Revolution (Autograph) Annulus Volume of Revolution (Excel) Annulus6 (video) Sunday Times Teaser |

###### 14. Integration Stretch

A parabola is stretched - calculate the stretch factor to achieve
the area bounded between the two. All parabolas are
different but the stretch factors are the same.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Integration Stretch | Integration Stretch | Integration
Stretch (Excel) Integration Stretch (Geogebra) |

22. The Shortest Race

You have to run from tree A to tree B whilst touching a
fence. Where should you touch the fence to minimise the
distance you have to run?

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

The Shortest Race | The Shortest Race | The
Shortest Race (Geogebra) The Shortest Race (Excel) |

#### Core 4*keyboard_arrow_up*
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###### 27. Skew Lines

Skew lines in 3D space in vector form. How close do they
get?

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Skew Lines | Skew Lines | Skew
Lines (Excel) Skew Lines (Autograph) |

###### 29. Volume of Tetrahedron

Given the 3D coordinates of four points what is the volume of the
tetrahedron that is formed by joining them with line
segments? This brings together just about everything they
need to know about vectors and so is a good revision task.
Using the triple product to solve it takes some of the fun out
(hence suitable up to C4). Of course, the volumes are all
the same.... but why?

#### Mechanics*keyboard_arrow_up*
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###### 26. Shoot the Monkey!

Projectiles. You want to shoot a monkey hanging in a
tree. But he's a cheeky monkey and at the exact moment you
fire, he lets go and falls to the ground. Knowing this,
where should you aim to be sure of hitting your target?

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Shoot the Monkey! | Shoot the Monkey! | Shoot
the Monkey! (Geogebra) Shoot the Money! (Excel) |

###### 33. Target Range

Simple projectiles problem. Given two points on a
trajectory the pupils need to ascertain range, angle of elevation
and muzzle velocity. A good test to see if they can remember
how to do it from first principles.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Target Range | Target
Range |
Target Range (Excel) |

###### 34. L-centre

Hands-on activity to find the centre of mass of an L-shaped plane
figure by construction only. But there is something special
about them all - each centre of mass is at the inner elbow of the
L. Can the (Year 13) pupils work out why this is?

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

L-centre | L-centre | L-centre
1 (Geogebra) L-centre 2 (Geogebra) |

#### Statisitics*keyboard_arrow_up*
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###### 10. Magic Bag

nRich. An algebra task thinly disguised as a probability
question. The pupils have to determine the number of white
balls in a bag. There are two answers! Triangular
numbers are involved.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Magic Bag | Magic Bag | None |

###### 50. Misprints do Hapen

Two proof-readers find errors in the same manuscript - but not
all the same ones. How many are likely to have been missed
by both? This problem seems intractable at first but
is quite straightforward if you keep a clear head. The
pupils should be challenged to create a new worksheet with the
same answer, 2.

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Misprints do Hapen | Misprints do Hapen | None |

#### Further Pure*keyboard_arrow_up*
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###### 44. Sea Shell Product

Pupils discover an intriguing result - the product of the lengths
of the chords equals the number of equally spaced dots on the
circle. Quite simple trigonometry and lengths of sides of
triangles but the proof is hard!

Worksheet | PowerPoint | Supporting Materials |
---|---|---|

Sea Shell Product | Sea Shell Product | None |