## Geometry and Measure

### Prism Volumes

Age 13+ Time: 1-2hrs This is a very practical activity to help students develop a sense of volume/capacity and how to calculate it for prisms. Through pouring sand or water into and between a range of millimetre precise relational prisms, students discover that the volumes of prisms are proportional to the areas of their cross-sections! Plenty of hands-on challenge for all abilities.

### Similar Triangles

Age 14+ Time: 1 hr+ A great triangle mystery! These 24 triangles can be split in to 8 groups of 3 similar triangles by matching the angles. Students can use proportional reasoning to work out the missing lengths on the similar triangles. Present the triangles and say 'Find out the missing lengths' or add some structure depending on the class. The activity can be extended in to Pythagoras's theorem and trigonometry.

### Olympic Rings Logo

Age 14+ Time: 1-2hr In this activity students are set the task of producing their own logo for the Olympic Games. The initial challenge is to create four beautiful dynamic images using dynamic geometry. Students are led through each construction with individual help videos to get to grips with the equation of circles, controlling the centre, radius and path of the circles. Opportunities for a class competition.

### Renaissance Mathematics

Age: 12+   Time: 1h  What changed during the Renaissance? This activity looks at the revolution in using algebra to describe geometries, graphs, 3D perspective and the introduction of decimal notation. It can be used as part of a Renaissance School Day where students make links between subjects and then present their findings in a whole school assembly. Overview of this day, lead by History department, available here.

### Vector Translations

Age 11+ Time: 1hr An introduction to the concept of vectors or for teaching translation. Students use arrow diagram vectors to reconstruct a picture and also record the vector in column format. They can then use Autograph or Geogebra to make their own challenge for a partner to reconstruct. The activity closes with a freekick, vector shoot-out applet.

Age 11+ Time: 1-3hr This activity is all about establishing and working with the defining properties of quadrilaterals. Students work with dynamic constructions and move them around to try and establish what is always, sometimes and never true about them so that they can identify wich shape it is. Students are then invited to construct their own before working on problems about the sets and subsets.

Age 11+ Time: 1-2hr By copying and pasting multiple triangles students experiment with what quadrilaterals, and polygons, can and can’t be made with different triangles. They then classify triangles using a flowchart, before creating their own flowchart to identify the special quadrilaterals. The activity closes with a triangular jigsaw puzzle that students have to fit into a sheet of quadrilaterals then, a work of art!

### Interior Angles

Age 13+ Time: 2hr This activity gets students to discover the formula for the sum of interior angles in polygons.  The investigation can be completed using dynamic geometry software and is followed by an opportunity to justify the results geometrically in two different ways.  Finally, it is completed with a lovely application using knowledge of interior angles to create semi-tessellations. A very complete resource.

### Re-arranging SOHCAHTOA

Age: 15+ Time 1hr This activity is all about helping students to understand how to solve right-angled trigonometry problems. The activity encourages students to make 'correct statements' about the situations/diagrams given and become fluent in re-arranging them to make unkowns the subject and ths solve the problem.  Students can do this on their own computers or it can be done from a central computer as a whole class activity.

### Which Rule?

Age: 15+ Time 1hr This activity is designed to help students solve trigonometry problems by encouraging them to 'Speculate' about what might be possible. Students are asked to state different truths or complete different equations for a given diagram without being told what to solve for. Having completed the equations they are asked to think about which of them is most useful for solving for a particular variable. [Show][Hide]

So often students feel that they must know 'the right thing to do' before they proceed and are afraid to try things out to see what happens. Yes, it is possible to learn how to recognise certain types of problems but it is equally important to learn that problems can be solved by trying to use the different pieces of knowledge you have to make new ones. The sine rule and cosine rule can both be applied to any given triangle it is just that often only one of them generates an equation that can be solved. We can either learn to spot types of problems or to speculate with both. In practice, one often leads to the other and then we are better equipped to solve more problems.

### Making a Trig Calculator

Age: 14+ Time 1hr Use dynamic geometry and the fundamental principles of trigonometry to construct and program a trig ratio calculator. The essence of this acitivity is in students constructing their own. The logical steps required to do so involve real enquiry in to the nature of trig ratios. The result is really pleasing!

### Paper Baubles

Age: 12+ Time 1-2hrs  Ideal for Christmas celebrations, this activity gets students to design and create Platonic and Archimedean polyhedra and make them into decorations.  No messy glue is required, as tabs are folded and stapled on the outside, making them easy and quick to make, as well as attractive and decoratively original. The shapes are strong enough to be made on paper, although [Show][Hide]

using coloured paper makes them even more attractive.There are suggestions to get started, but the nets are not provided for the students, as such they will have to think carefully about the number of shapes they will need and how they will fit together.  The Archimedean solids (semi-regular polyhedra) require two or more different shapes to fit together. To this end, dynamic geometry software, like freely available Geogebra could be used to construct polygons.  Students will love the beautiful results and could be encouraged to search for Euler’s Characteristic.  The names of the polyhedra are purposely not mentioned in this activity to enable students to discover the nets for themselves.

### 3D Perception

Age: 12+ Time 1h  The aim of this resource is to develop student’s association of nets, hence surface area, with 3D solids, hence volume. The activity starts with a matching activity, nets and solids, some of which work, some don’t, students can cut and fold to check. Two virtual manipulative websites are then used, one aimed at inspiring them with a wide, and unusual range of 3D shapes.

### Rectangular Relations

Age: 11+ Time 1-2 hrs  Students are given A3 templates of parallelograms and trapeziums to cut out, fold, rotate, reflect, paste in an attempt to fit them inside a template rectangle. No dimensions are given. Students have to decide what dimensions of the original parallelogram and trapezium correspond to the length and width of the rectangle - the activity's emphasis is on mathematical process.

### Sine Cosine: Model Waves

Age: 15+  Time: 30mins-1h.  Students use geometry software to model wave pictures from real-life objects and situations.  In doing so, students will investigate the effects of the coefficients for sine and cosine waves e.g. y = a cos[b(x-c)]+d asking themselves: “What Changes?”, “What Stays the Same?”. No software is required

### Sine Cosine Transformations

Age: 15+ Time 1-2 hrs  Using Autograph or the free Geogebra or Microsoft Maths 4.0, students investigate the functions of the sine and cosine graph. Students record the key, defining points in a pre-prepared table: coordinates of the maximum and minimum and x-intercepts, as they change different parameters using the constant controller or sliders. Without technology, students then have to predict[Show][Hide]

these key points for different functions.

### Sine and Cosine: Triangle, Circle, Wave!

Age: 15+ Time 1h This activity introduces sine and cosine graphs using the video of the construction of a Ferris wheel that demonstrates the link with triangles. Students then sketch the graph of their movement on the Big Wheel. The aim is to link the sine and cosine ratios to a circle. Students use calculators to plot the graphs exactly (spotting symmetries to save them calculation time!). VM also available.

### Around Circles

Age 11+ :Time 40mins+  This investigation enables students to discover pi for themselves through a practical activity.  The classic method of measuring the circumference of a circle with string is enhanced with a lovely applet and a chance to use dynamic geometry to make very accurate measurements.  This activity is best attempted before the students have any knowledge of pi.  It is an alternative to the  Discovering Pi

### Pyramid Model

Age: 13+ Time 2 hrs This is a lovely practical activity to help students visualise and derive the formula for the volume of a pyramid.  By constructing square based pyramids (10cm by 10cm) with height 5cm then fitting six of them together to make a cube of edge 10cm they realise the volume of the pyramid is 1000/6cm².  The activity is supported with videos and practice questions.

### Prism People

Age: 13+ Time 2 hrs Exploring prisms by making them! Students are asked to build model robots from different types of prisms. The practical element of building a prism is used to help students discover the related features of these shapes. Following this, students areasked to look in more detail at the structure and surface area of prisms and test out what they have learned on some examples and challenges. [Show][Hide]

There is nothing quite like having to build a triangular prism for helping to understand the structure of the shape. If the pairs of equal length sides are not correct then the shape doesn't work. This practical is fun and the learning objectives are an inherent part of the task. The follow up task leads nicely on by asking students to correctly identify the equal side lengths on the nets of prisms and this is key to working out the various surface areas! Its fun and takes advantage of students natural intuition. I particularly like the challenge of designing prisms whose surface area is 100cm², because of its apparent ease at the outset and creeping complexity.

### Body Surface Area

Age: 13+ Time 40mn - 1 hr. Working in small groups students are asked to find an approximation for the surface area of their bodies. This is a great practical investigation using surface area of prisms and spheres, etc. with real life applications. It encourages students to think critically about area.  Which 3D shapes will best approximate the shapes of the different parts of the body?  [Show][Hide]

The formulae required are not included and students are expected to find them out for them for themselves. To help verify the ‘accuracy’ of their estimate a formula used in the pharmaceutical industry that estimates their body surface area from their mass and height is provided.

### Dancing Vectors

Age: 15+ Time 2 hrs. Introduce vectors through dancing! This is a great fun and effective activity where students imagine displacement vectors as dance moves! The vectors are combined to make a dance routine. Get the whole class up and dancing this routine to Donna Summer's hotstuff! It is a memorable experience and really helps students get to grips with this concept.

### 3D Uncovered

Age: 15+ Time 1-2 hrs. This activity uses technology to help students with the notoriously difficult idea of working with 2D planes within 3D situations and thus solve problems with trigonometry in 3 dimensions.Google Sketchup is a simple and clear visual aid that encourages students to literally look at the problem from a different angle. The medium is both helpful and engaging and lends a [Show][Hide]

dynamic reality to a 3 dimensional object that can't really be achieved with paper! This activity is best done with students at computers, but it can be done all together from the teachers computer if technology is not available.

### Escher Symmetry

Age: 13+ Time: 2h. Observing symmetrical objects comes quite naturally to students because we are surrounded with symmetry.  Here’s an activity that takes that skill a giant step further.  A great example of how the use of computer software can create a whole new type of activity, this investigation gets students to analyse the symmetry in Escher tiling patterns by reflecting, rotating and translating[Show][Hide]

them using Geogebra. Geogebra is dynamic geometry software that can be downloaded for free.  For this activity it is fairly intuitive, so even students new to it will have no problems.  There is a help video provided should technical assistance be required.

This is a great example of how beautiful images, from the Artist Escher, can provide instant engagement for students.  Scaffolding is provided to help guide students through the early parts of this investigation, and some real challenges are posed at the end where a good deal of creativity and critical thinking will be required.

Symmetry patterns like these can be described in terms of rotations or gyrations, reflections, translations, glides and glide reflections.  Apparently differently looking symmetry patterns can have the same symmetry properties and there are 17 different wallpaper groups of symmetry in total.  Six of these groups are analysed in this investigation, but, if inspired, students could be encouraged to take this further and find and describe them all.  Inspired by this work, a recent student of mine wrote a research paper about all the different symmetries.

Age: 11+  Time: 1h. When describing rotations, students often forget that coordinates define a centre of rotation. Through the designing and playing of the “Around the World” and “Jungle Obstacle” games students get lots of experience of defining rotations in a fun and creative environment. The software forces students to consider angle and direction and displays the coordinates of the centre of rotation. Lots of[Show][Hide]

scope for extension challenging able 14-15yr olds as a starter activity

### Nature's Symmetry

Students explore the rotational and reflective symmetry occurring in nature and have fun producing their own flower symmetry patterns. This instantly engaging activity makes use of dynamic geometry software geogebra which is freely available requiring no software installing at geogebra.org Age: 11+ Time: 1-2hrs

### Around the World

Age: 13+Time: 1 to 1.5 hr This activity provides an interesting context to make some proportion and ratio calculations. Google Earth is used to make measurements about distance and angles of longitude and latitude.  These are then combined to make calculations about the circumference of the Earth. No previous knowledge of Google Earth is required.

### Equation Reflections

Age: 11+ Time: 1-2hrs Students often think reflection is easy, but the big change at this level is the need to define, using equations, the position of the mirror line. This activity uses Geogebra and/or Autograph to explore points on different lines to remind students why lines can be defined using equations. Three further[Show][Hide]
activities oblige students to use equations to perform reflections and create a reflective, art masterpiece!

### Modelling Music

Age: 15+ Time: 1h. This activity helps students see the connection between mathematics and music.  Students see the shape of music (pure tone sound appears as trig functions) thanks to sound recording software (Audacity).  No need to install the software, as short videos and images are ready to use. Students would be expected to have seen the graph of a sine curve before, and have some understanding of transforming graphs.

### Dr Who

Age: 13+ Time: 1 hr+. This is a really absorbing and engaging puzzle that is rich with interesting mathematical behaviour. The essential learning objective for this activity is about scale factors of enlargement and repeated enlargement. The medium for the challenge really appeals and this type of dynamic question represents a whole new genre of questions that are offered by technology... [Show][Hide]

The image of space as the backdrop and the effect of disappearing down a tunnel are really cool and makes students really want to be able to do it for themselves. To do so requires some critical mathematical thinking and problem solving. Students have to piece the observations together to make coherent explanations of what is happening and why it is happening before they can recreate it and it is here that the richness of this activity lies. I do seriously recommend that teachers try this for themselves before looking at the solution and discussion screencasts found in the teacher notes! Its great fun and an excellent challenge!

### Kaleidoscope

Age: 13+ Time: 1 hr+ Explore the concept of rotation with this dynamic problem. Students are shown an animation created from a construction in dynamic geometry and asked to recreate it by examining its properties. The construction is all based on repeated rotation. This is an absorbing problem combining some critical thinking with creativity.

### Plans and Elevations

Age: 11/12+   Time: 1h This activity uses the excellent Freudenthal Institute's Wisweb interactive "Rotating and Building Houses" to help students develop their ability to visualise 2D representations of 3D shapes. The activities on this page are carefully designed to get the most out of the virtual manipulative, with students using it to test their 2D written representations (digitally in word) in a 3D [Show][Hide]

environment and to set challenges for their partner to solve.

### Making Cones

Age: 15+   Time: 1 hr Explore cones by making one! This activity helps students understand where the formula for the surface area of a cone comes from and play with the associated mathematics. A great practical task that seems easy and works out to be more of a challenge. In making the cone students will confront some great mathematical reasoning and maybe even some algebraic proof!

### Polygons & Stars

Age: 11+ Time: 2 hours This activity gets students to construct regular polgons and stars using protactor and ruler.  It quickly moves onto a powerful online microworld to make the constructions.  Students can then discover the rules for a) the sum of exterior angles of a regular polygon and b) the exterior angle of any regular polygon or star.

### Skewey Squares

Age: 14+   Time: 2 - 3 hrs Discovering Pythagoras theorem by investigating the areas of different 'Skewey' squares. A great investigation, involving lots of thought and lateral thinking. This is followed by some powerful links and discoveries that really helps students think about Pythagoras theorem.

### Circle Theorems

Age: 14+   Time: 2 - 3 hrs The power of dynamic geometry is used here to help students discover the properties of angles and circles. Students are given constructions to build and investigate and then asked to generalise about their findings!

### Discovering Pi

Age: 12+   Time: 2  hrs Help students to understand that Pi is a geometric phenomenon. This investigation looks at regular polygons and the longest lines that fit inside them. It leads to the relationship between Pi and the circumference of a circle.

### Piece of Cake

Age: 14+   Time: 1 hr Which is the biggest piece? Give students this selection of parts of circles and ask them to put them in order of size. The result is an intuitive need to work out the area of sectors of circles!

### 3-2-1 Blast off!

Age: 14+   Time: 40mins This short activity asks students to calculate the height of a water powered rocket.  This practical lesson gets students out of the classroom and gives them a genuine application of right-angled triangle trigonometry.  It is also a great oppportunity to explore error bounds.  You may even wish to embark upon a joint project with the science department at your school.

### Spherical Cylinders

Age: 14+   Time: 1h This activity uses a useful interactive website animation to help students work out for themselves the relationship between the volume of a sphere and the volume of a surrounding cylinder of equal height and diameter to that of the sphere's. Once they understand where the formula comes from, they then apply it, in pairs or small groups, to a series of interesting, if unusual, problems!

### Human Loci

Age: 14+   Time: 2 hrs A chance to get your class outside, Human Loci is a fun and revealing activity that gets students representing various loci by positioning themselves according to the rules given. Best played in teams!

### Discovering SOHCAHTOA

Age: 13+   Time: 1.5 - 2 hrs This investigation gets students to discover the three trigonometric ratios for right-angled triangles.  Dynamic geometry software replaces a classic pencil and paper method for constructing and measuring sides in triangles.  Conjectures about ratios are quickly made and tested.