### 1 | Use Geometry to Define the Attributes of the Shapes Around Us, Like Parallelograms and Rhombuses

Remind your students that maths isn’t just about numbers with this detailed lesson from Colin Foster that encourages pupils to take a closer look at what we know about geometry, and polygons in particular.

What makes a parallelogram a parallelogram? Can the attributes of a shape be mathematically defined and used to decide whether, say, something that looks like a rhombus actually is one?

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### 2 | Boost 3D Shape Understanding By Making A Children’s Toy

Can you make a shape sorter, with at least four different shapes and holes, so that each shape will go through just one of the holes?

This lesson plan looks at shape sorter toys – which will be familiar to most students, particularly if they have a younger sibling – as designing and making one can be a good challenge for those in Key Stage 3. It requires a lot of careful thought, particularly if we want to avoid any shapes fitting through the ‘wrong’ hole.

Students will have to decide which shapes to include (they can go for easy or hard options), how big to make them, and how to draw out the required nets accurately. An additional constraint can be that all of the shapes should be able to fit inside the finished box at the same time.

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### 3 | Use Creative Approaches to Calculating Angles of Overlapping Polygons to Deepen Mathematical Understanding

The angles created by overlapping regular polygons can be easy or difficult to calculate, depending on what information you are given. So, taking a creative approach to important mathematical processes like this can deepen students’ understanding of the principles involved and develop vital skills of reasoning and deduction.

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### 4 | Explore The Properties Of Triangles And Spot Patterns In A Sequence

In this open ended lesson, learners explore a sequence of gradually larger shapes and consider what mathematical questions they can ask about them. What is the same and what is different about a group of triangles, and how can they be drawn accurately. What measurements make an impossible triangle, and can students work out which triangles come next in a sequence? This KS4 lesson also gives students the chance to use Cosine.

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### 5 | Use A Clock Face To Investigate Angles

Although we live in a digital age, analogue timepieces are still all around us and provide a helpful context for exploring angles.

Students often approach the topic of angles by memorising lots of vocabulary (acute, obtuse, right angle, etc) and ‘facts’ about angles around a point, on a straight line, inside and outside a triangle and other polygons and between a transversal and parallel lines.

Then they have to choose the appropriate rule to get the right answer to each calculation.

In this lesson, a clock face is used to stimulate some calculations involving angles. First, students are invited to estimate the size of the angle and then to calculate it exactly. Initially, some of the calculations are quite easy, while others are more complicated as the lesson progresses. Provided that students know that there are 360° in a whole turn, they should be able to work out everything else for themselves.

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### 6 | Show how Line and Rotational Symmetry are Linked by Inventing and Drawing Your Own Shapes

‘Shape Up’ teaches students about line and rotational symmetry and how they are related, getting them to invent and draw their own shapes with particular symmetries.

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### 7 | Use Leaning Ladders To Explore Triangles

Ladders leaning against walls is a classic scenario for mathematics problems and is often used to address topics such as Pythagoras’ theorem and trigonometry.

In this lesson, the situation of two crossing ladders is used as an opportunity to work on ideas relating to similar triangles.

How high will the crossing point be above the ground and how far will it be from each wall? What happens for different lengths of ladders? What happens if the walls are further apart or closer together?

Students need to identify triangles that are similar and then construct equal ratios to work out their answers. They will notice interesting properties, such as the fact that the distance between the walls does not affect the height of the crossing point. There are also connections with an average called the harmonic mean.

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### 8 | Work Out The Topic Area By Finding The Three Angles Of A Triangle

Learners are often asked to practise finding the third angle of a triangle when the other two are given. In this lesson they get the opportunity to work on the topic area by finding all three angles according to different constraints.

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### 9 | Use Perimeters And Pythagoras’ Theorem to Calculate the Shortest Route on a Map

Applying their knowledge of scale and Pythagoras’ Theorem, students are tasked with calculating the shortest route between two destinations on a map and explaining their reasoning as clearly as possible in this detailed lesson plan devised by Colin Foster.

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### 10 | Examine the Effects that Margins of Error Have on Time, Distance and Estimation

‘Hopping Along’ sees students examine the effect that margins of error can have on otherwise seemingly clear results. Starting with the seemingly clear result of a race, students look at how multiple inaccuracies in the measurement of distance and time can combine to produce a very different – and even counter-intuitive – outcome.

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### 11 | Use Pythagoras’ Theorem To Find Accurate Values For Triangle Perimeters

In this lesson, students use Pythagoras’ theorem to find accurate values for the perimeters of some triangles. Although the values of the perimeters are so close that it would be very hard to distinguish them by measurement, mathematics makes it possible to be absolutely sure about the order in which they come.

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### 12 | Develop Spatial Awareness and Problem-Solving Skills by Exploring Two Successive Geometrical Transformations

Traditionally, pupils are taught geometrical transformations by learning a set of skills they can apply to reflecting, rotating and enlarging shapes without giving the process too much thought. But what if they were to called upon to use those skills to solve a mathematical problem?

In this lesson by Colin Foster, that’s precisely what they’re tasked with doing…

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### 13 | Use Spatial Reasoning in Geometry to Understand How Shapes Move when Broken Down into Points and Lines

In geometry students need to use their spatial reasoning and visualisation skills to understand the shapes and how they will move when broken down into points and lines. This lesson allows pupils to look at two fixed points and how they relate to other points.

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