3D solids - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

3D solids

Transformations and constructions

Lesson 1 of 8

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Transformations and constructions

Lesson 1 of 8

Answer each one, then check.

1

How many faces does a cube have?

2

What is the name of a 3D shape with a circular base and one curved surface ending at a point?

3

What is the volume of a cuboid 2 by 3 by 4?

4

What shape is the cross-section of a cylinder cut straight across?

5

What is a polygon with 6 sides called?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Transformations and constructions

Lesson 1 of 8

1

How many faces does a cube have?

6

2

What is the name of a 3D shape with a circular base and one curved surface ending at a point?

A cone

3

What is the volume of a cuboid 2 by 3 by 4?

24

4

What shape is the cross-section of a cylinder cut straight across?

A circle

5

What is a polygon with 6 sides called?

A hexagon

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Transformations and constructions

Lesson 1 of 8

1

Name common 3D solids.

2

Count faces, edges and vertices.

3

Draw and interpret plans and elevations.

4

Use a 3D drawing to work out how many cubes make a solid.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Faces, Edges and Vertices

Transformations and constructions

Lesson 1 of 8

A face is a flat surface, an edge is where two faces meet, and a vertex is a corner.

Solid

Faces

Edges

Vertices

Cuboid

6

12

8

Triangular prism

5

9

6

Square-based pyramid

5

8

5

Cylinder

3

2

0

Cone

2

1

1

Sphere

1

0

0

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Naming Solids

Transformations and constructions

Lesson 1 of 8

The name comes from the shape of the base or cross-section.

Prism

The same cross-section all the way along: a triangular prism, a hexagonal prism.

Pyramid

A polygon base with triangular faces that meet at one apex.

Cylinder, cone, sphere

Solids with curved surfaces; a sphere is a perfect ball.

Tetrahedron

A pyramid with four triangular faces.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 1 of 8

EULER'S FORMULA

For any solid with flat faces, faces plus vertices minus edges equals 2.

Cuboid: 6 + 8 − 12 = 2. Square-based pyramid: 5 + 5 − 8 = 2.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Plan and Elevations

Transformations and constructions

Lesson 1 of 8

The plan is the view from above; elevations are views from the front and the side.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Three Views

Transformations and constructions

Lesson 1 of 8

Plan

The view from directly above.

Front elevation

The view from the front, looking straight at the face you choose to call the front.

Side elevation

The view from one side, usually the right, looking straight at that face.

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Drawing the Elevations

Transformations and constructions

Lesson 1 of 8

A solid is made from four cubes: a row of three, with one more cube on top of the right-hand cube. Draw the front elevation, the plan and the side elevation viewed from the right.

1

Front elevation: what you see facing the row

Three squares wide; the right column is two high

2

Plan: look down from above

Three squares in a row (the top cube is hidden above the right one)

3

Side elevation from the right: you see the end

One square wide and two squares high

ANSWER

Front elevation: an L-shape of 4 squares. Plan: 3 squares in a row. Side elevation: 2 squares in a column.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Tips for Elevations

Transformations and constructions

Lesson 1 of 8

Draw only what you can see, with straight lines.

Hidden edges

Do not draw edges you cannot see from that direction.

Use a grid

One square on the grid stands for one edge of a cube.

Check your views agree

The front elevation and plan must have the same width; the front and side must have the same height.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Transformations and constructions

Lesson 1 of 8

Face

A flat surface of a solid.

Edge

A line where two faces meet.

Vertex

A corner where edges meet (plural: vertices).

Plan

The view of a solid from above.

Elevation

The view of a solid from the front or side.

Prism

A solid with the same cross-section all the way through.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Transformations and constructions

Lesson 1 of 8

YOUR TASK

Build and Draw

12 minutes

Build a solid from 5 cubes: a row of three cubes, a fourth cube placed directly behind the middle cube of the row, and a fifth cube on top of the left-hand cube of the row. Draw its plan and its front elevation.

1

Build the solid.

2

Look from each direction.

3

Draw only what you see.

WHAT A GOOD ANSWER SHOWS

Plan: a row of three squares with a fourth square behind the middle one (a T-shape). Front elevation: three squares wide, with the left column two squares high and the other two columns one square high.

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Can I...?

Transformations and constructions

Lesson 1 of 8

Name the common solids.

Count faces, edges and vertices.

Check with Euler's formula.

Say what a plan is.

Draw a front elevation.

Draw a side elevation.

Match views to a solid.

Work out the volume of a cube solid from its views.

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Summary & Exam Focus

Faces, edges, vertices: count carefully and check with F + V − E = 2.

Plan is from above; elevations are from the sides.

Draw with straight lines on a grid.

Compare your views for consistency.

EXAM FOCUS

A solid is made of 4 cubes: a row of three cubes with one cube on top of the right-hand cube. Draw the front elevation of the solid on a square grid. (2 marks)

Look at the solid from the front only and draw what is visible: an L-shape here. Do not add depth lines.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: 3D Solids

Transformations and constructions

Lesson 1 of 8

Answer all questions. · 20 minutes

Question 1 · 2 marks · Non-calculator

Write down the number of faces, edges and vertices of a square-based pyramid.

Question 2 · 2 marks · Non-calculator

A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles. Write down the name of the solid.

Question 3 · 2 marks · Non-calculator

Write down the shape of the plan of a cylinder standing on a flat table.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: 3D Solids (cont.)

Transformations and constructions

Lesson 1 of 8

Question 4 · 2 marks · Non-calculator

The diagram shows a solid made from four identical cubes. On the grid, draw the front elevation of the solid, viewed from the direction of…

Question 5 · 2 marks · Non-calculator

For the solid in the last question, draw the plan.

Question 6 · 3 marks · Non-calculator

For the same solid, draw the side elevation viewed from the right.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 2 marks · Non-calculator

Transformations and constructions

Lesson 1 of 8

“

Write down the number of faces, edges and vertices of a square-based pyramid.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

Transformations and constructions

Lesson 1 of 8

2 marks available. Award a mark for each point made.

Two correct

B1

All three correct

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Non-calculator

Transformations and constructions

Lesson 1 of 8

“

A solid has 5 faces, 9 edges and 6 vertices. Two of its faces are triangles. Write down the name of the solid.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

Transformations and constructions

Lesson 1 of 8

2 marks available. Award a mark for each point made.

Triangular prism

B2

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 2 marks · Non-calculator

Transformations and constructions

Lesson 1 of 8

“

Write down the shape of the plan of a cylinder standing on a flat table.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

Transformations and constructions

Lesson 1 of 8

2 marks available. Award a mark for each point made.

Circle

B2

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Non-calculator

Transformations and constructions

Lesson 1 of 8

The diagram shows a solid made from four identical cubes. On the grid, draw the front elevation of the solid, viewed from the direction of the arrow F. (2 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

Transformations and constructions

Lesson 1 of 8

2 marks available. Award a mark for each point made.

Correct number of columns and heights

M1

A fully correct front elevation

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 2 marks · Non-calculator

Transformations and constructions

Lesson 1 of 8

“

For the solid in the last question, draw the plan.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

Transformations and constructions

Lesson 1 of 8

2 marks available. Award a mark for each point made.

Three squares in a row

B2

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 3 marks · Non-calculator

Transformations and constructions

Lesson 1 of 8

“

For the same solid, draw the side elevation viewed from the right.

HOW TO ANSWER IT

Command word: Non-calculator. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

Transformations and constructions

Lesson 1 of 8

3 marks available. Award a mark for each point made.

One square wide

M1

Two squares high

M1

A fully correct drawing

A1

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