Exploring Some Geometric Themes
Class 8 · Mathematics
5 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
The Sierpinski Carpet
Part of: Fractals and solids
A fractal is a shape that has the same pattern again and again at smaller and smaller scales. To build the Sierpinski carpet, cut a square into 9 equal small squares and remove the middle one. Then repeat on each of the 8 squares that remain, and so on. The number of remaining squares is multiplied by 8 at each step. The number of holes at the next step is the old holes plus the squares that were removed.
Why it matters. Fractals show how a simple rule repeated many times builds complex patterns. Counting them trains children to see patterns and write them as rules.
Try it in two minutes
- Draw a square on squared paper and divide it into 9 equal squares. Colour the middle one.
- Repeat the same on each of the 8 remaining squares and count how many squares remain.
- Ask your child how many squares will remain at the next step and how they know.
Try at home. Ask your child to predict the number of squares at step 3 and then check by multiplying.
Sierpinski Gasket and Koch Snowflake
Part of: Fractals and solids
For the Sierpinski gasket, join the midpoints of an equilateral triangle to cut it into 4 identical triangles and remove the middle one. Repeat on the 3 remaining triangles, so the number of triangles is multiplied by 3 each step. For the Koch snowflake, divide each side of an equilateral triangle into 3 equal parts and build a small equilateral bump on the middle part. Each side becomes 4 shorter sides.
Why it matters. These shapes show repeated rules that grow in surprising ways. Children practise finding patterns in counts and lengths and describing them in words.
Try it in two minutes
- Draw an equilateral triangle, join the midpoints of its sides and shade the middle triangle.
- Repeat on one of the three corner triangles and count how many are not shaded.
- For a snowflake, draw a triangle and make a bump on the middle third of each side. Count the sides.
Try at home. Ask your child to find how many sides the snowflake has after the third step.
Faces, Edges and Vertices of Solids
Part of: Fractals and solids
A solid has flat faces, edges where two faces meet and vertices where edges meet. A cuboid has 6 faces, 12 edges and 8 vertices. A prism has two identical polygons at the ends joined by rectangles or parallelograms; a prism with n-sided ends has n + 2 faces, 3n edges and 2n vertices. A pyramid has an n-sided base and a point above it, with n + 1 faces, 2n edges and n + 1 vertices.
Why it matters. Counting parts of solids builds spatial thinking and prepares children for net drawing and surface area.
Try it in two minutes
- Hold a box and count its faces, edges and corners with your child.
- Look at a tent shaped like a triangular prism and count its parts.
- Ask what happens to the counts if the end shape had one more side.
Try at home. Look for pyramid-shaped and prism-shaped objects in the house and count their faces and corners.
Nets of Solids
Part of: Fractals and solids
A net is a flat shape that folds into a solid. A cube has 6 square faces, and a cube has 11 different nets if we count a rotated or flipped net as the same. A regular tetrahedron has 4 triangles in its net and has only 2 nets. A cylinder's net has two circles and a rectangle whose one side equals the circle's circumference. A triangular prism has 2 triangles and 3 rectangles.
Why it matters. Nets connect flat drawings with solid objects. They are used in packaging and model making, and they prepare children for surface area.
Try it in two minutes
- Cut open an empty cereal box along its edges and lay it flat. Count the rectangles.
- Ask your child to roll a sheet of paper into a tube and describe the shapes in the net of a cylinder.
- Ask your child to draw a net of a cube and check it by cutting and folding.
Try at home. Challenge your child to find two different nets of a cube using squares of paper.
Shortest Paths and Views of Solids
Part of: Fractals and solids
To find the shortest path between two points on the surface of a cuboid, unfold the cuboid into a net and draw a straight line. A path on the net has the same length as the matching path on the cuboid. Different ways of unfolding give different lengths, so check them all. A solid has different outlines from different directions: a cylinder looks like a rectangle from the side and like a circle from above.
Why it matters. Looking at an object from different sides trains spatial thinking, which is needed in drawing, design and engineering.
Try it in two minutes
- Take a box and mark two spots on different faces. Ask your child which route along the surface is shortest.
- Ask them to unfold the box on paper and draw the straight line between the spots.
- Ask what shape a can of juice makes as a shadow from the side and from above.
Try at home. Use a torch to make shadows of a book, a ball and a cup, and ask your child what outline each makes from different sides.
What your child will practise
- Faces, Edges and Vertices of Solids: Lab Explorer Quest (Match It Up)
- Nets of Solids: Desert Quick Fire (Quick Fire)
- Shortest Paths and Views of Solids: Space Puzzle (Clue Puzzle)
- Sierpinski Gasket and Koch Snowflake: Ocean Quick Fire (Quick Fire)
- The Sierpinski Carpet: Number Quest (Collect the Answer)
- Quiz: Faces, Edges and Vertices of Solids (8 questions)
- Quiz: Nets of Solids (8 questions)
- Quiz: Shortest Paths and Views of Solids (8 questions)
- Quiz: Sierpinski Gasket and Koch Snowflake (8 questions)