Shapes and Patterns

Given: Triangular pieces cut from a rhombus (each piece is an isosceles triangle with 2 equal sides).

Activity: When you fit 2 such triangles together along different edges, you can form different triangles.

Types of triangles you can make:

1. Isosceles triangle – by placing two triangles base-to-base or side-to-side so that the resulting triangle has exactly 2 equal sides.
2. Equilateral triangle – by arranging two pieces so that all three sides of the resulting triangle are equal.

Observation about sides:
Each small triangular piece has 2 equal sides (and one different side). Such triangles are called isosceles triangles.

On folding: When you fold an isosceles triangle in half along the line of symmetry, the two base angles coincide, showing that each isosceles triangle has 2 equal angles.

Given: Triangular pieces from the rhombus (isosceles triangles).

Concept: An equilateral triangle has all three sides equal.

Answer: Yes, it is possible to make an equilateral triangle by fitting 2 of the isosceles triangular pieces together along their equal sides in the correct orientation, so that the resulting triangle has all three sides equal.

Conclusion: An equilateral triangle can be formed. All angles of an equilateral triangle are also equal (each = $60°$), as verified by paper folding.

Given: Triangular pieces from the rhombus.

Concept: A scalene triangle has no two sides equal.

Answer: Yes, it is possible to make a scalene triangle by joining two triangular pieces along a shorter or longer edge in a way that the three resulting sides are all of different lengths.

Note: Triangles that have no equal sides are called scalene triangles. In a scalene triangle, no two angles are equal either.

Conclusion: A scalene triangle can be formed from the pieces.

Given: Triangular pieces from the rhombus.

Activity: By fitting 2 triangular pieces together along different edges, you can form various quadrilaterals (4-sided shapes).

Different quadrilaterals possible:
1. Kite – two pairs of adjacent (neighbouring) sides are equal.
2. Parallelogram – opposite sides are equal and parallel.
3. Rectangle – a special parallelogram where all angles are right angles ($90°$).

About the Kite:
- Side 1 = Side 2 (one pair of adjacent sides are equal)
- Side 3 = Side 4 (another pair of adjacent sides are equal)
- These equal pairs are called adjacent sides (sides that share a common vertex/corner).

Conclusion: At least 3 different types of quadrilaterals (kite, parallelogram, rectangle) can be made.

Given: Two quadrilaterals A and B formed from the triangular pieces (both are parallelograms, with B being a rectangle).

Concept: In a parallelogram, opposite sides are equal and parallel.

Observations on measuring:

Quadrilateral A (Parallelogram):
- The pair of opposite sides are equal: top side = bottom side, and left side = right side.
- The equal pairs are opposite sides (not adjacent).
- Opposite angles are equal; adjacent angles are supplementary (add up to $180°$).

Quadrilateral B (Rectangle):
- Again, opposite sides are equal: top = bottom, left = right.
- The equal pairs are opposite sides.
- All four angles are equal and are right angles ($90°$ each).
- A rectangle is a special type of parallelogram.

Conclusion: In both A and B, opposite sides are equal. Quadrilaterals whose opposite sides are equal are called parallelograms. Parallelogram B, having all right angles, is called a rectangle.

Given: A dot/square grid.

Concept:
- A kite has two pairs of adjacent sides equal.
- A parallelogram has two pairs of opposite sides equal and parallel.

How to draw a Kite on the grid:
1. Mark a centre point.
2. Draw two pairs of line segments from adjacent vertices such that Side 1 = Side 2 and Side 3 = Side 4 (adjacent pairs equal).
3. Example: Choose vertices at grid points $(0,2)$, $(1,0)$, $(0,-1)$, $(-1,0)$ — this gives a kite shape.
4. Draw a second kite with different proportions (e.g., a taller or wider kite).

How to draw a Parallelogram on the grid:
1. Choose a starting point, draw a horizontal base of 3 units.
2. From each end, draw a slanted side of 2 units in the same direction.
3. Connect the tops — this gives a parallelogram with opposite sides equal and parallel.
4. Draw a second parallelogram with a different slant or different side lengths.

Note: Students should draw these on the actual grid provided in the textbook. The key properties to maintain are:
- Kite: two pairs of adjacent equal sides.
- Parallelogram: two pairs of opposite equal and parallel sides.

Given: 3 triangular pieces from the rhombus.

Concept: By arranging 3 triangles in different ways, we can create polygons with different numbers of sides.

(a) 3-sided shape (Triangle):
Arrange all 3 triangles so that they fit together to form a larger triangle. Place two triangles to form a parallelogram/rhombus, then attach the third on one slanted side. The result is a larger triangle (3 sides).

(b) 4-sided shape (Quadrilateral):
Arrange 3 triangles so that the outer boundary has exactly 4 sides. For example, place two triangles to form a parallelogram, then attach the third triangle on one of the shorter sides pointing outward — this can give a trapezium or parallelogram (4 sides).

(c) 5-sided shape (Pentagon):
Arrange 3 triangles so that the outer boundary has 5 sides. Place two triangles side by side and attach the third at an angle so that 5 outer edges are visible — this gives a pentagon (5 sides).

Conclusion: Using 3 triangular pieces, shapes with 3, 4, or 5 sides can be formed depending on how the pieces are arranged.

Given: 4 triangular pieces from the rhombus (the rhombus is divided into 4 congruent isosceles triangles).

Concept: By rearranging all 4 pieces, we try to form each shape.

(a) Square: Yes, it is possible. Arrange all 4 triangles so that their right-angle corners (or appropriate angles) meet at the centre, forming a square with all sides equal and all angles $90°$.

(b) Rectangle: Yes, it is possible. Arrange the 4 triangles in two pairs, each pair forming a parallelogram/rectangle, placed side by side to give a longer rectangle.

(c) Triangle: Yes, it is possible. Arrange all 4 pieces to form a larger equilateral or isosceles triangle.

(d) Pentagon (5-sided): Yes, it is possible with a suitable arrangement where 5 outer edges are visible.

(e) Hexagon (6-sided): Yes, it is possible. Arrange the 4 triangles so that the outer boundary forms a 6-sided figure.

(f) Octagon (8-sided): This is difficult/not straightforward with only 4 pieces from this particular rhombus, as the pieces may not produce 8 distinct outer edges easily. Students should try and verify.

Note: The actual results depend on the exact shape of the triangular pieces. Students are encouraged to physically try each arrangement and trace the outlines to verify.

Given: A standard tangram set (7 pieces cut from a square): 2 large right-angled triangles, 1 medium right-angled triangle, 2 small right-angled triangles, 1 square, 1 parallelogram.

Similarities:
- All pieces are flat (2-dimensional) shapes.
- All pieces are made up of right angles or $45°$ angles.
- All pieces can be combined to form a large square.

Differences:
- They differ in shape: some are triangles, one is a square, one is a parallelogram.
- They differ in size: there are large, medium, and small triangles.
- The parallelogram is the only piece that is not a triangle or square.

Conclusion: The pieces are similar in that they all have straight sides and specific angles ($45°$ and $90°$), but differ in shape and size.

Given: Tangram pieces — triangles, square, parallelogram.

Observation:
- All triangles in the tangram are right-angled isosceles triangles: they have one $90°$ angle and two $45°$ angles.
- The square has all four angles equal to $90°$.
- The parallelogram has two angles of $45°$ and two angles of $135°$.

Conclusion: The angles in all tangram pieces are either $45°$, $90°$, or $135°$. The triangles all have the same set of angles ($45°$-$45°$-$90°$).

Given: Tangram pieces.

Observation:
- The sides of all tangram pieces are related to each other — they are either equal in length or one is $\sqrt{2}$ times the other (due to the $45°$-$45°$-$90°$ triangle property).
- The small triangles have the shortest sides.
- The medium triangle's hypotenuse equals the side of the large triangle's leg.
- The square's side equals the leg of the medium triangle.
- The parallelogram's sides equal the legs of the small triangles.

Conclusion: The sides of the tangram pieces are all related — they are multiples or $\sqrt{2}$ multiples of the smallest unit length, showing a consistent proportional relationship throughout the set.

Given: Three views of a cube (front, top, side — each showing a different face with a pattern or marking).

Concept: A net of a cube is a 2D cross-shaped (or T-shaped) layout of 6 squares that folds into a cube. Each square in the net represents one face of the cube.

Method:
1. Identify which face is the front, which is the top, and which is the side from the three given views.
2. In the net, the faces are arranged in a cross pattern. The centre square is typically the bottom; the square above it is the front; the square to the left is the left side; the square to the right is the right side; the square above the front is the top; the square below the bottom is the back.
3. Place the markings/patterns from each view onto the corresponding square in the net.

Note: Since the actual images of the cube views are not visible here, students should physically look at the three views provided in the textbook, identify the pattern on each face, and draw/write the correct pattern on each face of the net in the correct position and orientation.

Conclusion: The net should have all 6 faces correctly labelled/drawn so that when folded, the cube matches all three given views.

Given: Large cube frames made of small cubes, with some small cubes removed from the interior or corners.

Concept:
- First find the total number of small cubes in the complete solid cube.
- Then count the small cubes remaining in the frame.
- Cubes removed = Total cubes − Remaining cubes.

Note: Since the actual images (a), (b), (c) are not visible in the OCR, the method is as follows:

General Method:
1. Determine the size of the large cube (e.g., $3 \times 3 \times 3 = 27$ small cubes total).
2. Count the small cubes visible in the frame structure.
3. Subtract: Removed = Total − Remaining.

Example for a $3 \times 3 \times 3$ cube frame (only edges remaining, all interior and face-centre cubes removed):
- Total = $27$
- Edge cubes only = $8$ corners $+ 12$ edges $\times 1 = 20$
- Removed = $27 - 20 = 7$

Students should apply this method to each figure (a), (b), (c) as shown in their textbook and count accordingly.

Given: A large cube made of $27$ small cubes ($3 \times 3 \times 3$). The outside of the large cube is painted red.

Concept: Depending on the position of each small cube in the large cube, different numbers of its faces are exposed (painted).

Analysis of positions:

(a) Three faces painted — Corner cubes:
Corner cubes are at the 8 corners of the large cube. Each corner cube has exactly 3 faces on the outside.
$$\boxed{8 \text{ small cubes have 3 faces painted red}}$$

(b) Two faces painted — Edge cubes (non-corner):
Each edge of the large cube has $3 - 2 = 1$ small cube in the middle (not at corners). There are 12 edges in a cube.
$$12 \times 1 = \boxed{12 \text{ small cubes have 2 faces painted red}}$$

(c) One face painted — Face cubes (centre of each face):
Each face of the large cube has a $3 \times 3$ grid, with $1$ centre cube that is not on any edge. There are 6 faces.
$$6 \times 1 = \boxed{6 \text{ small cubes have 1 face painted red}}$$

(d) No faces painted — Interior cube:
The cube completely inside, not touching any outer face.
$$\boxed{1 \text{ small cube has no faces painted red}}$$

Verification: $8 + 12 + 6 + 1 = 27$ ✓ (matches total number of small cubes)

Given: 7 shapes in a line — 2 squares, 2 triangles, 1 circle, 1 hexagon, 1 rectangle.

Clues analysis (step by step):

Step 1 — Clue (e): Circle is to the left of a square.
$$\text{Circle} \rightarrow \ldots \rightarrow \text{Square}$$

Step 2 — Clue (a): A square is between the circle and the rectangle.
$$\text{Circle} - \text{Square} - \text{Rectangle}$$

Step 3 — Clue (b): The rectangle is between the square and the triangle.
$$\text{Circle} - \text{Square} - \text{Rectangle} - \text{Triangle}$$

Step 4 — Clue (c): The two triangles are next to the square. We already have one triangle to the right of the rectangle. The second triangle must also be next to a square. Since we have 2 squares, the second square must have a triangle next to it.

From clues so far: $\text{Circle} - \text{Square}_1 - \text{Rectangle} - \text{Triangle}_1$

The second triangle must be next to the second square. Place: $\text{Triangle}_2 - \text{Square}_2$ or $\text{Square}_2 - \text{Triangle}_2$.

Step 5 — Clue (d): The hexagon is to the right of a triangle.

Combining all clues, the arrangement that satisfies everything is:

$$\boxed{\text{Circle} - \text{Square} - \text{Rectangle} - \text{Square} - \text{Triangle} - \text{Triangle} - \text{Hexagon}}$$

Verification:
- (a) Square (position 2) is between Circle (1) and Rectangle (3). ✓
- (b) Rectangle (3) is between Square (2) and Square (4) — and Square (4) leads to Triangle. ✓
- (c) Both triangles (positions 5 and 6) are next to Square (position 4). ✓
- (d) Hexagon (7) is to the right of Triangle (6). ✓
- (e) Circle (1) is to the left of Square (2). ✓

Final arrangement (left to right):
$$\text{Circle} - \text{Square} - \text{Rectangle} - \text{Square} - \text{Triangle} - \text{Triangle} - \text{Hexagon}$$

Given: 3D solid models — Icosahedron and Dodecahedron (made from nets at the end of the book).

Concept: These are regular polyhedra (Platonic solids).

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Icosahedron:

- Shape of faces: Equilateral triangles
- Number of faces: 20
- Do all faces look the same? Yes — all 20 faces are congruent equilateral triangles.
- Faces meeting at each vertex: 5 triangles meet at every vertex.
- Same number at each vertex? Yes — 5 faces meet at each vertex.
- Number of edges: 30

*Counting edges:* Each triangle has 3 edges; $20 \times 3 = 60$ edge-sides, but each edge is shared by 2 faces, so edges $= 60 \div 2 = 30$.

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Dodecahedron:

- Shape of faces: Regular pentagons
- Number of faces: 12
- Do all faces look the same? Yes — all 12 faces are congruent regular pentagons.
- Faces meeting at each vertex: 3 pentagons meet at every vertex.
- Same number at each vertex? Yes — 3 faces meet at each vertex.
- Number of edges: 30

*Counting edges:* Each pentagon has 5 edges; $12 \times 5 = 60$ edge-sides, but each edge is shared by 2 faces, so edges $= 60 \div 2 = 30$.

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Other solid shapes with all faces the same (Platonic Solids):
- Tetrahedron (4 equilateral triangular faces)
- Cube (6 square faces)
- Octahedron (8 equilateral triangular faces)

In all Platonic solids, the same number of faces meet at each vertex. ✓

How to count edges without missing or repeating: Go face by face, mark each edge as you count it, ensuring each edge is counted only once (since it is shared between exactly 2 faces).