Question 1
Describe the basic two-row paper weave.
Steps only
Row 1 repeats under, over.
Row 2 repeats over, under.
Repeat these two rows for all eight strips.
Answer
Odd rows: 1 under, 1 over; even rows: 1 over, 1 under.
CLASS 5 · MATHS MELA · CHAPTER 7
Shapes and Patterns
Create repeating weaves, test tessellations, combine geometric pieces and reason about circles, cubes and regular polyhedra.
70 worked answersAll 6 exercise sets
What you will learn
- Describe and extend visual patterns with rules.
- Classify triangles and quadrilaterals by sides and angles.
- Reason about tessellations, cube structure and polyhedron counts.
70 worked answers · short steps first, explanations when you need them
Weaving Mats and Grid Patterns
Textbook pages 92–93
Question 2
How is the second displayed mat woven?
Steps only
Row 1 repeats 2 over, 1 under.
Row 2 repeats 2 under, 1 over.
Answer
Alternate those two row rules until the mat is filled.
Question 3 · Row 1
State the weaving instruction for row 1.
Steps only
2 over, 1 under, repeated.
Answer
2 over, 1 under, repeated.
Question 3 · Row 2
State the weaving instruction for row 2.
Steps only
begin with 1 under once, then repeat 3 over, 3 under.
Answer
begin with 1 under once, then repeat 3 over, 3 under.
Question 3 · Row 3
State the weaving instruction for row 3.
Steps only
2 under, 1 over, repeated.
Answer
2 under, 1 over, repeated.
Question 3 · Row 4
State the weaving instruction for row 4.
Steps only
begin with 1 over once, then repeat 3 under, 3 over.
Answer
begin with 1 over once, then repeat 3 under, 3 over.
Question 3 · continuation
How should the four-row weave continue?
Steps only
After Row 4, return to the Row 1 instruction.
Answer
Repeat Rows 1–4 in the same order.
Question 4
How can a pictured weave be converted into written steps?
Steps only
Read one row left to right and group identical consecutive crossings.
Write counts such as “2 over, 1 under” until the block repeats.
Repeat for later rows until a previous row rule returns.
Answer
A correct answer lists the shortest repeating over/under block for each distinct row.
Question Grid design
Complete the partly drawn design on grid paper.
Steps only
Find the smallest block that repeats horizontally and vertically.
Copy that block square-for-square, keeping the same orientation and colour cycle.
Answer
The finished grid repeats the supplied motif without a break at any boundary.
Tiling and Tessellation
Textbook pages 94–96
Question Pentagons
Can regular pentagons cover the plane by themselves?
Steps only
A regular pentagon’s interior angle is 108°.
Three give 324°, leaving 36°; four exceed 360°.
Answer
No; regular pentagons leave gaps.
Question Triangles
How many equilateral triangles meet at a point?
Steps only
Each angle is 60°; 360÷60=6.
Answer
6 triangles; they tessellate.
Question Squares
How many squares meet at a point?
Steps only
Each corner is 90°; 360÷90=4.
Answer
4 squares; they tessellate.
Question Five squares
Can five squares meet without gap or overlap?
Steps only
5×90°=450°, which is 90° more than a full turn.
Answer
No; they must overlap.
Question Hexagons
How many regular hexagons meet at a point?
Steps only
Each interior angle is 120°; 360÷120=3.
Answer
3 hexagons; they tessellate.
Question Octagon pattern
What shapes are used in the common regular-octagon tiling?
Steps only
A regular octagon leaves a right-angle gap at each meeting.
A square fills that gap.
Answer
Regular octagons and squares.
Question Octagons alone
Do regular octagons tessellate by themselves?
Steps only
Each interior angle is 135°; 360÷135 is not a whole number.
Answer
No; gaps remain.
Question Marked vertices
How do you check the shapes at marked points in a mixed tiling?
Steps only
Name every corner touching the point and add its angle.
Repeat at the other marked points.
Answer
Each valid marked point has the same repeating vertex rule or another rule whose angles still total 360°.
Question Squares and triangles
In a square-and-triangle tiling, must the triangles be equilateral?
Steps only
Check whether all three sides or all three angles match.
Triangles filling a square-based gap may have unequal sides.
Answer
No. They are equilateral only if all three sides are equal; the pictured triangles should be checked by measurement.
Question Continue tiling
How should either mixed tiling be continued?
Steps only
Copy the smallest cluster by sliding it horizontally and vertically.
At every new vertex, preserve the same order of shapes and avoid gaps.
Answer
Repeat the original vertex arrangement and colour cycle.
Triangles and Quadrilaterals from Pieces
Textbook pages 97–99
Question 1
What triangles can be made by joining two triangular pieces from the rhombus?
Steps only
Join equal edges in different orientations and trace only the outside boundary.
The constructions shown have a line of symmetry.
Answer
The distinct traced triangles are isosceles: each has two equal sides.
Question Isosceles angles
What happens when an isosceles triangle is folded through its middle?
Steps only
The two equal sides coincide.
The two base angles also coincide.
Answer
An isosceles triangle has two equal angles.
Question 2
Can two supplied pieces make a triangle with all three sides equal?
Steps only
Try matching the long sides so the outside boundary has three equal segments.
Answer
Yes, an equilateral triangle can be formed in the intended cut-out activity.
Question 3
Can the two congruent rhombus pieces make a triangle with all sides unequal?
Steps only
Every triangular joining that mirrors the two congruent pieces creates a pair of equal outer sides.
Answer
No in the intended two-piece constructions; the resulting triangles are isosceles or equilateral.
Question Equilateral angles
Are all angles of an equilateral triangle equal?
Steps only
Fold or superpose its three corners.
Answer
Yes; all three angles are equal, each 60°.
Question Half equilateral
After cutting an equilateral triangle in half, how many sides of each half are equal?
Steps only
A half has lengths equal to half the base, the altitude, and an original side.
These three lengths differ.
Answer
No sides are equal; each half is a scalene right triangle.
Question Scalene angles
Can a scalene triangle have two equal angles?
Steps only
Equal angles would face equal opposite sides.
Answer
No; all three angles are unequal.
Question 4 · kite
What side property identifies the kite made from pieces?
Steps only
Measure neighbouring sides.
Answer
It has two pairs of equal adjacent sides.
Question 5 · A
Describe quadrilateral A.
Steps only
Opposite sides match in length and opposite angles match.
Answer
A is a parallelogram; opposite sides and opposite angles are equal.
Question 5 · B
Describe quadrilateral B.
Steps only
Opposite sides match and all four angles are right angles.
Answer
B is a rectangle, which is a special parallelogram.
Question 6
Draw two different kites and two different parallelograms on a grid.
Steps only
For a kite, reflect two points across a diagonal to get adjacent equal pairs.
For a parallelogram, choose two direction vectors and repeat each on the opposite side.
Answer
Any four grid drawings satisfying those side properties are valid.
Question 7(a)
Can three triangular pieces form a 3-sided shape?
Steps only
Arrange them so internal edges disappear and three straight outer chains remain.
Answer
Yes, a triangle can be formed.
Question 7(b)
Can three triangular pieces form a 4-sided shape?
Steps only
Join the pieces to leave four direction changes on the boundary.
Answer
Yes, a quadrilateral can be formed.
Question 7(c)
Can three triangular pieces form a 5-sided shape?
Steps only
Use an offset arrangement with five outer direction changes.
Answer
Yes, a pentagon can be formed.
Question 8(a)
Can all four rhombus triangles make a square?
Steps only
Test edge-to-edge arrangements and count only changes of direction on the outside.
Answer
Yes.
Question 8(b)
Can all four rhombus triangles make a rectangle?
Steps only
Test edge-to-edge arrangements and count only changes of direction on the outside.
Answer
Yes.
Question 8(c)
Can all four rhombus triangles make a triangle?
Steps only
Test edge-to-edge arrangements and count only changes of direction on the outside.
Answer
Yes.
Question 8(d)
Can all four rhombus triangles make a pentagon?
Steps only
Test edge-to-edge arrangements and count only changes of direction on the outside.
Answer
Yes.
Question 8(e)
Can all four rhombus triangles make a hexagon?
Steps only
Test edge-to-edge arrangements and count only changes of direction on the outside.
Answer
Yes.
Question 8(f)
Can all four rhombus triangles make a octagon?
Steps only
Test edge-to-edge arrangements and count only changes of direction on the outside.
Answer
No in the intended edge-to-edge four-piece set.
Tangram and Shape Properties
Textbook pages 99–100
Question Tangram names
Name the seven standard tangram pieces.
Steps only
Group by shape and size.
Answer
Five right isosceles triangles (2 large, 1 medium, 2 small), 1 square and 1 parallelogram.
Question Tangram angles
Compare the angles of tangram pieces.
Steps only
The triangles have 45°,45°,90°; the square has four 90° angles; the parallelogram has 45° and 135° angles.
Answer
All angles are multiples of 45°; right angles occur in the triangles and square.
Question Tangram sides
Compare the sides of tangram pieces.
Steps only
The triangles are similar but have different scales.
The square has four equal sides; the parallelogram has equal opposite sides.
Answer
Pieces share several matching edge lengths, which allows them to join without gaps.
Question Tangram figures
Make the pictured shapes using some or all tangram pieces.
Steps only
Begin with the largest pieces to form the outline.
Rotate or flip the parallelogram when needed, then fill gaps with smaller triangles.
Answer
More than one arrangement may be correct if pieces do not overlap and the outline matches.
Question Match 1
Match: All angles right, but not all sides equal.
Steps only
Check angle conditions and then side conditions.
Answer
Rectangle.
Question Match 2
Match: All sides equal, but not all angles equal.
Steps only
Check angle conditions and then side conditions.
Answer
Rhombus.
Question Match 3
Match: Opposite angles equal and no right-angle sides.
Steps only
Check angle conditions and then side conditions.
Answer
Parallelogram or a non-square rhombus.
Question Match 4
Match: Two pairs of equal sides that do not meet at right angles.
Steps only
Check angle conditions and then side conditions.
Answer
Kite.
Question Match 5
Match: All sides equal and every angle right.
Steps only
Check angle conditions and then side conditions.
Answer
Square.
Question Match 6
Match: Opposite angles equal and all sides equal.
Steps only
Check angle conditions and then side conditions.
Answer
Rhombus; a square also satisfies it.
Question Match 7
Match: Opposite angles equal and sides meet at right angles.
Steps only
Check angle conditions and then side conditions.
Answer
Rectangle; a square also satisfies it.
Question Paper kite
What shapes are visible after folding a square into a paper kite?
Steps only
The crease divides the kite into two mirror halves.
Additional fold lines divide those halves.
Answer
Triangles are visible inside the kite, arranged symmetrically.
Circles, Cubes and a Logic Puzzle
Textbook pages 101–102
Question Diameters
Join endpoints of two diameters. What quadrilateral forms?
Steps only
Each corner subtends a diameter on the opposite arc, so each is a right angle.
Opposite chord lengths also match.
Answer
A rectangle; perpendicular equal-position diameters give the special case of a square.
Question Different diameters
Does changing the two diameters change the shape type?
Steps only
The side lengths can change, but every corner remains a right angle.
Answer
It is always a rectangle, sometimes a square.
Question Other quadrilateral
Can this two-diameter process make a non-rectangle quadrilateral?
Steps only
Every constructed corner is 90°.
Answer
No.
Question Circle design
Continue the 24-point circle rule 1→11→2→12 and so on.
Steps only
After k from 1–10, connect to k+10; then connect to k+1.
Continue 10→20→11→21→…→14→24→1.
Answer
Follow alternating links n→n+10 and then to the next low-numbered point until returning to 1.
Question Cube net
Place three shown cube views onto the net.
Steps only
Choose one face as fixed.
Use shared edges to locate neighbouring faces, then fold mentally and check all three views.
Answer
A valid labelling makes every pair shown adjacent in each view and places opposite faces where they never share an edge.
Question Frame cubes
How can removed small cubes in each large frame be counted?
Steps only
Find the full cube’s side length n and calculate n³.
Count remaining corner, edge, face and interior cubes without double-counting.
Removed=n³−remaining.
Answer
Apply removed=n³−remaining separately to frames (a), (b) and (c); the printed perspective drawing supplies the remaining-cube count.
Question Painted (a)
In a painted 3×3×3 cube, how many small cubes have 3 faces painted?
Steps only
Only the 8 corner cubes touch three outside faces.
Answer
8.
Question Painted (b)
How many have exactly 2 faces painted?
Steps only
Each of 12 edges has one non-corner cube in a 3×3×3 cube.
Answer
12.
Question Painted (c)
How many have exactly 1 face painted?
Steps only
Each of 6 faces has one centre cube.
Answer
6.
Question Painted (d)
How many have no face painted?
Steps only
Only the central cube is completely inside.
Answer
1.
Question Line puzzle
Arrange 2 squares, 2 triangles, a circle, hexagon and rectangle using the clues.
Steps only
Use C for circle, S for square, T for triangle, R for rectangle and H for hexagon.
C–S–T–R–S–T–H satisfies: a square is between C and R, R is between a square and a triangle, each triangle touches a square, and H is right of a triangle.
Answer
Circle, Square, Triangle, Rectangle, Square, Triangle, Hexagon.
Icosahedron and Dodecahedron
Textbook pages 103
Question Icosahedron faces
What shapes form an icosahedron and are they alike?
Steps only
Count the net’s faces.
Answer
20 congruent equilateral triangular faces; all look the same.
Question Dodecahedron faces
What shapes form a dodecahedron and are they alike?
Steps only
Count the net’s faces.
Answer
12 congruent regular pentagonal faces; all look the same.
Question Icosahedron vertex
How many faces meet at each icosahedron vertex?
Steps only
Inspect one vertex and verify another.
Answer
5 triangular faces, the same at every vertex.
Question Dodecahedron vertex
How many faces meet at each dodecahedron vertex?
Steps only
Inspect one vertex and verify another.
Answer
3 pentagonal faces, the same at every vertex.
Question Icosahedron edges
How many edges has an icosahedron?
Steps only
20 faces×3 edges=60 face-edge counts.
Every edge belongs to 2 faces, so 60÷2=30.
Answer
30 edges.
Question Dodecahedron edges
How many edges has a dodecahedron?
Steps only
12 faces×5 edges=60 face-edge counts.
60÷2=30.
Answer
30 edges.
Question Other regular solids
Name other solids with congruent regular faces and identical vertices.
Steps only
Examples include a tetrahedron, cube and octahedron.
Answer
Tetrahedron, cube and octahedron.
Question Straw models
Which 3-D shapes can be built with equal sticks and clay joints?
Steps only
Join sticks as edges and clay as vertices.
Answer
Examples: tetrahedron, cube, octahedron or triangular prism; state the model actually built.