Constructions and Tilings Class 7 (Ganita Prakash II)

Constructions and Tilings is Chapter 6 of NCERT Class 7 Ganita Prakash Part 2. It combines classical ruler-and-compass constructions with geometric design, tangrams and plane tiling. Students construct perpendicular bisectors, right angles, angle bisectors, copied angles, parallel lines, arches, regular hexagons and six-pointed stars. They then investigate how shapes fit together without gaps or overlaps. Teachoo explains the purpose and sequence of each construction and provides solutions to the Figure it out activities.

Geometric constructions

The chapter begins with Constructing an Eye, using arcs and intersections to create a recognisable design. This shows how simple geometric rules can generate complex visual forms. A perpendicular bisector crosses a line segment at its midpoint and forms a 90° angle. Every point on it is equidistant from the segment’s endpoints, a property that explains the compass construction.

Students construct a 90° angle and examine methods found in the Śulba-Sūtras, connecting geometry with India’s mathematical heritage. An angle bisector divides an angle into two equal angles. Arcs from the vertex and equal-radius arcs from the two arms create a point that determines the bisector.

Copying an angle transfers its size to a new location without using a protractor. This idea is then used to construct a line parallel to a given line through an appropriate point. Students also create arch designs and construct a regular hexagon by marking the circle’s radius repeatedly around its circumference. The hexagon provides 60° angles and supports the construction of a six-pointed star.

Tangrams and tiling the plane

Tangrams develop spatial reasoning by rearranging a fixed set of pieces into different figures. Tiling, or tessellation, covers a region with shapes without gaps or overlaps. Students investigate which shapes or combinations can tile the entire plane and relate success to the angles meeting at a point. A full turn of 360° must be filled exactly.

Topics covered on Teachoo

Teachoo covers:

• Constructing an Eye;

• perpendicular bisectors;

• construction of a 90° angle;

• construction methods in the Śulba-Sūtras;

• angle bisectors;

• copying an angle;

• constructing a parallel line;

• arch designs;

• regular hexagon, 60° angle and six-pointed star constructions;

• tangrams;

• tiling and tessellation;

• tiling the entire plane; and

• Figure it out solutions for pages 144–145, 154–155 and 160.

Learning outcomes

Students should be able to complete the listed constructions, state why each intersection or arc is used and distinguish construction from estimation by eye. They should build designs from circles and repeated angles, rearrange tangram pieces and test whether shapes tile a region. They should explain a tessellation using angles around a point and identify a repeat unit capable of extending across the plane.

Why is this chapter important?

Geometric construction turns properties into procedures. Instead of trusting a measurement alone, students create equality or perpendicularity through equal radii and intersections. This develops accuracy, logical sequencing and understanding of loci.

Tilings connect geometry with art, architecture, flooring, textiles, crystallography and computer graphics. They require students to combine angle knowledge with visual experimentation. Tangrams also encourage flexible thinking: the same pieces can produce many wholes.

How Teachoo supports learning

Teachoo divides the chapter by construction, making it easy to revise a specific method. Step-by-step solutions help students understand what each arc establishes. For best results, perform every construction physically with a sharp pencil, ruler and compass while reading the explanation.

Write the construction as a short sequence and label all points. Keep arcs visible until the work is checked. For tiling questions, test a small repeating unit, calculate the angles around a meeting point and then explain why the pattern can continue across the plane.

Common mistakes to avoid

Do not change the compass radius when a step requires equal arcs. Do not estimate a midpoint or angle by eye when the question asks for construction. A copied angle must reproduce both the arc radius and the chord between its intersections.

In a tiling, shapes must cover the region with neither gaps nor overlaps. A pattern that works near the centre may still fail at other vertices, so verify the repeating arrangement.

Quick revision checklist

Repeat the main constructions without looking at the steps, then annotate each arc with its purpose. Verify the midpoint, right angle or equal angle created instead of judging the result only by appearance. Construct a regular hexagon inside a circle and use it to form a 60° angle or star. For tiling, test at least two arrangements, calculate the angle total around a vertex and explain whether the chosen repeat unit can continue indefinitely.

Deeper reasoning and concept connections

The strongest way to learn Constructions and Tilings (Ganita Prakash Part 2) is to separate three layers: the object being studied, the rule that describes it and the reason the rule works. A correct numerical result is useful, but a complete mathematical answer also explains the relationship used. Students should compare examples and non-examples, change one condition at a time and observe whether the conclusion still holds.

This chapter is part of a longer progression. Its vocabulary and representations will appear again in algebra, geometry, data, measurement or higher problem-solving. Build links deliberately: translate pictures into statements, statements into operations and operations back into a sensible interpretation. If the final result cannot be explained in ordinary language, the method has probably been followed mechanically rather than understood.

How to solve unfamiliar and competency-based questions

When a question looks new, do not search memory for an identical example. Classify it. Decide whether it asks for recognition, calculation, representation, comparison, explanation or proof. Write the relevant definition or property first. Next, organise the data and select the shortest valid method. This converts an unfamiliar surface story into a familiar mathematical structure.

Use estimation and special cases as quality checks. Test zero, one, equal values, endpoints or a simple symmetric figure whenever they are permitted. A result that violates the diagram, scale, sign, unit or expected range is a signal to recheck the setup. In multi-part cases, carry forward only verified results so one early error does not silently contaminate every later answer.

What complete mastery looks like

For Constructions and Tilings (Ganita Prakash Part 2), a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.

Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.

Additional frequently asked questions

What should a student know before starting Constructions and Tilings (Ganita Prakash Part 2)?

Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.

How can a student check an answer in Constructions and Tilings (Ganita Prakash Part 2)?

Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.

How many questions are enough for strong preparation?

There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.

How should Teachoo solutions be used without becoming dependent on them?

Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.

Frequently asked questions

What is a perpendicular bisector?

It is a line that divides a segment into two equal parts at a right angle.

Why can a regular hexagon be constructed using a circle’s radius?

The radius fits six times as equal chords around the circle, forming six equilateral triangles and six 60° central angles.

What is a tiling or tessellation?

It is a repeating arrangement of shapes that covers a surface completely without gaps or overlaps.

Treat each compass arc as geometric evidence. Accurate construction and clear reasoning should appear together.