# IGCSE Higher: Construction

## Scheme of work: IGCSE Higher: Year 10: Term 2: Constructions

#### Prerequisite Knowledge

1. Understanding Basic Geometric Terms and Symbols
• Concept: Knowing basic geometric terms such as line segment, midpoint, and angle, as well as the ability to recognize and use geometric symbols.
• Example: $\text{Line segment } AB, \quad \text{Midpoint of } AB = M, \quad \angle ABC$
2. Using a Compass to Draw Circles and Arcs
• Concept: Understanding how to use a compass to draw circles and arcs, which is fundamental for creating geometric constructions.
• Example: Drawing a circle with center $$A$$ and radius $$r$$: $\text{Circle: } (x – A_x)^2 + (y – A_y)^2 = r^2$
3. Understanding the Properties of Perpendicular and Angle Bisectors
• Concept: Knowing that a perpendicular bisector of a line segment divides the segment into two equal parts at a right angle, and that an angle bisector divides an angle into two equal parts.
• Example: $\text{Perpendicular bisector of } AB: \text{ Line that intersects } AB \text{ at } M \text{ and forms a } 90^\circ \text{ angle.} \\ \text{Angle bisector of } \angle ABC: \text{ Line that divides } \angle ABC \text{ into two equal angles } \angle ABD \text{ and } \angle DBC.$

#### Success Criteria

1. Construct the Perpendicular Bisector of a Line Segment
• Objective: Students should be able to use a straight edge and compass to construct the perpendicular bisector of a given line segment accurately.
• Example: Given a line segment $$AB$$, students should be able to: $\text{1. Place the compass at point } A \text{ and draw an arc above and below the line segment.} \\ \text{2. Without changing the compass width, place the compass at point } B \text{ and draw arcs that intersect the first pair of arcs.} \\ \text{3. Draw a straight line through the points of intersection of the arcs. This line is the perpendicular bisector of } AB.$
2. Construct the Bisector of an Angle
• Objective: Students should be able to use a straight edge and compass to bisect a given angle accurately.
• Example: Given an angle $$\angle ABC$$, students should be able to: $\text{1. Place the compass at the vertex } B \text{ and draw an arc that intersects both rays of the angle. Let the intersection points be } D \text{ and } E. \\ \text{2. Place the compass at point } D \text{ and draw an arc within the angle.} \\ \text{3. Without changing the compass width, place the compass at point } E \text{ and draw another arc that intersects the arc drawn from } D. \\ \text{4. Draw a straight line from the vertex } B \text{ through the intersection of the arcs. This line bisects } \angle ABC.$
3. Verify the Accuracy of Constructions
• Objective: Students should be able to verify the accuracy of their constructions by checking the properties of the bisected line segments and angles.
• Example: After constructing the perpendicular bisector of $$AB$$ and the bisector of $$\angle ABC$$, students should be able to: $\text{1. Measure the lengths of } AM \text{ and } MB \text{ to ensure they are equal, verifying the perpendicular bisector.} \\ \text{2. Measure the angles } \angle ABD \text{ and } \angle DBC \text{ to ensure they are equal, verifying the angle bisector.}$

#### Key Concepts

1. Construct the Perpendicular Bisector of a Line Segment
• Key Concept: Understanding that the perpendicular bisector of a line segment is a line that divides the segment into two equal parts at a 90-degree angle.
• Example: For a line segment $$AB$$ with midpoint $$M$$: $\text{If } M \text{ is the midpoint, then } AM = MB \text{ and } \angle AMB = 90^\circ.$
2. Construct the Bisector of an Angle
• Key Concept: Understanding that the angle bisector is a line that divides an angle into two equal smaller angles.
• Example: For an angle $$\angle ABC$$: $\text{If a line bisects } \angle ABC, \text{ then } \angle ABD = \angle DBC.$
3. Verify the Accuracy of Constructions
• Key Concept: Ensuring the accuracy of geometric constructions by verifying that the constructed bisectors maintain the intended properties (equality of segments and angles).
• Example: After constructing the perpendicular bisector of $$AB$$ and the bisector of $$\angle ABC$$: $\text{Measure to confirm } AM = MB \text{ and } \angle ABD = \angle DBC.$

#### Common Misconceptions

1. Construct the Perpendicular Bisector of a Line Segment
• Common Mistake: Students might incorrectly adjust the compass width when drawing the arcs, leading to inaccurate bisectors.
• Example: Given a line segment $$AB$$, a student might: $\text{1. Place the compass at point } A \text{ and draw an arc.} \\ \text{2. Adjust the compass width and place it at point } B, \text{ resulting in arcs that do not intersect correctly.}$
2. Construct the Bisector of an Angle
• Common Mistake: Students might incorrectly place the compass points or draw arcs that do not intersect within the angle, leading to inaccurate angle bisectors.
• Example: Given an angle $$\angle ABC$$, a student might: $\text{1. Place the compass at the vertex } B \text{ and draw an arc intersecting the rays at points } D \text{ and } E. \\ \text{2. Place the compass at point } D \text{ and draw an arc.} \\ \text{3. Incorrectly place the compass at a different point instead of } E, \text{ resulting in non-intersecting arcs.}$
3. Verify the Accuracy of Constructions
• Common Mistake: Students might not measure the constructed segments and angles accurately, leading to incorrect verification of the bisectors.
• Example: After constructing the perpendicular bisector of $$AB$$ and the bisector of $$\angle ABC$$: $\text{1. Students might fail to measure } AM \text{ and } MB \text{ precisely, leading to incorrect verification of equal segments.} \\ \text{2. Students might inaccurately measure } \angle ABD \text{ and } \angle DBC, \text{ leading to incorrect verification of equal angles.}$

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