Lesson 1
Build It
1.1: The Right Tool (10 minutes)
Warm-up
The purpose of this warm-up is for students to familiarize themselves with the straightedge and compass.
They will learn to:
- draw a circle
- draw a line segment
- transfer a distance
Launch
In this unit, students start with a small set of tools for construction and editing in a custom applet, called Constructions, which can be found in the Math Tools menu or at ggbm.at/C9acgzUx. These are the GeoGebra tools in that app, those that do the same jobs as a pencil, a compass, and a straightedge.
Three pencil tools:
free point
point plotted on object
point of intersection of objects
Four straightedge tools:
line
segment
ray
polygon
Two compass tools:
circle with center through point
compass
To begin the activity, give students 2 minutes of quiet work time.
Pause the class to:
- demonstrate how to use the Circle tool by creating a circle centered at a given point and passing through another point
- demonstrate how to use the Compass tool by selecting a circle, segment, or distance to define its radius, and then choosing a point for its center
- demonstrate how to use a straightedge by marking a point on the circle and connecting it to the center to make a radius
- note that segment \(PQ\) is the part of the line through \(P\) and \(Q\) that has the endpoints \(P\) and \(Q\)
- note that length \(PQ\) is the distance from point \(P\) to point \(Q\)
Invite students to use their digital straightedge and compass tools to complete the remaining questions.
Student Facing
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Copy this figure using only the Pen tool and no other tools.
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Familiarize yourself with your digital straightedge and compass tools by drawing a few circles of different sizes, drawing a few line segments of different lengths, and extending some of those line segments in both directions.
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Copy the figure by completing these steps with the Line, Segment, and Ray tools and the Circle and Compass tools:
- Draw a point and label it \(A\).
- Draw a circle centered at point \(A\) with a radius equal to length \(PQ\).
- Mark a point on the circle and label it \(B\).
- Draw another circle centered at point \(B\) that goes through point \(A\).
- Draw a line segment between points \(A\) and \(B\).
Student Response
For access, consult one of our IM Certified Partners.
Launch
Give students 2 minutes of quiet work time.
Pause the class to:
- demonstrate how to use a compass by marking a point and creating a circle centered at that point
- demonstrate how to use a straightedge by marking a point on the circle and connecting it to the center to make a radius
- note that segment \(PQ\) is the part of the line through \(P\) and \(Q\) that has the endpoints \(P\) and \(Q\)
- note that length \(PQ\) is the distance from point \(P\) to point \(Q\)
Invite students to use their tools to complete the remaining questions.
Student Facing
- Copy this figure using only a pencil and no other tools.
- Familiarize yourself with your straightedge and compass by drawing a few circles of different sizes, a few line segments of different lengths, and extending some of those line segments in both directions.
- Complete these steps with a straightedge and compass:
- Draw a point and label it \(A\).
- Draw a circle centered at point \(A\) with a radius of length \(PQ\).
- Mark a point on the circle and label it \(B\).
- Draw another circle centered at point \(B\) that goes through point \(A\).
- Draw a line segment between points \(A\) and \(B\).
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If using rulers as straightedges, some students may wish to use the ruler to measure the length of \(PQ\). Emphasize that our straightedge can only be used to create lines or line segments between two marked points, but that your compass can be set to the length between two points and then moved to create a circle with that radius at any marked point.
Activity Synthesis
The goal is to make sure students understand the straightedge and compass moves that will be allowed during activities that involve constructions and why it is important to agree on standard construction moves. Ask students, “What is the difference between your attempt in the first question and what you came up with using the straightedge and compass?” (Sample response: Without the tools, it was difficult to make circles and straight lines. The compass makes it easier to make circles, and the straightedge makes it easier to make straight lines.)
Make one class display that incorporates all valid moves. This display should be posted in the classroom for the remaining lessons within this unit. It should include:
- If starting from a blank space, start by marking two points.
- You can create a line or line segment between two marked points.
- You can create a circle centered at a marked point going through another marked point.
- You can set your compass to the length between two marked points and create a circle with that radius centered at any marked point.
- You can mark intersection points.
- You can mark a point on a circle.
- You can mark a point on a line or line segment.
Tell students that using these moves guarantees a precise construction. Conversely, eyeballing where a point or segment should go means that there is no guarantee someone will be able to reproduce it accurately.
1.2: Illegal Construction Moves (15 minutes)
Activity
The purpose of this activity is for students to explore why straightedge and compass constructions can be used to communicate geometric information precisely and consistently.
Identify a student who places point \(C\) closer to point \(A\), and another student who places point \(C\) closer to point \(B\) to compare during discussion.
Launch
Arrange students in groups of 2.
Supports accessibility for: Memory; Language
Student Facing
- Create a circle centered at \(A\) with radius \(AB\).
- Estimate the midpoint of segment \(AB\), mark it with the Point on Object tool, and label it \(C\).
- Create a circle centered at \(B\) with radius \(BC\). Mark the 2 intersection points with the Intersection tool. Label the one toward the top of the page as \(D\) and the one toward the bottom as \(E\).
- Use the Polygon tool to connect points \(A\), \(D\), and \(E\) to make triangle \(ADE\).
Student Response
For access, consult one of our IM Certified Partners.
Launch
Arrange students in groups of 2.
For students using the digital Constructions tool, recommend that students begin by drawing a segment \(AB\).
Supports accessibility for: Memory; Language
Student Facing
- Create a circle centered at \(A\) with radius \(AB\).
- Estimate the midpoint of segment \(AB\) and label it \(C\).
- Create a circle centered at \(B\) with radius \(BC\). This creates 2 intersection points. Label the one toward the top of the page as \(D\) and the one toward the bottom as \(E\).
- Use your straightedge to connect points \(A\), \(D\), and \(E\) to make triangle \(ADE\) and lightly shade it in with your pencil.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students do not remember how to find a midpoint, break the word down and explain that it is a point in the middle of the segment.
Activity Synthesis
The key point for discussion is that with constructions, it is possible to investigate geometry without numbers. Instead, students can use construction tools to transfer distances without measuring.
Ask students to trace triangle \(ADE\) onto tracing paper and compare their triangle with their partners. Here are some questions for discussion:
- “Which steps in the instructions made it possible for these triangles to look so different?” (Estimating the location of the midpoint.)
- “What is identical in every diagram?” (The first circle.)
- “Writing \(AD=AE\) means the length of segment \(AD\) is equal to the length of segment \(AE\). Is that true?” (Yes, they are both radii of the same circle.)
- “Writing \(AB=2AC\) means the length of segment \(AB\) is equal to twice the length of segment \(AC\). Is that true?” (It looks like they might be, but we estimated the midpoint, so not necessarily.)
- “Why do valid straightedge and compass moves guarantee everyone will produce the same construction?” (There is never any estimating or eyeballing required. You are only ever using your tools to do one specific move.)
If question 2 were replaced with a method of finding the midpoint precisely with a straightedge and compass, then triangle \(ADE\) would be guaranteed to be consistent regardless of which student constructed it, up to the small error allowed by the tools. To be sure that a construction is valid, it must not include any estimation or eyeballing.
1.3: Can You Make a Perfect Copy? (10 minutes)
Activity
The purpose of this activity is to let students determine how to use straightedge and compass moves to construct a regular hexagon precisely. Students should play with construction moves until they reach their goal rather than follow an explicit demonstration of construction steps. While the term regular appears in the task, it is not important for students to know the precise definition of regular polygon at this time.
Identify students whose explanations that the sides are congruent use tracing paper, or compare the radii of the different circles in the construction. Tracing paper connects to the idea of rigid motions, while comparing radii references the precise definition of a circle, which students will use throughout this unit and subsequent units.
Launch
Arrange students in groups of 2. Give students 10 minutes of work time followed by a whole-class discussion.
Supports accessibility for: Language; Social-emotional skills; Conceptual processing
Student Facing
Here is a hexagon with all congruent angles and all congruent sides (called a regular hexagon).
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Try to draw a copy of the regular hexagon using only the pen tool. Draw your copy next to the hexagon given, and then drag the given one onto yours. What happened?
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Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Drag the given one onto yours and confirm that it fits perfectly onto itself.
- How do you know each of the sides of the shape are the same length? Show or explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
Student Response
For access, consult one of our IM Certified Partners.
Launch
Arrange students in groups of 2. Provide access to tracing paper. Give students 10 minutes of work time followed by a whole-class discussion.
Supports accessibility for: Language; Social-emotional skills; Conceptual processing
Student Facing
Here is a hexagon with all congruent angles and all congruent sides (called a regular hexagon).
- Draw a copy of the regular hexagon using only your pencil and no other tools. Trace your copy onto tracing paper. Try to fold it in half. What happened?
- Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Trace it onto tracing paper and confirm that when you fold it in half, the edges line up.
- How do you know each of the sides of the shape are the same length? Show or explain your reasoning.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students spend more than a few minutes without significant progress, tell them the segment given in the figure is one of the 6 sides of the hexagon. Invite students to compare the given hexagon to the start of the construction. Then ask if they can draw another segment to make an adjacent side of the hexagon.
Activity Synthesis
The purpose of this discussion is to build toward the concept of a proof by asking students to informally explain why a fact about a geometric object must be true. Ask previously identified students to share their responses to “How do you know each of the sides of the shape are the same length?”
Lesson Synthesis
Lesson Synthesis
Point out the display of straightedge and compass moves again. Ask students to identify and define the geometric terms in the display.
- If starting from a blank space, start by marking 2 points.
- Create a line or line segment between 2 marked points.
- Create a circle centered at a marked point going through another marked point.
- Set your compass to the length between 2 marked points and create a circle with that radius centered at any marked point.
- Mark intersection points.
- Mark a point on a circle.
- Mark a point on a line or line segment.
After several students share, tell the class that point, line, and distance (or length) are undefined terms. We can use these undefined terms to define other terms. It is important to know that:
- points are infinitesimally small
- lines are infinitely long, extending in both directions
- part of a line with one endpoint is called a ray, and it extends in one direction
- part of a line with two endpoints is called a segment, and it has a measurable length
- a circle is made up of all the points a set distance from a point
- the point is called the center, and the set distance is called the radius
Tell students that, in this course, they will build on their previous understanding of these terms and others to use precise definitions to describe geometric figures.
1.4: Cool-down - Build It (5 minutes)
Cool-Down
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
To construct geometric figures, we use a straightedge and a compass. These tools allow us to create precise drawings that someone else could copy exactly.
- We use the straightedge to draw a line segment, which is a set of points on a line with 2 endpoints.
- We name a segment by its endpoints. Here is segment \(AB\), with endpoints \(A\) and \(B\).
- We use the compass to draw a circle, which is the set of all points the same distance from the center.
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We describe a circle by naming its center and radius. Here is the circle centered at \(F\) with radius \(FG\).
Early mathematicians noticed that certain properties of shapes were true regardless of how large or small they were. Constructions were used as a way to investigate what has to be true in geometry without referring to numbers or direct measurements.