Lesson 7
Construction Techniques 5: Squares
7.1: Which One Doesn’t Belong: Polygons (5 minutes)
Warmup
This is the first Which One Doesn't Belong routine in the course. In this routine, students are presented with four figures, diagrams, graphs, or expressions with the prompt “Which one doesn’t belong?”. Typically, each of the four options “doesn’t belong” for a different reason, and the similarities and differences are mathematically significant. Students are prompted to explain their rationale for deciding that one option doesn’t belong and given opportunities to make their rationale more precise.
This warmup prompts students to compare four polygons. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.
Launch
Arrange students in groups of 2–4. Display the polygons for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together, find at least one reason each item doesn't belong.
Student Facing
Which one doesn’t belong?
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class whether they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology they use, such as shape names, regular, equilateral, or equiangular. Also, press students on unsubstantiated claims.
7.2: It’s Cool to Be Square (15 minutes)
Activity
In this activity, students construct a square, given a side. This is similar to how students constructed a parallel line with two successive perpendicular lines, except they also have to pay attention to marking equal distances along the perpendicular lines.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
If students find that the diagram becomes too cluttered, encourage them to hide objects that are not needed to continue the construction. To do so, click on the last tool in the Toolbar—the Move Graphics Window tool. Beneath it is a dropdown menu of editing tools, including the Show/Hide Object tool. Select the tool and click on each element you want hidden. The selected objects will be faded. Select any other tool, and the faded objects will disappear. The same tool undoes the hiding.
Supports accessibility for: Conceptual processing; Visualspatial processing
Student Facing
Use straightedge and compass tools to construct a square with segment \(AB\) as one of the sides.
Student Response
For access, consult one of our IM Certified Partners.
Launch
Suggest students use a pencil to lightly draw the straightedge and compass moves and then use a colored pencil to emphasize the sides of the square.
Supports accessibility for: Conceptual processing; Visualspatial processing
Student Facing
Use straightedge and compass moves to construct a square with segment \(AB\) as one of the sides.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Some students may struggle more than is productive. Ask these students what they know about squares and what previous construction techniques they might use to tackle this problem.
Activity Synthesis
Ask students, “How do you know that what you constructed is a square?” (From the construction of perpendicular lines, we know the shape has 4 right angles. From the compass, we know the 4 sides have length \(AB\).)
Design Principle(s): Optimize output (for explanation); Cultivate conversation
7.3: Trying to Circle a Square (15 minutes)
Activity
The purpose of this activity is for students to construct a square inscribed in a circle. Just like the construction of the equilateral triangle inscribed in a circle, this construction provides an opportunity to preview rotation symmetry.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Give students 5 minutes to answer questions about square \(ABCD\) and then pause the class for a brief wholeclass discussion.
Students should come away with two key conjectures:
 The diagonals of a square are perpendicular bisectors of each other.
 In order to inscribe a square in a circle, the diagonals need to be diameters of the circle.
Give students 5 minutes to finish the activity, and follow with a wholeclass discussion.
Supports accessibility for: Organization; Attention
Student Facing
 Here is square \(ABCD\) with diagonal \(BD\) drawn:
 Construct a circle centered at \(A\) with radius \(AD\).
 Construct a circle centered at \(C\) with radius \(CD\).
 Draw the diagonal \(AC\) and write a conjecture about the relationship between the diagonals \(BD\) and \(AC\).
 Label the intersection of the diagonals as point \(E\) and construct a circle centered at \(E\) with radius \(EB\). How are the diagonals related to this circle?
 Use your conjecture and straightedge and compass tools to construct a square inscribed in a circle.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Use straightedge and compass moves to construct a square that fits perfectly outside the circle, so that the circle is inscribed in the square. How do the areas of these 2 squares compare?
Student Response
For access, consult one of our IM Certified Partners.
Launch
Give students 5 minutes to answer questions about square \(ABCD\) and then pause the class for a brief, wholeclass discussion.
Students should come away with two key conjectures:
 The diagonals of a square are perpendicular bisectors of each other.
 In order to inscribe a square in a circle, the diagonals need to be diameters of the circle.
Give students 5 minutes to finish the activity, and follow with a wholeclass discussion.
Supports accessibility for: Organization; Attention
Student Facing
 Here is square \(ABCD\) with diagonal \(BD\) drawn:
 Construct a circle centered at \(A\) with radius \(AD\).
 Construct a circle centered at \(C\) with radius \(CD\).
 Draw the diagonal \(AC\) and write a conjecture about the relationship between the diagonals \(BD\) and \(AC\).

Label the intersection of the diagonals as point \(E\) and construct a circle centered at \(E\) with radius \(EB\). How are the diagonals related to this circle?
 Use your conjecture and straightedge and compass moves to construct a square inscribed in a circle.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Use straightedge and compass moves to construct a square that fits perfectly outside the circle, so that the circle is inscribed in the square. How do the areas of these 2 squares compare?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Some students may struggle with the fact that when starting with the circle, we do not have two points marked to either construct a line or set a radius for a circle. Ask them how we may mark new points that can be used in our construction.
Activity Synthesis
“How was this construction different from the square in the previous activity?” (I started with the diagonal rather than a side.)
Conjecture that the entire construction remains the same even when rotated \(\frac14\) of a full turn (90 degrees) around the center. This means that each side can be rotated onto the other sides, and each angle can be rotated onto the other angles.
Lesson Synthesis
Lesson Synthesis
Remind students that they have now constructed an equilateral triangle, a regular hexagon, and a square, each inscribed in a circle. Each of these is an example of a regular polygon, which is a polygon with all congruent sides and all congruent angles. Ask students, “Starting with any of these shapes, which construction techniques would help you make other regular polygons inscribed in circles?” (Starting from any of them, we can make twice as many sides by bisecting the angles and marking the points where the angle bisectors intersect with the circle. We could repeat this process.)
7.4: Cooldown  Build a House (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
We can use what we know about perpendicular lines and congruent segments to construct many different objects. A square is made up of 4 congruent segments that create 4 right angles. A square is an example of a regular polygon since it is equilateral (all the sides are congruent) and equiangular (all the angles are congruent). Here are some regular polygons inscribed inside of circles: