Lesson 7

Construction Techniques 5: Squares

7.1: Which One Doesn’t Belong: Polygons (5 minutes)

Warm-up

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 warm-up 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?

A

A green square with all sides marked 2.

B

A rectangle with sides marked 3 and 1.

C

A parallelogram with all sides marked 1.

D

An octagon with all sides marked 1.

Student Response

Student responses to this activity are available at 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 drop-down 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.

Representation: Internalize Comprehension. Begin with a physical demonstration of the construction of a perpendicular line through a point on the given line, to support connections between new situations and prior understandings. Ask students how the construction of a perpendicular line could be used to construct a square.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Use straightedge and compass tools to construct a square with segment \(AB\) as one of the sides.

 

Student Response

Student responses to this activity are available at 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. 

Representation: Internalize Comprehension. Begin with a physical demonstration of the construction of a perpendicular line through a point on the given line, to support connections between new situations and prior understandings. Ask students how the construction of a perpendicular line could be used to construct a square.
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Use straightedge and compass moves to construct a square with segment \(AB\) as one of the sides.

Line segment with endpoints A and B.

 

Student Response

Student responses to this activity are available at 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 \(AB\).)

Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to prepare students for the whole-class discussion by providing them with multiple opportunities to clarify their explanations through conversation. Before the whole-class discussion begins, give students time to meet with 2–3 partners to share their response to the question, “How do you know that what you constructed is a square?” Invite listeners to ask questions for clarity and reasoning, and to press for details and mathematical language. 
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 whole-class 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 whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to aide students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step.
Supports accessibility for: Organization; Attention

Student Facing

  1. Here is square \(ABCD\) with diagonal \(BD\) drawn:
    1. Construct a circle centered at \(A\) with radius \(AD\).
    2. Construct a circle centered at \(C\) with radius \(CD\).
    3. Draw the diagonal \(AC\) and write a conjecture about the relationship between the diagonals \(BD\) and \(AC\).
    4. 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?

     
  2. Use your conjecture and straightedge and compass tools to construct a square inscribed in a circle.

Student Response

Student responses to this activity are available at 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

Student responses to this activity are available at 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, whole-class 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 whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to aide students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step.
Supports accessibility for: Organization; Attention

Student Facing

  1. Here is square \(ABCD\) with diagonal \(BD\) drawn:
    1. Construct a circle centered at \(A\) with radius \(AD\).
    2. Construct a circle centered at \(C\) with radius \(CD\).
    3. Draw the diagonal \(AC\) and write a conjecture about the relationship between the diagonals \(BD\) and \(AC\).
    4. 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?

      Square ABCD with diagonal BD drawn.
  2. Use your conjecture and straightedge and compass moves to construct a square inscribed in a circle.
    Circle with point at center.

Student Response

Student responses to this activity are available at 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

Student responses to this activity are available at 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: Cool-down - Build a House (5 minutes)

Cool-Down

Cool-downs for this lesson are available at 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:

 A row of growing regular polygons inscribed inside of 5 identical circles, starts with 3 sides, adds 1 side every circle and ends with 7 sides.