Lesson 7

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.

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.

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.

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).

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.

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 \(AB\).)

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).

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.

“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.