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Which Three Go Together?
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This is the first Which Three Go Together? routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together? Why do they go together?”
Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepens their awareness of connections across representations.
Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
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. If time allows, after 30 seconds of quiet think time, invite 2–3 students to briefly share what they notice that all of the figures have in common (for example, they are all polygons with labeled edge lengths). The purpose of this initial share out is to support all students in naming common attributes before they work to identify more specific features that a group of only three out of the four have.
Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell each student to share their response with their group, then together find as many sets of three as they can.
Which three go together? Why do they go together?
A
B
C
D
Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, 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 “regular,” “equilateral,” or “equiangular.” Also, press students on unsubstantiated claims.
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 circles:
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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).
Invite students to locate their catalog of constructions that they know and to use it throughout the lesson. 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.
Use straightedge and compass moves to construct a square with segment
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
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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).
Ask students:
Conjecture that the entire construction remains the same even when rotated
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.
Label the intersection of the diagonals as point
Use your conjecture and straightedge and compass moves to construct a square inscribed in a circle.
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.