Sign in to view assessments and invite other educators
Sign in using your existing Kendall Hunt account. If you don’t have one, create an educator account.
Which Three Go Together?
None
This Warm-up prompts students to compare the graphs of four quadrilaterals. 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 figures. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three figures that go together and can explain why they go together. Next, tell students to share their response with their group and then together to 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 go together. Record and display the responses. 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 that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “equilateral” or “diagonal,” and to clarify their reasoning, as needed. Consider asking:
If possible, leave the list of responses displayed until the end of class. Students will return to these images during the Lesson Synthesis.
Display the images from the Warm-up again.
A
B
C
D
Arrange students in groups of 2–4. Invite students to each choose a different quadrilateral and to classify the quadrilateral as precisely as they can. Then ask them to calculate their quadrilateral’s area and perimeter. Suggest that they refer to the list of their responses from the Warm-up for ideas of properties to use in their classifications.
After a few minutes of quiet work time, invite students to share, with their group, some of their classifications and reasons. Ask the other group members to listen and critique the reasoning of the person who is sharing. Repeat as time allows.
Sample responses:
| area (square units) | perimeter (units) | |
|---|---|---|
| A | 8 |
|
| B | 4 |
|
| C | 4 |
|
|
D |
4 | 8 |
What kind of shape is quadrilateral
We did just show that two pairs of opposite sides of
Can we find the area of triangle
To see if
Help us improve by sharing suggestions or reporting issues.
To Gather
Graph paper
Students are presented with four points and are asked to fully describe the quadrilateral that has those points as its vertices. Slopes will show that all pairs of adjacent sides are perpendicular, making the shape a rectangle. Then students will use the Pythagorean Theorem to calculate both the area and the perimeter of the quadrilateral.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Make graph paper available to students who would like to use it.
A quadrilateral has vertices
Some students may state that the quadrilateral is a rectangle simply because it looks like one. Remind these students that we need to back up our reasoning with mathematics. Suggest that students review their reference charts for definitions and properties of rectangles.
Invite students to share their reasoning for each question. Highlight students who carried information from one question to the next, such as recognizing that in a rectangle, opposite sides have equal length, so they only need to calculate two distances (rather than all four).
To Gather
Geometry toolkits (HS)
Students observe that the angle formed by connecting endpoints of a diameter to a third point on the circle appears to be a right angle. They confirm that this is true for a few particular points, and they then write a conjecture. This conjecture will be generalized in an upcoming unit. The term "chord" will be defined in a subsequent unit. It is not necessary to define it here. As students make a conjecture and later listen to each other's reasoning about triangles and quadrilaterals, they create viable arguments and critique the arguments of others (MP3).
In the digital version of the activity, students use an applet to make their conjectures about the right angle that is made from two endpoints of a diameter and another point on the circle. The applet allows students to move the third point around on the circle, calculate the slopes of the chords, and calculate the product of the slopes.
This activity works best when each student has access to devices that can use the embedded applet, because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the Activity Synthesis.
Direct students’ attention to the reference created using Collect and Display. Ask students to share their conjectures. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “The lines make 90 degrees because they have opposite slopes” already on it and can be updated with the more precise phrase “Segments
There are many ways to correctly word the conjecture. As students work to refine the wording, be sure that the word "diameter" is included, that it’s clear that two points are at the endpoints of the diameter, and that the third point is clearly somewhere else on the circle. Students may describe the result as two segments that form a right angle or three segments that form a right triangle.
Once the conjecture is finalized, ask students if they have proven it (they have not; they’ve just shown it’s true for a few particular cases). Tell students that we will look at a more general case of this assertion in an upcoming unit.
The image shows a circle with several points plotted on the circle.