This Warm-up prompts students to compare four images. 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 images for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three images that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.
Student Lesson in Spanish
Activity
None
Which three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go 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 that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any terminology they use, such as “right angle,” “acute angle,” and “obtuse angle” and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
9.2
Activity
Standards Alignment
Building On
Addressing
7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
In this activity, students continue creating a triangle from given conditions and seeing if it will match a given triangle. This activity transitions from students just noticing things about triangles already drawn to students drawing triangles themselves to test whether conditions result in unique triangles.
As students work on the task, monitor for students who draw triangles that differ from those of their classmates.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time, followed by time to check with their partner. Discuss whether any of the triangles they drew are identical copies. Follow with a whole-class discussion. Provide access to geometry toolkits that include a ruler marked with centimeters and a protractor.
Activity
None
Three students have each drawn a triangle. For each description:
Draw a triangle with the given measurements.
Measure and label the other side lengths and angle measures in your triangle.
Decide whether the triangle you drew must be an identical copy of the triangle that the student drew. Explain your reasoning.
Jada’s triangle has one angle measuring .
Andre’s triangle has one angle measuring and one angle measuring .
Lin’s triangle has one angle measuring , one angle measuring , and one side measuring 5 cm.
Student Response
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Building on Student Thinking
Students may have trouble recognizing that Lin’s triangle could have the pieces described in different orders. They are likely to immediately think of the side being between the two angles and not visualize other arrangements. Remind students of the task from the previous day and some of the triangles they saw there.
Activity Synthesis
Select previously identified students to share their triangles.
To highlight the fact that there could be different triangles drawn, ask:
“Did anybody draw a triangle that was identical to one drawn by their partner?”
“Do we know enough about Jada’s triangle to draw an identical copy of it? Andre’s triangle? Lin’s triangle?” (no)
If not mentioned by students, explain that it could be possible that we all drew identical copies for Lin’s triangle (because it is most straightforward to draw the 5-cm side in between the 75° and 45° angles). However, that does not mean that we were given enough information about Lin’s triangle to draw an identical copy of it. The problem did not say that we had to put the 5-cm side between those two angles.
Display the image of Lin’s triangle for all to see. Invite students to confirm that it matches the description of Lin’s triangle. Ask whether any student drew an identical copy of Lin’s triangle.
Introduce the word “unique.” Explain to students that in all three cases, the information given is not enough to determine a unique triangle, not even for Lin’s triangle, because there is more than 1 way we can draw a triangle with those given conditions. Ask students, “What information would Lin have to give us to make the triangle unique (so we knew our drawing would be an identical copy of her triangle)?”
Before moving on to the next activity, it would be helpful to model how Lin drew her triangle:
Draw the 5 cm segment.
Draw the angle on one end of the segment, with a very long ray.
Place a protractor along the ray.
Line up a ruler at the mark on the protractor.
Keeping the ruler and protractor together, slide them along the ray until the edge of the ruler intersects with the other end of the 5 cm segment.
Keeping the ruler in place on the paper, remove the protractor from underneath it.
Draw a line along the ruler from the ray to the segment.
MLR8 Discussion Supports. Display sentence frames to help students explain whether any of the triangles they drew are identical copies to Jada’s, Andre’s or Lin’s triangles. Examples: “I noticed so I . . . .” “This triangle is/isn’t identical because . . . .” Advances: Speaking, Conversing
9.3
Activity
Standards Alignment
Building On
Addressing
7.G.A.2
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
In this activity, students are asked to draw as many different triangles as they can with the given conditions. The purpose of this activity is to provide an opportunity for students to see the three main results for this unit: a situation in which only a unique triangle can be made, a situation in which it is impossible to create a triangle from the given conditions, and a situation in which multiple triangles can be created from the conditions. Students make use of the structure of triangle side lengths and angles as they explore these conditions (MP7).
Students are not expected to remember which conditions lead to which results, but should become more familiar with some methods for attempting to create different triangles. They will practice including various conditions into the triangles, including the conditions in different combinations, and practice recognizing when the resulting triangles are identical copies or not.
In the digital version of the activity, students use an applet to create triangles with given conditions. The applet allows students to change the conditions dynamically.
Launch
Keep students in the same groups. Tell students they must try at least two different times to draw a triangle with the measurements given in each problem. Give students 5 minutes of quiet work time followed by time to discuss their different triangles with a partner. Follow with a whole-class discussion. Provide access to geometry toolkits.
Action and Expression: Internalize Executive Functions. To support organization, provide students with a printed graphic organizer to categorize the different triangles by condition. Supports accessibility for: Language, Organization
Activity
None
Draw as many different triangles as you can with each of these sets of measurements:
Two angles measure , and one side measures 4 cm.
Two angles measure , and one side measures 4 cm.
One angle measures , one angle measures , and one side measures 4 cm.
Which of these sets of measurements determine one unique triangle? Explain or show your reasoning.
Activity Synthesis
Ask students to indicate how many different triangles (triangles that are not identical copies) they could draw for each set of conditions. Select students to share their drawings and reasoning about the uniqueness of each problem. Discuss the methods that students used as they thought about other triangles that might fit the conditions.
Consider asking some of the following questions:
“Which conditions produced a unique triangle?” (the first set of conditions)
“Were there conditions that produced more than one triangle?” (the third set of conditions)
“Were there conditions you could not draw a triangle for?” (the second set of conditions)
“Why could you not draw a triangle for the second set of conditions?” (because two sides are parallel and will never intersect)
If not mentioned by students, explain to students that for the third set of conditions it is possible that all students drew identical copies using the 4-cm length as the side between the and angles. Consider asking them to think of the previous activity and to try to draw the triangle the way Lin would.
In grade 7, students do not need to know that the angle measures within triangle have a sum of . Tell them that next year they will learn more about why these different conditions determine different numbers of triangles.
Lesson Synthesis
Here are some questions for discussion:
Sometimes a set of conditions result in a unique triangle. What other results can come from a set of conditions? (It could be impossible, or they could be used to make multiple triangles.)
If you are given a side length and two angles, what would you do to try to get started making different triangles? (Draw a line segment with the given length and put the two angles on each end. Then I would try leaving one angle on one end, but using Lin’s method of using a protractor and sliding it along for the other angle to create a triangle. Finally, I would do something similar, but switch which angle is next to the given length.)
Student Lesson Summary
Sometimes, we are given two different angle measures and a side length, and it is impossible to draw a triangle. For example, there is no triangle with side length 2 and angle measures and :
In the figure a horizontal line segment is drawn and labeled 2. On the left end of the line segment, a dashed line is drawn upward and to the left. The angle formed between the dashed line and the horizontal line is labeled 120 degrees. On the right end of the horizontal line, a dashed line is drawn upward and to the right. The angle formed between the dashed line and horizontal line is labeled 100 degrees.
Sometimes, we are given two different angle measures and a side length between them, and we can draw a unique triangle. For example, if we draw a triangle with a side length of 4 between angles and , there is only one way in which they can meet up and make a triangle:
Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and the same angle measures.
Standards Alignment
Building On
Addressing
7.G.A
Draw, construct, and describe geometrical figures and describe the relationships between them.
Some students may draw two different orientations of the same triangle for the third set of conditions, with the 4-cm side in between the and angles. Prompt them to use tracing paper to check whether their two triangles are really different (not identical copies).
Some students may say the third set of measurements determines one unique triangle, because they assume the side length must go between the two given angle measures. Remind them of the discussion about Lin’s triangle in the previous activity.
