Lesson 9

Drawing Triangles (Part 1)

9.1: Which One Doesn’t Belong: Triangles (5 minutes)

Warm-up

This warm-up prompts students to compare four images. It encourages students to explain their reasoning, hold mathematical conversations, and gives you the opportunity to hear how they use terminology and talk about characteristics of the images in comparison to one another. To allow all students to access the activity, each image has one obvious reason it does not belong. Encourage students to move past the obvious reasons (e.g., Figure A has 3 equal angles) and find reasons based on geometrical properties (e.g., Figure A is the only figure whose sides seem to have equal length). During the discussion, listen for important ideas and terminology that will be helpful in upcoming work of the lesson.

Launch

Arrange students in groups of 2–4. Display the image for all to see. Ask students to indicate when they have noticed one image that doesn’t belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask the group to offer at least one reason each image doesn’t belong.

Student Facing

Which one doesn’t belong?

Four triangles, A, B, C, D.

 

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 image does not belong. Record and display the responses for all to see. After each response, poll the class if they agree or disagree. Since there is no single correct answer to the question of 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. Also, press students on unsubstantiated claims.

9.2: Does Your Triangle Match Theirs? (15 minutes)

Activity

In this activity, students continue the work from the previous lesson by creating triangles 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 student work on the task, monitor for students who draw different triangles than each other.

Launch

Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by time to check with their partner and discuss whether any of the triangles they drew are identical copies. Follow with whole-class discussion. Provide access to geometry toolkits, including rulers marked with centimeters and protractors.

Students using the digital version can create new triangles by dragging the vertices of the equilateral triangle in the applet. The measurements will be made for them, allowing them to focus on the new ideas.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts: Remind students to be measuring with centimeters and demonstrate how to use a protractor to draw the given angle.
Supports accessibility for: Conceptual processing; Memory

Student Facing

Three students have each drawn a triangle. For each description:

  1. Drag the vertices to create a triangle with the given measurements.

  2. Make note of the different side lengths and angle measures in your triangle.  
  3. Decide whether the triangle you made must be an identical copy of the triangle that the student drew. Explain your reasoning.

Jada’s triangle has one angle measuring 75°.

Andre’s triangle has one angle measuring 75° and one angle measuring 45°.

Lin’s triangle has one angle measuring 75°, one angle measuring 45°, and one side measuring 5 cm.
 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by time to check with their partner and discuss whether any of the triangles they drew are identical copies. Follow with whole-class discussion. Provide access to geometry toolkits, including rulers marked with centimeters and protractors.

Students using the digital version can create new triangles by dragging the vertices of the equilateral triangle in the applet. The measurements will be made for them, allowing them to focus on the new ideas.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts: Remind students to be measuring with centimeters and demonstrate how to use a protractor to draw the given angle.
Supports accessibility for: Conceptual processing; Memory

Student Facing

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.
  1. Jada’s triangle has one angle measuring \(75^\circ\).

  2. Andre’s triangle has one angle measuring \(75^\circ\) and one angle measuring \(45^\circ\).

  3. Lin’s triangle has one angle measuring \(75^\circ\), one angle measuring \(45^\circ\), and one side measuring 5 cm.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Three triangles, varied attributes.

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:

  1. Draw the 5 cm segment.
  2. Draw the \(75^\circ\) angle on one end of the segment, with a very long ray.
  3. Place a protractor along the ray.
  4. Line up a ruler at the \(45^\circ\) measure on the protractor.
  5. 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.
  6. Keeping the ruler in place on the paper, remove the protractor from underneath.
  7. Draw a line along the ruler from the ray to the segment.
An angle, a protractor, a ruler.
Speaking, Listening: MLR8 Discussion Supports. Use this routine to help students explain whether any of the triangles they drew are identical copies to Jada’s, Andre’s or Lin’s triangles. Provide sentence frames such as: “I noticed ___ so I …” ; “This triangle is/isn’t identical because….” These help students use mathematical language related to triangles (e.g., angle, side) to reason about whether their triangle is identical to a given triangle.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

9.3: How Many Can You Draw? (15 minutes)

Activity

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 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 recognizing when the resulting triangles are identical copies or not.

Launch

Keep students in 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. Provide students with a printed graphic organizer to categorize the different triangles by condition.
Supports accessibility for: Language; Organization

Student Facing

  1. Draw as many different triangles as you can with each of these sets of measurements:

    1. Two angles measure \(60^\circ\), and one side measures 4 cm.
    2. Two angles measure \(90^\circ\), and one side measures 4 cm.
    3. One angle measures \(60^\circ\), one angle measures \(90^\circ\), and one side measures 4 cm.
  2. Which sets of measurements determine one unique triangle? Explain or show your reasoning.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Keep students in 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. Provide students with a printed graphic organizer to categorize the different triangles by condition.
Supports accessibility for: Language; Organization

Student Facing

  1. Draw as many different triangles as you can with each of these sets of measurements:

    1. Two angles measure \(60^\circ\), and one side measures 4 cm.

    2. Two angles measure \(90^\circ\), and one side measures 4 cm.

    3. One angle measures \(60^\circ\), one angle measures \(90^\circ\), and one side measures 4 cm.

  2. Which of these sets of measurements determine one unique triangle? Explain or show your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Nine toothpicks are arranged to make 3 equilateral triangles, their bases form a horizontal line.

In the diagram, 9 toothpicks are used to make three equilateral triangles. Figure out a way to move only 3 of the toothpicks so that the diagram has exactly 5 equilateral triangles.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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 \(60^\circ\) and \(90^\circ\) 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.

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 methods students used to try to think 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 \(60^\circ\) and \(90^\circ\) 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 angles within a triangle sum to \(180^\circ\). Tell them that next year they will learn more about why these different conditions determine different numbers of triangles.

Speaking: MLR7 Compare and Connect. Use this routine to compare and contrast the different ways students reasoned about the uniqueness of the constructed triangles. Ask students to consider what is the same and what is different about the triangles produced for each condition. Draw students’ attention to the association between the conditions given and the ability to construct unique, many, or no triangles. In this discussion, model the language used to make sense of the conditions that resulted in the three different scenarios. These exchanges strengthen students’ mathematical language use and supports them to compare geometric shapes.
Design Principle(s): Maximize meta-awareness; Support sense-making

Lesson Synthesis

Lesson Synthesis

  • 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 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.)

9.4: Cool-down - Checking Diego’s Triangle (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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 \(120^\circ\) and \(100^\circ\):

Figure of a horizontal line segment with dashed line segments forming angles at each end.

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 \(90^\circ\) and \(60^\circ\), there is only one way they can meet up and complete to a triangle:

A segment 4 units long is drawn.  A dotted line is drawn perpendicular to one end of the segment, a dotted line is drawn at a 60 degree angle to the other end of the segment.  The dotted lines meet.

Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and same angle measures.