Download Drawing Triangles AAS

Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 68424
Drawing Triangles AAS
Students are asked to draw a triangle given the measures of two angles and a non-included side and to explain if these conditions determine a
unique triangle.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, triangle, angle, non-included side
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_DrawingTrianglesAAS_Worksheet.docx
MFAS_DrawingTrianglesAAS_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Drawing Triangles AAS worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to draw a triangle with the given conditions.
Examples of Student Work at this Level
The student:
Draws a figure that is not a triangle (e.g., an open figure with three sides).
page 1 of 4 Draws a triangle with incorrect angle or side measures.
Questions Eliciting Thinking
What are the features of a triangle?
What is a nonincluded side?
What strategies would you use to draw a triangle given the two angles and a non-included side? Where is a good place to begin drawing?
Instructional Implications
Define a triangle as a polygon with three sides. Make clear that an open figure with three sides is not a triangle (since it is not a polygon). Describe the parts of a triangle
and how to name them (e.g., the vertices, sides, and angles). Be sure the student understands how to measure angles.
Provide the student with a manipulative such as
or software such as Geogebra (www.geogebra.org) to assist in building triangles with given conditions. The
student may be more adept in drawing triangles with given conditions after working with a hands-on manipulative or software.
Guide the student to draw a triangle with given conditions. Assist the student in using the ruler and protractor to construct the triangle. Explain that a good way to begin is
by drawing a working line and “building” the triangle on it. Ask the student to measure the given length on the working line. Using one endpoint as a vertex, have the
student draw one of the given angles. Next, guide the student to observe that if two angle measures are known, the third measure is also known since the angles in a
triangle must always total 180°. Have the student calculate the third angle measure (60°) and draw the angle using the other endpoint of the known side as its vertex.
Next, have the student extend the sides of these angles to intersect in the third vertex of the triangle. If needed, model how to properly label the angles and sides of a
triangle. Finally, have the student verify that the drawn triangle fits the given conditions.
Provide the student with opportunities to practice constructing triangles given two angle measures and a non-included side.
Moving Forward
Misconception/Error
The student is unable to correctly determine if the given conditions form a unique triangle.
Examples of Student Work at this Level
The student is able to draw a triangle with the given conditions, but says it is possible to draw a different triangle with the same conditions. For example, the student
concludes:
A different triangle can be made by “swapping” angles R and T.
The side lengths can be made longer and the angles can be kept the same.
The triangles are not the same (e.g., congruent) because they are oriented differently.
Questions Eliciting Thinking
What will be different in the new triangle?
What would swapping the angles achieve?
Can you show me how you would make the side lengths longer while the angles stay the same?
Instructional Implications
Use tracing paper to show the student that two triangles can be oriented differently but still be the same (e.g., congruent). If the student did not attempt to construct a
second triangle with the given conditions, ask the student to do so. Have the student confirm that the measures of the side and angles correspond to the given measures.
page 2 of 4 Then have the student use tracing paper to determine if the triangles are congruent.
Another option is to have the student imagine changing the length of side
and the effect this would have on the measure of the opposite angle,
student to observe that changing the measure of the side of a triangle causes the measure of the opposite angle to change as well. Since
R. Guide the
R must measure 45°, the
length of the opposite side cannot take on a different measure. Use this exercise as an opportunity to conclude:
Two angle measures and a non-included side determine a unique triangle.
Two angle measures determine the measure of the third angle of the triangle.
Provide the student with another set of AAS conditions, and encourage the student to further experiment and confirm this conclusion.
Almost There
Misconception/Error
The student does not adequately explain why the given conditions form a unique triangle.
Examples of Student Work at this Level
The student is able to draw a triangle with sides of the given lengths and says it is not possible to draw more than one triangle with these conditions, but does not provide
a clear explanation. The student explains:
“If one side has to be 7 cm, then the rest would still have to be the same.”
“One of the sides has to be 7 cm, so you wouldn’t be able to make any of the other sides bigger.”
Questions Eliciting Thinking
How are the angles and sides of a triangle related?
Can you change the measure of an angle and not affect the length of the opposite side?
Can you change just two side lengths and keep all three angle measures the same?
What conditions must be true for triangles to be congruent?
Instructional Implications
Help the student confirm his or her conclusion by constructing another triangle with the same three measurements. Have the student directly measure the angles and
compare the measurements to the angle measures of the original triangle. Guide the student in discussing the relationship among the sides and angles within a triangle.
Model constructing a triangle with angles and a nonincluded side of the given measures as described in the Getting Started Instructional Implications. Show the student that
the measure of the third angle is fixed by the size of the given two angles. Since one of the side lengths is also fixed by the given conditions, there is only one way to draw
the triangle. Model a concise explanation using mathematical terminology. For example, if m R is 45° and m T is 75°, then m S must be 60°. Because one side length is
also given, the lengths of the other two sides are determined. Therefore, only one triangle is possible.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student is able to draw a triangle with the two given angles and one non-included side measure and says it is not possible to draw more than one triangle with these
conditions. The student explains in terms of:
The uniqueness of the third side (see Instructional Implications for Almost There).
The relationship between the length of a side and the opposite angle measure.
page 3 of 4 Questions Eliciting Thinking
What does congruent mean? How do you know that only one triangle can be drawn given the stated conditions?
Can you describe your strategy in drawing triangle RST?
If
R and
T were switched, would the triangle be congruent to the original?
Does it matter which angles are drawn on the endpoints of the given side?
Instructional Implications
Pair the student with a Moving Forward partner to share strategies for drawing triangles.
Consider implementing the MFAS tasks Drawing Triangles SAS, Drawing Triangles ASA, Drawing Triangles SSS, Drawing Triangles SSA, Drawing Triangles AAA, and/or Sides of
Triangles (7.G.1.2), if not done previously.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Drawing Triangles AAS worksheet
Ruler
Protractor
Technology such as Geogebra (optional)
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.7.G.1.2:
Description
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.
page 4 of 4