options: 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.

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
problem_activity: Optional activity
This activity reviews the work students did previously drawing shapes with given conditions. Students draw as many different triangles as they can that could be the base of the triangular prism, given two side lengths and one angle measure for the triangle.
In preparation for calculating surface area and volume in the next activity, students select one of their triangles and find its area. This will require them to draw and measure the height of the triangle. As needed, remind students that the height must be perpendicular to whichever side they are using as the base of their triangle. Also, prompt students to measure the height as precisely as possible, because it will influence the accuracy of their later calculations.
Launch
Provide access to geometry toolkits and compasses.
Student Facing
The base of a triangular prism has one side that is 7 cm long, one side that is 5.5 cm long, and one angle that measures
\(45^\circ\)
.
Draw as many different triangles as you can with these given measurements.
Select one of the triangles you have drawn. Measure and calculate to approximate its area. Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
Students may try to multiply two side lengths of the triangle to calculate the area. Remind them that the height must be perpendicular to the base. If necessary, demonstrate using an index card to draw in the height of the triangle.
Activity Synthesis
Ask students to share triangles they drew so that everyone has an opportunity to see all four triangles. If any of the four triangles are not presented by students, demonstrate how to construct it. Ensure the class agrees that 4 unique triangles have the given measurements. The third side length of the triangle could be 5.0 cm, 7.3 cm, 2.6 cm, or 9.7 cm.
Make sure students have calculated the area of their selected triangle correctly, because this will affect their volume and surface area calculations in the next activity.
| If the third side of the triangle is | then the area of the triangle should be about |                                                                                              possible strategies                                                                                              |
|--------------------------------------|-----------------------------------------------|---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|                5.0 cm                |          13.7 cm                   2          |                                          \(\frac12 \boldcdot 7 \boldcdot (3.9)\)                  or                    \(\frac12 \boldcdot (5.5) \boldcdot (5.0)\)                                           |
|                7.3 cm                |          18.2 cm                   2          | \(\frac12 \boldcdot 7 \boldcdot (5.2)\)                  or                   \(\frac12 \boldcdot (5.5) \boldcdot (6.6)\)                  or                    \(\frac12 \boldcdot (7.3) \boldcdot (5.0)\)  |
|                2.6 cm                |          6.3 cm                   2           | \(\frac12 \boldcdot 7 \boldcdot (1.8)\)                  or                    \(\frac12 \boldcdot (5.5) \boldcdot (2.3)\)                  or                    \(\frac12 \boldcdot (2.6) \boldcdot (5.0)\) |
|                 9.7                  |          18.9 cm                   2          | \(\frac12 \boldcdot 7 \boldcdot (5.4)\)                  or                    \(\frac12 \boldcdot (5.5) \boldcdot (6.9)\)                  or                    \(\frac12 \boldcdot (9.7) \boldcdot (3.9)\) |
Representing, Conversing: MLR7 Compare and Connect.
Invite students to prepare a visual display that shows how they measured and calculated the approximate area for their chosen triangle. Students should consider how to display their calculations so that another student can interpret them. For example, some students may wish to add notes or details to their drawings to help communicate their thinking. Arrange students in groups of 2–4. Give 2–3 minutes of quiet think time for students to read and interpret each other's calculations before they begin to discuss them. Display a list of questions that students can ask each other about their work. For example, "Can you show me how you measured the side lengths?", “How did you calculate the area of the triangle?”, etc. During the whole-class discussion, draw students’ attention to the relationship between measuring and approximating area. Emphasize the language used to make sense of the strategies used to measure and approximate area. These exchanges strengthen students’ mathematical language use and reasoning based on the relationship between measures and area.
Design Principle(s): Maximize meta-awareness


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