few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Example 2:
Concept/skill: 
Solve quadratic equations in one variable.

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
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
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
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


aligns: no