few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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.

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


Concept/skill: 
Solve quadratic equations in one variable.

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

Example 3:
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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

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 variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
problem_activity: Warm-up
The purpose of this warm-up is for students to begin to see the need for samples of data when the population is too large. In this activity, students are asked to think about a question involving all the students at their school and compare the question to an earlier lesson in which the population was small and it was easy to obtain data for the entire population.
Teacher Notes for IM 6–8 Accelerated
The activity referring to song lengths in the second question is not included in IM 6–8 Math Accelerated. If necessary, tell students that the albums each contained either 7 or 8 songs.
Launch
Give students 2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
Consider the question: In general, do the students at this school have more letters in their first name or last name? How many more letters?
What are some ways you might get some data to answer the question?
The other day, we compared the heights of people on different teams and the lengths of songs on different albums. What makes this question about first and last names harder to answer than those questions?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of the discussion is to highlight the methods of getting data for the school more than it is the method of computing the answer.
Select some students to share their responses.
Students who have elected to sum all the letters in the first names in the school and all the letters in the last names in the school may note that it is a simple comparison to tell whether there are more in first or last names, since you get one single large number for each group. (Comparing data sets.)
Students who have elected to calculate the mean for each group and use MAD as a method of comparison may note that while the calculations may take more time, they give you more precise information, such as knowing about how long first names and last names are, as well as a way to compare the two sets. (Using the general rule from the previous lesson.)
Students who suggest surveying a small group of students may point out that it would be easier to do the calculation with a smaller group.  The information would not be as accurate, but it would take a lot less time and might give a good general idea. It would depend on how accurate you needed your answer to be.  (Introduction to sampling.)


aligns: no