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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

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: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
In this activity, students determine the length of a “diagonal” line segment on a grid. Students can give an exact value for the length of the line segment by finding the area of a square and writing the side length using square root notation. The goal of this activity is for students to connect values expressed using square roots with values expressed in decimal form—a form they are more familiar with.
Monitor for students who draw a tilted square for the first problem during the first two minutes of work time. Then monitor for students who use the following strategies to find the length of the segment:
drawing a square and finding the area
using tracing paper
using a compass to make a circle
Launch
For this activity, it is best if students do not have access to a calculator with a square root button. If student calculators do have a square root button that students are familiar with, tell students that their explanations about their answers to the second problem need to dig deeper than pressing a button. In later lessons, however, they will be able to use it.
Before students begin, remind students that “exact length” means it can’t just be an approximation, so if it is not a rational number, we should write it with square root notation. For example, a square with area 17 has a side length of exactly
\(\sqrt{17}\)
, which is a little larger than 4, since
\(4^2=16\)
.
Begin by displaying the diagram for all to see. Ask students how this diagram is similar and how it is different from the diagram in the warm-up. Then 2–3 minutes after students begin working, pause the class and select a previously identified student who drew a square on the grid to share what they did and why. Give 2–3 minutes work time to finish the problems followed by a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Provide students with access to tracing paper, straight edges, and compasses to support information processing in estimating the length of the line segment.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
[IMG]
Estimate the length of the line segment to the nearest tenth of a unit (each grid square is 1 square unit).
Find the exact length of the segment.
Student Response
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Activity Synthesis
Select students to present in this sequence:
Someone who drew a square and used the area to find the exact side length.
Someone who used tracing paper. This is essentially like the number line as a ruler. Ask students what this tells us about the exact value we found with the square. (
\(\sqrt{10}\)
is about 3.1.)
Someone who used a compass to find the approximate side length. This is a more formal geometric construction, but it is just another way to use the number line as a ruler.
Speaking, Listening: MLR8 Discussion Supports.
As students share their methods for finding the exact length of the line segment, press for details in students’ reasoning by asking how they know the figure they drew in the coordinate plane is a square. Listen for and amplify the language students use to describe the important features of the square (e.g., opposite sides are equal, opposite sides are parallel, each angle is
\(90^\circ\)
). Then ask students to explain why the side length of the tilted square must be
\(\sqrt{10}\)
. This will support rich and inclusive discussion about strategies for finding the exact length of a line segment in the coordinate plane.
Design Principle(s): Support sense-making


aligns: yes