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 following problem or activity 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

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im_task_001160_neg_0

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standard_description: Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.
problem_activity: Narrative
The purpose of this activity is for students to practice identifying fractional intervals along a number line. This is Stage 2 of the center activity, Number Line Scoot. This activity encourages students to count by the number of intervals (the numerator). Students have to land exactly on the last tick mark, which represents 4, to encourage them to move along different number lines. While this activity does not focus on equivalence, it gives students exposure to this idea before they work more formally with it in the next section. In the synthesis, students relate counting on a number line marked off in whole numbers to their number lines marked off in fractional-sized intervals.
It may be helpful to play a few rounds with the whole class to be sure students are clear on the rules of the game. Keep the number line game boards for center use.
Engagement: Develop Effort and Persistence.
Check in and provide each group with feedback that encourages collaboration and community. For example, check in after the first round of Number Line Scoot.
Supports accessibility for: Attention, Social-Emotional Functioning
Required Materials
Materials to Gather
Base-ten blocks
Number cubes
Materials to Copy
Number Line Scoot Stage 2 Gameboard
Number Line Scoot Stage 2 Directions
Required Preparation
Each group of 2 students needs a number cube.
Each student needs at least 5 base-ten cubes to use as game pieces.
Launch
Groups of 2
Give each group of 2 a gameboard, two recording sheets, a number cube, and at least 10 base-ten cubes.
“Now you will play a game where you move, by fractions, along different number lines. To start, each player places a small cube on zero on each number line. The goal of the game is to get as many small cubes as you can to the 4 on any of the number lines.”
Roll the number cube, demonstrate where to record the rolled number and move that fraction along one of the number lines.
Activity
10 minutes: partner work time
As students work, monitor for students who count by the numerator once they have chosen a number line.
Activity Synthesis
Display a gameboard with a marker on
\(\frac{3}{4}\)
.
“If I rolled a 4, and chose to move
\(\frac{4}{4}\)
, how would you count the move?” (I would count 1, 2, 3, 4.)
“How did you know you have moved
\(\frac{4}{4}\)
?” (Because each space is
\(\frac{1}{4}\)
, so I need to move 4 times.)
Display a number line marked with only 0, 1, 2, 3, 4.
“How is counting along this number line the same and different than counting along your number lines?” (On the whole number one each space is 1 so we just count 1, 2, 3, 4. On our number lines we still count the jumps, but now each space is smaller than 1 so we need the denominator to tell us the size of each space.)


aligns: no