few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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

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: 
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, 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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Fluently multiply multi-digit whole numbers using the standard algorithm.
problem_activity: Narrative
The purpose of this activity is for students to find quotients of a whole number by tenths or hundredths. This activity focuses on divisors of 0.2 and 0.02 to build on students' previous work dividing by 0.1 and 0.01. The strategies, diagrams and place value reasoning used in the previous lessons still apply. Students may also think about the problems using multiplication. For example
\(1 \div 0.2 = 5\)
because
\(5 \times 0.2 = 1\)
.
Required Materials
Materials to Copy
Small Grids
Launch
Groups of 2
Give students access to blackline masters with grids.
Activity
1–2 minutes: quiet think time
8–10 minutes: partner work time
Monitor for students who represent their reasoning with:
diagrams
expressions or words using multiplication
Student Facing
Find the value of each expression. Explain or show your reasoning.
\(1 \div 0.2\)
\(2 \div 0.2\)
\(3 \div 0.2\)
\(4 \div 0.2\)
Find the value of each expression. Explain or show your reasoning.
\(1 \div 0.02\)
\(2 \div 0.02\)
\(3 \div 0.02\)
\(4 \div 0.02\)
What patterns do you notice?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If a student is not able to use the value of the first expression in each set to find the value of the later expressions, ask them to describe what they notice and wonder about each set of expressions. Use numbers and symbols when possible to record the relationships they describe.
Activity Synthesis
Ask selected students to share the representations listed above.
“How are the representations the same? How are they different?” (They both figured out how many tenths or hundredths are in one whole. Then they multiplied that by the number of wholes. One of them used a picture and one of them reasoned about the size of tenths and hundredths to find how many 0.2's and 0.02's are in one whole.)
Display a student's completed work showing the values of the quotients.
“What patterns do you see across the sets of equations?” (The quotients with a divisor of 0.01 are 10 times larger than those with a divisor of 0.1. The quotients with a divisor of 0.1 are a tenth of the value of those with a divisor of 0.01.)
Record the patterns as students describe them.


aligns: no