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: Use place value understanding to round decimals to any place.
problem_activity: Narrative
The purpose of this activity is for students to consider possible mistakes when multiplying large numbers. Monitor for students who:
revise their answer after examining Kiran’s mistake.
recognize that
\(650 \times 10 = 6,\!500\)
so
\(650 \times 27\)
has to be much greater than 5,850.
can explain why Kiran should be multiplying
\(650 \times 2 \times 10\)
.
recognize that
\(20 \times 50 = 1,\!000\)
so there should be three zeros in the second partial product.
When students determine Kiran's error and make sense of his work, they interpret and critique the work of others (MP3).
MLR1 Stronger and Clearer Each Time.
Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “what parts of Kiran’s work do you agree and disagree with?” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening
Launch
Display or write for all to see.
\(650 \times 27\)
Display each number in a different corner of the room:
14,000
18,000
13,000
19,000
“When I say go, stand in the corner with the number that you think is the most reasonable estimate for
\(650 \times 27\)
. Be prepared to explain your reasoning.”
1 minute: quiet think time
Ask a representative from each corner to explain their reasoning.
“Does anyone want to switch corners?”
Ask a student who switched corners to explain their reasoning.
“Now you are going to find this product and analyze some work.”
Activity
Groups of 2
5–7 minutes: partner work time
Student Facing
Find the value of the product.
[IMG]
Below is Kiran’s work finding the value of the product
\(650 \times 27\)
. Is his answer reasonable? Explain your reasoning.
[IMG]
What parts of the work do you agree with? Be prepared to explain your reasoning.
What parts of the work do you disagree with? Be prepared to explain your reasoning.
Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.
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
Ask previously identified students to share their thinking.
Display Kiran's work.
“Why doesn’t 5,850 make sense?” (
\(650 \times 10 = 6,\!500 \)
so
\(650 \times 27\)
should be a lot larger than 6,500.)
“What makes sense about Kiran’s work?” (
\(650 \times 2 =1,\!300\)
, but he needs to multiply
\(650\times2\times10\)
.)
Display a student's solution or the image from the student solution.
“How do we know that 17,550 is a reasonable value for the product?” (Because
\(600 \times 30 =18,\!000\)
.)


aligns: no