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 addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
problem_activity: Narrative
The purpose of this activity is for students to identify two-digit numbers or a part of a number represented in different ways, with different amounts of tens and ones. Students determine how many connecting cubes are in each bag, given clues about how many tens and ones are in the bag. Students also determine how many tens or ones are in a bag, given the total number of cubes and either the number of tens or ones. Students may use connecting cubes to make sense of the problems, and show their thinking using drawings, numbers, or words.
MLR8 Discussion Supports.
During partner work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Required Materials
Materials to Gather
Connecting cubes in towers of 10 and singles
Launch
Groups of 4
Give students access to connecting cubes in towers of 10 and singles.
Activity
“You are going to solve problems about connecting cubes in mystery bags. You can use connecting cubes if they will help you. Show your thinking using drawings, numbers, or words.”
“You will begin by working on your own. Then you will share your thinking with a partner at your table.”
6 minutes: independent work time
“Share your thinking for problem 1 with a partner at your table.”
1 minute: partner discussion
“Share your thinking for problem 2 with a different partner at your table.”
1 minute: partner discussion
Repeat for problems 3 and 4.
Monitor for students who use connecting cubes in these ways to solve for mystery bag C:
Shows 49 as 4 tens 9 ones and moves 2 tens over to the ones cubes to have 29, shows 2 tens left.
Shows 29 ones, adds towers of 10 to get to 49.
Student Facing
Bag A has 2 ones and 5 tens.
How many cubes are in Bag A?
Show your thinking using drawings, numbers, or words.
Bag B has 4 tens and 25 ones.
How many cubes are in Bag B?
Show your thinking using drawings, numbers, or words.
Bag C has 49 cubes.
If there are 29 ones, how many tens are in the bag?
Show your thinking using drawings, numbers, or words.
Bag D has 36 cubes.
If there are only 2 tens, how many ones are in the bag?
Show your thinking using drawings, numbers, or words.
If you have time: Write a mystery bag problem about tens and ones.
Switch with your partner.
Solve.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Advancing Student Thinking
If students show they are adding the total number of cubes and the known tens or ones for Bag C or Bag D, consider asking:
“How could you act out this problem?”
“What do you know? What don’t you know?”
“How could you use what we’ve learned about making numbers with different amounts of tens and ones to solve the problem?”
Activity Synthesis
Invite previously identified students to share.
“How are these ways for finding the mystery number of tens the same? How are they different?” (They both used tens and ones. One person started with 29 ones and added the tens, the other person started with 4 tens 9 ones and broke apart tens until there were 29 ones.)


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