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: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only 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_neg_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
problem_activity: Narrative
In this activity, students use rectangular diagrams and similar reasoning as in earlier activities to represent the multiplication of 2 two-digit numbers. They analyze a progression of diagrams, starting with those that represent multiplication of two-digit and one-digit numbers (18 and 6), a two-digit number and a ten (18 and 10), and then 2 two-digit numbers (18 and 16).
Students may decompose factors in different ways. For example, those who are familiar with multiples of 25 may find it intuitive to decompose
\(25 \times 46\)
as
\(25 \times 40\)
and
\(25 \times 6\)
, rather than decomposing 25 into
\(20 + 5\)
.
MLR8 Discussion Supports.
Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms such as partial product.
Advances: Listening, Representing
Action and Expression: Develop Expression and Communication.
Give students access to base-ten blocks. If students use the blocks for the last question, ask them to also draw a diagram that represents their work with the blocks.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Display the three diagrams in the activity.
“What do you notice? What do you wonder?”
1 minute: quiet think time
1 minute: partner discussion
Activity
“Take a few quiet minutes to work on the first problem. Then, share your thinking with your partner.”
3 minutes: independent work time
3 minutes: partner discussion
Invite students to share their responses: “What multiplication expression can be represented by each diagram?”
“Complete the rest of the activity.”
3 minutes: independent or partner work time
Monitor for students who decompose both factors into tens and ones and those who choose to keep one of the factors intact.
Student Facing
For each diagram, write a multiplication expression that the diagram can represent. Then, find the value of the expression. Use equations to show or explain your reasoning.
[IMG]
[IMG]
[IMG]
How are the diagrams alike? How are they different? Discuss with your partner.
Use a diagram to find each product.
\(13 \times 21\)
\(25 \times 46\)
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Activity Synthesis
Select two students to share their diagrams, solutions, and reasoning for the last problem.
“How did you decompose the factors and the diagram?” (I decomposed the 13 into 10 and 3 and the 21 into 20 and 1.)
“What expression could we write to show the partial product represented by each part of the diagram?”
“How do these partial products help us find the value of
\(13 \times 21\)
?” (Adding them gives us the value of
\(13 \times 21\)
:
\(200 + 10 + 3 + 60 = 273\)
)


aligns: yes