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: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)
problem_activity: Narrative
The purpose of this activity is for students to compare three-digit numbers based on their understanding of place value. They are invited to explain or show their thinking in any way that makes sense to them. A number line is provided. Students may revise their thinking after locating the numbers on the number line, or may choose to draw diagrams to represent their thinking. During the activity synthesis, methods based on comparing the value of digits by place are highlighted.
For the last problem, students persevere in problem solving as there are many ways to make most of inequalities true but students will need to think strategically in order to fill out all of them. In particular, 810 can be used in the first, second, or fourth inequality but it needs to be used in the fourth because it is the
only
number on the list that is larger than 793.
Representation: Develop Language and Symbols.
Represent the problem in multiple ways to support understanding of the situation. For example, have one group member place the numbers on a number line to verify which number is greater. Another option is to have one group member make the numbers using base-ten blocks to prove which is greater.
Supports accessibility for: Conceptual Processing, Organization
Launch
Groups of 2
“In the warm-up, you saw that different forms of writing a number can help you think about the value of each digit.”
Write
\(564\phantom{3} \boxed{\phantom{33}}\phantom{3}504\)
on the board.
“What symbol would make this expression true? Explain.” (>, because they both have 500, but the first number has 6 tens or 60 and the second number has 0 tens.)
Activity
“Today you will be comparing three-digit numbers by looking at place value.”
“If it helps, you can use base-ten diagrams or expanded form to help you think about place value.”
“Try it on your own and then compare with your partner.”
5 minutes: independent work time
5 minutes: partner work time
Monitor for students who compare the numbers without drawing base-ten diagrams or using a number line.
Student Facing
Compare the numbers.
\(>\)
,
\(=\)
, or
\(<\)
521
\(\boxed{\phantom{33}}\)
523
Explain or show your thinking. If it helps, use a diagram or number line.
\(>\)
,
\(=\)
, or
\(<\)
889
\(\boxed{\phantom{33}}\)
878
Explain or show your thinking. If it helps, use a diagram or number line.
Place the numbers in the blanks to make each comparison true. Use each number only once. Use base-ten diagrams or a number line if it helps.
810
529
752
495
\(\underline{\hspace{1 cm}} > 519\)
\(687 < \underline{\hspace{1 cm}}\)
\(\underline{\hspace{1 cm}} < 501\)
\(\underline{\hspace{1 cm}} > 793\)
Student Response
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Advancing Student Thinking
If students write comparison statements that are not true, consider asking:
“Could you read each statement aloud?”
“How did you know your statement is true?”
“How could you use the base-ten diagram or number line to help you show whether your statement is true or false?”
Activity Synthesis
Display
\(564 > 504\)
.
Display
\(500 + 60 + 4 > 500 + 4\)
.
“We decided that this was a true statement. How does the expanded form of these numbers help justify our thinking?” (We can see the value of each place, so we can compare each digit.)
Invite previously selected students to share how they compared numbers without drawing diagrams or using the number line.


aligns: no