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: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

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: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

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_pos_0

HS_2
standard_description: Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
problem_activity: Narrative
This optional activity gives students an additional opportunity to practice using “jumps” on number lines to subtract fractions, decomposing any whole numbers as needed along the way. (It is similar in structure to an optional activity in an earlier lesson on addition.)
Students are given four number lines with a point marked on each. They then draw a card with a fraction on it. All fractions on the cards, shown here, are greater than the values of the points. Students make one or more jumps to find the difference of the two points and then represent it with a subtraction equation.
\(\frac{8}{5} \qquad \frac{9}{5} \qquad \frac{10}{5} \qquad \frac{11}{5} \qquad \frac{12}{5} \qquad \frac{13}{5} \qquad \frac{14}{5} \qquad \frac{15}{5}\)
Required Materials
Materials to Copy
Make a Jump, Subtraction Edition
Required Preparation
Create a set of cards from the blackline master for each group of 2.
Launch
Groups of 2
Give each group a set of fraction cards from the blackline master.
If students did not complete the optional activity in the previous lesson (which had a similar structure):
Display the four number lines.
“What do you notice? What do you wonder?”
30 seconds: quiet think time
30 seconds: partner discussion
Activity
“Label each point on the number line with a fraction it represents. This is your target.”
1 minute: independent work time
“You will now make one or more jumps from another fraction to your target and write an equation to represent the difference between the two.”
Explain how to use the cards and how to complete the task.
Monitor for students who facilitate subtraction by:
rewriting a fraction on their card as a whole number or a mixed number
labeling their number lines with whole numbers beyond 1
Student Facing
Here are four number lines, each with a point on it. Label each point with a fraction it represents.
[IMG]
[IMG]
[IMG]
[IMG]
The point you labeled is your target.
Pick a card from the set given to you. Locate and label the fraction on the number line.
From that point, draw one or more jumps to reach the target. What do you need to subtract? Label each jump you draw.
Write an equation to represent the difference of your two fractions.
Student Response
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Activity Synthesis
Select students to share their responses to the first couple of diagrams (or more if time permits).


aligns: no