few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Narrative
The purpose of this activity is for students to use the weights from the previous activity to support place value understanding, specifically to see the multiplicative relationships between different decimal place values (MP7). These relationships will be taken up in greater detail in the next unit but the weights provide a convenient way to see these relationships which complements the diagrams students used in earlier lessons.
Students first compare weights of two gold nuggets, one weighed using 0.1 ounce weights and the other using 0.01 ounce weights. The two nuggets have the same weight because ten 0.01 ounce weights are equivalent to one 0.1 ounce weight, as students saw in the warm-up. Students move from here to making multiplicative comparisons between place values. They can use the weights to help visualize or calculate or they might use a diagram like those in the previous lesson.
Representation: Internalize Comprehension.
Synthesis: Provide students with a partially completed graphic organizer, such as a two column table, to record the relationship between tenths and hundredths, hundredths and thousandths, and tenths and thousandths.
Supports accessibility for: Conceptual Processing, Memory
Launch
Groups of 2
Activity
2 minutes: independent work time
8 minutes: partner work time
Monitor for students who use different strategies to compare the values of the 6s in the gold nugget weights:
using the value of each place in the decimal
thinking about fractions or representing the decimals as fractions
using a diagram such as a hundredths grid
Student Facing
How many 0.01 ounce weights will balance one 0.1 ounce weight? Explain or show your reasoning.
How many 0.001 ounce weights will balance a 0.1 ounce weight? Explain or show your reasoning.
The table shows the weights of 3 of the gold nuggets Diego and his friends found panning for gold.
Fill in the blanks. Explain or show your reasoning.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"style": "text-align:center"}, "_value": "gold"}, {"_attributes": {"style": "text-align:center"}, "br": [{}], "_values": ["weight", "(grams)"]}]}, {"td": [{"_value": "nugget A"}, {"_value": "0.6"}]}, {"td": [{"_value": "nugget B"}, {"_value": "0.06"}]}, {"td": [{"_value": "nugget C"}, {"_value": "0.006"}]}]}]}]}
Nugget A weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget B.
Nugget A weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget C.
Nugget C weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget B.
Nugget C weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget A.
Student Response
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Activity Synthesis
“How are the weights of the nuggets the same? How are they different?” (They all have a 6 in them. They each balance with 6 of one of the unit weights. The value of the 6 is different for each nugget.)
“How do hundredths, the weights for B, compare to tenths, the weights for A?” (There are 10 hundredths in each tenth or a tenth of a tenth is a hundredth.)
“Nugget B weighs how many times as much as Nugget A? How do you know?” (
\(\frac{1}{10}\)
because Nugget A is 6 tenths and a tenth of that is 6 hundredths since a tenth of a tenth is a hundredth.)
“How do thousandths, the weights for C, compare to tenths, the weights for A?” (There are 100 thousandths in a tenth and one hundredth of a tenth is a thousandth.)
“Nugget A weighs how many times as much as Nugget C?” (100, there are 100 thousandths in a tenth.)


aligns: no