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 square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.
problem_activity: Activity
This task is analogous to a previous activity with positive exponents. The number line strongly encourages students to think about how to change expressions so they all take the form
\(b \boldcdot 10^k\)
, where
\(b\)
is between 1 and 10, as in the case of scientific notation. The number line also is a useful representation to show that, for example,
\(29 \boldcdot 10^\text{-7}\)
is about half as much as
\(6 \boldcdot 10^\text{-6}\)
. The last two questions take such a comparison a step further, asking students to estimate relative sizes using numbers expressed with powers of 10.
As students work, look for different strategies they use to compare expressions that are not written as a product of a number and
\(10^{\text-6}\)
. Also look for students who can explain how they estimated in the last two problems. Select them to share their strategies later.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time, followed by partner discussion and whole-class discussion. During partner discussion, ask students to share their responses for the first two questions and reach an agreement about where the numbers should be placed on the number line.
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch take time to review the visual display of rules for exponents.
Supports accessibility for: Memory; Language
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students as they describe how they compared and estimated the difference in magnitude between the pairs of values
\(29⋅10^{\text-7}\)
and
\(6⋅10^{\text-6}\)
or
\(7⋅10^{\text-8}\)
and
\(3⋅10^{\text-9}\)
. For example, “First I ____ , then I ____ .” or “_____ is (bigger/smaller) because ______ .”
Design Principle(s): Optimize output (for explanation)
Student Facing
[IMG]
Label the tick marks on the number line.
Plot the following numbers on the number line:
A.
\(6 \boldcdot 10^{\text -6}\)
B.
\(6 \boldcdot 10^{\text -7}\)
C.
\(29 \boldcdot 10^{\text -7}\)
D.
\((0.7) \boldcdot 10^{\text -5}\)
Which is larger,
\(29 \boldcdot 10^{\text -7}\)
or
\(6 \boldcdot 10^{\text -6}\)
? Estimate how many times larger.
Which is larger,
\(7 \boldcdot 10^{\text -8}\)
or
\(3 \boldcdot 10^{\text -9}\)
? Estimate how many times larger.
Student Response
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Anticipated Misconceptions
Students may have trouble comparing negative powers of 10. Remind these students that, for example,
\(10^\text{-5}\)
is 5 factors that are
\(\frac{1}{10}\)
and
\(10^\text{-6}\)
is 6 factors that are
\(\frac{1}{10}\)
, so
\(10^\text{-5}\)
is 10 times larger than
\(10^\text{-6}\)
.
Students may also have trouble estimating how many times one larger expression is than another. Offer these students an example to illustrate how representing numbers as a single digit times a power of 10 is useful for making rough estimations. We have that
\(9 \boldcdot 10^\text{-12}\)
is roughly 50 times as much as
\(2 \boldcdot 10^\text{-13}\)
because
\(10^\text{-12}\)
is 10 times as much as
\(10^\text{-13}\)
and 9 is roughly 5 times as much as 2. In other words,
\(\displaystyle \frac{9 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} \approx \frac{10 \boldcdot 10^\text{-12}}{2 \boldcdot 10^\text{-13}} = 5 \boldcdot 10^{\text-12 - (\text-13)} = 5 \boldcdot 10^1 = 50\)
Activity Synthesis
Select previously identified students to explain how they used the number line and powers of 10 to compare the numbers in the last two problems.
One important concept is that it’s always possible to change an expression that is a multiple of a power of 10 so that the leading factor is between 1 and 10. For example, we can think of
\(29 \boldcdot 10^\text{-7}\)
as
\((2.9) \boldcdot 10 \boldcdot 10^\text{-7}\)
or
\((2.9) \boldcdot 10^\text{-6}\)
. Another important concept is that powers of 10 can be used to make rough estimates. Make sure these ideas are uncovered during discussion.


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