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: 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: Activity
Students learn the definition of scientific notation and practice using it. Students attend to precision when determining whether or not a number is in scientific notation and converting numbers into scientific notation (MP6).
Throughout the activity, students use the usual
\(\boldcdot \)
symbol to indicate multiplication, but the discussion establishes the standard way to show multiplication in scientific notation with the
\(\times\)
symbol. Although these materials tend to avoid the
\(\times\)
symbol because it is easy to confuse with
\(x\)
, the ubiquitous use of
\(\times\)
for scientific notation outside of these materials necessitates its use here.
Launch
Tell students, “Earlier, we examined the speed of light through different materials. We zoomed into the number line to focus on the interval between
\(2.0 \times 10^8\)
meters per second and
\(3.0 \times 10^8\)
meters per second as shown in the figure.” Display the following image notation for all to see.
[IMG]
Tell students, “We saw that the speed of light through ice was
\(2.3 \times 10^8\)
meters per second. This way of writing the number is called
scientific notation
. Scientific notation is useful for understanding very large and very small numbers.”
Display and explain the following definition of scientific notation for all to see.
A number is said to be in scientific notation when it is written as a product of two factors:
The first factor is a number greater than or equal to 1, but less than 10, for example 1.2, 8, 6.35, or 2.008.
The second factor is an integer power of 10, for example
\(10^8\)
,
\(10^\text{-4}\)
, or
\(10^{22}\)
.
Carefully consider the first question and go through the list of numbers as a class, frequently referring to the definition to decide whether the number is written in scientific notation. When all numbers written in scientific notation have been circled, consider demonstrating or discussing how a number that was not circled could be written in scientific notation. Then, ask students to complete the second question (representing the other numbers in scientific notation). Leave 3–4 minutes for a whole-class discussion.
Student Facing
The table shows the speed of light or electricity through different materials.
|       material       | speed (meters per second) |
|----------------------|---------------------------|
|        space         |        300,000,000        |
|        water         |   \(2.25 \times 10^8\)    |
| copper (electricity) |        280,000,000        |
|       diamond        |    \(124 \times 10^6\)    |
|         ice          |    \(2.3 \times 10^8\)    |
|      olive oil       |    \(0.2 \times 10^9\)    |
Circle the speeds that are written in scientific notation. Write the others using scientific notation.
[IMG]
Student Response
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Activity Synthesis
Tell students that almost all books and information about scientific notation use the
\(\times\)
symbol to indicate multiplication between the two factors, so from now on, these materials will use the
\(\times\)
symbol in this same way. Display
\((2.8) \boldcdot 10^8\)
for all to see, and then rewrite it as
\(2.8 \times 10^8\)
. Emphasize that using
\(\boldcdot \)
is not incorrect, but that
\(\times\)
is the most common usage.
Ask students to come up with at least two examples of numbers that are
not
in scientific notation. Select responses that highlight the fact that the first factor must be between 1 and 10 and other responses that highlight that one of the factors must be an integer power of 10. Make sure students recognize what does and does not count as scientific notation.
Also make sure students understand how to write an expression that may use a power of 10 but is not in scientific notation as one that is in scientific notation. Consider using the speed of light through diamond as an example. Ask a series of questions such as:
“In
\(124 \times 10^6\)
, how must we write the first factor for the expression to be in scientific notation?” (A number between 1 and 10, so 1.24 in this case)
“How can we rewrite 124 as an expression that has 1.24?” (Write it as
\(1.24 \times 100\)
or
\(1.24 \times 10^2\)
)
“What is the equivalent expression in scientific notation?” (
\(1.24 \times 10^2) \times 10^6\)
, which is
\(1.24 \times 10^8\)
)
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary.  Include the following term and maintain the display for reference throughout the unit: scientific notation. Invite students to suggest language or diagrams to include on the display that will support their understanding of this term.
Supports accessibility for: Memory; Language
Writing: MLR3 Clarify, Critique, Correct.
Display a hypothetical student statement that represents a misunderstanding about how to write values in scientific notation, such as: “To write the speed of light through a diamond is
\(12.4 \times 10^7\)
.” Ask pairs of students to critique the response by asking, “Do you agree with the author? Why or why not?”Invite students to write feedback to the author that identifies the reasoning error and how to improve the statement. Listen for students who include in their feedback a need for the first factor to be between 1 and 10. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: no