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: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
problem_activity: Activity
This activity serves as a brief and light introduction to how
\(e\)
is visible in the behaviors of certain functions. The standards addressed in this course don’t require a deep understanding of the meaning of
\(e\)
, so the observations from this activity should not be assessed.
\(e\)
has a special connection to the expression
\(\left(1+\frac{1}{x}\right)^x\)
. Jumping to this expression would likely be an opaque process for most students, so they first investigate the range of behavior of its component pieces, for very small and very large positive values of
\(x\)
.
Entering increasingly larger or smaller values of
\(x\)
in the functions and making observations about them is a chance to look for and express regularity through repeated reasoning (MP8).
Launch
Tell students that they are to analyze the behavior of some functions and see what observations could be made about
\(e\)
along the way. Provide access to graphing technology and spreadsheet tools.
Arrange students in groups of 2. Tell students that they each should analyse the functions in different ways and then compare results. For example, one student could examine the functions numerically, using a table or a spreadsheet, while another student could examine them graphically.
Consider naming some tiny positive values of
\(x\)
(for example, 0.1, 0.01, and 0.001) and large values of
\(x\)
(for example, 100, 1,000, and 100,000) to students who may benefit from concrete examples.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to capture their observations. Provide seven columns for students to organize their work. The first column label should be “function,” while the remaining six columns are divided into two sets; three for the three “tiny input” numbers and three for the “large input numbers.” Ask students to select three tiny input numbers and three large input numbers (either from the numbers given, or their own). Demonstrate sequencing the numbers for the column headers from left to right getting tinier, and from left to right getting larger. Explain to students that some may not need all six columns to draw conclusions. Students may especially want to capture observations and calculations for all six input values for
\(f(x)\)
and
\(g(x)\)
.
Supports accessibility for: Language; Organization
Student Facing
Here are some functions. For each function, describe, in words, the outputs for very tiny, positive values of
\(x\)
and for very large values of
\(x\)
.
\(a(x) = 1^x\)
\(b(x) = \text-x\)
\(d(x) = \frac{1}{x}\)
\(f(x) = \left(\frac{1}{x}\right)^x\)
\(g(x) = \left(1 + \frac{1}{x}\right)^x\)
\(h(x) = e^{x}\)
\(k(x) = 1 + x\)
Remember that
\(e \approx 2.718\)
. What does the function
\(g\)
have to do with the number
\(e\)
?
What do you notice about the relationship between
\(h\)
and
\(k\)
for very small, positive values of
\(x\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Complete the table to show the value of each expression to the nearest hundred-thousandth. Two entries have already been completed as an example.
| \(x\)  | \(2^x\) | \(e^x\) |        \(3^x\)        |
|--------|---------|---------|-----------------------|
|  0.1   | 1.07177 | 1.10517 | \(\phantom{1.11612}\) |
|  0.01  |         |         |                       |
| 0.001  |         |         |                       |
| 0.0001 |         |         |                       |
What do you notice about the values in the table?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may substitute
\(0\)
for
\(x\)
in order to see what happens for small positive values of
\(x\)
. Focus their attention on functions
\(d\)
,
\(f\)
, and
\(g\)
where the
\(0\)
does not belong to the domain. For the other functions, encourage students to look at a graph (using technology for example). Does the value when
\(x = 0\)
help figure out what values each function takes for small positive values of
\(x\)
?
Activity Synthesis
Invite students to share their observations for the first question and the supporting work (a table, a spreadsheet, or graphs) that led to these observations, if available.
Focus the conversation on the responses to the last two questions. If not already observed by students, highlight that:
The value of function
\(g\)
gets closer and closer to
\(e\)
as
\(x\)
gets larger and larger. Display a graph representing
\(g(x)= \left(1 + \frac{1}{x}\right)^x\)
and trace it out for large values of
\(x\)
. The curve essentially flattens out as
\(x\)
increases.
[IMG]
Here is a graph of
\(g\)
along with a graph of the horizontal line given by
\(y = e\)
.
[IMG]
As the input grows, the two graphs are difficult to distinguish. The function
\(g\)
has a horizontal asymptote of
\(e\)
. (The exact value of the horizontal asymptote for
\(g\)
is one way to
define
the number
\(e\)
.)
Point out that, when
\(x\)
is a very large value,
\(\frac 1x\)
is very small (as we saw in function
\(d\)
). So in the expression
\(\left(1 + \frac{1}{x}\right)^x\)
, we are expressing
\(e\)
as
\(e \approx(1 + \text{tiny})^{\text{huge}}\)
. (It is critical that the "huge" number is the multiplicative inverse of the "small" number.)
The values of
\(h\)
and
\(k\)
are very close for very small, positive values of
\(x\)
. In other words, raising
\(e\)
to an increasingly small positive exponent
\(x\)
gives essentially the same values as adding 1 to
\(x\)
. Display a graph such as the following, or use graphing technology and zoom in toward smaller and smaller values of
\(x\)
.
[IMG]
Tell students that these behaviors turn out to be useful in mathematical and scientific applications, and they will learn a lot more about them in future math courses.
Conversing, Representing: MLR 7 Compare and Connect.
Invite students to make connections between the various representations and approaches. Ask students compare how the number
\(e\)
is represented in the table or graph of the function
\(g\)
. Listen for and amplify the language students use to describe how the value of
\(g\)
approaches the number
\(e\)
in the table or graph.
Design Principle(s): Maximize meta-awareness; Optimize output (for comparison)


aligns: no