few_shots: 
Example 1:
Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

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


Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
problem_activity: Activity
Students continue to compare quadratic and exponential functions in this activity. This time, they decide how to compare the functions.
Monitor for different strategies students may use to compare the functions. Identify students who:
Create one or more tables of values and compare the values of
\(p(x)\)
and
\(q(x)\)
at increasingly large values of
\(x\)
.
Graph the functions defined by
\(p(x) = 6x^2\)
and
\(q(x) = 3^x\)
and compare the graphs.
Create one or more tables of values, calculate the growth factors at equal intervals of input, and then compare the growth factors.
If students choose graphing or spreadsheet tools strategically to perform comparisons, they practice MP5.
Launch
Ask students to observe the equations representing the two functions and determine which function is exponential and which is quadratic. Invite students to share how they know. Make sure students recognize that
\(p\)
is quadratic and
\(q\)
is exponential before they begin the activity.
Provide access to graphing technology and spreadsheet tools. This may be a good opportunity for students to experiment with the graphing window. If the horizontal dimension is very small (for example,
\(0<x<5\)
) or the vertical dimension is very large (for example,
\(0<y<3,\!000\)
), the two graphs will be hard to distinguish. As needed, remind students to think about adjusting the graphing window to make the graphs more informative.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to adjust the graphing window to experiment with the horizontal and vertical dimensions.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Here are two functions:
\(p(x) = 6x^2\)
and
\(q(x) = 3^x\)
.
Investigate the output of
\(p\)
and
\(q\)
for different values of
\(x\)
. For large enough values of
\(x\)
, one function will have a greater value than the other. Which function will have a greater value as
\(x\)
increases?
Support your answer with tables, graphs, or other representations.
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?
Jada says that some exponential functions grow more slowly than the quadratic function
\(f(x)=x^2\)
as
\(x\)
increases. Do you agree with Jada? Explain your reasoning.
Let
\(f(x) = x^2\)
. Could you have an exponential function
\(g(x)=b^x\)
so that
\(g(x) < f(x)\)
for all values of
\(x\)
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to present their strategies in the sequence listed in the Activity Narrative. If no students chose to graph the functions, consider displaying the graphs for all to see.  By presenting strategies that involved calculation of growth factors last, students will deepen their understanding of why exponential values will eventually overtake the quadratic values.
Again, one way to make sense of why exponential functions eventually grow faster than quadratic functions is by thinking of the growth factors. The output of an exponential function always increases by the same factor when the input increases by 1 (for example, 3 for the exponential function
\(q\)
studied here). The output of a quadratic function, on the other hand, increases by smaller and smaller amounts when the input increases by 1. So even though a quadratic function may take larger values than an exponential function for many inputs, the values of the exponential function will eventually overtake the quadratic.
Conversing: MLR2 Collect and Display.
As students share their analysis with the class, listen for and collect the language students use to identify and describe how the output of the exponential function eventually outgrows that of the quadratic function. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Maximize meta-awareness


aligns: no