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


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.

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:
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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
problem_activity: Activity
In this activity, students practice constructing an exponential function and evaluating it at a fractional input value as they translate between decades and years. They also think about how to express a percent increase over a fractional interval of input—specifically, how to express the annual growth rate of a population given the growth rate for one decade. They analyze and articulate why the growth rate for one year is
not
one tenth of that of one decade (MP3).
Launch
Tell students that, as of 2017, Nigeria is the seventh-most populous country in the world. It has grown at a fairly steady (and relatively high) rate since 1980.
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to orient students to the language of exponential functions in context. Ask students to keep their books or devices closed, and display only the image of the map and the problem statement, without revealing the questions that follow. Arrange students in groups of 2, and invite them to create a list of mathematical questions that could be asked about the situation.  Listen for and amplify questions about growth and decay factors, percent increase or decrease, fractional intervals, and comparisons of growth rates. Invite 2–3 groups to share their questions with the class.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
[IMG]
In 1990, Nigeria had a population of about 95.3 million. By 2000, there were about 122.4 million people, an increase of about 28.4%. During that decade, the population can be reasonably modeled by an exponential function.
Express the population of Nigeria
\(f(d)\)
, in millions of people,
\(d\)
decades since 1990.
Write an expression to represent the population of Nigeria in 1996.
A student said, “The population of Nigeria grew at a rate of 2.84% every year.”
Explain or show why the student’s statement is incorrect.
Find the correct annual growth rate. Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students struggle to identify the input to their function where
\(d\)
is not an integer, remind them that there are 10 years in a decade. Then ask them how many decades is 6 years? What about one year?
Activity Synthesis
Focus the discussion on why the annual growth rate is not
\(\frac1{10}\)
of 28.4%. Consider asking students:
“Why might someone think the annual population growth rate is 2.84%?” (Because 28.4 divided by 10 is 2.84, and they assume that population was growing linearly, or by the same amount each year, rather than by the same factor each year.)
“How can we determine the growth factor for one year?” (We know that when the annual growth factor, say
\(b\)
, is raised to an exponent of 10, it must equal 1.284, or
\(b^{10} = 1.284\)
, so
\(b\)
must be
\(1.284^\frac{1}{10}\)
or
\(\sqrt[10]{1.284}\)
because raising that number to an exponent of 10 gives 1.284.)
“How do we express the factor as a growth rate? (
\(1.284^\frac{1}{10} \approx 1.0258\)
, so the growth rate is 2.58%.)
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about the last question, scribe their thinking on a visible display. To illustrate why the growth factor for one year is not
\(\frac{1}{10}\)
of that of a decade, it may be helpful to return to a diagram that is configured like a double number line (especially if used in the last lesson). Use the double number line of years and decades to annotate and show the growth factor. When students explain that the growth factor for one decade is 1.284, draw an arrow across the decade line and label it
\(\boldcdot 1.284\)
. Then draw small arrows for the years labeled with
\(\boldcdot b\)
for each segment. Label the decade line with the corresponding equation. Encourage students to refer to the diagram when explaining why
\(b\)
is not one-tenth the growth rate of a decade. When students first explain the use of the fractional exponent to calculate the yearly population growth, label the line representing years with the corresponding equation.
Supports accessibility for: Visual-spatial processing; Conceptual processing


aligns: yes