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: 
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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: 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: Activity
Having just seen an example of the meaning of
\(a\)
and
\(b\)
in an exponential expression
\(a \boldcdot b^x\)
, students now focus on interpreting these numbers using graphs. They graph the equations from the previous task, noticing that
\(a\)
is the vertical intercept of the graph while the number
\(b\)
determines how quickly the graph grows (since in these cases
\(b>1\)
). A larger value of
\(b\)
corresponds to a more rapid rate of growth for the bacteria population. The axes for the graphs have been labeled here, but in future activities, students will have to think strategically about how to label the axes to most effectively plot the points.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Display the equations from the previous activity for all to see (if they are not already visible).
Conversing: MLR8 Discussion Supports.
Use this routine to help students describe the meaning of variables in exponential expressions. Arrange students in groups of 2. Invite Partner A to share their observations for the first graph and Partner B to share their observations for the second graph. Provide sentence frames such as “I can connect the equation to the graph by….” and “I notice that….” Invite the listener to press for additional details referring to features of the graphs, such as the vertical intercept, specific points, and the steepness of the curve. This will help students justify how features of the graph can be used to identify matching equations.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
Refer back to your work in the table of the previous task. Use that information and the given coordinate planes to graph the following:
a. Graph
\((t,n)\)
when
\(t\)
is 0, 1, 2, 3, and 4.
[IMG]
b. Graph
\((t,p)\)
when
\(t\)
is 0, 1, 2, 3, and 4. (If you get stuck, you can create a table.)
[IMG]
On the graph of
\(n\)
, where can you see each number that appears in the equation?
On the graph of
\(p\)
, where can you see each number that appears in the equation?
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
Students may have trouble graphing the points, particularly finding the appropriate vertical (
\(n\)
or
\(p\)
) values. Ask them to find the coordinates of the grid points on the vertical axis and use that to estimate the vertical position of their points.
When calculating values by hand, many students may mistakenly write an expression like
\(100 \boldcdot 3^2\)
as
\(300^2\)
. Remind them that the expression
\(100 \boldcdot 3^2\)
means
\(100 \boldcdot 3 \boldcdot 3\)
.
Activity Synthesis
Make sure students recognize two key takeaways from this activity:
The vertical intercept of the graph is the initial bacteria population size. It is the size of the population when first measured, or when
\(t\)
, the number of hours since measurement, is 0.
The growth factor of the populations is represented by how quickly the graph increases.


aligns: no