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


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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:
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_pos_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
The purpose of this lesson is to increase fluency in solving equations. Students will solve equations individually and then compare differing, though accurate, solution paths in order to compare their work with others. This will help students recognize that while the final solution will be the same, there is more than one path to the correct answer that uses principles of balancing equations learned in previous lessons.
Launch
Arrange students in groups of 3–4. Give students quiet think time to complete the activity and then tell groups to share how they solved the equations for
\(x\)
and discuss the similarities and differences in their solution paths.
Representation: Internalize Comprehension.
Activate or supply background knowledge about solving equations. Encourage students to use previously solved equations as guidelines to determine appropriate steps.
Supports accessibility for: Memory; Conceptual processing
Representing, Conversing: MLR2 Collect and Display.
As groups discuss their work, circulate and listen for the language students describe the similarities and differences in their solution paths. Write down different solution paths that led to the same result in a visual display. Consider grouping words and phrases used for each step in different areas of the display (e.g., “multiply first”, “divide first”, “subtract
\(x\)
from the right”, “add
\(x\)
from the left”). Continue to update the display as students move through the activity, and remind them to borrow from the display while discussing with their group. This will help students develop their mathematical language around explaining different solution paths to solving equations.
Design Principle(s); Maximize meta-awareness
Student Facing
Solve these equations for
\(x\)
.
1.
\(\frac{12+6x}{3}=\frac{5-9}{2}\)
2.
\(x-4=\frac13(6x-54)\)
3.
\(\text-(3x-12)=9x-4\)
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?
I have 24 pencils and 3 cups. The second cup holds one more pencil than the first. The third holds one more than the second. How many pencils does each cup contain?
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
Students should take away from this activity the importance of using valid steps to solve an equation over following a specific solution path. Invite students to share what they discussed in their groups. Consider using some of the following prompts:
“How many different ways did your group members solve each problem?”
“When you compared solution paths, did you still come up with the same solution?” (Yes, even though we took different paths, we ended up with the same solutions.)
“How can you make sure that the path you choose to solve an equation is a valid path?” (I can use the steps we discovered earlier when we were balancing: adding the same value to each side, multiplying (or dividing) by the same value to each side, distributing and collecting like terms whenever it is needed.)
“What are some examples of steps that will not result in a valid solution?” (Performing an action to only one side of an equation and distributing incorrectly will give an incorrect solution.)


aligns: no