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: 
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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
problem_activity: Activity
This activity represents the first time students solve a system of equations using algebraic methods. They first match systems of equations to their graphs and then calculate the solutions to each system. The purpose of matching is so students have a way to check that their algebraic solutions are correct, but not to shortcut the algebraic process since the graphs themselves do not include enough detail to accurately guess the coordinates of the solution.
Digital
Launch
Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and encourage them to do so.
If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so the students have another way of checking their solutions.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. Invite students to illustrate connections between slopes and
\(y\)
-intercepts of each line to the corresponding parts of each equation using the same color.
Supports accessibility for: Visual-spatial processing
Conversing: MLR8 Discussion Supports.
Use this routine to support small-group discussion as students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin with this sentence frame: “Figure ____ matches with the system of equations ____, because ____.” Invite the listener, Partner B, to press for additional details referring to specific features of the graphs (e.g. positive slope, negative y-intercept, coordinates of the intersection point, etc). Students should switch roles for each figure. 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
Here are three systems of equations graphed on a coordinate plane:
[IMG]
Match each figure to one of the systems of equations shown here.
\(\begin{cases} y=3x+5\\ y=\text- 2x+20 \end{cases}\)
\(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)
\(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)
Find the solution to each system and then check that your solution is reasonable on the graph.
Notice that the sliders set the values of the coefficient and the constant term in each equation.
Change the sliders to the values of the coefficient and the constant term in the next pair of equations.
Click on the spot where the lines intersect and a labeled point should appear.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in groups of 2. Give 2–3 minutes of quiet work time for the first problem and then ask students to pause their work. Select 1–2 students per figure to explain how they matched it to one of the systems of equations. For example, a student may identify the system matching Figure A as the only system with an equation that has negative slope. Give students 5–7 minutes of work time with their partner to complete the activity followed by a whole-class discussion. If students finish early and have not already done so on their own, ask them how they could check their solutions and encourage them to do so.
If using the digital activity, implement the lesson as indicated above. The only difference between the print and digital version is the digital lesson has an applet that will simulate the graphs so the students have another way of checking their solutions.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. Invite students to illustrate connections between slopes and
\(y\)
-intercepts of each line to the corresponding parts of each equation using the same color.
Supports accessibility for: Visual-spatial processing
Conversing: MLR8 Discussion Supports.
Use this routine to support small-group discussion as students describe the reasons for their matches. Arrange students in groups of 2. Invite Partner A to begin with this sentence frame: “Figure ____ matches with the system of equations ____, because ____.” Invite the listener, Partner B, to press for additional details referring to specific features of the graphs (e.g. positive slope, negative y-intercept, coordinates of the intersection point, etc). Students should switch roles for each figure. 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
Here are three
systems of equations
graphed on a coordinate plane:
[IMG]
Match each figure to one of the systems of equations shown here.
\(\begin{cases} y=3x+5\\ y=\text-2x+20 \end{cases}\)
\(\begin{cases} y=2x-10\\ y=4x-1 \end{cases}\)
\(\begin{cases} y=0.5x+12\\ y=2x+27 \end{cases}\)
Find the solution to each system and check that your solution is reasonable based on the graph.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is to deliberately connect the current topic of systems of equations to the previous topic of solving equations with variables on both sides. For each of the following questions, give students 30 seconds of quiet think time and then invite students per question to explain their answer. The final question looks ahead to the following activity.
“Do you need to see the graphs of the equations in a system in order to solve the system?" (No, but the graphs made me feel more confident that my answer was correct.)
“How do you know your solution doesn’t contain any errors?" (I know my solution does not have errors because I substituted my values for
\(x\)
and
\(y\)
into the equations and they made both equations true.)
“How does solving systems of equations compare to solving equations with variables on both sides like we did in earlier lessons?" (They are very similar, only with a system of equations you are finding an
\(x\)
and a
\(y\)
to make both equations true and not just an
\(x\)
to make one equation true.)
“When you solved equations with variables on both sides, some had one solution, some had no solutions, and some had infinite solutions. Do you think systems of equations can have no solutions or infinite solutions?" (Yes. We have seen some graphs of parallel lines where there were no solutions and some graphs of lines that are on top of one another where there are infinite solutions.)


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