few_shots: 
Example:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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_000972_neg_0

6
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
In this activity, students solve a system of equations by graphing then include a third line through the solution point that is either vertical or horizontal. In the associated Algebra lesson, students will graph equations in a system and additional equations that arise from the elimination method to show that they all intersect at the solution.
To focus on the work of the activity, it is recommended that students use the digital version of the activity or have access to Desmos or another graphing tool to more quickly and accurately find the solutions to the system of equations through graphing. If students need practice graphing lines representing equations by hand or if digital access is not available, the print version provides a graph given one of the lines and asks students to graph the second line by hand.
Launch
Display the instructions for all to see. Ask students,
"What do you know about equations that are represented by vertical lines?" (They are of the form
\(x = a\)
for a number
\(a\)
.)
"What do you know about equations that are represented by horizontal lines?" (They are of the form
\(y = a\)
for a number
\(a\)
.)
Student Facing
For each system of equations:
Solve the system of equations by graphing. Write the solution as an ordered pair.
Write an equation that would represented by a vertical or horizontal line that also passes through the solution of the system of equations.
Graph your new equation along with the system.
\(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)
The line representing
\(y = 3x+5\)
is shown
[IMG]
\(\begin {cases} \begin {align}y = \frac{1}{3}x-2\\ y=x-6\end{align} \end {cases}\)
The line representing
\(y = \frac{1}{3}x-2\)
is shown
[IMG]
\(\begin {cases} \begin {align} 2x+3y=10\\ x+y=3\end{align} \end {cases}\)
The line representing
\(2x+3y=10\)
is shown
[IMG]
Student Response
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Activity Synthesis
The purpose of the discussion is for students to recognize that, for a system of equations represented by graphs that intersect at
\((a,b)\)
, the lines
\(x = a\)
and
\(y = b\)
also go through the solution. Select students to share their solutions including students who found vertical as well as horizontal lines through the intersection point.
Some questions for discussion:
"How could you use your answers to these questions to write another system of equations that have the same solution as the original system of equations?" (The system
\(\begin {cases} \begin {align} x = \text{-}1\\ y=2\end{align} \end {cases}\)
has the same solution as the original equation
\(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)
since all 4 lines go through the point
\((\text{-}1,2)\)
.)
"Two lines intersect at the point
\((3,5)\)
. How could you know whether the equation
\(3x - 2y = \text{-}1\)
is represented by a line that goes through that same point?" (Substitute the value 3 for
\(x\)
and 5 for
\(y\)
. If the equation is still true, then the associated line will go through the same point. If the equation is false, the associated line will not go through the same point.)


aligns: no