few_shots: 
Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

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_001392_neg_0

8
standard_description: Understand ordering and absolute value of rational numbers.
problem_activity: Activity
The purpose of this activity is for students to extend the vertical and horizontal axes to include 4 quadrants just as they extended the number line to include negative numbers. Students are introduced to the term
quadrant.
Students plot and label coordinates using ordered pairs and identify their quadrants.
Digital
Launch
Introduce the concept of the 4-region coordinate plane by explaining that, just like we extended the number line include negative numbers, we can extend both the number lines of the coordinate plane (the axes) to include negative coordinates. Use the word
quadrant
to describe the four regions of the coordinate plane. It may be helpful to explain that the prefix "quad-" means 4 and give other examples from English and other languages that use the prefix (quadriceps, quadrilateral, cuatro). Give students 10 minutes to work followed by whole-class discussion.
Classes using the digital version have an interactive applet to use. Instead of naming the coordinates, students enter them into the Input Bar. If correct, the new points will hit the targets and turn them black.
Representation: Develop Language and Symbols.
Activate or supply background knowledge. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: vertical number line, horizontal number line.
Supports accessibility for: Conceptual processing; Language
Student Facing
[IMG]
The colored points on the coordinate plane are like targets. Hit each point by entering its coordinates as an ordered pair in the Input Bar, like this:
What do you notice about the locations and ordered pairs of
\(B\)
,
\(C\)
, and
\(D\)
? How are they different from those for point
\(A\)
?
Plot a point at
\((\text-2, 5)\)
. Label it
\(E\)
. Plot another point at
\((3, \text-4.5)\)
. Label it
\(F\)
.
The coordinate plane is divided into four quadrants, I, II, III, and IV, as shown here.
[IMG]
\(G = (5, 2)\)
\(H=(\text-1, \text-5)\)
\(I=(7,\text-4)\)
In which quadrant is
\(G\)
located?
\(H\)
?
\(I\)
?
A point has a positive
\(y\)
-coordinate. In which quadrant could it be?
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
Introduce the concept of the 4-region coordinate plane by explaining that, just like we extended the number line include negative numbers, we can extend both the number lines of the coordinate plane (the axes) to include negative coordinates. Use the word
quadrant
to describe the four regions of the coordinate plane. It may be helpful to explain that the prefix "quad-" means 4 and give other examples from English and other languages that use the prefix (quadriceps, quadrilateral, cuatro). Give students 10 minutes to work followed by whole-class discussion.
Classes using the digital version have an interactive applet to use. Instead of naming the coordinates, students enter them into the Input Bar. If correct, the new points will hit the targets and turn them black.
Representation: Develop Language and Symbols.
Activate or supply background knowledge. During the launch, take time to review terms that students will need to access for this activity. Invite students to suggest language or diagrams to include that will support their understanding of: vertical number line, horizontal number line.
Supports accessibility for: Conceptual processing; Language
Student Facing
Label each point on the coordinate plane with an ordered pair.
[IMG]
What do you notice about the locations and ordered pairs of
\(B\)
,
\(C\)
, and
\(D\)
? How are they different from those for point
\(A\)
?
Plot a point at
\((\text-2, 5)\)
. Label it
\(E\)
. Plot another point at
\((3, \text-4.5)\)
. Label it
\(F\)
.
The coordinate plane is divided into four
quadrants
, I, II, III, and IV, as shown here.
[IMG]
\(G = (5, 2)\)
\(H=(\text-1, \text-5)\)
\(I=(7,\text-4)\)
In which quadrant is point
\(G\)
located? Point
\(H\)
? Point
\(I\)
?
A point has a positive
\(y\)
-coordinate. In which quadrant could it be?
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 confuse the order of the coordinate pairs when plotting points. Direct students to look at their work with vertical and horizontal lines in the warm up to remind them which coordinate is vertical.
Students may locate -4.5 between -3 and -4, rather than between -4 and -5. Direct them to the number lines used in earlier lessons to remind them of locating non integer numbers on number lines.
Activity Synthesis
The most important idea for students to understand is that by extending the two number lines that form the coordinate axes for the first quadrant, we now have 4 quadrants. We describe points in these quadrants using negative and positive numbers as the
\(x\)
and
\(y\)
coordinates. Focus on responses from question 4 for discussion. Invite students to share their reasoning about how to identify the quadrants for the points
\(G\)
,
\(H\)
, and
\(I\)
. As time allows, consider asking the following questions:
“If a point has a negative
\(x\)
-coordinate, what quadrant could it be in?”
“If a point has a negative
\(y\)
-coordinate, what quadrant could it be in?”
“If a point has a positive
\(x\)
-coordinate, what quadrant could it be in?”
To involve more students in the conversation, consider asking:
“Do you agree or disagree? Why?”
“Who can restate ___’s reasoning in a different way?”
“Does anyone want to add on to _____’s reasoning?”
Speaking: MLR8 Discussion Supports.
To support students in producing statements about features of the quadrants that have negative and/or positive numbers provide sentence frames for students to use such as: “Point _____ is in quadrant _____ because _____.” This will support class discussion in providing low entry points for students to start speaking about their reasoning.
Design Principle(s): Support sense-making


aligns: no