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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_pos_0

8
standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Activity
In this activity students use the structure of the coordinate plane to examine the observation that the altitudes of a triangle all intersect at a single point. Students practice writing equations for perpendicular lines as they represent altitudes algebraically. Then they solve the system of equations (a fairly simple system with
\(x=2\)
as one equation) to show that the altitudes all intersect at a single point.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Representation: Internalize Comprehension.
Activate or supply background knowledge about perpendicular bisectors. Display the following information: the perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. Provide a visual example to accompany this definition.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Triangle
\(ABC\)
is graphed.
[IMG]
Find the slope of each side of the triangle.
Find the slope of each altitude of the triangle.
Sketch the altitudes. Label the point of intersection
\(H\)
.
Write equations for all 3 altitudes.
Use the equations to find the coordinates of
\(H\)
and verify algebraically that the altitudes all intersect at
\(H\)
.
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?
Any triangle
\(ABC\)
can be translated, rotated, and dilated so that the image
\(A’\)
lies on the origin,
\(B’\)
lies on the point
\((1,0)\)
, and
\(C’\)
has position
\((a,b)\)
. Use this as a starting point to prove that the altitudes of all triangles all meet at the same point.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students aren’t sure how to find the slopes of the altitudes, ask them about the relationship between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.)
If students struggle to verify algebraically that
\(H\)
is the point of intersection, ask them what’s true if a particular point is on 3 different lines (the lines must intersect at that point, unless they coincide). How can students test if
\(H\)
is on each line?
As in the previous activity, an index card can be a useful too to help visualize the altitudes.
Activity Synthesis
Ask students what the relationship is between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.) Invite students to share strategies for verifying their coordinates of
\(H\)
. (Sample response: Start with
\(x=2\)
, find the
\(y\)
-value using one equation, and test that point in the other equation.)
Speaking: MLR8 Discussion Supports.
As students discuss the relationship between the slope of a side and the slope of the altitude through that side, press for details by asking students how they know that the product of the slopes is
\(\text-1\)
. Also ask how they determined the equations for all three altitudes. Show concepts multi-modally by drawing and labeling a slope triangle for segment
\(AC\)
or
\(BC\)
and the altitude through that side. This will help students justify the slopes and equations for all three altitudes of the triangle.
Design Principle(s): Support sense-making; Optimize output (for justification)


aligns: no