few_shots: 
Example 1:
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.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
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: 
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_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
problem_activity: Activity
The goal of this activity is to practice applying the definition of rotation. In this activity, students rotate a segment around its endpoint to create an isosceles triangle. By rotating a segment around one endpoint, students can observe that the distance between the center of rotation and a point is the same as the distance between the center of rotation and the image of that point. This observation is later captured in the formal definition of rotation.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to aide students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step.
Supports accessibility for: Organization; Attention
Student Facing
Draw a segment. Label the endpoints
\(A\)
and
\(B\)
.
Rotate segment
\(AB\)
clockwise around center
\(B\)
by 90 degrees. Label the new endpoint
\(A’\)
.
Use the Polygon tool to draw triangle
\(ABA'\)
.
What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
Draw a segment. Label the endpoints
\(C\)
and
\(D\)
.
Rotate segment
\(CD\)
counterclockwise around center
\(D\)
by 30 degrees. Label the new endpoint
\(C’\)
.
Rotate segment
\(C’D\)
counterclockwise around center
\(D\)
by 30 degrees. Label the new endpoint
\(C’’\)
.
Use the Polygon tool to draw triangle
\(CDC’’\)
.
What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
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?
You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a
regular polygon
: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Briefly demonstrate how to create an angle of a given measure with a protractor and ask students to use their protractors along with you.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5). You may wish to demonstrate using GeoGebra Geometry in Math Tools.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to aide students who benefit from support with organizational skills in problem solving. For example, present one question at a time and ensure students complete each step correctly before moving on to the next step.
Supports accessibility for: Organization; Attention
Student Facing
Draw a segment. Label the endpoints
\(A\)
and
\(B\)
.
Rotate segment
\(AB\)
clockwise around center
\(B\)
by 90 degrees. Label the new endpoint
\(A’\)
.
Connect
\(A\)
to
\(A’\)
and lightly shade in the resulting triangle.
What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
Draw a segment. Label the endpoints
\(C\)
and
\(D\)
.
Rotate segment
\(CD\)
counterclockwise around center
\(D\)
by 30 degrees. Label the new endpoint
\(C’\)
.
Rotate segment
\(C’D\)
counterclockwise around center
\(D\)
by 30 degrees. Label the new endpoint
\(C’’\)
.
Connect
\(C\)
to
\(C’’\)
and lightly shade in the resulting triangle.
What kind of triangle did you draw? What other properties do you notice in the figure? Explain your reasoning.
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?
You constructed an equilateral triangle by rotating a given segment around one of its endpoints by a specific angle measure. An equilateral triangle is an example of a
regular polygon
: a polygon with all sides congruent and all interior angles congruent. Try to construct some other regular polygons with this method.
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
Students may not notice any properties other than that rotation preserves distances to the center of rotation. Encourage these students to use tracing paper to make other conjectures about side lengths or angles.
Activity Synthesis
Display an isosceles triangle
\(ABC\)
with base
\(BC\)
.
Here are some key conjectures and observations students should come away from the discussion with:
Rotations preserve the distance to the center of rotation.
The angle bisector of angle
\(A\)
is also the perpendicular bisector of the base (
\(BC\)
) of an isosceles triangle.
The two base angles (angle
\(B\)
and angle
\(C\)
) of an isosceles triangle are congruent.
An isosceles triangle where angle
\(A\)
is 60 degrees is also equilateral.
It is not necessary at this time to formalize the language of these ideas. This will come later as students prove some of these statements in a subsequent unit.


aligns: no