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: 
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_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
problem_activity: Activity
The purpose of this activity is to let students determine how to use straightedge and compass moves to construct a regular hexagon precisely. Students should play with construction moves until they reach their goal rather than follow an explicit demonstration of construction steps. While the term
regular
appears in the task, it is not important for students to know the precise definition of
regular polygon
at this time.
Identify students whose explanations that the sides are congruent use tracing paper, or compare the radii of the different circles in the construction. Tracing paper connects to the idea of rigid motions, while comparing radii references the precise definition of a circle, which students will use throughout this unit and subsequent units.
Digital
Launch
Arrange students in groups of 2. Give students 10 minutes of work time followed by a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their reasoning. For example, “Each of the sides of the shape I constructed are the same length because....”
Supports accessibility for: Language; Social-emotional skills; Conceptual processing
Student Facing
Here is a hexagon with all congruent angles and all congruent sides (called a
regular
hexagon).
Try to draw a copy of the regular hexagon using only the pen tool. Draw your copy next to the hexagon given, and then drag the given one onto yours. What happened?
Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Drag the given one onto yours and confirm that it fits perfectly onto itself.
How do you know each of the sides of the shape are the same length? Show or 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?
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
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
Arrange students in groups of 2. Provide access to tracing paper. Give students 10 minutes of work time followed by a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their reasoning. For example, “Each of the sides of the shape I constructed are the same length because....”
Supports accessibility for: Language; Social-emotional skills; Conceptual processing
Student Facing
Here is a hexagon with all congruent angles and all congruent sides (called a
regular
hexagon).
[IMG]
Draw a copy of the regular hexagon using only your pencil and no other tools. Trace your copy onto tracing paper. Try to fold it in half. What happened?
Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Trace it onto tracing paper and confirm that when you fold it in half, the edges line up.
[IMG]
How do you know each of the sides of the shape are the same length? Show or 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?
Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?
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 spend more than a few minutes without significant progress, tell them the segment given in the figure is one of the 6 sides of the hexagon. Invite students to compare the given hexagon to the start of the construction. Then ask if they can draw another segment to make an adjacent side of the hexagon.
Activity Synthesis
The purpose of this discussion is to build toward the concept of a proof by asking students to informally explain why a fact about a geometric object must be true. Ask previously identified students to share their responses to “How do you know each of the sides of the shape are the same length?”
Writing, Speaking, Conversing: MLR 1 Stronger and Clearer Each Time
. Use this with successive pair shares to give students a structured opportunity to revise and refine their response to “How do you know each of the sides of the shape are the same length?” Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “Can you explain how…?”, “How do circles help with the construction?”, “What do you know about radii that helps here?”, or “What do you mean by…?”. Students can borrow ideas and language from each partner to strengthen their final explanation.
Design Principle(s): Optimize output (for justification); Support sense-making


aligns: no