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: Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
problem_activity: Narrative
Students begin the activity by looking at the problem displayed, rather than in their books. At the end of the launch, students work on the problem. This activity prompts students to use what they know about multiplication, division, factors, and multiples to solve problems. The problem does not have a question, so students will need to make sense of the context and generate potential questions that might be answered (MP2). Students are encouraged in the task to attend to the details of the situation and to engage in genuine curiosity about the mathematics that is embedded within it.
This activity uses MLR5 Co-craft Questions. Advances: writing, reading, representing
Representation: Develop Language and Symbols.
Represent the problem in multiple ways to support understanding. For example, invite students to represent the situation as a comic strip or a collage. Offer relevant images such as a movie theater, a sign showing admissions prices, a cash register, and a calendar. Alternatively, invite students to act out the situation. Consider using play money, signs with two days of the week written on them, and the action of “fast forwarding” through days and a night. Consider asking, “How might you represent the situation in a mathematical diagram?”
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention
Launch
Groups of 2
MLR5 Co-Craft Questions
Read only the first two paragraphs without revealing the question(s).
“Write a list of mathematical questions that could be asked about this situation.”
2 minutes: independent work time
2–3 minutes: partner discussion
Invite several students to share one question with the class. Record responses for use later in the task.
“What information from the situation can be used to answer this question?” (The number of days tickets were purchased, the total number of money earned by the theater, the price of movie tickets)
Reveal the task (students open books), and invite additional connections.
Activity
“Choose a question from the list to answer.”
“Work with a partner to complete the activity.”
Remind students of the list of questions generated during the launch as a reference during the activity.
Student Facing
Movie tickets are \$9 each. The theater sold the same number of tickets two days in a row.
The theater made \$3,132 from ticket sales on the first day.
[IMG]
Record and answer one question of your choice from the list the class generated. Discuss your strategy with your partner.
Use the given information about movie tickets to complete the following statement:
__________ tickets were sold on the first and second days.
A medium drink is \$7 and small popcorn is \$5. If each ticket holder purchases popcorn and a drink, how much money will the theater collect from the sales of popcorn and drink?
Student Response
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Advancing Student Thinking
Students may think of questions that cannot be answered using the information provided. Consider asking: “What would we need to know to be able to answer this question?” and “How might we find out this information?” The result of these questions may not enable the students to answer the questions, but will support them in making sense of the problem and identifying information necessary to solve the problems.
Activity Synthesis
Select 1–2 students to share their reasoning and responses.
If not clarified in students’ explanations, discuss a possible path for finding out the number of tickets sold over the two days using the given information. (For instance: Each ticket is \$9 and we know the total amount of money earned by selling tickets in one day, \$3,132. If we divide the total amount earned by the price of each ticket, we can find out how many tickets were sold on one day.
\(3,\!132\div 9 = 348\)
. If 348 tickets were sold on one day, then
\(348 \times 2\)
or 696 tickets were sold in the two days. We can also multiply \$3,132 by 2 first then divide by \$9 to get the total number of tickets.)


aligns: no