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: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
problem_activity: Narrative
The purpose of this activity is for students to equally divide sandwiches in situations where the number of portions does not evenly divide the number of sandwiches. Students select the number of people sharing and the number of sandwiches from a small set of deliberately chosen numbers. Depending on their choice, each share may be less than a full sandwich or more than a full sandwich. This encourages students to make sense of the problem and persevere in solving it (MP1). The focus of the synthesis is on different ways students choose to make equal shares and how they know that the shares are equal. Many of the strategies for dividing the sandwiches equally are examined in detail in the next activity. In this activity, students explain how they know the shares are equal. Student representations may include coloring to show the equal shares. Monitor and select students who use the following strategies to share in the synthesis:
choose numbers so each person gets one or two whole sandwiches and a fraction of a sandwich
choose numbers so each person gets less than a full sandwich
MLR1 Stronger and Clearer Each Time.
Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to how much sandwich each person gets. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.
Advances: Writing, Speaking, Listening
Launch
Groups of 2
Display the prompt without the questions.
__________ sandwiches are shared equally by __________ people.
“What do you notice? What do you wonder?” (There are no numbers. How many sandwiches are there? How many people are there? What kind of sandwiches are they?)
“What are some numbers that would make sense? Why do those numbers make sense?” (One sandwich per person because that is a familiar serving size. Many people and many sandwiches because there is a party or event. More people than sandwiches because the people are sharing big sandwiches.)
Activity
2 minutes: independent think time
5–8 minutes: partner work time
As students work, consider asking:
“Where are the sandwiches in your drawing or diagram?”
“How does your diagram or drawing show how much sandwich each person will get?”
Student Facing
__________ sandwiches are shared equally by __________ people.
Choose numbers to fill in the blanks. You can only use each number once: 2, 3, 5.
Represent the situation with a diagram or drawing.
Explain or show how you know that each person will get the same amount of sandwich.
Student Response
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Advancing Student Thinking
Students may recognize that the natural shape of their chosen sandwich impacts the size of certain pieces and, therefore, some people are getting a little bit more sandwich if they get a certain piece. When they recognize the constraints of the shape of a sandwich in real life, students are making sense of problems. Ask these students, “If you were going to share these sandwiches with your friends, which pieces would you want? Why would you want those pieces? Why do you think someone might choose the end pieces? Is it fair to say that each of you got about the same amount of sandwich?”
Activity Synthesis
Ask the selected students to display their responses side by side for all to see.
“Does each person get more than a full sandwich or less? How do you know?” (Less because there are more people than sandwiches or more because there are more sandwiches than people.)
“This situation is about sharing. We can represent equal sharing situations with division expressions. What division expressions can we write to represent each of the situations we represented?” (Answers vary depending on number choices:
\(2\div3\)
,
\(3\div2\)
,
\(2\div5\)
,
\(5\div2\)
,
\(3\div5\)
,
\(5\div3\)
)


aligns: yes