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 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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
problem_activity: Activity
Earlier, in the backpack example, students saw a distribution described in terms of where data points are clustered on a dot plot and which values have a large number of occurrences. The shape of that distribution was approximately symmetric. In this activity, they continue to analyze distributions in those terms, and try to identify and interpret the
center
and
spread
of a distribution that is not symmetric. The two distributions used here allow students to contrast a narrow spread and a wide spread and develop a deeper understanding of variability.
As students work, notice how students identify a general location for the center of a data set and the descriptions they use to talk about the spread (e.g., “wide,” “narrow,” or “something in between”). Identify students who connect the size of a spread to how different or alike the data points are; ask them to share later. Additionally, identify students who measure spread as the range of the entire data as well as those who use the distance to the center.
Launch
Arrange students in groups of 2. Give students 5–6 minutes of quiet work time for the first three sets of questions, and another 4–5 minutes to share their responses and then discuss the last two questions with a partner.
Students are asked to find a percentage. If necessary, briefly review how to find a percentage.
Representation: Internalize Comprehension.
Activate or supply background knowledge about calculating percentages. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Representing, Writing: MLR5 Co-Craft Questions.
Invite pairs of students to create one or two mathematical questions that could be answered by the data displayed in the dot plot. Note questions that vary in complexity, making sure to have students share examples that ask about percent, center, or spread, if available. Allow students to ask clarifying questions of their peers regarding how the dot plot could be used to answer the questions, if needed. This will help students to make sense of the data using informal language prior to connecting their existing understanding to formal language.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Twenty-five sixth-grade students were asked to estimate how many hours a week they spend talking on the phone. This dot plot represents their reported number of hours of phone usage per week.
[IMG]
How many of the students reported not talking on the phone during the week? Explain how you know.
What percentage of the students reported not talking on the phone?
What is the largest number of hours a student spent talking on the phone per week?
What percentage of the group reported talking on the phone for this amount of time?
How many hours would you say that these students typically spend talking on the phone?
How many minutes per day would that be?
How would you describe the
spread
of the data? Would you consider these students’ amounts of time on the phone to be alike or different? Explain your reasoning.
Here is the dot plot from an earlier activity. It shows the number of hours per week the same group of 25 sixth-grade students reported spending on homework.
[IMG]
Overall, are these students more alike in the amount of time they spend talking on the phone or in the amount of time they spend on homework? Explain your reasoning.
Suppose someone claimed that these sixth-grade students spend too much time on the phone. Do you agree? Use your analysis of the dot plot to support your answer.
Student Response
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Anticipated Misconceptions
Students may neglect to change the rate given (from minutes per week to hours per week, or to minutes per day) and may draw incorrect conclusions as a result. Ask them to think about the unit they are using in their responses.
Activity Synthesis
The purpose of the discussion is to help students find good ways to describe a distribution based on center and spread.
Select a few students to answer the questions about the first dot plot. Tell students that the “typical” value for the data is generally considered the center. Ask, “What would you consider the center for the two dot plots shown in this activity?”
Ask students how they thought about the spread of the data. If possible, select students who thought of spread as the range of the entire data and those who thought of it as an interval around the center. Ask students to share their interpretation of the what the spread means in the context of using the phone. Make sure to include previously identified students who connect spread to how alike or different the data points are.
Tell students that distributions are generally described using the center and spread. Select a few students to describe the distributions of the two data sets shown in this activity.


aligns: no