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: Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^(2) + y^(2))^(2) = (x^(2 )– y^(2))^(2) + (2xy)^(2 )can be used to generate Pythagorean triples.
problem_activity: Activity
The mathematical purpose of this activity is for students to gain an understanding of the relationship between a two-way frequency table and a two-way relative frequency table. Students have the same information in a two-way table and a segmented bar graph, then are asked to interpret the information given in context. In some cases, it makes sense to view the raw data in the table, and in other cases, it can make sense to understand the data as percentages of various larger groups. Students examine the meaning of data when they are presented as a percentage of the entire group or either of the 2 subgroups.
Launch
Arrange students in groups of 2.
Help orient students by having them notice:
The sum of all 4 percentages in the second table is 100%.
For each of the next two tables, ask students to find percentages that sum to 100%. (The third table has columns that sum to 100% and the fourth table has rows that sum to 100%.)
Action and Expression: Internalize Executive Functions.
To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, instead of presenting tables and graphs all at once, provide 4 different pieces. Ask students to place the table with the actual values at the top of their workspace, and line up the other tables and graphs in a new row below. This allows students to see all of the information while reading and answering the questions without flipping back and forth. Encourage students to move or cover tables to focus on one table at a time.
Supports accessibility for: Memory; Organization
Student Facing
200 people were asked if they prefer dogs or cats, and whether they live in a rural or urban setting.
The actual values collected from the survey are in the first table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "urban"}, {"_attributes": {"scope": "col"}, "_value": "rural"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "cat"}], "td": [{"_value": "54"}, {"_value": "42"}, {"_value": "96"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "dog"}], "td": [{"_value": "80"}, {"_value": "24"}, {"_value": "104"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "total"}], "td": [{"_value": "134"}, {"_value": "66"}, {"_value": "200"}]}]}]}]}
The next table shows what percentage of the 200 total people included are represented by each combination of categories. The segmented bar graph represents the same information graphically.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "urban"}, {"_attributes": {"scope": "col"}, "_value": "rural"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "cat"}], "td": [{"_value": "27%"}, {"_value": "21%"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "dog"}], "td": [{"_value": "40%"}, {"_value": "12%"}]}]}]}]}
[IMG]
The next table shows the percentage of each column that had a certain pet preference in a column relative frequency table. The segmented bar graph represents the same information graphically.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "urban"}, {"_attributes": {"scope": "col"}, "_value": "rural"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "cat"}], "td": [{"_value": "40%"}, {"_value": "64%"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "dog"}], "td": [{"_value": "60%"}, {"_value": "36%"}]}]}]}]}
[IMG]
The last table shows the percentage of each row that live in a certain area in a row relative frequency table. The segmented bar graph represents the same information graphically.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "urban"}, {"_attributes": {"scope": "col"}, "_value": "rural"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "cat"}], "td": [{"_value": "56%"}, {"_value": "44%"}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "dog"}], "td": [{"_value": "77%"}, {"_value": "23%"}]}]}]}]}
[IMG]
For each relative frequency table, select a percentage and explain how numbers from the original table were used to get the percentage.
What percentage of those surveyed live in an urban area and prefer dogs?
Among the people surveyed who prefer dogs, what percentage of them live in an urban setting?
What percentage of people surveyed who live in an urban setting prefer dogs?
How many of the people responded that they prefer dogs and live in an urban setting?
Among the people surveyed, are there more people who prefer dogs or cats?
Your pet food company has access to a billboard in a rural setting. Would you recommend advertising dog food or cat food on this billboard? Which table did you use to make this decision? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Students may not understand the difference between what each relative frequency table represents. You can have a whole-class discussion about the data presented before students begin the task. You can also individually ask students questions concerning how the different representations relate to one another. For example: ”Which percentage in the relative frequency table corresponds to a specific portion of the bar graph?” Ensure students understand that each table uses a different total to get the percentages. Ask them, “Which tables are the questions referring to when it says, ‘Among the people surveyed’?”
Activity Synthesis
The purpose of this discussion is to make sure students understand how to interpret information in a relative frequency table and to prime student thinking about associations between categorical variables.
After selecting a student to share a response, select another student to share which table or graph can be used to answer the question, then select a third student to explain the connection between the question and the table or graph.
Ask students:
“What information can be understood from each type of relative frequency table?” (The first table can explain how each combination of categories fits into the whole group. The second table divides the entire group by their location and shows the pet preference in those locations. The third table divides the entire group by pet preference and shows the location of people for each pet preference.)
“Do you notice any interesting patterns in the data? Explain your reasoning.” (It is okay if students struggle with this question at this time. The focus of the next lesson is on associations between categorical variables. Students may notice that dog lovers tend to live in urban settings, since 77% of dog lovers are urban.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their ideas, press for details by requesting students to challenge an idea, elaborate on an idea, or give an example. Revoice student responses to demonstrate mathematical language use by applying appropriate language. For instance, revoice the statement: “The percentage of people living in a rural setting who love cats,” as “The percentage of cat lovers who are rural.” Call students’ attention to any words or phrases that helped clarify the original statement. If needed, practice phrases or words through choral response. This provides more students with an opportunity to produce language as they share their ideas about data represented in relative frequency tables.
Design Principle(s): Support sense-making


aligns: no