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: 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.
problem_activity: Activity
Students design districts in three towns to “gerrymander” the results of elections. In two of the towns, the election results can be skewed to either color. In the third, it isn’t possible to skew the results.
Launch
Arrange students in groups of 2–4.
Explain the history of gerrymandering: Sometimes people in charge of designing districts make them in strange shapes to give the election results they want. One of the first was Elbridge Gerry (governor of Massachusetts in 1812), whose party designed a district that many people thought looked like a salamander. They called a Gerrymander, and the name stuck. It means a very strangely shaped, spread-out district designed to produce a certain result.
Usually districts are required to be connected: a person traveling to all parts of the district should be able to stay inside the district. There should be no “islands” that are separated by parts of other districts.
Engagement: Internalize Self-Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to four out of six task statements.
Supports accessibility for: Organization; Attention
Student Facing
Smallville’s map is shown, with opinions shown by block in green and gold. Decompose the map to create three connected, equal-area districts in two ways:
Design three districts where
green
will win at least two of the three districts. Record results in Table 1.
[IMG]
Table 1:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Design three districts where
gold
will win at least two of the three districts. Record results in Table 2.
[IMG]
Table 2:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Squaretown’s map is shown, with opinions by block shown in green and gold. Decompose the map to create five connected, equal-area districts in two ways:
Design five districts where
green
will win at least three of the five districts. Record the results in Table 3.
[IMG]
Table 3:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}, {}, {}]}]}]}]}
Design five districts where
gold
will win at least three of the five districts. Record the results in Table 4.
[IMG]
Table 4:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}, {}, {}]}]}]}]}
Mountain Valley’s map is shown, with opinions by block shown in green and gold. (This is a town in a narrow valley in the mountains.) Can you decompose the map to create three connected, equal-area districts in the two ways described here?
Design three districts where
green
will win at least 2 of the 3 districts. Record the results in Table 5.
[IMG]
Table 5:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Design three districts where
gold
will win at least 2 of the 3 districts. Record the results in Table 6.
[IMG]
Table 6:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Student Response
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Anticipated Misconceptions
The white streets on the maps do not disconnect the districts.
However, probably squares that are only connected at their corners should not be considered connected to each other. (But you or the students can make their own rule, since they are in charge of making districts.)
Activity Synthesis
This set of questions is similar to the previous activity, except that the patterns are irregular, so more planning and computation is needed to design districts.
Ask several students with different maps to show and explain their work.
The important result for this problem is that it’s possible to design both fair districts (where the result of the vote is similar to the vote if all individual votes were counted) and unfair districts. According to the rules (equal area and connected), these sets of districts are legal. Encourage students to try to devise some rules for drawing districts that would be more fair.


aligns: yes