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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_pos_0

8
standard_description: Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
problem_activity: Activity
Students have already looked at measurement conversions that can be represented by proportional relationships, both in grade 6 and in previous lessons in this unit. This task introduces a measurement conversion that is not associated with a proportional relationship. The discussion should start to move students from determining whether a relationship is proportional by examining a table, to making the determination from the equation.
Note that some students may think of the two scales on a thermometer like a double number line diagram, leading them to believe that the relationship between degrees Celsius and degrees Fahrenheit is proportional. If not mentioned by students, the teacher should point out that when a double number line is used to represent a set of equivalent ratios, the tick marks for 0 on each line need to be aligned.
Launch
Arrange students in groups of 2. Provide access to calculators, if desired. Give students 5 minutes of quiet work time followed by students discussing responses with a partner, followed by whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
The other day you worked with converting meters, centimeters, and millimeters. Here are some more unit conversions.
Use the equation
\(F =\frac95 C + 32\)
, where
\(F\)
represents degrees Fahrenheit and
\(C\)
represents degrees Celsius, to complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "temperature", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "41140"}, "_value": "\\((^\\circ\\text{C})\\)"}]}, {"_attributes": {"scope": "col"}, "_value": "temperature", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "41172"}, "_value": "\\((^\\circ\\text{F})\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "20"}, {}]}, {"td": [{"_value": "4"}, {}]}, {"td": [{"_value": "175"}, {}]}]}]}]}
Use the equation
\(c = 2.54n\)
, where
\(c\)
represents the length in centimeters and
\(n\)
represents the length in inches, to complete the table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "length (in)"}, {"_attributes": {"scope": "col"}, "_value": "length (cm)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "10"}, {}]}, {"td": [{"_value": "8"}, {}]}, {"td": [{"_value": "3", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "238"}, "_value": "\\(\\frac12\\)"}]}, {}]}]}]}]}
Are these proportional relationships? Explain why or why not.
Student Response
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Anticipated Misconceptions
Some students may struggle with the fraction
\(\frac95\)
in the temperature conversion. Teachers can prompt them to convert the fraction to its decimal form, 1.8, before trying to evaluate the equation for the values in the table.
Activity Synthesis
After discussing the work students did, ask, “What do you notice about the forms of the equations for each relationship?” After soliciting students’ observations, point out that the proportional relationship is of the form
\(y = kx\)
, while the nonproportional relationship is not.
Speaking: MLR8 Discussion Supports.
To provide a support students in producing statements when they compare and contrast equations with proportional and nonproportional relationships. Provide sentence frames such as: “I noticed that _____,” “What makes _____ different from the others is _____,” or “The patterns show _____.” Invite students to use these sentence frames when students comprehend orally the patterns they notice with proportional equations.
Design Principle(s): Support sense-making


aligns: yes