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 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: 
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_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
problem_activity: Activity
This activity prompts students to create expressions to represent the quantities and relationships in a situation and engages them in mathematical modeling.
Students plan a pizza party and present a cost estimate. To do so, they need to consider relevant variables, make assumptions and estimates, perform calculations, and adjust their thinking along the way (MP4). Some students may choose to perform research and revise their models as they gather new information. There are many possible solutions to the task.
As students discuss their ideas, monitor for those who:
find and use actual data or exact values (for example, count the number of students in the class, research the cost of a large pizza at a nearby shop, or quickly survey the class for topping preferences)
estimate quantities based on prior knowledge (for example, the cost of a large pizza in a recent purchase, or the number of slices they and their friends generally consume at lunch time)
make assumptions about behaviors, preferences, or quantities (for instance, assume that a certain percentage of the class prefers a certain topping)
Launch
Ask students if they have ever been in charge of planning a party. Solicit a few ideas of what party planners need to consider. Ask students to imagine being in charge of a class pizza party. Explain that their job is to present a plan and a cost estimate for the party.
Arrange students in groups of 4. Provide access to calculators and, if feasible and desired, access to the internet for researching pizza prices. Students can also make estimates based on prior experience, refer to printed ads, or use their personal device to look up pricing information.
Limit the time spent on the first question to 7–8 minutes and pause the class before students move on to subsequent questions. Give groups of students 1–2 minutes to share their proposals with another group. Then, select a few groups who used contrasting strategies (such as those outlined in the Narrative) to briefly share their plans with the class. Record or display their plans for all to see.
Next, ask students to complete the remaining questions. If needed, give an example of an expression that can be written to represent a cost calculation.
Student Facing
Imagine your class is having a pizza party.
Work with your group to plan what to order and to estimate what the party would cost.
[IMG]
Record your group’s plan and cost estimate. What would it take to convince the class to go with your group’s plan? Be prepared to explain your reasoning.
Write down one or more expressions that show how your group’s cost estimate was calculated.
In your expression(s), are there quantities that might change on the day of the party? Which ones?
Rewrite your expression(s), replacing the quantities that might change with letters. Be sure to specify what the letters represent.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Find a pizza place near you and ask about the diameter and cost of at least two sizes of pizza. Compare the cost per square inch of the sizes.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students do not understand what is meant by “quantities that might change,” ask them if it is more likely that the cost of a pepperoni pizza increases on the day of the party, or that some students are absent that day. In a model that incorporates both of these quantities, they may wish to use a number for the cost of each type of pizza and a letter for the number of students present that day.
Activity Synthesis
Invite groups who did not previously share their plans to share the expressions they wrote and explain what the expressions represent. After each group shares, ask if others calculated the costs the same way but wrote different expressions.
As students present their expressions, record the quantities that they mention and display them for all to see. Some examples:
the number of students in the class
the number of pizza slices per person
the cost of delivery
the price per topping
Briefly discuss the quantities that students anticipate would change (and therefore would replace with letters).
Explain to students that the expressions they have written are examples of mathematical
models
. They are mathematical representations of a situation in life that can be used to make sense of problems and solve them. We will look more closely at how expressions could represent the quantities in a situation like party planning, which involves certain conditions or requirements.
Representing, Conversing: MLR7 Compare and Connect.
As students share their plans with the class, call students’ attention to the different ways the quantities are represented in expressions, and to the different expressions that represent the same quantity. Take a close look at the contrasting strategies used within the context of the situation. Wherever possible, amplify student words and actions that students use to revise their models by estimating quantities and making assumptions about the situation.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: yes