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 volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
problem_activity: Activity
In previous grades, students learned how to calculate the volume of a cylinder. In this activity, they revisit that process. Students are prompted to think about volume in 1-unit layers. This will be helpful in upcoming lessons when students learn about Cavalieri’s Principle, or the idea that if two solids have equal-area cross sections at all heights, the solids have equal volumes. Cavalieri’s Principle will be used to derive a formula for the volume of a pyramid.
Note that in this activity, “volume of a container” is used as shorthand for “the volume of the region enclosed by the container.”
Monitor for students who calculate the volume of the water in the prism then work backward to find the height of the water in the cylinder, and for those who use deductive logic (for example, reasoning that because the two bases have equal area, the height of the water must be equal in both solids) to arrive at the same answer.
Students reason abstractly and quantitatively (MP2) when they compare the prism and cylinder volumes.
Launch
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about the volumes of solids. Before revealing the questions in this activity, display the image of both solids along with the statement: “Here are two solids. All measurements are in centimeters,” and ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the volume of both solids. For example, “Does the cylinder have the same volume as the prism?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because _____. Then, I….”, “I noticed _____ so I….”, and “I tried _____, and what happened was….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are two containers. All measurements are in centimeters.
[IMG]
Suppose the prism contains water that reaches a height of 1 cm.
Draw a representation of this situation.
The water is poured from the prism into the cylinder. What is the height of the water in the cylinder? Explain your reasoning.
Suppose the prism contained water that reached a height of 3 cm instead of 1 cm. If the water were poured into the empty cylinder, what would the height of the water in the cylinder be?
Student Response
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Anticipated Misconceptions
Students may be unsure how to work with the prism measurement
\(4\pi\)
cm. Invite them to write out the area calculation as
\(9\boldcdot4\pi\)
. Alternatively, they can use an approximation of
\(\pi\)
to get rounded answers.
If students struggle to find the height of the water in the cylinder, ask them what they already know about the cylinder, and to consider what additional information they would need to find the height. Prompt them to use their responses to the first question to help them answer this one.
Activity Synthesis
The goal is to make connections between methods of calculating volumes for prisms and cylinders. Ask previously identified students to share their reasoning, starting with those who did volume calculations and worked backward, and ending with those who used deductive reasoning. Connect these methods by asking these questions:
“Describe what volume formulas you used. Do they apply to just one of the containers, or to both?” (Sample responses: The expression
\(\ell wh\)
applies to the prism. The expression
\(\pi r^2 h\)
applies to the cylinder. The expression
\(BH\)
applies to both.)
“Consider the volume formula
\(V=Bh\)
. How does this formula help explain why the heights were the same for the same volume of water in each container?” (The area of the base,
\(B\)
, is the same for each container:
\(36\pi\)
square centimeters. The volume formula can be rewritten
\(h=V\div B\)
, so if the containers have the same volume of water in each and have bases with the same area, the height will be the same.)
Invite students to describe how the volumes of the two containers compare (they are equal), and challenge them to describe another solid that has the same volume as these two.


aligns: no