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: 
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.

Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_pos_0

8
standard_description: Find inverse functions.
problem_activity: Problem 1
Part A: Predict the shape of the graph at 2,000 trials.
{"table": [{"_attributes": {"class": ["table"], "style": "width:100%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "10 trials"}, {"_attributes": {"style": "text-align:center"}, "_value": "100 trials"}, {"_attributes": {"style": "text-align:center"}, "_value": "533 trials"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "img": [{"_attributes": {"alt": "", "src": "https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l7_at1.png", "style": "height:122px; width:300px"}}]}, {"_attributes": {"style": "text-align:center"}, "img": [{"_attributes": {"alt": "", "src": "https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l7_at1a.png", "style": "height:115px; width:300px"}}]}, {"_attributes": {"style": "text-align:center"}, "img": [{"_attributes": {"alt": "", "src": "https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l7_at1b.png", "style": "height:120px; width:300px"}}]}]}]}]}]}
Part B: At 1,799 trials, the graph looks like this:
[IMG]
The mean of the data is approximately 6, and the standard deviation is 1.7. The blue lines around the graph are what should be the "ideal" normal distribution.
Describe what the mean and the standard deviation mean in terms of this situation.
Problem 2
Match the following distributions with the histograms. Explain your reasoning.
{"table": [{"_attributes": {"align": "center", "class": ["table"], "style": "width:80%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Distribution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Shape"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Mean"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Standard Deviation"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "A"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Skewed to the right"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "100"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "10"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "B"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Approximately symmetric, mound shaped"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "100"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "10"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "C"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Approximately symmetric, mound shaped"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "100"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "40"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "D"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Skewed to the right"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "100"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "40"}]}]}]}]}]}
[IMG]


aligns: no