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: Explain and use the relationship between the sine and cosine of complementary angles.
problem_activity: Task
Students were asked to report how far (in miles) they each live from school. The following distances were recorded.
{"table": [{"tbody": [{"tr": [{"td": [{"p": [{}]}, {"p": [{"_value": "Student"}]}, {"p": [{"_value": "Distance"}]}]}, {"td": [{"p": [{"_value": "1"}]}, {"p": [{"_value": "Zane"}]}, {"p": [{"_value": "0.4"}]}]}, {"td": [{"p": [{"_value": "2"}]}, {"p": [{"_value": "Jackson"}]}, {"p": [{"_value": "0.5"}]}]}, {"td": [{"p": [{"_value": "3"}]}, {"p": [{"_value": "Benjamin"}]}, {"p": [{"_value": "1.0"}]}]}, {"td": [{"p": [{"_value": "4"}]}, {"p": [{"_value": "Bethany"}]}, {"p": [{"_value": "1.0"}]}]}, {"td": [{"p": [{"_value": "5"}]}, {"p": [{"_value": "Joe"}]}, {"p": [{"_value": "2.4"}]}]}, {"td": [{"p": [{"_value": "6"}]}, {"p": [{"_value": "Noelle"}]}, {"p": [{"_value": "2.7"}]}]}, {"td": [{"p": [{"_value": "7"}]}, {"p": [{"_value": "Tianye"}]}, {"p": [{"_value": "3.0"}]}]}, {"td": [{"p": [{"_value": "8"}]}, {"p": [{"_value": "Anthony"}]}, {"p": [{"_value": "3.2"}]}]}, {"td": [{"p": [{"_value": "9"}]}, {"p": [{"_value": "Amanda"}]}, {"p": [{"_value": "4.1"}]}]}, {"td": [{"p": [{"_value": "10"}]}, {"p": [{"_value": "Michaela"}]}, {"p": [{"_value": "4.2"}]}]}, {"td": [{"p": [{"_value": "11"}]}, {"p": [{"_value": "Miranda"}]}, {"p": [{"_value": "4.4"}]}]}, {"td": [{"p": [{"_value": "12"}]}, {"p": [{"_value": "Joseph"}]}, {"p": [{"_value": "5.0"}]}]}, {"td": [{"p": [{"_value": "13"}]}, {"p": [{"_value": "John"}]}, {"p": [{"_value": "7.5"}]}]}]}]}]}
1.  Summary statistics for the distances are given below.
{"table": [{"tbody": [{"tr": [{"td": [{"p": [{"_value": "Min"}]}, {"p": [{"_value": "Q1"}]}, {"p": [{"_value": "Median"}]}, {"p": [{"_value": "Q3"}]}, {"p": [{"_value": "Max"}]}, {"p": [{"_value": "Mean"}]}]}, {"td": [{"p": [{"_value": "0.4"}]}, {"p": [{"_value": "1.0"}]}, {"p": [{"_value": "3.0"}]}, {"p": [{"_value": "4.3"}]}, {"p": [{"_value": "7.5"}]}, {"p": [{"_value": "3.03"}]}]}]}]}]}
Construct a box plot for the distances and describe the main features of the distribution.
2.  John currently lives furthest from school. Would you consider his distance from school (7.5 miles) to be ”unusual"? Explain.
3.  John’s family is considering moving to a house that is 10 miles from school. How will this move affect the summary statistics? Change John’s distance from 7.5 miles to 10 miles and complete the table based on this new data.
{"table": [{"tbody": [{"tr": [{"td": [{"p": [{"_value": "Min"}]}, {"p": [{"_value": "Q1"}]}, {"p": [{"_value": "Median"}]}, {"p": [{"_value": "Q3"}]}, {"p": [{"_value": "Max"}]}, {"p": [{"_value": "Mean"}]}]}, {"td": [{"p": [{}]}, {"p": [{}]}, {"p": [{}]}, {"p": [{}]}, {"p": [{}]}, {"p": [{}]}]}]}]}]}
Has the mean increased, decreased or remained unchanged? Has the median increased, decreased or remained unchanged? Explain how John’s move has affected these measures of center.
4.  Construct a boxplot of the distances after John's move. Would you consider John's new distance from school (10 miles) to be "unusual" now? Explain.
5.  A data point can be considered an “outlier” if it is more than 1.5 times the IQR above Q3 or more than 1.5 times the IQR below Q1. Using this description of an outlier, was John’s distance from school considered an outlier before the move? How about after the move? Support your answer with appropriate work.
6.  Zane lives closest to school. Using the description of an outlier given in question 5, is his distance considered an outlier?


aligns: no