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: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
problem_activity: Optional activity
In this activity, students compare and contrast different measures of center and variability for data sets that have gaps and are not symmetrical. They interpret mean, MAD, median, and IQR in the context of a situation. Unlike many of the data sets students have seen so far, this one shows values that could roughly divide into three parts: the days when there is little or no homework, the days when there is a moderate number of homework problems, and the days when the assignment is relatively large. Because of this distribution, finding a typical number of homework problems (or whether it would be helpful to identify a typical number) is not obvious, prompting students to interpret measures of center and spread more carefully (MP2).
As students work and discuss, identify at least one student or group that decides that the mean and MAD are appropriate measures of center and spread and can explain their reasoning, and another that decides to go with the median and IQR and could support their choice. Invite them to share during whole-class discussion.
Digital
Launch
Keep students in groups of 2. Give students a moment of quiet time to look at the data on homework problems and identify at least one thing they notice and one thing they wonder. Give them another brief moment to share their observation and question with their partner. Then, ask a few students to share their responses with the class.
Students are likely to notice that the data values are quite different, that there are some days with no homework and others with quite a few problems, that there is not an obvious cluster, or that the number of problems could be roughly grouped into three kinds (a little, moderate, and a lot). They are likely to wonder why the numbers are so spread out and varied.
Briefly discuss the following questions to encourage students to think about the data contextually:
“Why might the homework assignment data show this distribution? What are some possible explanations?” (When only one problem was assigned, the problem might be particularly challenging or might require considerable work or collaboration. Another possibility: there might be an upcoming exam, so the homework load was reduced. When many problems were assigned, the problems might be quick exercises with short answers, or the assignment might be review materials for an entire chapter.)
“How might we describe ‘a typical number of homework problems’ in this case?”
“Which do you predict would be higher: the mean or the median number of problems? Why?”
Next, give students 8–10 minutes to complete the task, either independently or collaboratively. Ask students to think quietly about the last question before discussing their response with their partner.
If students are using the digital activities, they will need to enter the data points in the column “A” for the applet to “list”, “sort”, etc. The applet allows for students to populate their own mean, Q1 and Q3 values.
Student Facing
Over a two-week period, Mai recorded the number of math homework problems she had each school day.
2
15
20
0
5
25
1
0
10
12
Calculate the following. Show your reasoning.
The mean number of math homework problems.
The mean absolute deviation (MAD).
Interpret the mean and MAD. What do they tell you about the number of homework problems Mai had over these two weeks?
Find or calculate the following values and show your reasoning.
The median, quartile, maximum, and minimum of the same data on Mai’s math homework problems.
The interquartile range (IQR).
Which pair of measures of center and variability—mean and MAD, or median and IQR—do you think summarize the distribution of Mai’s math homework assignments better? Explain your reasoning.
You may use the applet below to help if you choose to. Begin by dragging the left edge across the screen until you see only one column in the spreadsheet. Enter the values needed to calculate the IQR and the mean when prompted.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Keep students in groups of 2. Give students a moment of quiet time to look at the data on homework problems and identify at least one thing they notice and one thing they wonder. Give them another brief moment to share their observation and question with their partner. Then, ask a few students to share their responses with the class.
Students are likely to notice that the data values are quite different, that there are some days with no homework and others with quite a few problems, that there is not an obvious cluster, or that the number of problems could be roughly grouped into three kinds (a little, moderate, and a lot). They are likely to wonder why the numbers are so spread out and varied.
Briefly discuss the following questions to encourage students to think about the data contextually:
“Why might the homework assignment data show this distribution? What are some possible explanations?” (When only one problem was assigned, the problem might be particularly challenging or might require considerable work or collaboration. Another possibility: there might be an upcoming exam, so the homework load was reduced. When many problems were assigned, the problems might be quick exercises with short answers, or the assignment might be review materials for an entire chapter.)
“How might we describe ‘a typical number of homework problems’ in this case?”
“Which do you predict would be higher: the mean or the median number of problems? Why?”
Next, give students 8–10 minutes to complete the task, either independently or collaboratively. Ask students to think quietly about the last question before discussing their response with their partner.
If students are using the digital activities, they will need to enter the data points in the column “A” for the applet to “list”, “sort”, etc. The applet allows for students to populate their own mean, Q1 and Q3 values.
Student Facing
Over a two-week period, Mai recorded the number of math homework problems she had each school day.
2
15
20
0
5
25
1
0
10
12
Calculate the following. Show your reasoning.
The mean number of math homework problems
The mean absolute deviation (MAD)
Interpret the mean and MAD. What do they tell you about the number of homework problems Mai had over these two weeks?
Find or calculate the following values and show your reasoning.
The median, quartiles, maximum, and minimum of Mai’s data
The interquartile range (IQR)
Which pair of measures of center and variability—mean and MAD, or median and IQR—do you think summarizes the distribution of Mai’s math homework assignments better? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Briefly discuss students’ interpretations of the measures they just calculated:
“What do the mean of 9 and MAD of 7.4 tell us? How can we interpret them in this context?”
“What do the median of 7.5 and IQR of 14 tell us?”
Then, select two or more previously identified students to share their responses about which measures of center and spread are appropriate for summarizing the data set. After each person shares, briefly poll the class to see if others reasoned about the measures the same way. Sum up by asking:
“Now that you have two pairs of measures of center and spread, how would you respond if someone asked you, ‘What is a typical number of homework problems for Mai’s class?’ Is the question easier to answer now?”
Students should walk away with increased awareness that, in some cases, measures of center and spread do not always paint a full picture of what the actual data set entails, and that the measures should be interpreted with care.


aligns: no