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: Add, subtract, and multiply matrices of appropriate dimensions.
problem_activity: Narrative
In this activity, students create a line plot using measurements to the nearest
\(\frac{1}{4}\)
and
\(\frac{1}{8}\)
inch. This task prompts students to use their understanding of fraction equivalence to plot and partition the horizontal axis.
Representation: Access for Perception.
Provide access to fraction strips that show fourths and eighths, and invite students to use them to answer question 4. Ask students to identify correspondences between the fraction strips and the horizontal axis of the line plot.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Groups of 2
“The table lists many different lengths.”
“What do you notice about the pencil lengths?” (Sample responses:
Some repeat more than one time.
The numbers are mixed numbers.
There are no whole numbers.)
1 minute: quiet think time
“There are some lengths that are more common or occur more often than others.”
“Tell a partner the length that is most common.” (
\(6\frac{3}{4}\)
)
Activity
Groups of 2
5 minutes: independent work time
Monitor for students who use equivalence to plot measurements to the nearest eighth inch.
“Share your line plots with your partner and make revisions as needed.”
2 minutes: partner discussion
Student Facing
Andre’s class measured the length of some colored pencils to the nearest
\(\frac{1}{4}\)
inch. The data are shown here:
\(1\frac{3}{4}\)
\(2\frac{1}{4}\)
\(5\frac{1}{4}\)
\(5\frac{1}{4}\)
\(4\frac{2}{4}\)
\(4\frac{2}{4}\)
\(6\frac{1}{4}\)
\(6\frac{3}{4}\)
\(6\frac{3}{4}\)
\(6\frac{3}{4}\)
[IMG]
Plot the colored-pencil data on the line plot.
[IMG]
Which colored-pencil length is the most common in the data set?
Write 2 new questions that could be answered using the line plot data.
Next, Andre’s class measured their colored pencils to the nearest
\(\frac{1}{8}\)
inch. The data are shown here:
\(1\frac{6}{8}\)
\(2\frac{2}{8}\)
\(5\frac{2}{8}\)
\(5\frac{3}{8}\)
\(4\frac{4}{8}\)
\(4\frac{4}{8}\)
\(6\frac{6}{8}\)
\(6\frac{6}{8}\)
\(6\frac{6}{8}\)
\(6\frac{3}{8}\)
Plot the colored-pencil data on the line plot.
[IMG]
Which colored-pencil length is the most common in the line plot?
Why did some colored-pencil lengths change on this line plot?
What is the difference between the length of the longest colored pencil and the shortest colored pencil? Show your reasoning.
Student Response
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Advancing Student Thinking
If students do not yet identify “most common” length as the length with the most data points on the line plot, consider asking: “Which measurement occurred more often than others?” and “Which pencil measurement was plotted most often?”
Activity Synthesis
“How many colored pencils were measured in Andre’s class?” (10, because there were ten data points and each represented a pencil that was measured.)
“What was the most common measurement in the first set of data? In the second set of data?” (In the first set of data:
\(6\frac{3}{4}\)
inches. In the second set of data:
\(6\frac{6}{8}\)
inches.)
“How did you use equivalence to help as you plotted measurement data in eighths of an inch?” (I know that two eighths are equivalent to one fourth and this helped to find eighths on the line plots.)
Use the line plot image to clearly label
\(\frac{1}{2}\)
,
\(\frac{1}{4}\)
and
\(\frac{1}{8}\)
with help from students.


aligns: no