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: 
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_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Read, write, and compare decimals to thousandths.
problem_activity: Narrative
The purpose of this activity is for students to use the weights from the previous activity to support place value understanding, specifically to see the multiplicative relationships between different decimal place values (MP7). These relationships will be taken up in greater detail in the next unit but the weights provide a convenient way to see these relationships which complements the diagrams students used in earlier lessons.
Students first compare weights of two gold nuggets, one weighed using 0.1 ounce weights and the other using 0.01 ounce weights. The two nuggets have the same weight because ten 0.01 ounce weights are equivalent to one 0.1 ounce weight, as students saw in the warm-up. Students move from here to making multiplicative comparisons between place values. They can use the weights to help visualize or calculate or they might use a diagram like those in the previous lesson.
Representation: Internalize Comprehension.
Synthesis: Provide students with a partially completed graphic organizer, such as a two column table, to record the relationship between tenths and hundredths, hundredths and thousandths, and tenths and thousandths.
Supports accessibility for: Conceptual Processing, Memory
Launch
Groups of 2
Activity
2 minutes: independent work time
8 minutes: partner work time
Monitor for students who use different strategies to compare the values of the 6s in the gold nugget weights:
using the value of each place in the decimal
thinking about fractions or representing the decimals as fractions
using a diagram such as a hundredths grid
Student Facing
How many 0.01 ounce weights will balance one 0.1 ounce weight? Explain or show your reasoning.
How many 0.001 ounce weights will balance a 0.1 ounce weight? Explain or show your reasoning.
The table shows the weights of 3 of the gold nuggets Diego and his friends found panning for gold.
Fill in the blanks. Explain or show your reasoning.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"style": "text-align:center"}, "_value": "gold"}, {"_attributes": {"style": "text-align:center"}, "br": [{}], "_values": ["weight", "(grams)"]}]}, {"td": [{"_value": "nugget A"}, {"_value": "0.6"}]}, {"td": [{"_value": "nugget B"}, {"_value": "0.06"}]}, {"td": [{"_value": "nugget C"}, {"_value": "0.006"}]}]}]}]}
Nugget A weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget B.
Nugget A weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget C.
Nugget C weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget B.
Nugget C weighs
\(\underline{\hspace{0.9cm}}\)
times as much as Nugget A.
Student Response
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Activity Synthesis
“How are the weights of the nuggets the same? How are they different?” (They all have a 6 in them. They each balance with 6 of one of the unit weights. The value of the 6 is different for each nugget.)
“How do hundredths, the weights for B, compare to tenths, the weights for A?” (There are 10 hundredths in each tenth or a tenth of a tenth is a hundredth.)
“Nugget B weighs how many times as much as Nugget A? How do you know?” (
\(\frac{1}{10}\)
because Nugget A is 6 tenths and a tenth of that is 6 hundredths since a tenth of a tenth is a hundredth.)
“How do thousandths, the weights for C, compare to tenths, the weights for A?” (There are 100 thousandths in a tenth and one hundredth of a tenth is a thousandth.)
“Nugget A weighs how many times as much as Nugget C?” (100, there are 100 thousandths in a tenth.)


aligns: yes