few_shots: 
Example 1:
Concept/skill: 
Recognize and represent proportional relationships between quantities.

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?


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:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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
$$-\frac{1}{4}$$


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:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

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?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, assess whether the following problem or activity is 'aligned' with the given Common Core standard in Mathematics.

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
problem_activity: Narrative
The purpose of this activity is for students to convert between measurements in milliliters and liters, providing practice multiplying and dividing by 1,000. Students work with numbers in many forms including whole numbers, decimals, fractions, and numbers in exponential form.
Engagement: Internalize Self-Regulation.
Synthesis: Provide students an opportunity to self-assess and reflect on their own progress. For example, provide students with a partially completed chart to compare their answers and talk about the similarities and differences with their partners.
Supports accessibility for: Attention; Memory; Social-Emotional Functioning
Launch
Groups of 2
Display the image.
“What do you notice? What do you wonder?” (There are different sized water bottles. There are numbers underneath the water bottles. How much more water does the big one hold than the little one? How many milliliters are in a liter?)
Display a table like this one:
{"table": [{"_attributes": {"border": "1", "style": "width:40%"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "_value": "L"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "_value": "mL"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "1"}, {"_attributes": {"style": "text-align:center"}, "_value": "1,000"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "10"}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "0.1"}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "100,000"}]}, {"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "10"}]}]}]}]}
“What numbers go in the empty boxes in the table?”
30 seconds quiet think time
“Explain your thinking to a partner.”
Fill in the table and leave it displayed for students to refer to during the lesson.
{"table": [{"_attributes": {"border": "1", "style": "width:40%"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "_value": "L"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "_value": "mL"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "1"}, {"_attributes": {"style": "text-align:center"}, "_value": "1,000"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "10"}, {"_attributes": {"style": "text-align:center"}, "_value": "10,000"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "0.1"}, {"_attributes": {"style": "text-align:center"}, "_value": "100"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "100"}, {"_attributes": {"style": "text-align:center"}, "_value": "100,000"}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "0.01"}, {"_attributes": {"style": "text-align:center"}, "_value": "10"}]}]}]}]}
Activity
3–5 minutes: independent work time
1–2 minutes: partner discussion
Monitor for students who:
compare the quantities by converting milliliters to liters
compare by converting liters to milliliters
Student Facing
[IMG]
Complete the table.
{"table": [{"_attributes": {"border": "1", "style": "width:50%"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "_value": "L"}, {"_attributes": {"scope": "col", "style": "text-align:center; width:50%"}, "_value": "mL"}]}]}], "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}, "_value": "5"}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "6.3"}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "_value": "0.95"}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(10^2\\)"}]}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "800,000"}]}, {"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(10^6\\)"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "_value": "65"}]}]}]}]}
Decide if the two measurements are equal. If not, choose which one is greater. Explain or show your reasoning.
15 mL and 0.15 L
2,500 mL and 2.5 L
200 mL and
\(\frac{1}{4}\)
L
1 mL and
\(\frac{1}{1,000}\)
L
15,600 mL and 15.5 L
Student Response
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Advancing Student Thinking
If students confuse the operations they need to use to convert milliliters to liters or liters to milliliters, refer to the table from the launch and ask, “What patterns do you notice?”
If necessary, add rows to the table and ask them to explain how many milliliters of water are in 2 liters of water and 3 liters of water, and 0.001 liter of water.
Activity Synthesis
Invite selected students to share how they compared the measurements.
Display: 15,600 mL and 15.5 L
“How many liters are 15,600 milliliters? How do you know?” (15.6 since I divide by 1,000)
“How many milliliters are 15.5 liters? How do you know?” (15,500 since there are 1,000 milliliters in each liter so that’s 15,000 and half of a thousand which is 500.)
“Which is greater? 15,600 milliliters or 15.5 liters? How do you know?” (15,600 mL because I can compare them using either liters or milliliters.)
“When you solved this problem, did you convert from milliliters to liters or from liters to milliliters? Why?” (I converted from liters to milliliters because multiplying by 1,000 is more comfortable than dividing by 1,000.)


aligns: yes