few_shots: 
Example:
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_001392_neg_0

8
standard_description: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.
problem_activity: Optional activity
The purpose of this activity is for students to choose how they can apply math concepts and strategies to a problem arising in a real-world context: predicting whether a restaurant will make a profit.
After students have estimated the monthly cost of their ongoing expenses on their list, poll the class on their total estimated monthly costs.
If students have completed the previous lesson about using spreadsheets to calculate the cost of ingredients for one serving of their recipe, they can use that as the basis for calculating the percentage of the markup. If students have not calculated the cost of ingredients in a previous lesson, then tell students that the
ingredients
for one serving of a meal typically cost about:
$
2 for a fast-food restaurant
$
5 for a casual sit-down restaurant
$
10 to
$
25 for a formal sit-down restaurant
Launch
Point out that restaurants have many more expenses than just the cost of the food. Give students quiet work time followed by partner discussion.
Consider having students pause their work after the first question so that you can record their answers displayed for all to see, and all students will have access to this same information for the rest of the activity.
Speaking, Listening, Writing: MLR1 Stronger and Clearer Each Time.
Use this routine after students have assessed the profitability of Restaurant A. Ask students to share their thinking with 2–3 consecutive partners. With each share, encourage listeners to push students for clarity in mathematical language (e.g., “How do you know the restaurant is making a profit?”). With each share, students’ organization of steps should get stronger and the explanations of reasoning should get clearer. This helps students to use mathematical language as they explain their profit analysis.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Student Facing
Restaurants have many more expenses than just the cost of the food.
Make a list of other items you would have to spend money on if you were running a restaurant.
Identify which expenses on your list depend on the number of meals ordered and which are independent of the number of meals ordered.
Identify which of the expenses that are independent of the number of meals ordered only have to be paid once and which are ongoing.
Estimate the monthly cost for each of the ongoing expenses on your list. Next, calculate the total of these monthly expenses.
Tell whether each restaurant is making a profit or losing money if they have to pay the amount you predicted in ongoing expenses per month. Organize your thinking so it can be followed by others.
Restaurant A sells 6,000 meals in one month, at an average price of
$
17 per meal and an average cost of
$
4.60 per meal.
Restaurant B sells 8,500 meals in one month, at an average price of
$
8 per meal and an average cost of
$
2.20 per meal.
Restaurant C sells 4,800 meals in one month, at an average price of
$
29 per meal and an average cost of
$
6.90 per meal.
Predict how many meals your restaurant would sell in one month.
How much money would you need to charge for each meal to be able to cover all the ongoing costs of running a restaurant?
What percentage of the cost of the ingredients is the markup on your meal?
Student Response
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Activity Synthesis
Poll the class on whether they think each of the restaurants A, B, and C made money for the month. Select students to share their reasoning.
Note: The way the problem is written, there is not just one correct answer to the question. Whether or not each restaurant made money depends on how much the students estimated for the total monthly expenses (excluding food).
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "if students estimated the total ongoing expenses (excluding food) to be:"}, {"_attributes": {"scope": "col"}, "_value": "restaurants that would have made money"}, {"_attributes": {"scope": "col"}, "_value": "restaurants that would have lost money"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "below $49,300"}], "td": [{"_value": "A, B, and C"}, {"_value": "none"}]}, {"th": [{"_attributes": {"scope": "col"}, "span": [{"_value": "$"}, {"_value": "$"}], "_values": ["between", "49,300 and", "74,400"]}], "td": [{"_value": "A and C"}, {"_value": "B"}]}, {"th": [{"_attributes": {"scope": "col"}, "span": [{"_value": "$"}, {"_value": "$"}], "_values": ["between", "74,400 and", "106,080"]}], "td": [{"_value": "C"}, {"_value": "A and B"}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "above $106,080"}], "td": [{"_value": "none"}, {"_value": "A, B, and C"}]}]}]}]}
Next, ask students to trade with a partner, and check their work for calculating the percentage of the mark up.
Ask students to discuss:
“How does the amount you plan to charge for each meal compare to your partner’s amount?”
“How does your mark up percentage compare to your partner’s percentage?”
Consider telling students that many restaurant owners use 300% as an estimate of a good percentage for the mark up on their meals to be able to make a profit. That means the price of the meal is 4 times what the cost of the ingredients were.


aligns: no