few_shots: 
Example 1:
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.

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: 
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:
Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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_pos_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
problem_activity: Narrative
The purpose of this activity is for students to generate two patterns from rules and then graph them on the coordinate grid. Students first identify a point on the coordinate grid with one of the pairs of numbers from the patterns and then plot the rest of the points. Students may notice that the points on the graph are regularly spaced. They are invited to share this and other observations in the synthesis.
MLR8 Discussion Supports.
Display sentence frames to support partner discussion: “First, I _____ because . . .” and “I noticed _____ so I . . . .”
Advances: Speaking, Writing, Conversing, Representing
Action and Expression: Develop Expression and Communication.
Synthesis: Develop fluency with connecting rules, tables, and a coordinate grid to the same pattern. Provide access to blank or partially completed tables.
Supports accessibility for: Conceptual Processing, Attention
Launch
Groups of 2
“You and your partner will each start some problems about patterns and the coordinate grid independently. After a couple minutes, work with your partner to complete the problems.”
Activity
2 minutes: independent time
5 minutes: partner work time
Student Facing
Partner A
Rule 1: Start at 0. Keep adding 8.
Rule 2: Start at 0. Keep adding 2.
Use the rules to complete the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "tbody": [{"tr": [{"th": [{"_attributes": {"style": "text-align:center; width:14%"}}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "A"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "B"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "C"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "D"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "E"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "F"}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 1"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 2"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}]}]}]}
Which table column represents the point on the coordinate grid? Label the point with the appropriate letter.
[IMG]
Plot and label the rest of the points.
Partner B
Rule 1: Start at 0. Keep adding 10.
Rule 2: Start at 0. Keep adding 40.
Use the rules to complete the table.
{"table": [{"_attributes": {"border": "1", "style": "width:100%"}, "tbody": [{"tr": [{"th": [{"_attributes": {"style": "text-align:center; width:14%"}}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "A"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "B"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "C"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "D"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "E"}, {"_attributes": {"style": "text-align:center; width:14%"}, "_value": "F"}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 1"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"th": [{"_attributes": {"style": "text-align:center"}, "_value": "rule 2"}], "td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}]}]}]}
Which table column represents the point on the coordinate grid? Label the point with the appropriate letter.
[IMG]
Plot and label the rest of the points.
Student Response
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Activity Synthesis
“How did you decide where to place the points on the grid?” (I used the top row to decide how far over to go on the horizontal axis and the second rule to decide how far up to go on the vertical axis.)
Invite students to share completed graphs for parts A and B.
“How are the two graphs the same?” (There is a point at the bottom left,
\((0,0)\)
, on each. The points are regularly spaced and go up and to the right. They all lie at the intersection of gridlines.)
“How are the two graphs different?” (The numbers on the axes are different. The ones for partner B get bigger really quickly.)


aligns: no