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: 
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.

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:
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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
In this activity, students continue to identify points on a scatter plot as representatives of a single month when two variables are measured. Students are also asked to explain the meaning of some abstract points in the context of the problem and identify when it does not make sense to extrapolate information from the graph based on the context (MP2).
Monitor for students who understand that 60 degrees Celsius is unreasonable for an average monthly temperature in most parts of the world, as well as those who identify the trend to predict negative sales which does not make sense in this context either.
Launch
Arrange students in groups of 2. Allow students 5 minutes quiet work time followed by 5 minutes partner discussion and whole-class discussion.
Ask students to predict what a scatter plot might look like when the temperature is along the
\(x\)
-axis and the sales of coats at a store is along the
\(y\)
-axis.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, ask students to use the different colors to represent and annotate temperature and coat sales in the graph and table.
Supports accessibility for: Visual-spatial processing
Student Facing
A clothing store keeps track of the average monthly temperature in degrees Celsius and coat sales in dollars.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "temperature (degrees Celsius)"}, {"_attributes": {"scope": "col"}, "_value": "coat sales (dollars)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "-5"}, {"_value": "1,550"}]}, {"td": [{"_value": "-3"}, {"_value": "1,340"}]}, {"td": [{"_value": "3"}, {"_value": "1,060"}]}, {"td": [{"_value": "8"}, {"_value": "1,070"}]}, {"td": [{"_value": "15"}, {"_value": "680"}]}, {"td": [{"_value": "21"}, {"_value": "490"}]}, {"td": [{"_value": "23"}, {"_value": "410"}]}, {"td": [{"_value": "21"}, {"_value": "510"}]}, {"td": [{"_value": "17"}, {"_value": "600"}]}, {"td": [{"_value": "11"}, {"_value": "740"}]}, {"td": [{"_value": "6"}, {"_value": "940"}]}, {"td": [{"_value": "-2"}, {"_value": "1,390"}]}]}]}]}
[IMG]
What does the point
\((15, 680)\)
represent?
For the month with the lowest average temperature, estimate the total amount made from coat sales. Explain how you used the table to find this information.
For the month with the smallest coat sales, estimate the average monthly temperature. Explain how you used the scatter plot to find this information.
If there were a point at
\((0, A)\)
what would it represent? Use the scatter plot to estimate a value for
\(A\)
.
What would a point at
\((B, 0)\)
represent? Use the scatter plot to estimate a value for
\(B\)
.
Would it make sense to use this trend to estimate the value of sales when the average monthly temperature is 60 degrees Celsius? Explain your reasoning.
Student Response
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Activity Synthesis
​The purpose of the discussion is for students to understand that each point in the graph represents a single month in which two measurements are made. Select students to share their responses, including those identified for the last problem.
Consider asking some of the following questions:
“When discussing functions, what is a point of the form
\((0,A)\)
called? What about a point of the form
\((B,0)\)
?” (The
\(y\)
-intercept. The
\(x\)
-intercept.)
“In a scatter plot is there a
\(y\)
-intercept?” (It depends on the context. For some situations, it may not make sense for the variable represented along the
\(x\)
-axis to be zero. For example if the variable is the height of a building. On other situations, there may be more than one point along the
\(y\)
-axis. For example, in a situation like the one in the task, there may be multiple months where the average temperature is 0 degree Celsius.)
“When the average monthly temperature goes up, what seems to be happening to sales?” (They are decreasing.)
Writing, Speaking: MLR1 Stronger and Clearer Each Time.
Use this routine to give students a structured opportunity to revise and refine their response to the last question. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help students strengthen their ideas and clarify their language (e.g., “Can you give an example?”, “Why do you think…?”, “How do you know…?”, etc.). Students can borrow ideas and language from each partner to strengthen their final version.
Design Principle(s): Optimize output (for explanation)


aligns: no