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: 
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.

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, 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_neg_0

8
standard_description: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
problem_activity: Activity
A scatter plot is shown and the points interpreted in context. Later, a linear model is graphed with the scatter plot to help students see an obvious outlier (MP7). In the discussion, the term outlier is introduced.
Digital
Launch
[IMG]
Ask students, “What do you notice? What do you wonder?” Ideally, they will notice that these are pictures of a foot of three different people. All three feet are approximately the same length but different widths. Tell students that all of these feet are a size 8. However, they wouldn’t all necessarily find the same shoe equally comfortably, because of the varying widths. (Besides the numerical size, some shoes also come in different widths.) Students should understand that human feet can vary in both length and width.
If using the digital activity, after this discussion, clarify for students that a scatter plot has been created for them to anaylze feet dimensions, using technology. Students can then work to complete the digital task which includes using a line of best fit to identify outliers.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Speaking: MLR8 Discussion Supports.
Use this routine to clarify the meaning of “width” and “length” in the context of the task. Connect the terms width and length multi-modally by utilizing different types of sensory inputs, such as demonstrating the width and length on the images of each foot using gestures or inviting students to do so. Invite students to chorally repeat phrases that include these words in context. This will help students produce and make sense of the language needed to understand the context of the task.
Design Principle(s): Support sense-making
Student Facing
Here is a scatter plot that shows lengths and widths of 20 different left feet. Use the double arrows to show or hide the expressions list.
Estimate the widths of the longest foot and the shortest foot.
Estimate the lengths of the widest foot and the narrowest foot.
Click on the gray circle next to the words “The Line” in the expressions list. The graph of a linear model should appear. Find the data point that seems weird when compared to the model. What length and width does that point represent?
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
[IMG]
Ask students, “What do you notice? What do you wonder?” Ideally, they will notice that these are pictures of a foot of three different people. All three feet are approximately the same length but different widths. Tell students that all of these feet are a size 8. However, they wouldn’t all necessarily find the same shoe equally comfortably, because of the varying widths. (Besides the numerical size, some shoes also come in different widths.) Students should understand that human feet can vary in both length and width.
If using the digital activity, after this discussion, clarify for students that a scatter plot has been created for them to anaylze feet dimensions, using technology. Students can then work to complete the digital task which includes using a line of best fit to identify outliers.
Speaking: MLR8 Discussion Supports.
Use this routine to clarify the meaning of “width” and “length” in the context of the task. Connect the terms width and length multi-modally by utilizing different types of sensory inputs, such as demonstrating the width and length on the images of each foot using gestures or inviting students to do so. Invite students to chorally repeat phrases that include these words in context. This will help students produce and make sense of the language needed to understand the context of the task.
Design Principle(s): Support sense-making
Student Facing
Here is a scatter plot that shows lengths and widths of 20 different left feet.
[IMG]
Estimate the widths of the longest foot and the shortest foot.
Estimate the lengths of the widest foot and the narrowest foot.
Here is the same scatter plot together with the graph of a model for the relationship between foot length and width.
[IMG]
Circle the data point that seems weird when compared to the model. What length and width does that point represent?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Introduce the term
outlier
. An outlier is a point that is separated from the rest of the data. Sometimes data sets have outliers. Sometimes that’s because there really is a data point that is very different than the others. Sometimes it is because there was an error in collecting the data. Sometimes it is because there was an error in entering the data. When there are outliers, one has to make a judgement about whether to include it in the analysis or not.


aligns: yes