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 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_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Write, read, and evaluate expressions in which letters stand for numbers.
problem_activity: Activity
Previously students analyzed distributions to identify center and spread. In this activity, they continue to practice finding reasonable values for centers of data and describing variability. The focus, however, is on making use of the structure of distributions (MP7) to compare groups in those terms and interpreting their analyses in the context of a situation (MP2).
By comparing distributions, seeing how center and spread for the same population could change, and making sense of what these changes mean, students deepen their understanding of these concepts before learning about more formal measures of center and variability.
Launch
Give students a brief overview on keyboarding courses. Explain that these are classes designed to help people improve their typing speed and accuracy, which they may need for their jobs. Typing proficiency is usually measured in terms of number of words typed per minute; the more words typed correctly per minute, the faster or more proficient one's typing is.
Keep students in groups of 2. Give them 5–6 minutes of quiet time to work on the first two questions, and then 2–3 minutes to discuss their responses and complete the last question together.
Representing, Writing, Conversing: MLR8 Discussion Supports.
When explaining the context of keyboarding courses, ask students to complete the sentence frame, “A person’s typing speed increases when they type (more / fewer) words per minute because....” This will help students to connect the language of the questions to the meaning of keyboarding improvement.
Design Principle(s): Support sense-making; Cultivate conversation
Student Facing
A keyboarding teacher wondered: “Do typing speeds of students improve after taking a keyboarding course?” Explain why her question is a statistical question.
The teacher recorded the number of words that her students could type per minute at the beginning of a course and again at the end. The two dot plots show the two data sets.
[IMG]
Based on the dot plots, do you agree with each of the following statements about this group of students? Be prepared to explain your reasoning.
Overall, the students’ typing speed did not improve. They typed at the same speed at the end of the course as they did at the beginning.
20 words per minute is a good estimate for how fast, in general, the students typed at the beginning of the course.
20 words per minute is a good description of the
center
of the data set at the end of the course.
There was more variability in the typing speeds at the beginning of the course than at the end, so the students’ typing speeds were more alike at the end.
Overall, how fast would you say that the students typed after completing the course? What would you consider the center of the end-of-course data?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Use one of these suggestions (or make up your own). Research to create a dot plot with at least 10 values. Then, describe the center and spread of the distribution.
Points scored by your favorite sports team in its last 10 games
Length of your 10 favorite movies (in minutes)
Ages of your favorite 10 celebrities
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students might find it challenging to tell where the center of a distribution could be just by looking at a single dot plot. The idea of center might be more apparent when presented in comparative terms. For example, ask them to describe in their own words how the distribution of the first dot plot differs from that of the second dot plot. Students are likely able to say that, compared to the first dot plot, the aggregation of dots in the second dot plot is overall higher on the number line. Ask them if there is a location on each dot plot around which data points seem to congregate.
Activity Synthesis
The purpose of the discussion is for students to deepen their understanding of distributions and using the descriptions to compare two groups.
Focus the whole-class discussion on two ideas:
The distinctions between the two distributions: Students should see that, overall, the cluster of data points have both shifted up toward a greater number of words per minute (moving its center up) and become more compressed in its spread by the end of the course. Because the center moved up in location, the value we use to describe that center would also increase.
What the changes in the center and spread tell us in this situation: Students should recognize that a higher center means that, overall, the group has improved in their typing speed. They should see that a narrower spread at the end of the course suggests that there's now less variability in the typing speeds of different students (compared to a much larger variability initially).
Representation: Internalize Comprehension.
Use color coding and annotations to highlight the distinctions between the two distributions. For example, make visible the changes in the center and spread, and note connections to the situation.
Supports accessibility for: Visual-spatial processing


aligns: no