few_shots: 
Example:
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).

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_001097_pos_0

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standard_description: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
problem_activity: Optional activity
In this optional activity, students compare two populations using samples again, but this time only one sample is given. For the other sample, the characteristics have either been computed already or are the focus of the question. This type of analysis is useful when comparing two similar populations as in this activity or when comparing a group against a standard.
Launch
Keep students in groups of 2.
Tell students that sometimes it is useful to compare one group to a standard or another group where the important characteristics have already been computed. In these problems, a random sample from one group is given and characteristics of the second group is either given or sought.
Allow students 10 minutes partner work time followed by a whole-class discussion.
Student Facing
A college graduate is considering two different companies to apply to for a job. Acme Corp lists this sample of salaries on their website:
\$45,000
\$55,000
\$140,000
\$70,000
\$60,000
\$50,000
What typical salary would Summit Systems need to have to be meaningfully different from Acme Corp? Explain your reasoning.
A factory manager is wondering whether they should upgrade their equipment. The manager keeps track of how many faulty products are created each day for a week.
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The new equipment guarantees an average of 4 or fewer faulty products per day. Is there a meaningful difference between the new and old equipment? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of the discussion is to help students understand how to compare groups when one set of characteristics are known and the other group is represented by sample data.
Select some groups to share their answers and reasoning for the two problems.
Consider asking these discussion questions:
“How did you determine what to use as a typical value for the first problem?” (Since there was one value much greater than the others, the distribution would not be symmetric, so median is a more appropriate measure of center.)
“How did you determine what measure of center to use for the second problem?” (Since the data were all close, either value could be used, but the new equipment reported the “average” or mean, so man should be used for the sample as well.)
“The manufacturer for the new equipment guarantees 4 flaws
or fewer
per day with the new equipment. If the new equipment produces only 3 flaws per day does that change the answer for the second problem?” (No. There is an even greater difference between the current and new equipment, so it is even more meaningful.)
“What other factors would the college graduate want to consider other than the meaningful difference in median salary between the two companies?” (In addition to the other factors for a job such a benefits, relationship with coworkers, type of work being done at each company, etc., the graduate should consider the salary for the type of job he will get at the company. For example, if his degree is in computer science, he may be looking at a job with computers rather than sales or some other department within the company, so he might be able to get a better comparison of salaries that way.)
“What other factors would the factory manager want to consider other than the meaningful difference in flaws for the equipment?” (The cost of the frequent flaws as well as the cost of the new equipment will probably factor into her decision to buy new equipment. The age of the current equipment and maintenance for older equipment compared to new equipment may also be important.)
Representation: Internalize Comprehension.
Use color and annotations to illustrate connections between representations. As students share their diagrams and reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker whether their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


aligns: no