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: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
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 following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
problem_activity: Activity
This activity continues the work begun in the previous activity for this lesson. Students compute the means for sample ages to determine what shows might be associated with each sample (MP2). They also consider the variability to assess the accuracy of population estimates. A sample from a population with less variability should provide a more accurate estimate than a sample that came from a population with more spread in the data. In the discussion, students think about why a sample is used and why an estimate of the mean is helpful, but it may miss some important aspects of the data. The discussion following the activity also asks students to think again about why different samples from the same population may produce different results.
Launch
Keep students in groups of 3.
Tell students that advertisers are interested in age groups for certain television shows so that they can try to sell appropriate items to the audience. For example, it does not make sense to advertise tricycles during a nighttime crime drama show nor to show an ad for an expensive sports car during a children’s cartoon.
The samples given in this activity are related to the shows mentioned in the previous activity for this lesson.
Tell students to divide samples 4 through 6 among the group members so that each person only needs to find one mean and share their answer with the group so that the group has access to all 3 answers.
Give students 3 minutes quiet work time for the first 3 problems, then pause the class after the third problem for a quick discussion and to assign items to groups before they continue.
After students have completed the first 3 problems, ask students to indicate which of the shows seem to go with each of the 6 samples (samples 1 through 3 from the previous activity as well as 4 through 6 in this activity). Discuss any disagreements until the class can agree on which samples correspond to which shows. Tell half of the groups that they will use samples 1 through 3 from the previous activity for the last 3 problems and the other half of the groups that they will use samples 4 through 6 for the last 3 problems.
Give students another 3 minutes of quiet work time to finish the activity followed by a whole-class discussion.
Representation: Internalize Comprehension.
Use color and annotations to illustrate student strategies. As students describe how they calculated the measures of centers, use color and annotations to scribe their thinking on a display. Ask students how they matched samples and television shows, and label each representation accordingly.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Here are three more samples of viewer ages collected for these same 3 television shows.
sample 4
57
71
5
54
52
13
59
65
10
71
sample 5
15
5
4
5
4
3
25
2
8
3
sample 6
6
11
9
56
1
3
11
10
11
2
Calculate the mean for
one
of these samples. Record all three answers.
Which show do you think each of these samples represents? Explain your reasoning.
For each show, estimate the mean age for all the show's viewers.
Calculate the mean absolute deviation for
one
of the shows' samples. Make sure each person in your group works with a different sample. Record all three answers.
|               | Learning                       to Read | Science                       Experiments                       YOU Can Do | Trivia the                       Game Show |
|---------------|----------------------------------------|----------------------------------------------------------------------------|--------------------------------------------|
| Which sample? |                                        |                                                                            |                                            |
|      MAD      |                                        |                                                                            |                                            |
What do the different values for the MAD tell you about each group?
An advertiser has a commercial that appeals to 15- to 16-year-olds. Based on these samples, are any of these shows a good fit for this commercial? Explain or show your reasoning.
Student Response
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Activity Synthesis
The purpose of the discussion is to understand why it might be helpful to estimate the mean of a population based on a sample.
Some questions for discussion:
“Why do you think a sample was used in this situation rather than data from the population?” (There are probably millions of people who watch these shows and it would be difficult to collect data about their ages from all of them.)
“How could we improve the estimate of the mean for the populations?” (Include more viewers in the sample.)
“If a sample has a large MAD, what does that imply about the population?” (That the data in the population is very spread out.)
“If a sample has a small MAD, what is the relationship between the data and the mean?” (Most of the data is close to the mean.)
“Which estimate of the mean for the population do you expect to be more accurate: the mean from a sample with a large MAD or the mean from a sample with a small MAD? Explain or show your reasoning.” (The mean from a sample with a small MAD. If the data in the sample is close to the mean, then most of the data from the population is also probably close to the mean. Therefore the data in the sample is probably close to the mean of the population and will provide a good estimate.)
“What do you notice about the different answers for the same show, but the different samples?”  (The means are close, but the MADs can be different by a lot.)
“Why were the answers different for the same show, but different samples?” (Different people were included in the samples, so the numbers may change some, but if they are representative they should be close. In these examples, even though the MADs may seem very different, the relative size compared to the other shows is similar.)
Some students may wonder why they need to calculate the mean when it might be obvious how to match the titles by just looking at the data. This example included 10 ages in each sample so that the important information could be calculated quickly. In a more realistic scenario, the sample may include hundreds of ages. A computer could still calculate the mean quickly, but scanning through all of the data may not make the connection to the correct show as obvious.
“Notice that there is a 56 year old in sample 6. What are some reasons you think they might be watching this show?” (Maybe a grandmother is watching with her grandchild. Maybe an older person is interested in how science is shown on TV.)
“The questions asked you to consider means, but are there any data sets for which median might be a better measure of center? Explain your reasoning.” (Yes, sample 6 has a wide range of data ranging from a 1 year old to a 56 year old, but most of the data are around 10, so median might be better to use for that sample.)
“A lot of families might be watching ‘Learning to Read’ with their children or older people may be using the show to learn English. How might this affect the mean? How could you recognize that there are two main age groups that watch this show?” (It would bring the mean up from just the kids who watch the show. The mean would not make the two age groups obvious, so looking at a dot plot or histogram might be more helpful with this group.)


aligns: yes