few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Summarize, represent, and interpret data on two categorical and quantitative variables.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Summarize, represent, and interpret data on a single count or measurement variable.

Understand independence and conditional probability and use them to interpret data.

Draw informal comparative inferences about two populations.

Use probability to evaluate outcomes of decisions.

Summarize and describe distributions.

Develop understanding of statistical variability.

Investigate patterns of association in bivariate data.

Understand and evaluate random processes underlying statistical experiments.

Use random sampling to draw inferences about a population.

Interpret linear models.

Calculate expected values and use them to solve problems.

Investigate chance processes and develop, use, and evaluate probability models.
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.)


correct_option_index: 5

11