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

Use random sampling to draw inferences about a population.

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

Investigate patterns of association in bivariate data.

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

Develop understanding of statistical variability.

Calculate expected values and use them to solve problems.

Summarize and describe distributions.

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

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

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

Interpret linear models.

Draw informal comparative inferences about two populations.

Understand and evaluate random processes underlying statistical experiments.

Use probability to evaluate outcomes of decisions.
problem_activity: Activity
The mathematical purpose of this activity is for students to gain an understanding of the relationship between a two-way frequency table and a two-way relative frequency table. Students have the same information in a two-way table and a segmented bar graph, then are asked to interpret the information given in context. In some cases, it makes sense to view the raw data in the table, and in other cases, it can make sense to understand the data as percentages of various larger groups. Students examine the meaning of data when they are presented as a percentage of the entire group or either of the 2 subgroups.
Launch
Arrange students in groups of 2.
Help orient students by having them notice:
The sum of all 4 percentages in the second table is 100%.
For each of the next two tables, ask students to find percentages that sum to 100%. (The third table has columns that sum to 100% and the fourth table has rows that sum to 100%.)
Action and Expression: Internalize Executive Functions.
To support development of organizational skills in problem-solving, chunk this task into more manageable parts. For example, instead of presenting tables and graphs all at once, provide 4 different pieces. Ask students to place the table with the actual values at the top of their workspace, and line up the other tables and graphs in a new row below. This allows students to see all of the information while reading and answering the questions without flipping back and forth. Encourage students to move or cover tables to focus on one table at a time.
Supports accessibility for: Memory; Organization
Student Facing
200 people were asked if they prefer dogs or cats, and whether they live in a rural or urban setting.
The actual values collected from the survey are in the first table.
|       | urban | rural | total |
|-------|-------|-------|-------|
|  cat  |       |       |       |
|  dog  |       |       |       |
| total |       |       |       |
The next table shows what percentage of the 200 total people included are represented by each combination of categories. The segmented bar graph represents the same information graphically.
|     | urban | rural |
|-----|-------|-------|
| cat |       |       |
| dog |       |       |
[IMG]
The next table shows the percentage of each column that had a certain pet preference in a column relative frequency table. The segmented bar graph represents the same information graphically.
|     | urban | rural |
|-----|-------|-------|
| cat |       |       |
| dog |       |       |
[IMG]
The last table shows the percentage of each row that live in a certain area in a row relative frequency table. The segmented bar graph represents the same information graphically.
|     | urban | rural |
|-----|-------|-------|
| cat |       |       |
| dog |       |       |
[IMG]
For each relative frequency table, select a percentage and explain how numbers from the original table were used to get the percentage.
What percentage of those surveyed live in an urban area and prefer dogs?
Among the people surveyed who prefer dogs, what percentage of them live in an urban setting?
What percentage of people surveyed who live in an urban setting prefer dogs?
How many of the people responded that they prefer dogs and live in an urban setting?
Among the people surveyed, are there more people who prefer dogs or cats?
Your pet food company has access to a billboard in a rural setting. Would you recommend advertising dog food or cat food on this billboard? Which table did you use to make this decision? Explain your reasoning.
Student Response
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Anticipated Misconceptions
Students may not understand the difference between what each relative frequency table represents. You can have a whole-class discussion about the data presented before students begin the task. You can also individually ask students questions concerning how the different representations relate to one another. For example: ”Which percentage in the relative frequency table corresponds to a specific portion of the bar graph?” Ensure students understand that each table uses a different total to get the percentages. Ask them, “Which tables are the questions referring to when it says, ‘Among the people surveyed’?”
Activity Synthesis
The purpose of this discussion is to make sure students understand how to interpret information in a relative frequency table and to prime student thinking about associations between categorical variables.
After selecting a student to share a response, select another student to share which table or graph can be used to answer the question, then select a third student to explain the connection between the question and the table or graph.
Ask students:
“What information can be understood from each type of relative frequency table?” (The first table can explain how each combination of categories fits into the whole group. The second table divides the entire group by their location and shows the pet preference in those locations. The third table divides the entire group by pet preference and shows the location of people for each pet preference.)
“Do you notice any interesting patterns in the data? Explain your reasoning.” (It is okay if students struggle with this question at this time. The focus of the next lesson is on associations between categorical variables. Students may notice that dog lovers tend to live in urban settings, since 77% of dog lovers are urban.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their ideas, press for details by requesting students to challenge an idea, elaborate on an idea, or give an example. Revoice student responses to demonstrate mathematical language use by applying appropriate language. For instance, revoice the statement: “The percentage of people living in a rural setting who love cats,” as “The percentage of cat lovers who are rural.” Call students’ attention to any words or phrases that helped clarify the original statement. If needed, practice phrases or words through choral response. This provides more students with an opportunity to produce language as they share their ideas about data represented in relative frequency tables.
Design Principle(s): Support sense-making


correct_option_index: 10