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

Calculate expected values and use them to solve problems.

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

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

Use random sampling to draw inferences about a population.

Summarize and describe distributions.

Draw informal comparative inferences about two populations.

Investigate patterns of association in bivariate data.

Interpret linear models.

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

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

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

Use probability to evaluate outcomes of decisions.

Understand and evaluate random processes underlying statistical experiments.

Develop understanding of statistical variability.
problem_activity: Activity
The mathematical purpose of this activity is to make, critique, and justify claims using the data-generating process. Students flip a coin to determine the number of heads that show up for several groups of 20 flips. Students should recognize that some variability is expected from the expected 10 heads that are predicted for 20 flips, but that some results are very unlikely even if they are possible. Students should begin to recognize when a statistical model is consistent with results and when the data is so unusual that a new model might be more appropriate.
In later lessons, students will need to recognize when a normal curve is an appropriate model for data so that the area under a normal curve can be used to get information about the data.
Digital
Launch
Tell students that they will be flipping coins, recording the data, and reporting their results so that a dot plot can be created from class data. Monitor students as they flip their coins in the applet and collect data from the class to be included in a dot plot.
Action and Expression: Internalize Executive Functions.
Provide students with a template so they can keep track of the data for each trial of coin flipping.
Supports accessibility for: Language; Organization
Student Facing
What is the probability that you will flip heads when using the coin in the applet?
Estimate the number of heads you will get when you flip the coin 20 times.
Flip your coin 20 times and record the number of heads you get. Repeat this process 4 more times.
|  trial number   | 1 | 2 | 3 | 4 | 5 |
|-----------------|---|---|---|---|---|
| number of heads |   |   |   |   |   |
Create a dot plot that shows the number of heads in 20 flips using data from the class.
What is the least number of heads flipped by the class in 20 flips? What is the greatest number of heads flipped by the class in 20 flips?
Based on the class dot plot, describe a range of values that represent a reasonable number of heads to flip when flipping 20 times.
Priya flips her coin 20 times and it lands showing heads twice.
Is it possible for this to happen with a fair coin?
Based on the class distribution, should she be suspicious of this being an unfair coin? What can she do to provide evidence that it’s not a fair coin?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Tell students that they will be flipping coins, recording the data, and reporting their results so that a dot plot can be created from class data. Monitor students as they flip their coins and collect data from the class to be included in a dot plot.
Action and Expression: Internalize Executive Functions.
Provide students with a template so they can keep track of the data for each trial of coin flipping.
Supports accessibility for: Language; Organization
Student Facing
What is the probability that you will flip heads when using the coin you have?
Estimate the number of heads you will get when you flip the coin 20 times.
Flip your coin 20 times and record the number of heads you get. Repeat this process 4 more times.
|  trial number   | 1 | 2 | 3 | 4 | 5 |
|-----------------|---|---|---|---|---|
| number of heads |   |   |   |   |   |
Create a dot plot that shows the number of heads in 20 flips using data from the class.
What is the fewest number of heads flipped by the class in 20 flips? What is the greatest number of heads flipped by the class in 20 flips?
Based on the class dot plot, describe a range of values that represent a reasonable number of heads to flip when flipping 20 times.
Priya flips her coin 20 times and it lands showing heads twice.
Is it possible for this to happen with a fair coin?
Based on the class distribution, should she be suspicious of this being an unfair coin? What can she do to provide evidence that it’s not a fair coin?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
This time instead of flipping the coin 20 times and counting how many heads result, consider how many coin flips it takes to get a heads.
Before flipping estimate the average number of flips it will take to get your fist heads.
Flip! Record how many flips it takes to get your first heads. Then start again and record how many flips it takes to get your next heads. Keep flipping and recording how many flips it takes to get successive heads until you get 30 heads. Create a dot plot that shows how many flips it took each time to get a heads and compute the mean number of times it took.
Based on the dot plot, describe a range of values that represent a reasonable number of coin flips it takes to get a heads.
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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 may be uncomfortable with the less precise answers for the work in this unit and this activity in particular. Remind students that the actual world is messy and the work of statistics is to help make sense of that messiness. Math is not proving Priya has a fair coin or not, but can give us evidence to suspect that it is not and a decision can be made based on that evidence and anything else that is known about the situation.
Activity Synthesis
The purpose of this discussion is to talk about making and justifying claims based on data. Here are some questions for discussion:
“You performed 20 coin flips 5 times. Based on your data alone, what range of values would you predict for the number of times the coin would land on heads if it was flipped 20 times?” (I got 8, 9, 10, 12, 13 heads so I would predict that a coin would land on heads between 7 and 13 times. I included seven in my estimate because 13 is 3 above ten and 7 is 3 less than 10. The coin should land on heads an average of 10 times.)
“How does your prediction compare to the range of values you described from the class's data? Why do you think it was the same or different? Explain your reasoning.” (My range was smaller than the range from the whole class data. Since there was much more data for the whole class, I expected there to be a broader range of values since there was likely to be more variability.)
“If 1,000 people flipped a coin 20 times, how might that change your prediction from the rage for the class's data?” (If we had 1,000 data values, I would feel really confident about the range of values that I would expect from 20 coin flips. I know that it is possible to get between 0 and 20 heads, but I wonder how unlikely it really is to get 0 or 20. I think the 1,000 data values would help me to know a typical range of values.)


correct_option_index: 13