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: Develop understanding of statistical variability.

Use probability to evaluate outcomes of decisions.

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

Calculate expected values and use them to solve problems.

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

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

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

Use random sampling to draw inferences about a population.

Summarize and describe distributions.

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

Understand and evaluate random processes underlying statistical experiments.

Investigate patterns of association in bivariate data.

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

Interpret linear models.

Draw informal comparative inferences about two populations.
problem_activity: Optional activity
To close the unit, students practice using the methods they have learned. Students begin by collecting data in small groups and then analyzing the data for the entire class. They create a scatter plot and two-way tables to look for patterns in the data (MP2). Although it is expected there is a very weak, if any, association in the data, the discussion asks students to think about what it would mean if there were a positive or negative association in context.
Launch
Arrange students in groups of 4. Select one student from each group to submit the data from the group when they finish recording their data. Prepare a table for all to see that is large enough to include the data from all of the students in the class.
Tell students that the same data is treated as both numerical and categorical data at different times within this activity. This is sometimes done when there is a threshold that makes sense for the context. In this case, it makes sense to look at data for which the second time is greater or less than the first time. Although the actual values may be lost when viewing the data as categorical, it can provide a different type of insight. In this example, we can more easily see if there was an improvement in reaction time for the second attempt.
Engagement: Internalize Self Regulation.
Provide a project checklist that chunks the various steps of the project into a set of manageable tasks.
Supports accessibility for: Organization; Attention
Representing, Conversing: MLR8 Discussion Supports.
Demonstrate the process for one round of the experiment, in which a group of students and staff play an example round while the rest of the class observes. This will help invite more student participation, conversations, and meta-awareness of language involving collecting and analyzing the data.
Design Principle(s): Support sense-making; Maximize meta-awareness.
Student Facing
In this activity, you’ll get two chances to guess at how long 30 seconds is, then look for an association between the two guesses of all students.
Work with a partner. Follow the instructions listed here to gather your data.
One of you will hold a stopwatch where the other person cannot see it.
The person holding the stopwatch says “go” and starts the timer.
The other person says “stop” when they think 30 seconds have passed.
The person holding the stopwatch will stop the timer, then report and record the time to the nearest second.
The person holding the stopwatch will give a second chance, repeating the experiment.
After
both
times are recorded, switch roles.
Record the group data in this table. When you finish, a group member should give the data to the teacher.
| name | time 1 | time 2 |
|------|--------|--------|
|      |        |        |
|      |        |        |
|      |        |        |
|      |        |        |
Look at your data. Comparing Time 1 to Time 2, do you think there is a positive association, a negative association, or no association? Discuss your thinking with your group.
What are some ways you could organize and represent the entire class's data?
Make a scatter plot of the entire class’s data and look for patterns. Identify any outliers and the type of any association you observe.
[IMG]
Draw two lines on your scatter plot: a vertical line and a horizontal line, each representing 30 seconds for one trial. Use the table for the class’s data to complete this two-way table.
|                 | time 2 < 30 sec | time 2 = 30 sec | time 2 > 30 sec | total |
|-----------------|-----------------|-----------------|-----------------|-------|
| time 1 < 30 sec |                 |                 |                 |       |
| time 1 = 30 sec |                 |                 |                 |       |
| time 1 > 30 sec |                 |                 |                 |       |
|      total      |                 |                 |                 |       |
Use the two-way table to decide whether there is an association between Time 1 and Time 2. Explain how you know.
Student Response
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Activity Synthesis
The purpose of the discussion is to solidify understanding of the strategies and tools from the unit. Specifically, how to organize and analyze categorical data to determine if there is an association between two variables.
Some questions for discussion:
“What representation (data table, scatter plot, two-way table, or something else) helped the most in determining any relationships between time 1 and time 2?”
“If someone's time was slower on the second attempt, how would that appear on the scatter plot?” (The point would be above and to the left of the line
\(y = x\)
, since the value for the
\(y\)
-coordinate would be greater than the value for the
\(x\)
-coordinate like the point
\((1.2, 1.4)\)
.)
“If there was a positive association between the variables, what would that mean?” (In general, the slower the first time, the slower the second time. The faster the first time, the faster the second time as well.)
“If there was a negative association between the variables, what would that mean?” (In general, the slower the first time, the faster the second time. The faster the first time, the slower the second time.)
"If the class showed a tendency to overcompensate by going from a time that was longer than 30 seconds to one that was shorter or the other way around, how would that appear in the two-way table?" (The numbers in the table would be greatest on the bottom left and top right of the table with smaller numbers elsewhere.)


correct_option_index: 11