few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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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: Model periodic phenomena with trigonometric functions.

Analyze functions using different representations.

Interpret functions that arise in applications in terms of the context.

Interpret expressions for functions in terms of the situation they model.

Extend the domain of trigonometric functions using the unit circle.

Build new functions from existing functions.

Use functions to model relationships between quantities.

Build a function that models a relationship between two quantities.

Construct and compare linear and exponential models and solve problems.

Define, evaluate, and compare functions.

Understand the concept of a function and use function notation.

Prove and apply trigonometric identities.
problem_activity: Activity
In this activity, students examine relationships shown in graphs. They consider different types of events that cause repeated outputs (MP2). They examine one situation which is periodic (the distance of a car from the start line as it goes around a racetrack) and two situations which are not periodic, but do rise and fall at regular intervals. The long-term goal in this unit is to develop mathematical tools which can help model these complex phenomena.
Monitor for students who connect the repetitive nature of these graphs to the context of the clock hands or Ferris wheel from earlier lessons.
Launch
Engagement: Provide Access by Recruiting Interest.
Leverage choice around perceived challenge. Invite students to select 2 of 3 to complete. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
Here are some relationships that produce graphs that have a repetitive nature. For each situation, describe the dependent and independent variable. How does the dependent variable change? What might cause this change?
This is the graph of the distance of a race car from the starting line as it goes around a track.
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This is the graph of the temperature in a city in Australia over 21 days.
[IMG]
This is the graph of two populations over time.
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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 Student Response.
Student Facing
Are you ready for more?
A ladybug is resting at the tip of a clock’s minute hand, which is 1 foot long. When it is 12:15, the ladybug is 10 feet above the ground.
Calculate how far above the ground the ladybug is at 12:00, 12:30, 12:45, and 1:00.
Estimate how far above the ground the ladybug is at 12:10, 12:20, and 12:40.
Plot the distances of the ladybug from the ground from 12:00 to 1:00.
[IMG]
If the ladybug stays on the minute hand, predict how its distance from the ground will change over the next hour (from 1:00 to 2:00). What about from 2:00 to 3:00?
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 unsure how to read the axes for all but the first relationship, as they look different than what students have seen so far. Encourage these students to focus on a specific coordinate on a graph to help them make sense of the axis labels as a way to get started.
Activity Synthesis
Begin the discussion by selecting previously identified students to share their observations about connections between the repeating nature of these graphs and the clock hands or Ferris wheel contexts from earlier lessons. If no groups make this connection, invite students to do so now—for example, by comparing the period of the race car to the period of the minute hand on a clock or by contrasting how the height of a point on a Ferris wheel always moves between the same high and low while the city high and low temperatures change from day to day.
Display the graphs for all students to see. Here are some questions for discussion:
“What similarities do you see between the graphs representing these situations?” (There are highs and lows in all the graphs, and these highs and lows come at pretty regular intervals.)
“Do these graphs all represent functions? How can you tell?” (Yes, in these situations for each time, there is a single value.)
“What do you think it means for a function to be
periodic
?” (The output values follow a repeating pattern.)
“What other words could you use to describe this type of behavior?” (repeating, cyclic)
Conclude the discussion by reminding students that previously, we named functions whose values repeat at regular intervals
periodic functions
. The graph of the race track is an example of what a periodic function can look like. The other two situations shown, temperature and population over time, are not quite periodic functions since the output values are not the same over and over again, but they have a definite periodic pattern to them.


correct_option_index: 2