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.


Your response:
{
  "explanation": "This task teaches students to understand addition as putting together counters, or adding to already placed counters.",
  "answer": "AB"
}

Example 2:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

Mathematical concept/skill options: 
A. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
B. Extend understanding of fraction equivalence and ordering.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
C. Use equivalent fractions as a strategy to add and subtract fractions.
D. Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
E. Understand decimal notation for fractions, and compare decimal fractions.
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
F. Develop Understanding of fractions as numbers.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Your response:
{
  "explanation": "This problem involves understanding fractions as numbers by comparing their quantities.",
  "answer": "F"
}

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Mathematical concept/skill options: 
A. Use place value understanding and properties of operations to perform multi-digit arithmetic. (A range of algorithms may be used.)
B. Understand place value.
C. Understand the place value system.
D. Extend the counting sequence.
E. Work with numbers 11–19 to gain foundations for place value.
F. Use place value understanding and properties of operations to perform multi-digit arithmetic.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
G. Perform operations with multi-digit whole numbers and with decimals to hundredths.
H. Generalize place value understanding for multi-digit whole numbers.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
I. Use place value understanding and properties of operations to add and subtract.


Your response:
{
  "explanation": "This set of problems asks students to perform multiplication, which is an operation, on multi-digit whole numbers.",
  "answer": "G"
}

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

K

im_task_000157

3

fl_problem_000674

5
options: Interpret functions that arise in applications in terms of the context.

Define, evaluate, and compare functions.

Prove and apply trigonometric identities.

Model periodic phenomena with trigonometric functions.

Use functions to model relationships between quantities.

Extend the domain of trigonometric functions using the unit circle.

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

Understand the concept of a function and use function notation.

Build new functions from existing functions.

Build a function that models a relationship between two quantities.

Analyze functions using different representations.

Construct and compare linear and exponential models and solve problems.
problem_activity: Activity
The purpose of this activity is for students to consider the height of points on a circle not centered at
\((0,0)\)
. Working within the context of a Ferris wheel, students use their understanding of the unit circle to identify the height of a point on the wheel relative to the center of the wheel and then combine this value with the given height of the center of the Ferris wheel to calculate the actual height off the ground. This work builds on the previous activity and builds toward transformations of trigonometric functions, which will be used in future lessons.
While approximate answers are acceptable and can be easier to think about than an expression with sine in it, encourage students to write out the exact value for the heights using sine. This practice will help in the future when students write and algebraically transform equations for trigonometric functions.
Digital
Launch
If students have access to devices that can run a GeoGebra applet, suggest that the Unit Circle Slider applet might be a helpful tool in this activity. Consider demonstrating how to use the applet. The slider works to change the angle that the given segment creates with the positive
\(x\)
-axis. The displayed coordinates indicate the coordinates of the intersection point of the line segment with the unit circle.
Display the image of the Ferris wheel for all to see. Have students estimate the height of each position as the Ferris wheel moves counterclockwise. In the activity, they will precisely calculate the value.
\(\frac \pi {12}\)
radians
\(\frac \pi 2\)
radians
\(\frac {5\pi}6\)
radians
\(\frac {5\pi}3\)
radians
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (A Ferris wheel with point
\(P\)
on the circle). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the distance off of the ground, the radius, and the angle of rotation in radians). After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making
Student Facing
The center of a Ferris wheel is 40 feet off the ground, and the radius of the Ferris wheel is 30 feet. Point
\(P\)
is shown at 0 radians.
[IMG]
​​​​​​
Calculate how high off the ground point
\(P\)
is as the Ferris wheel rotates counterclockwise starting from 0 radians.
\(\frac \pi {12}\)
radians
\(\frac \pi 2\)
radians
\(\frac {5\pi}6\)
radians
\(\frac {5\pi}3\)
radians
As you go around on the Ferris wheel, at which position(s) would you be 60 feet off the ground? Explain your reasoning.
You may use this applet if you choose.
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
A Unit Circle Slider applet is available in the digital version of this activity.
Arrange students in groups of 2. Display the image of the Ferris wheel for all to see. Tell them the center of the wheel is 40 feet high and it has a radius of 30 feet. Ask students to give a rough estimate for these questions:
“How high off the ground will point
\(P\)
be after the wheel rotates
\(\frac \pi {12}\)
radians?” (That is a small rotation, so around 50 feet.)
“How high off the ground will point
\(P\)
be after the wheel rotates
\(\frac \pi {2}\)
radians?” (That’s a quarter turn, so
\(P\)
will be at the top, which is a height of 70 feet.)
Give students quiet work time to complete the actual calculations needed for the first question and then time to share their work with a partner before completing the rest of the activity.
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (A Ferris wheel with point
\(P\)
on the circle). After the second read, students list any quantities that can be counted or measured, without focusing on specific values (the distance off of the ground, the radius, and the angle of rotation in radians). After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem while keeping the cognitive demand of the task.
Design Principle: Support sense-making
Action and Expression: Internalize Executive Functions.
Provide students with a two-column table to organize the given angles of rotation (in radians) and corresponding height off the ground (in feet). The table will also provide a visual support for students to write out exact values using sine.
Supports accessibility for: Language; Organization
Student Facing
The center of a Ferris wheel is 40 feet off of the ground, and the radius of the Ferris wheel is 30 feet. Point
\(P\)
is shown at 0 radians.
[IMG]
Calculate how high off the ground point
\(P\)
is as the Ferris wheel rotates counterclockwise starting from 0 radians.
\(\frac \pi {12}\)
radians
\(\frac \pi 2\)
radians
\(\frac {5\pi}6\)
radians
\(\frac {5\pi}3\)
radians
As
\(P\)
goes around on the Ferris wheel, estimate which angle(s) of rotation put
\(P\)
60 feet off the ground. Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
For students having difficulty figuring out different rotations or heights on the Ferris wheel, suggest that they add more labels to the diagram, such as some of the known heights.
Activity Synthesis
Begin the discussion by asking, “Would you ever be 75 feet off the ground? Explain your reasoning.” (No, the highest you would ever be is 70 feet, which is directly above the center of the Ferris wheel rotated
\(\frac \pi 2\)
radians from 0 radians.)
Invite students to share how they estimated the angles of rotation that put
\(P\)
60 feet off the ground. Highlight students who:
labeled the diagram of the Ferris wheel in order to identify that there are 2 angles, one in quadrant 1 and one in quadrant 2
used the unit circle display to identify angles where the value of sine is close to
\(\frac{2}{3}\)
, since 60 feet is 20 feet higher than the center and 20 feet is
\(\frac{2}{3}\)
the radius of the wheel
wrote an equation to represent the situation, such as
\(60=40 + 30 \sin(\theta)\)
For any of these methods that were not used, ask students to consider them now. A key point here is recognizing that the angle needed is one where
\(\frac23\)
the length of the radius is added to the height of the center of the wheel.


correct_option_index: 3