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:
Thought: 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:
Thought: 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:
Thought: 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.
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fl_problem_000674

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options: Apply trigonometry to general triangles.

Find arc lengths and areas of sectors of circles.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Experiment with transformations in the plane.

Use coordinates to prove simple geometric theorems algebraically.

Understand and apply theorems about circles.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Make geometric constructions.

Classify two-dimensional figures into categories based on their properties.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Prove geometric theorems.

Understand congruence in terms of rigid motions.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Translate between the geometric description and the equation for a conic section.

Analyze, compare, create, and compose shapes.

Reason with shapes and their attributes.

Define trigonometric ratios and solve problems involving right triangles.

Understand and apply the Pythagorean Theorem.

Solve real-world and mathematical problems involving area, surface area, and volume.

Prove theorems involving similarity.

Apply geometric concepts in modeling situations.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Visualize relationships between two-dimensional and three-dimensional objects.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand similarity in terms of similarity transformations.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Explain volume formulas and use them to solve problems.
problem_activity: Activity
In this activity, students make sense of three different methods for calculating the surface area of a figure. Three different methods are described to students and they are asked to determine which one they agree with (if any) (MP3). They then think about generalizing the methods to figure out if they would work for any prism. This activity connects to work they did with nets in a previous grade and builds upon strategies students might have to calculate surface area.
As students work on the task, monitor for students who understand the different methods and can explain if any of them will work for any other prisms.
Note: It is not important for students to learn the term “lateral area.”
Launch
Arrange students in groups of 2. Display the prism assembled previously in the warm-up for all to see. Ask students: “how might we find surface area of this prism?” Invite students to share their ideas. Give students 1 minute of quiet think to read Noah’s method for calculating surface area followed by time discuss whether they agree with Noah or not. Repeat this process for the remaining two methods. Once all three methods have been discussed give students 1–2 minutes of quiet work time to answer the rest of the questions in the task statement.
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use different colors to represent each of Noah, Elena and Andre’s methods.
Supports accessibility for: Visual-spatial processing
Student Facing
Here is a picture of your teacher's prism:
[IMG]
Three students are trying to calculate the
surface area
of this prism.
Noah says, “This is going to be a lot of work. We have to find the areas of 14 different faces and add them up.”
Elena says, “It’s not so bad. All 12 rectangles are identical copies, so we can find the area for one of them, multiply that by 12 and then add on the areas of the 2 bases.”
Andre says, “Wait, I see another way! Imagine unfolding the prism into a net. We can use 1 large rectangle instead of 12 smaller ones.”
Do you agree with any of them? Explain your reasoning.
How big is the “1 large rectangle” Andre is talking about? Explain or show your reasoning. If you get stuck, consider drawing a net for the prism.
Will Noah’s method always work for finding the surface area of any prism? Elena’s method? Andre’s method? Be prepared to explain your reasoning.
Which method do you prefer? Why?
Student Response
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Anticipated Misconceptions
Students may think that Andre’s method will not work for all prisms, because it will not work for solids that have a hole in their base and therefore more lateral area on the inside. Technically, these solids are not prisms, because their base is not a polygon. However, students could adapt Andre’s method to find the surface area of a solid composed of a prism and a hole.
Activity Synthesis
Select previously identified students to share their reasoning. If not brought up in students' explanations, display the image for all to see and point out to students that the length of the “1 big rectangle” is equal to the perimeter of the base.
[IMG]
Students may have trouble generalizing which method would work for any prism. Here are some guiding questions:
“Which of the students’ methods will work for finding the surface area of this particular prism?” (all 3)
“Which of the students’ methods will work for finding the surface area of any prism?” (Noah’s and Andre’s)
“Which of the students’ methods will work for finding the surface area of other three-dimensional figures that are not prisms?” (only Noah’s)
If not mentioned by students, be sure students understand:
Noah’s method will always work, but it can be inefficient if there are a lot of faces.
Elena’s method will not always work because the rectangles will not always be the same size, but we can notice that some shapes are the same and not have to work them all out individually.
Andre’s method does always work even if the rectangles have different widths. The length of the rectangle will be the same as the perimeter of the base and the width of the rectangle will be the height of the prism.
Prisms can always be cut into three pieces: two bases and one rectangle whose length is the perimeter of a base and whose width is the height of the prism. This can be more efficient than the other methods because students only need to calculate two areas (since the two bases will be identical copies).
This method only works for prisms. For other shapes, such as pyramids, Noah’s method of finding all the faces individually or Elena’s method of combining those faces into identical copy groups will work. Solids with holes, such as the triangular prism with a square hole, can use a variation on Elena’s method: two congruent triangles with holes for the bases, one rectangle for the outside side faces, and another rectangle for the faces forming the hole.
Explain to students that they will have the opportunity in the next activity to practice using any of these strategies.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve a written responses to the question, “How big is the '1 large rectangle' Andre is talking about?” Give students time to meet with 2–3 partners, to share and get feedback on their responses.
Provide students with prompts for feedback that will help their partners strengthen their ideas and clarify their language (e.g., “Can you draw a picture to support your explanation?”, “You should expand on….”, “How does that match with Andre’s thinking?”, etc.). Invite students to go back and revise or refine their written explanation based on their peer feedback. These conversations will help students make sense of the different methods for calculating the surface area of a figure.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)


correct_option_index: 2