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:
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.


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.


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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options: Experiment with transformations in the plane.

Analyze, compare, create, and compose shapes.

Apply geometric concepts in modeling situations.

Reason with shapes and their attributes.

Understand congruence in terms of rigid motions.

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

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

Use coordinates to prove simple geometric theorems algebraically.

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

Explain volume formulas and use them to solve problems.

Prove theorems involving similarity.

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

Prove geometric theorems.

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

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

Define trigonometric ratios and solve problems involving right triangles.

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

Understand and apply theorems about circles.

Find arc lengths and areas of sectors of circles.

Understand and apply the Pythagorean Theorem.

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

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

Apply trigonometry to general triangles.

Understand similarity in terms of similarity transformations.

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

Make geometric constructions.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
problem_activity: Activity
In previous grades, students learned how to calculate the volume of a cylinder. In this activity, they revisit that process. Students are prompted to think about volume in 1-unit layers. This will be helpful in upcoming lessons when students learn about Cavalieri’s Principle, or the idea that if two solids have equal-area cross sections at all heights, the solids have equal volumes. Cavalieri’s Principle will be used to derive a formula for the volume of a pyramid.
Note that in this activity, “volume of a container” is used as shorthand for “the volume of the region enclosed by the container.”
Monitor for students who calculate the volume of the water in the prism then work backward to find the height of the water in the cylinder, and for those who use deductive logic (for example, reasoning that because the two bases have equal area, the height of the water must be equal in both solids) to arrive at the same answer.
Students reason abstractly and quantitatively (MP2) when they compare the prism and cylinder volumes.
Launch
Speaking, Reading: MLR5 Co-Craft Questions.
Use this routine to increase awareness of the language used to talk about the volumes of solids. Before revealing the questions in this activity, display the image of both solids along with the statement: “Here are two solids. All measurements are in centimeters,” and ask students to write down possible mathematical questions that could be asked about the image. Invite students to compare their questions before revealing the actual questions. Listen for and amplify any questions about the volume of both solids. For example, “Does the cylinder have the same volume as the prism?”
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because _____. Then, I….”, “I noticed _____ so I….”, and “I tried _____, and what happened was….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are two containers. All measurements are in centimeters.
[IMG]
Suppose the prism contains water that reaches a height of 1 cm.
Draw a representation of this situation.
The water is poured from the prism into the cylinder. What is the height of the water in the cylinder? Explain your reasoning.
Suppose the prism contained water that reached a height of 3 cm instead of 1 cm. If the water were poured into the empty cylinder, what would the height of the water in the cylinder be?
Student Response
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Anticipated Misconceptions
Students may be unsure how to work with the prism measurement
\(4\pi\)
cm. Invite them to write out the area calculation as
\(9\boldcdot4\pi\)
. Alternatively, they can use an approximation of
\(\pi\)
to get rounded answers.
If students struggle to find the height of the water in the cylinder, ask them what they already know about the cylinder, and to consider what additional information they would need to find the height. Prompt them to use their responses to the first question to help them answer this one.
Activity Synthesis
The goal is to make connections between methods of calculating volumes for prisms and cylinders. Ask previously identified students to share their reasoning, starting with those who did volume calculations and worked backward, and ending with those who used deductive reasoning. Connect these methods by asking these questions:
“Describe what volume formulas you used. Do they apply to just one of the containers, or to both?” (Sample responses: The expression
\(\ell wh\)
applies to the prism. The expression
\(\pi r^2 h\)
applies to the cylinder. The expression
\(BH\)
applies to both.)
“Consider the volume formula
\(V=Bh\)
. How does this formula help explain why the heights were the same for the same volume of water in each container?” (The area of the base,
\(B\)
, is the same for each container:
\(36\pi\)
square centimeters. The volume formula can be rewritten
\(h=V\div B\)
, so if the containers have the same volume of water in each and have bases with the same area, the height will be the same.)
Invite students to describe how the volumes of the two containers compare (they are equal), and challenge them to describe another solid that has the same volume as these two.


correct_option_index: 9