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: Use properties of operations to generate equivalent expressions.

Create equations that describe numbers or relationships

Represent and solve problems involving addition and subtraction.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand and apply properties of operations and the relationship between addition and subtraction.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Work with addition and subtraction equations.

Generate and analyze patterns.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Use polynomial identities to solve problems.

Represent and solve problems involving multiplication and division.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Analyze patterns and relationships.

Solve equations and inequalities in one variable.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand the relationship between zeros and factors of polynomials.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Work with radicals and integer exponents.

Solve systems of equations.

Reason about and solve one-variable equations and inequalities.

Understand properties of multiplication and the relationship between multiplication and division.

Gain familiarity with factors and multiples.

Understand the connections between proportional relationships, lines, and linear equations.

Work with equal groups of objects to gain foundations for multiplication.

Multiply and divide within 100.

Use the four operations with whole numbers to solve problems.

Represent and solve equations and inequalities graphically.

Represent and analyze quantitative relationships between dependent and independent variables.

Add and subtract within 20.

Write and interpret numerical expressions.

Rewrite rational expressions.
problem_activity: Activity
In this activity, students continue to develop their understanding of square roots. Students first find the areas of three squares, estimate the side lengths using tracing paper, and then write the exact side lengths. Then they make a table of side-area pairs. They then graph the ordered pairs from the table and use the graph to estimate the values of squares with non-integer side lengths, such as the ones they drew in the previous activity.
Digital
Launch
Arrange students in groups of 2. Provide access to calculators. Remind students about the meaning and use of square root notation. Have students work together on the first problem and check each other’s work. Have them make their graphs independently and then check with their partners that they look the same. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Student Facing
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.
[IMG]
Complete the tables with the missing side lengths and areas.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "0.5"}, {}, {"_value": "1.5"}, {}, {"_value": "2.5"}, {}, {"_value": "3.5"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "row 1"}, {"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "1"}, {}, {"_value": "4"}, {}, {"_value": "9"}, {}, {"_value": "16"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "4.5"}, {}, {"_value": "5.5"}, {}, {"_value": "6.5"}, {}, {"_value": "7.5"}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "_value": "row 1"}, {"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "25"}, {}, {"_value": "36"}, {}, {"_value": "49"}, {}, {"_value": "64"}]}]}]}]}
Plot the points,
\((s, a)\)
, on the coordinate plane shown here.
Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?
Use the graph to approximate
\(\sqrt{45}\)
.
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
Arrange students in groups of 2. Provide access to calculators. Remind students about the meaning and use of square root notation. Have students work together on the first problem and check each other’s work. Have them make their graphs independently and then check with their partners that they look the same. Follow with a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. Check in with students after the first 2-3 minutes of work time. Check to make sure students have attended to all parts of the original figures.
Supports accessibility for: Organization; Attention
Student Facing
Find the area of each square and estimate the side lengths using your geometry toolkit. Then write the exact lengths for the sides of each square.
[IMG]
Complete the tables with the missing side lengths and areas.
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "0.5"}, {}, {"_value": "1.5"}, {}, {"_value": "2.5"}, {}, {"_value": "3.5"}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "1"}, {}, {"_value": "4"}, {}, {"_value": "9"}, {}, {"_value": "16"}]}]}]}]}
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "side length,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "833"}, "_value": "\\(s\\)"}]}], "td": [{"_value": "4.5"}, {}, {"_value": "5.5"}, {}, {"_value": "6.5"}, {}, {"_value": "7.5"}, {}]}, {"th": [{"_attributes": {"scope": "col"}, "_value": "area,", "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "81"}, "_value": "\\(a\\)"}]}], "td": [{}, {"_value": "25"}, {}, {"_value": "36"}, {}, {"_value": "49"}, {}, {"_value": "64"}]}]}]}]}
Plot the points,
\((s, a)\)
, on the coordinate plane shown here.
[IMG]
Use this graph to estimate the side lengths of the squares in the first question. How do your estimates from the graph compare to the estimates you made initially using your geometry toolkit?
Use the graph to approximate
\(\sqrt{45}\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share their graphs from the second problem and display the graph for all to see. Ask: “What relationship does the graph display?” (The relationship between the side length and area of a square.) Select 2–3 students to share their answers to the last problem, including how they used the graph to estimate
\(s\)
for the tilted square they drew earlier.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. After a student shares their response to the last problem, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement.
Design Principle(s): Support sense-making


correct_option_index: 20