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.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Build new functions from existing functions.

Analyze functions using different representations.

Define, evaluate, and compare functions.

Prove and apply trigonometric identities.

Model periodic phenomena with trigonometric functions.

Understand the concept of a function and use function notation.

Use functions to model relationships between quantities.

Build a function that models a relationship between two quantities.

Extend the domain of trigonometric functions using the unit circle.

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

Construct and compare linear and exponential models and solve problems.

Interpret functions that arise in applications in terms of the context.
problem_activity: Activity
In this activity students are presented with a series of questions like, “A person is 60 inches tall. Do you know their height in feet?” For some of the questions the answer is "yes" (because you can convert from inches to feet by dividing by 12). In other cases the answer is "no" (for example, “A person is 14 years old. Do you know their height?”). The purpose is to develop students’ understanding of the structure of a function as something that has one and only one output for each allowable input. In cases where the answer is yes, students draw an input-output diagram with the rule in the box. In cases where the answer is no, they give examples of an input with two or more outputs. In the Activity Synthesis, the word function is introduced to students for the first time.
Identify students who use different ways to describe the rules and different notation for the input and outputs of the final two problems.
Launch
Display the example statement (but not the input-output diagram) for all to see.
Example: A person is 60 inches tall. Do you know their height in feet?
Give students 30 seconds of quiet think time, and ask them to be prepared to justify their response. Select students to share their answers, recording and displaying different justifications for all to see.
Display the following input-output diagrams for all to see. Ask students if the rules in the diagrams match the justifications they just heard:
[IMG]
[IMG]
or
Tell students that they will draw input-output diagrams like these as part of the task. Answer any questions students might have around the input-output diagrams. Be sure students understand that if they answer yes to the question they will need to draw the input-output diagram and if they answer no they need to give an example of why the question does not have one answer.
Give students 8–10 minutes of quiet work time for the problems followed by a whole-class discussion.
Representation: Access for Perception.
Read all statements aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language
Student Facing
Say yes or no for each question. If yes, draw an input-output diagram. If no, give examples of two different outputs that are possible for the same input.
A person is 5.5 feet tall. Do you know their height in inches?
A number is 5. Do you know its square?
The square of a number is 16. Do you know the number?
A square has a perimeter of 12 cm. Do you know its area?
A rectangle has an area of 16 cm
2
. Do you know its length?
You are given a number. Do you know the number that is
\(\frac15\)
as big?
You are given a number. Do you know its reciprocal?
Student Response
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Activity Synthesis
The goal of this discussion is for students to understand that functions are rules that have one distinct output for each input. For several problems, select previously identified students to share their rule descriptions. For example, some students might write the rule for finding the area of a square from its perimeter as "find the side length then square it" and others might write, "divide by four and then square it." Compare the different rules and ask students if they agree (or disagree) that the statements represent the same rule.
For the final two problems where the input is not a specific value, select previously identified students to share what language they used. For example, in the final problem some students may use a letter to stand for the input while others may just write "input" or "a number." Ask, "How can using a letter sometimes make it easier to represent the output?" (Writing "
\(\frac15 n\)
" is shorter than writing "
\(\frac15\)
of a number.")
Tell students that each time they answered one of the questions with a yes, the sentence defined a
function
, and that one way to represent a function is by writing a rule to define the relationship between the input and the output. Functions are special types of rules where each input has only one possible output. Because of this, functions are useful since once we know and input, we can find the single output that goes with it. Contrast this with something like a rolling a number cube where the input "roll" has many possible outputs. For the questions students responded to with no, these are not functions because there is no single output for each input.
To highlight how rules that are not functions do not determine outputs in a unique way, end the discussion by asking:
"Was the warm-up, where you have to square numbers, an example of a function?" (Yes, each input has only one output, even though some inputs have the same output.)
"Is the reverse, that is knowing what number was squared to get a specific number, a function?" (No, if a number squared is 16, we don't know if the number was 4 or -4.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Call on students to use mathematical language to restate and/or revoice the rule descriptions that are presented. 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 accurately restated their rule. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language to describe functions.
Design Principle(s): Support sense-making; Maximize meta-awareness


correct_option_index: 2