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.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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} $

Standard options: 
A. Understand a fraction as a number on the number line; represent fractions on a number line diagram.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
B. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
C. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.


Thought: By arranging the fractions in this problem, students learn to compare fractions by reasoning about their size.
Answer: B

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?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Thought: This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
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im_task_000157

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fl_problem_000674

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options: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

Recognize and represent proportional relationships between quantities.

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
problem_activity: Activity
This activity is intended to build confidence and facility in writing an equation for a proportional relationship. Students build on their understanding that measurement conversions can be represented by proportional relationships, which they studied in an earlier lesson and will revisit in future lessons. Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and use repeated reasoning (MP8) to arrive at an equation. In addition, they are expected to identify the relationship between the constant of proportionality when going from centimeters to meters and vice versa (reciprocals). Since students have already explored a similar relationship (centimeters and millimeters) in a previous lesson, this activity may go very quickly.
As students work, look for students writing an equation like
\(100k = 1\)
(for table 2) as a step to finding the constant of proportionality. Encourage them to say how they would solve the equation. Ask students to say why using this equation makes sense in the scenario.
Launch
Introduce the task: “In a previous lesson, you examined the relationship between millimeters and centimeters. Today we will examine the relationship between centimeters and meters.”
Student Facing
There are 100 centimeters (cm) in every meter (m).
| length (m) | length (cm) |
|------------|-------------|
|     1      |     100     |
|    0.94    |             |
|    1.67    |             |
|   57.24    |             |
|   \(x\)    |             |
| length (cm) | length (m) |
|-------------|------------|
|     100     |     1      |
|     250     |            |
|    78.2     |            |
|    123.9    |            |
|    \(y\)    |            |
Complete each of the tables.
For each table, find the constant of proportionality.
What is the relationship between these constants of proportionality?
For each table, write an equation for the proportional relationship. Let
\(x\)
represent a length measured in meters and
\(y\)
represent the same length measured in centimeters.
Student Response
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Student Facing
Are you ready for more?
How many cubic centimeters are there in a cubic meter?
How do you convert cubic centimeters to cubic meters?
How do you convert the other way?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Invite students to share their answers. These questions can guide the discussion:
"How can we find an equation for each table?"
"Where does the constant of proportionality occur in the table and equation?"
"What is the relationship between the two constants of proportionality? How can you use the equations to see why this should be true?"
The equations can help students see why the constants of proportionality are reciprocals:
\(\displaystyle y = 100x\)
\(\displaystyle \left(\frac{1}{100}\right)y = \frac{1}{100} \boldcdot (100x)\)
\(\displaystyle \left(\frac{1}{100}\right)y = x\)
\(\displaystyle x = \left(\frac{1}{100}\right)y\)
This line of reasoning illustrated above should be accessible to students, because it builds on grade 6 work with expressions and equations.
Ask students to interpret the meaning of the equations in the context: "What do the equations tell us about the conversion from meters to centimeters and back?"
Given the length in meters, to find the length in centimeters, multiply the number of centimeters by 100.
Given the length in centimeters, to find the the length in meters, multiply the number of meters by
\(\frac{1}{100}\)
.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because….,” “I noticed _____ so I….,” “Why did you . . .?,” and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion supports.
Provide students with sentence frames such as “I found the constant of proportionality by _____” to help explain their reasoning. Revoice student ideas to model mathematical language use by restating a response as a question in order to clarify, and apply appropriate language.
Design Principle(s): Maximize linguistic and cognitive awareness


correct_option_index: 1