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.


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.
problem_activity: Activity
In this activity, students continue to reason about half-lives and exponential functions to solve problems. Both questions provide possible reasoning for a calculation and then prompt students to construct an argument to justify their position (MP3).
The context continues to be the decay of carbon-14 and students investigate two different situations. In each case, the amount of time that has passed since the carbon-14 began to decay is not a multiple of the half-life. This means that students need to make estimates, either in terms of a fraction of a half-life or a whole number of half-lives.
Launch
In case students are unfamiliar with papyrus, explain that it is a material that was used as a writing surface in ancient times, similar to how paper is used today. Papyrus is made out of the papyrus plant and once it is harvested and turned into a type of paper, the amount of carbon-14 in the papyrus starts decreasing.
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. While students work, provide tools for creating a visual display for them to record information for sharing. Encourage students to record their group’s thinking by writing out explanations using equations, words, pictures, or other symbolic representations such as timelines, that explain their approaches.
Supports accessibility for: Language; Social-emotional skills; Attention
Student Facing
The half-life of carbon-14 is about 5,730 years.
Pythagoras lived between 600 BCE and 500 BCE. Explain why the age of a papyrus from the time of Pythagoras is about half of a carbon-14 half-life.
Someone claims they have a papyrus scroll written by Pythagoras. Testing shows the scroll has 85% of its original amount of carbon-14 remaining. Explain why the scroll is likely a fake.
Student Response
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Student Facing
Are you ready for more?
A copy of the Gutenberg Bible was made around 1450. Would more or less than 90% of the carbon-14 remain in the paper today? Explain how you know.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
As in the previous activity, students may mistakenly think that after about
\(\frac{1}{2}\)
of a half-life, only about
\(\frac{1}{4}\)
of the carbon-14 remains. Ask them if more or less than
\(\frac{1}{2}\)
of the carbon-14 remains after 2,600 years. (Less since this is less than a half-life.)
Activity Synthesis
Focus the discussion on the estimates students made and the properties of exponential functions that they used. Here are some questions for discussion:
“About how old is the papyrus? About how many half-lives of carbon-14 is that?” (2,600 years,
\(\frac{1}{2}\)
half-life)
“By what factor does the amount of carbon-14 decrease in one half-life? What about in half of a half-life?” (
\(\frac{1}{2}\)
and
\(\frac{1}{\sqrt{2}}\)
)
One thing to note to students is that carbon-14 dating, while improving, is not an exact calculation but rather a good approximation for dating items from long ago. Consider telling students that carbon-14 dating is only used for objects that are less than about 50,000 years old. Ask,
“About how many half lives is that?” (9)
“About what fraction of the carbon-14 would be left in a 50,000-year-old artifact?” (
\(\frac{1}{2^9}\)
or
\(\frac{1}{512}\)
)
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 0