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.
shot_info: im_task_001477

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im_task_000157

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fl_problem_000674

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options: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Write and evaluate numerical expressions involving whole-number exponents.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Write, read, and evaluate expressions in which letters stand for numbers.
problem_activity: Activity
In this activity, students use surface area as a context to extend the order of operations to expressions with exponents. The context provides a reason to evaluate the exponent before performing the multiplication.
Launch
Give students 10 minutes of quiet work time, followed by a class discussion.
Reading: MLR6 Three Reads.
Use this routine to support students’ comprehension of the situation. In the first read, students read the text with the goal of comprehending the situation (e.g., Jada and Noah have different solutions for the surface area of the same cube). In the second read, ask students to identify important quantities that can be counted or measured (e.g., the side length of the cube; the number of faces of a cube; the area of each face of the cube). In the third read, reveal the question, “Do you agree with either of them? Explain your reasoning.” Ask students to brainstorm possible strategies to answer the question (e.g., Find the area of each face of the cube. Multiply the area by 6 to find the surface area of the cube). This will help students concentrate on making sense of the situation before rushing to a solution or method.
Design Principle(s): Support sense-making
Student Facing
A cube has side length 10 inches. Jada says the surface area of the cube is 600 in
2
, and Noah says the surface area of the cube is 3,600 in
2
. Here is how each of them reasoned:
Jada’s Method:
\(6 \boldcdot 10^2\)
\(6 \boldcdot 100\)
\(600\)
Noah’s Method:
\(6 \boldcdot 10^2\)
\(60^2\)
\(3,\!600\)
Do you agree with either of them? Explain your reasoning.
Student Response
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Activity Synthesis
In finding the surface area, there is a clear reason to find
\(10^2\)
and then multiply by 6. Tell students that sometimes it is not so clear in which order to evaluate operations. There is an order that we all generally agree on, and when we want something done in a different order, brackets are used to communicate what to do first. When an exponent occurs in the same expression as multiplication or division, we evaluate the exponent first, unless brackets say otherwise. Examples:
\((3\boldcdot 4)^2 = 12^2 = 144\)
, since the brackets tell us to multiply
\(3\boldcdot 4\)
first. But
\(3\boldcdot  4^2 = 3\boldcdot  16 = 48\)
, because since there are no brackets, we evaluate the exponent before multiplying.
If students bring up PEMDAS or another mnemonic for remembering the order of operations, point out that PEMDAS can be misleading in indicating multiplication before division, and addition before subtraction. Discuss the convention that brackets or parentheses indicate that something should be evaluated first, followed by exponents, multiplication, or division (evaluated left to right), and last, addition or subtraction (evaluated left to right).


correct_option_index: 1

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