few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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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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options: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
problem_activity: Activity
This activity prompts students to create expressions to represent the quantities and relationships in a situation and engages them in mathematical modeling.
Students plan a pizza party and present a cost estimate. To do so, they need to consider relevant variables, make assumptions and estimates, perform calculations, and adjust their thinking along the way (MP4). Some students may choose to perform research and revise their models as they gather new information. There are many possible solutions to the task.
As students discuss their ideas, monitor for those who:
find and use actual data or exact values (for example, count the number of students in the class, research the cost of a large pizza at a nearby shop, or quickly survey the class for topping preferences)
estimate quantities based on prior knowledge (for example, the cost of a large pizza in a recent purchase, or the number of slices they and their friends generally consume at lunch time)
make assumptions about behaviors, preferences, or quantities (for instance, assume that a certain percentage of the class prefers a certain topping)
Launch
Ask students if they have ever been in charge of planning a party. Solicit a few ideas of what party planners need to consider. Ask students to imagine being in charge of a class pizza party. Explain that their job is to present a plan and a cost estimate for the party.
Arrange students in groups of 4. Provide access to calculators and, if feasible and desired, access to the internet for researching pizza prices. Students can also make estimates based on prior experience, refer to printed ads, or use their personal device to look up pricing information.
Limit the time spent on the first question to 7–8 minutes and pause the class before students move on to subsequent questions. Give groups of students 1–2 minutes to share their proposals with another group. Then, select a few groups who used contrasting strategies (such as those outlined in the Narrative) to briefly share their plans with the class. Record or display their plans for all to see.
Next, ask students to complete the remaining questions. If needed, give an example of an expression that can be written to represent a cost calculation.
Student Facing
Imagine your class is having a pizza party.
Work with your group to plan what to order and to estimate what the party would cost.
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Record your group’s plan and cost estimate. What would it take to convince the class to go with your group’s plan? Be prepared to explain your reasoning.
Write down one or more expressions that show how your group’s cost estimate was calculated.
In your expression(s), are there quantities that might change on the day of the party? Which ones?
Rewrite your expression(s), replacing the quantities that might change with letters. Be sure to specify what the letters represent.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Find a pizza place near you and ask about the diameter and cost of at least two sizes of pizza. Compare the cost per square inch of the sizes.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students do not understand what is meant by “quantities that might change,” ask them if it is more likely that the cost of a pepperoni pizza increases on the day of the party, or that some students are absent that day. In a model that incorporates both of these quantities, they may wish to use a number for the cost of each type of pizza and a letter for the number of students present that day.
Activity Synthesis
Invite groups who did not previously share their plans to share the expressions they wrote and explain what the expressions represent. After each group shares, ask if others calculated the costs the same way but wrote different expressions.
As students present their expressions, record the quantities that they mention and display them for all to see. Some examples:
the number of students in the class
the number of pizza slices per person
the cost of delivery
the price per topping
Briefly discuss the quantities that students anticipate would change (and therefore would replace with letters).
Explain to students that the expressions they have written are examples of mathematical
models
. They are mathematical representations of a situation in life that can be used to make sense of problems and solve them. We will look more closely at how expressions could represent the quantities in a situation like party planning, which involves certain conditions or requirements.
Representing, Conversing: MLR7 Compare and Connect.
As students share their plans with the class, call students’ attention to the different ways the quantities are represented in expressions, and to the different expressions that represent the same quantity. Take a close look at the contrasting strategies used within the context of the situation. Wherever possible, amplify student words and actions that students use to revise their models by estimating quantities and making assumptions about the situation.
Design Principle(s): Maximize meta-awareness; Support sense-making


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