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: Calculate expected values and use them to solve problems.

Use probability to evaluate outcomes of decisions.

Understand and evaluate random processes underlying statistical experiments.

Summarize, represent, and interpret data on a single count or measurement variable.

Develop understanding of statistical variability.

Interpret linear models.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Summarize and describe distributions.

Investigate chance processes and develop, use, and evaluate probability models.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Understand independence and conditional probability and use them to interpret data.

Draw informal comparative inferences about two populations.

Investigate patterns of association in bivariate data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Use random sampling to draw inferences about a population.
problem_activity: Activity
In this activity, students practice using their understanding of ways to calculate the number of outcomes in the sample space without writing out the entire sample space (MP7). Many situations with multiple steps have very large sample spaces for which it is not helpful to write out the entire sample space, but it is still useful to know the number of outcomes in the sample space. In this activity, students find the number of different sandwiches that can be made from available options.
Launch
Explain to students that the sandwich makers are instructed to put a certain amount of each item on the sandwich for each selection. Therefore, if a person really loves tomatoes, he should ask for tomatoes twice as his two veggie choices. Give students 5 minutes of quiet work time followed by a whole-group discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding of determining the number of outcomes after 3–5 minutes of work time.
Supports accessibility for: Organization; Attention
Student Facing
A submarine sandwich shop makes sandwiches with one kind of bread, one protein, one choice of cheese, and
two
vegetables. How many different sandwiches are possible? Explain your reasoning. You do not need to write out the sample space.
Breads: Italian, white, wheat
Proteins: Tuna, ham, turkey, beans
Cheese: Provolone, Swiss, American, none
Vegetables: Lettuce, tomatoes, peppers, onions, pickles
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Andre knows he wants a sandwich that has ham, lettuce, and tomatoes on it. He doesn’t care about the type of bread or cheese. How many of the different sandwiches would make Andre happy?
If a sandwich is made by randomly choosing each of the options, what is the probability it will be a sandwich that Andre would be happy with?
Student Response
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Student Facing
Are you ready for more?
Describe a situation that involves three parts and has a total of 24 outcomes in the sample space.
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
Some students may attempt to write out the entire sample space. Encourage them to write a few outcomes to get the idea of what is possible, but let them know that the answer for question 1 is well over 1,000, so finding a pattern or way to calculate the answer might be more efficient.
Some students may not count the “none” option for cheese as a distinct choice. Show these students that a sandwich like “Italian bread, tuna, provolone, lettuce, and tomatoes” is different than a sandwich like “Italian bread, tuna, no cheese, lettuce, and tomatoes” and should be counted separately.
Some students may notice that the order of the vegetable selection may not matter. For example, selecting lettuce then tomato would create a similar sandwich to one with tomato then lettuce selected. Some sandwich shops may offer more of the first option, so we could ask students to consider the sandwiches as different based on this idea. Otherwise, there are only 720 sandwiches possible since there are only 15 different options for 2 vegetables and
\(3 \boldcdot 4 \boldcdot 4 \boldcdot 15 = 720\)
.
Activity Synthesis
The purpose of the discussion is to help students understand the calculations behind the solutions of these problems.
Some questions for discussion:
“Describe how the tree of sandwich options would look without drawing it out.” (The first column would have the 3 options for bread. Coming out from each of those options would be 4 branches for each of the proteins. From each of these there would be 4 more branches for each of the cheese options. Each of those would have 5 branches for the veggies. Finally, there would be 5 branches for the veggies again since the sandwich has 2 of them.)
“How is the tree connected to the calculation of the size of the sample space?” (Since the first two choices [bread and protein] are 3 groups of 4 branches, there are 12 options for those two choices. When adding the cheese option, there are 12 groups of 4 or 48 options. For the first veggie, there are 48 groups of 5 things or 240 options. Finally, there are 240 groups of 5 veggies for the last option, giving a total of 1,200 outcomes.)
“If the two veggie choices had to be different, would there be a higher or lower total number of possible sandwiches? Explain your reasoning.” (Fewer, since an item like “Italian bread, ham, Swiss, onions, and onions” was an option before, but is not an option with the new restriction.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation or response that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. 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 was accurately able to restate their thinking. 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 as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 8