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.


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


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


Your response, in json format:
{
  "explanation": "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: Use properties of rational and irrational numbers.

Perform arithmetic operations with complex numbers.

Reason quantitatively and use units to solve problems.

Represent and model with vector quantities.

Perform operations on vectors.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Extend the properties of exponents to rational exponents.

Compute fluently with multi-digit numbers and find common factors and multiples.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of numbers to the system of rational numbers.

Perform operations on matrices and use matrices in applications.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
problem_activity: Optional activity
The purpose of this activity is for students to choose how they can apply math concepts and strategies to a problem arising in a real-world context: predicting whether a restaurant will make a profit.
After students have estimated the monthly cost of their ongoing expenses on their list, poll the class on their total estimated monthly costs.
If students have completed the previous lesson about using spreadsheets to calculate the cost of ingredients for one serving of their recipe, they can use that as the basis for calculating the percentage of the markup. If students have not calculated the cost of ingredients in a previous lesson, then tell students that the
ingredients
for one serving of a meal typically cost about:
$
2 for a fast-food restaurant
$
5 for a casual sit-down restaurant
$
10 to
$
25 for a formal sit-down restaurant
Launch
Point out that restaurants have many more expenses than just the cost of the food. Give students quiet work time followed by partner discussion.
Consider having students pause their work after the first question so that you can record their answers displayed for all to see, and all students will have access to this same information for the rest of the activity.
Speaking, Listening, Writing: MLR1 Stronger and Clearer Each Time.
Use this routine after students have assessed the profitability of Restaurant A. Ask students to share their thinking with 2–3 consecutive partners. With each share, encourage listeners to push students for clarity in mathematical language (e.g., “How do you know the restaurant is making a profit?”). With each share, students’ organization of steps should get stronger and the explanations of reasoning should get clearer. This helps students to use mathematical language as they explain their profit analysis.
Design Principle(s): Optimize output (for explanation); Cultivate conversation
Student Facing
Restaurants have many more expenses than just the cost of the food.
Make a list of other items you would have to spend money on if you were running a restaurant.
Identify which expenses on your list depend on the number of meals ordered and which are independent of the number of meals ordered.
Identify which of the expenses that are independent of the number of meals ordered only have to be paid once and which are ongoing.
Estimate the monthly cost for each of the ongoing expenses on your list. Next, calculate the total of these monthly expenses.
Tell whether each restaurant is making a profit or losing money if they have to pay the amount you predicted in ongoing expenses per month. Organize your thinking so it can be followed by others.
Restaurant A sells 6,000 meals in one month, at an average price of
$
17 per meal and an average cost of
$
4.60 per meal.
Restaurant B sells 8,500 meals in one month, at an average price of
$
8 per meal and an average cost of
$
2.20 per meal.
Restaurant C sells 4,800 meals in one month, at an average price of
$
29 per meal and an average cost of
$
6.90 per meal.
Predict how many meals your restaurant would sell in one month.
How much money would you need to charge for each meal to be able to cover all the ongoing costs of running a restaurant?
What percentage of the cost of the ingredients is the markup on your meal?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Poll the class on whether they think each of the restaurants A, B, and C made money for the month. Select students to share their reasoning.
Note: The way the problem is written, there is not just one correct answer to the question. Whether or not each restaurant made money depends on how much the students estimated for the total monthly expenses (excluding food).
|               if students estimated the total ongoing expenses (excluding food) to be:               | restaurants that would have made money | restaurants that would have lost money |
|------------------------------------------------------------------------------------------------------|----------------------------------------|----------------------------------------|
|                                            below $49,300                                             |                                        |                                        |
| between                   $                  49,300 and                   $                  74,400  |                                        |                                        |
| between                   $                  74,400 and                   $                  106,080 |                                        |                                        |
|                                            above $106,080                                            |                                        |                                        |
Next, ask students to trade with a partner, and check their work for calculating the percentage of the mark up.
Ask students to discuss:
“How does the amount you plan to charge for each meal compare to your partner’s amount?”
“How does your mark up percentage compare to your partner’s percentage?”
Consider telling students that many restaurant owners use 300% as an estimate of a good percentage for the mark up on their meals to be able to make a profit. That means the price of the meal is 4 times what the cost of the ingredients were.


correct_option_index: 13