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:
{
  "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:
{
  "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:
{
  "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: Work with radicals and integer exponents.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Create equations that describe numbers or relationships

Understand properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100.

Solve equations and inequalities in one variable.

Represent and solve problems involving multiplication and division.

Understand and apply properties of operations and the relationship between addition and subtraction.

Solve systems of equations.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Analyze patterns and relationships.

Represent and analyze quantitative relationships between dependent and independent variables.

Gain familiarity with factors and multiples.

Work with equal groups of objects to gain foundations for multiplication.

Interpret the structure of expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Represent and solve problems involving addition and subtraction.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with addition and subtraction equations.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Add and subtract within 20.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use properties of operations to generate equivalent expressions.

Write expressions in equivalent forms to solve problems.

Understand solving equations as a process of reasoning and explain the reasoning.

Write and interpret numerical expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Reason about and solve one-variable equations and inequalities.

Represent and solve equations and inequalities graphically.

Use the four operations with whole numbers to solve problems.

Rewrite rational expressions.

Generate and analyze patterns.

Perform arithmetic operations on polynomials.
problem_activity: In Class Launch
Use after Unit 2, Lesson 23
Solicit information that students already know about heating buildings, which they will use in this task. Here are some possible questions for discussion:
“How are buildings heated?” (fireplaces, boilers, gas furnaces)
“What kinds of energy can be used to heat buildings?” (electricity, natural gas, solar power)
“When in the year do you think our school uses the most power for heating?” (probably right before or after winter break, when it’s pretty cold and there are still people in the building for most of the day)
Tell students that in this task they will investigate energy costs for some different heating systems. They will begin by considering a homeowner’s current heating system, and compare some other systems the homeowner could use instead. Tell students that energy is measured in kilowatt-hours, which is abbreviated kWh.
Then give students a preview of some of the calculations they’ll need to do by showing why the homeowner currently pays
$
975/year to heat the house. Tell students that the current system is 60% efficient, which means that for every 100 kWh it uses, it only produces 60 kWh of heat. Display this statement for all to see:
“For every 100 kWh of energy a certain heating system uses, it produces 60 kWh of heat. If the system has to produce 11,700 kWh of heat to heat a house for the winter, how many kWh of energy will it use?”
Ask students to think about how they would find an answer to this question. They do not need to calculate an answer, only think of a strategy. After some quiet think time, ask students to share their thoughts with a partner. Then invite students to share strategies with the class. Use one of the suggested strategies to calculate the answer, and write the steps for all to see. Here is one possible way:
\(\displaystyle \begin{align*}{} \dfrac{100 \text{ kWh input}}{60 \text{ kWh output}} &= \dfrac{x \text{ kWh input}}{11,\!700 \text{ kWh output}} \\  \dfrac{100 \text{ kWh input}}{60 \text{ kWh output}} \boldcdot 11,\!700 \text{ kWh output} &= x \text{ kWh input} \\ 19,\!500 \text{ kWh} &= x \end{align*} \)
To find out how much this will cost the homeowner, we need to know how expensive the fuel is. If we assume natural gas costs
$
0.05 per kWh, then multiplying 19,500 by 0.05 will give us the total cost of
$
975.
Teacher Instructions
Students will need to do a lot of independent research for this task. One way to make the research less time-consuming would be to have students suggest other ways of heating the house, like solar panels or geothermal heating and cooling systems, and have each group investigate one of the ideas. Students do not have to limit themselves to alternative heating systems—if they wish, they can also look at ways the homeowner could reduce the amount of heat needed, for example by insulating their walls. Once the groups are done researching, they can report their results to the class so that everyone can use the same information to do their analysis, or new groups can form so that each member of the group researched a different idea, and the group can work together to use the information each member found.
Students will need to be very careful not to confuse the amount of energy a system uses with the amount it produces, and in their research they will see many ways of describing the energy that is used and produced, such as British thermal units, therms, and Joules. If students struggle with the calculations, it may help if they make a table with column headings like “energy used” and “energy produced” so they can fill in the values they know for each system and calculate the unknown ones.
Student-Facing Statement
A homeowner wants to replace their old heating system. Energy is measured in kilowatt-hours (kWh). It takes about 11,700 kWh of energy to heat the house for the winter. The current heating system uses natural gas and is 60% efficient, which means that for every 100 kWh of natural gas it uses, it produces 60 kWh of heat. With the homeowner’s current system, it costs
$
975 to heat the house.
Research at least two other options available in your area that this homeowner could replace their heating system with. Assume that natural gas costs
$
0.05/kWh and electricity costs
$
0.21/kWh. The house is 2,500 square feet.
The homeowner also has an air conditioner that uses 2,500 kWh of electricity per year and produces 290 kWh of cooling for every 100 kWh it uses. They also have a water heater that uses 4,300 kWh of electricity per year and produces 90 kWh of heat for every 100 kWh it uses. These systems could also be replaced if there is a cheaper option, but it isn’t necessary.
Which system would you recommend? Make a graph to convince the homeowner to switch to this system.
If the homeowner switches to the system you recommend, how long will it take them to save as much money as the new system cost?
Teacher Instructions
Students will need to be very careful not to confuse the amount of energy a system uses with the amount it produces. If students struggle with the calculations, it may help if they make a table with column headings like “energy used” and “energy produced” so they can fill in the values they know for each system and calculate the unknown ones.
Student-Facing Statement
A homeowner wants to replace their old heating system. Energy is measured in kilowatt-hours (kWh). It takes about 11,700 kWh of energy to heat the house for the winter. The current heating system uses natural gas and is 60% efficient, which means that for every 100 kWh of natural gas it uses, it produces 60 kWh of heat. With the homeowner’s current system, it costs
$
975 to heat the house. Assume that natural gas costs
$
0.05/kWh and electricity costs
$
0.21/kWh.
The homeowner also has an air conditioner that uses 2,500 kWh of electricity per year and produces 290 kWh of cooling for every 100 kWh it uses. They also have a water heater that uses 4,300 kWh of electricity per year and produces 90 kWh of heat for every 100 kWh it uses. These systems could also be replaced if there is a cheaper option, but it isn’t necessary.
Here are three other types of heating systems the homeowner could replace their current system with:
A new furnace which also runs on natural gas and is more efficient. For every 100 kWh of natural gas it uses, it produces 95 kWh of heat. This system costs
$
5,000 to install.
A geothermal heat pump. This system uses electricity instead of natural gas, but it produces 4 kWh of heat for every 1 kWh of electricity it uses. It costs
$
14,000, but it’s very low-maintenance and it also replaces the air conditioner and water heater.
A grid-tied solar array. This system is connected to the electrical grid so that when it generates more energy than the house needs, the extra electricity can be sold back to the grid. Over the whole year, the house will use as much electricity as is sold back, which means the homeowner would basically be heating and cooling the house for free. It costs
$
16,000 to install.
Which system would you recommend? Make a graph to convince the homeowner to switch to this system.
How long will it take to save as much money as the new system cost?


correct_option_index: 2

15