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: Interpret expressions for functions in terms of the situation they model.

Build new functions from existing functions.

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.

Use functions to model relationships between quantities.

Analyze functions using different representations.

Model periodic phenomena with trigonometric functions.

Understand the concept of a function and use function notation.

Extend the domain of trigonometric functions using the unit circle.

Build a function that models a relationship between two quantities.

Define, evaluate, and compare functions.

Interpret functions that arise in applications in terms of the context.
problem_activity: In Class Launch
Use after Unit 4, Lesson 16
Ask students to give examples of situations where there is exponential change (population growth, compounding interest, radioactive decay). Write down students’ ideas for all to see.
Then ask students for an example of an exponential function (
\(y=2e^{\text-t}\)
). Write it for all to see. Here are some possible questions for discussion:
“What situation could this function model?” (Since the exponent is negative, this models decay. The coefficient of the exponential part is 2, so the initial value is 2. The amount is divided by
\(e\)
every time
\(t\)
increases by 1. So it’s a small amount of something and it’s decreasing. How fast it decreases depends on what
\(t\)
represents.)
“What questions about a situation could this function help you answer?” (I could find out what
\(t\)
is when the value of the function is a specific number, like 0.01. If
\(t\)
represents time, I could find out how often the value of the function doubles.)
Tell students that in this task they will ask and answer their own questions about a situation that can be modeled with an exponential function.
Teacher Instructions
After students have finished the task, they should see what questions were asked about each situation. One way to do this would be to have groups that looked at the same situation get together to compare their questions and the assumptions they made about the situation. Groups that made up their own situations can get together and share what they did. Then one or two representatives from each large group can summarize the discussion for the whole class.
Student-Facing Statement
Choose one of these situations to think about, or write a similar one of your own:
There are currently 61 million cars in a certain country, increasing by 6% annually.
A vampire enters a town. He looks around for about an hour, and then chooses someone to bite. The person he bit is now also a vampire. Both of them take an hour to choose new victims, and then they each bite someone. All the vampires continue choosing victims in the same way.
The population in millions,
\(W\)
, of the country of West Mathlandia is modeled by
\(W(x) = 17.9e^{0.0015x}\)
, where
\(x\)
is the number of years since September 1998. The population in millions,
\(E\)
, of East Mathlandia is modeled by
\(E(x) = 13.5e^{0.0128x}\)
.
There are 10,000 bacteria in a petri dish. The population of bacteria declines to 4,000 in 4 hours.
An artifact is discovered. It has 57% of the carbon-14 it originally contained. Carbon-14 decays at the rate of 0.0125% per year.
The number of books,
\(B\)
, in a small library increases according to the function
\(B = 5,\!300e^{0.02t}\)
, where
\(t\)
is measured in years.
The growth in the population,
\(P\)
, of a certain species of rodent at a landfill is modeled by the exponential function
\(P(t)= 648e^{0.016t}\)
, where
\(t\)
is the number of years since 1987.
The initial mass of a certain compound is 448 g, and every 27 hours it decreases by half.
Write down questions you could ask about the situation you chose. Write as many questions as you can think of. As a group, you will choose some interesting questions to answer. If you would need to do calculations, make a sketch, or draw a graph in order to answer a question, then it’s probably an interesting question.
You should answer 2 questions for every person in your group. Multiple people can work on a question together—each person in the group does not have to answer 2 questions by themselves.
Write a report to explain your questions and answers. The report should include:
A context for your situation. Write a story that includes all the relevant information. If the description of the situation is missing some details, fill them in however you want.
An equation to describe the situation, with an explanation of what each part of the equation represents.
A labeled graph of the situation.
The questions you chose and your answers. Include your process for finding the answers. Say what the answers mean in the context of this situation.
Teacher Instructions
After students have finished the task, they should see what questions were asked about each situation. One way to do this would be to have groups that looked at the same situation get together to compare their questions and the assumptions they made about the situation. Groups that made up their own situations can get together and share what they did. Then one or two representatives from each large group can summarize the discussion for the whole class.
Student-Facing Statement
Choose one of these situations to think about, or write a similar one of your own:
A vampire enters a town. He looks around for about an hour, and then chooses someone to bite. The person he bit is now also a vampire. Both of them take an hour to choose new victims, and then they each bite someone. All the vampires continue choosing victims in the same way.
The population in millions,
\(W\)
, of the country of West Mathlandia is modeled by
\(W(x) = 17.9e^{0.0015x}\)
, where
\(x\)
is the number of years since September 1998. The population in millions,
\(E\)
, of East Mathlandia is modeled by
\(E(x) = 13.5e^{0.0128x}\)
.
There are 10,000 bacteria in a petri dish. The population of bacteria declines to 4,000 in 4 hours.
The number of books,
\(B\)
, in a small library increases according to the function
\(B = 5,\!300e^{0.02t}\)
, where
\(t\)
is measured in years.
The growth in the population,
\(P\)
, of a certain species of rodent at a landfill is modeled by the exponential function
\(P(t)= 648e^{0.016t}\)
, where
\(t\)
is the number of years since 1987.
Write down questions you could ask about the situation you chose. Write as many questions as you can think of. As a group, you will choose some interesting questions to answer. If you would need to do calculations, make a sketch, or draw a graph in order to answer a question, then it's probably an interesting question. To get you started, here are some example questions you could ask about each situation:
If there are 200 people in the town, how long will it be before all of them are vampires?
Will the populations of East and West Mathlandia ever be the same, and if so, when?
How much does the population of bacteria decline each hour?
How long will it be before the library has 1 million books?
How many rodents were born between 1987 and now?
You should answer 2 questions for every person in your group. Multiple people can work on a question together—each person in the group does not have to answer 2 questions by themselves.
Write a report to explain your questions and answers. The report should include:
A context for your situation. Write a story that includes all the relevant information. If the description of the situation is missing some details, fill them in however you want.
An equation to describe the situation, with an explanation of what each part of the equation represents.
A labeled graph of the situation.
The questions you chose and your answers. Include your process for finding the answers. Say what the answers mean in the context of this situation.


correct_option_index: 9

11