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: Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Extend the domain of trigonometric functions using the unit circle.

Analyze functions using different representations.

Define, evaluate, and compare functions.

Use functions to model relationships between quantities.

Understand the concept of a function and use function notation.

Interpret functions that arise in applications in terms of the context.

Model periodic phenomena with trigonometric functions.

Construct and compare linear and exponential models and solve problems.

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

Prove and apply trigonometric identities.
problem_activity: Activity
Students now explore the idea of repeated percent change in a banking or savings context. This may be a good time to more formally introduce the term
compounding
. They examine the effect of compounding 1% interest each month for a year. From the previous activity, students should recognize that the result will be
more
than 12% interest on the initial balance. This is because for each month after the first one, the amount used to calculate the interest includes previously earned interest, so that amount is higher than the initial balance. In future lessons, students will continue to investigate the effects of compounding interest in the long term.
To find the account balance at different times (the first question), students may choose to calculate the numerical values or to write expressions. Those who choose the latter will be in a better position to write a generalized expression for the account balance at any time. Select some students who chose different approaches to share their choice later.
Note that this activity simplifies what banks actually do to calculate interests. Financial institutions commonly calculate interest rates based on the number of days in the month or the year. The monthly interest rate for February, for instance, will be lower than that for March.
Launch
Give students a brief overview of savings accounts. Explain that when we deposit money in a savings account, we are allowing the bank to hold on to the money (and potentially use it to generate more revenue for the bank). In exchange, the bank pays us interest, that is a certain percentage of the money at a regular interval. The more money we put in the account, the more interest we earn. The interest calculation here works the same way as that for loans and credit cards, except that, in this case, higher interest rates and more-frequent interest calculations work in our favor, instead of against us.
Arrange students in groups of 2.
Representation: Internalize Comprehension.
Activate or supply background knowledge about generalizing a process to create an expression or equation. After students calculate the account balance after 6 months, 1 year, 2 years, and 5 years, pause and check to make sure students are using the correct units of time (months instead of years). Invite students to identify which parts of their process changed and which stayed the same, and to use this information to create an equation for the account balance (
\(a\)
) in terms of the number of months (
\(m\)
).
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
A bank account has a monthly interest rate of 1% and initial balance of
$
1,000. Any earned interest is added to the account and no other deposits or withdrawals are made.
What is the account balance after 6 months, 1 year, 2 years, and 5 years? Show your reasoning.
Write an equation expressing the account balance (
\(a\)
) in terms of the number of months (
\(m\)
). Assume that all interest earned continues to be added to the account and no other deposits or withdrawals are made.
How much interest will the account earn in 1 year? What percentage of the initial balance is that? Show your reasoning.
The term annual return refers to the percent of interest an account holder could expect to receive in one year. Discuss with your partner: If you were the bank, would you advertise the account as having a 12% annual return? Why or why not? Use your work so far to explain your reasoning.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
Students may miss that the interest is applied monthly rather than annually. Help them read the question carefully, note this difference, and adjust the units to months for the appropriate questions.
Activity Synthesis
Ask previously selected students to share their strategies for the first question (using numerical values or using expressions). Highlight that writing expressions enables us to generalize and find the account balance for any time frame.
Select students to share how they found the interest earned over one year and what percentage it was of the initial balance. Make sure students understand that twelve successive increases of 1% yield a balance that is more than a single 12% increase. Then, invite groups to share which annual interest rate they would advertise if they were the bank.
Conclude by explaining that banking institutions call the two different interest rates different names. The 12% is called the
nominal rate
and the 12.68% the
effective rate
. The latter reflects the actual interest amount earned over a year, taking into account the monthly interest payment. This process of generating new earnings from previous earnings is called
compounding.
(In the context of loans,
compounding
refers to the generation of new debt from previous debts.)
As in the previous activity, the difference between the one-year interest calculated using the nominal rate versus the actual rate is not large. The differences become more and more pronounced, however, as time goes on.
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to describe the situation involving a bank account that earns monthly interest. Call students’ attention to language such as “initial or starting balance,” “annual return,” or “initial earning plus the interest.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Be sure that the display includes new terms such as “nominal rate,” “effective rate,” and “compounding.” Remind students to borrow language from the display as needed. This will help students use mathematical language for describing a repeated percent change—compounding—in a banking context.
Design Principle(s): Maximize meta-awareness


correct_option_index: 0