few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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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 new functions from existing functions.

Extend the domain of trigonometric functions using the unit circle.

Define, evaluate, and compare functions.

Build a function that models a relationship between two quantities.

Analyze functions using different representations.

Use functions to model relationships between quantities.

Understand the concept of a function and use function notation.

Construct and compare linear and exponential models and solve problems.

Model periodic phenomena with trigonometric functions.

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

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

Prove and apply trigonometric identities.
problem_activity: Optional activity
The goal of this task is to review the connections between different ways of expressing a relationship: a description, a graph, and likely an equation. Students begin by analyzing different graphs, finding one that matches a given description, and explaining how they know the graph is a correct representation (MP3). They then use their analyses to evaluate the function at a different input value, compare it to another exponential function, and sketch a graph of the second function.
Monitor for students who:
Articulate clearly how the features of the graph enable them to choose the correct graph to represent the depreciation of the first van.
Write an expression to answer questions about the value of either van.
Launch
Clarify the meaning of
depreciation
for students who may not know what it means.
Arrange students in groups of 2. Give them a moment to think quietly about the first question, and then share their response with a partner. They should be ready to explain to each other how they know that certain graphs
cannot
represent the given function.
Representation: Internalize Comprehension.
Use color coding and annotations to highlight connections between representations in a problem. For example, invite students to highlight each important element in the problem statement in a different color (starting price, year, and percent depreciation). Invite students to use these same colors to identify corresponding elements on the graphs.
Supports accessibility for: Visual-spatial processing
Student Facing
A small business bought a van for $40,000 in 2008. The van depreciates by 15% every year after its purchase.
Which graph correctly represents the value of the van as a function of years since its purchase? Be prepared to explain why each of the other graphs could not represent the function.
Graph A
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Graph B
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Graph C
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Graph D
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Find the value of the van 8 years after its purchase. Show your reasoning.
In the same year (2008), the business bought a second van that cost $10,000 less than the first van and depreciates at 10% per year. Would the second van be worth more or less than the first van 8 years after the purchase? Explain or show your reasoning.
On the same coordinate plane as the graph you chose in the first question, sketch a graph that shows the value of the second van, in dollars, as a function of years since its purchase.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may not be sure how to start calculating the value of the van after 8 years. Ask these students to make a table showing years since 2008 and the value of the van each year.
Activity Synthesis
Invite previously identified students to explain how they chose the graph that represents the value of the first van. Make sure students see that:
The $40,000 price of the van appears as the vertical intercept of the graph.
The 15% rate of decay tells you that after 1 year, the van loses 15% of its value, retaining 85% of its original value.
\((0.85) \boldcdot 40,\!000 = 34,\!000\)
, so the point
\((1, 34,\!000)\)
must be on the correct graph.
If students haven’t already shown (in their partner discussions) that they understood why graphs A, C, and D cannot be the right representations, clarify the reasons (as shown in the Student Response).
Conclude the discussion by selecting previously identified students to share how they used expressions to answer questions about the value of the vans over the years. If no students wrote expressions, invite them to do so now. Highlight that:
Calculating the decay factor means subtracting the depreciation from 100%. For example, the first van loses 15% of its value each year, which means each year, it is worth only 85% of the amount of the previous year.
To find the value after 8 years, we could write an expression (
\(40,\!000 \boldcdot (0.85)^t\)
for the first van and
\(30,\!000 \boldcdot (0.9)^t\)
for the second van, where
\(t\)
represents time in years since purchase) and calculate its value when the input is 8.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole class discussion. At the appropriate time, invite students to create a visual display that shows how they determined the values of each van 8 years after purchase. Groups should consider how to display their calculations so that another student can interpret them. Some groups may wish to add notes or details to their displays to help communicate their thinking. Give students 3–4 minutes to quietly circulate and observe at least two other visual displays in the room. Invite groups to discuss what they noticed about what is the same and what is different between their display and the others they observed. Listen for and amplify the language students use to interpret the exponential expressions that represent the situation.
Design Principle(s): Cultivate conversation; Support sense-making


correct_option_index: 7