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.

Analyze functions using different representations.

Use functions to model relationships between quantities.

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

Construct and compare linear and exponential models and solve problems.

Define, evaluate, and compare functions.

Prove and apply trigonometric identities.

Model periodic phenomena with trigonometric functions.

Extend the domain of trigonometric functions using the unit circle.

Understand the concept of a function and use function notation.

Build a function that models a relationship between two quantities.
problem_activity: Activity
In this activity, students again apply what they learned about quadratic expressions and their graphs. This time, they solve a contextual problem. Students consider how to change the parameters of a quadratic expression that models the movement of a digitally animated object—a peanut—such that it travels in a parabolic path and meets certain restrictions (MP4).
Providing time to notice and wonder will elicit the idea that this equation shows the horizontal and vertical movement of the peanut, which will be useful when students adjust the equation in a later activity. While students may notice and wonder many things about the graphs and equation, connecting the vertex of the graph to the parameters in the equation are the important discussion points. This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).
For the second question, there are many possible values of
\(h\)
and
\(k\)
that students could use. As students work, notice those who choose to:
Shift the starting position of the peanut closer to the wall (translating the graph horizontally to the right).
Stay in the same starting position but shift the vertex to a point closer to the wall.
Shift both the peanut and the vertex of the parabolic path.
If students choose to use graphing technology to help them experiment with expressions or verify their predictions, they practice choosing tools strategically (MP5).
Digital
Launch
You may wish to use the applet during the synthesis to demonstrate student equations. There is a copy of the applet with an option to show the path of the peanut.
Representing, Conversing: MLR7 Compare and Connect.
As students share their values of
\(h\)
and
\(k\)
with the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the
\(h\)
and
\(k\)
represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to show the path of the main character, an animated peanut, jumping over a wall.
Mai uses the equation
\(y = \text-0.1(x-h)^2+k\)
to represent the path of the jump.
\(y\)
represents the height of the peanut as a function of the horizontal distance it travels
\(x\)
. On the screen, the base of the wall is located at
\((22, 0)\)
, with the top of the wall at
\((22,4.5)\)
.
The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.
[IMG]
What are the values of
\(k\)
and
\(h\)
in this equation?
This applet is programmed to use Mai’s equation. Experiment with the applet to find an equation to get the peanut over the wall, and keep it on the screen. Be prepared to explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Ask students to read the first couple of paragraphs in the statement. Then, display this graph for all to see. Tell students that when Mai used the equation
\(y = \text -0.1(x-16)^2+7\)
, this is the path of the peanut’s jump.
[IMG]
Ask students what they notice and wonder about the equation and the graph. Make sure students see the connection between the parameters in the equation and the vertex of the graph before they proceed with the activity.
Provide continued access to graphing technology, in case students choose to use it.
Representing, Conversing: MLR7 Compare and Connect.
As students share their values of
\(h\)
and
\(k\)
with the class, call students’ attention to the different ways the output is represented graphically within the context of the situation. Take a close look at the equation to distinguish what the
\(h\)
and
\(k\)
represent in each situation. Wherever possible, amplify student words and actions that describe the connections between a specific feature of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to model the path of the main character, an animated peanut, jumping over a wall.
[IMG]
Mai uses the equation
\(y = \text-0.1(x-h)^2+k\)
to represent the path of the jump.
\(y\)
represents the height of the peanut as a function of the horizontal distance it travels,
\(x\)
.
On the screen, the base of the wall is located at
\((22, 0)\)
, with the top of the wall at
\((22,4.5)\)
. The dashed curve in the picture shows the graph of 1 equation Mai tried, where the peanut fails to make it over the wall.
[IMG]
What are the values of
\(h\)
and
\(k\)
in this equation?
Starting with Mai’s equation, choose values for
\(h\)
and
\(k\)
that will guarantee the peanut stays on the screen but also makes it over the wall. Be prepared to explain your reasoning.
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
Some students may randomly choose values for
\(h\)
and
\(k\)
until they find a combination whose graph meets the requirements. Encourage them to reason about the problem more systematically, by considering what they learned about the vertex form.
Activity Synthesis
Select students with contrasting strategies to share their equations and explanations for the second question. To verify that students’ proposed equations meet the animation criteria, consider graphing students’ proposed equations for all to see. Highlight the three strategies mentioned in the Activity Narrative.
Point out one distinction between the graphs in this activity and previous graphs that represent projectile motions. In earlier cases, the output represented the height of an object as a function of
time
, so the graph of the function and physical trajectory of the object may be very different (for example, the object may be going straight up and then fall straight down, but the graph shows a parabola). Here, they see a model where the height of an object is a function of
horizontal distance
, so the path of the object in a projectile motion does resemble the graph representing the function.


correct_option_index: 1

2