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:
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.


Your response:
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.


Your response:
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.
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fl_problem_000674

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options: Analyze patterns and relationships.

Multiply and divide within 100.

Use properties of operations to generate equivalent expressions.

Write expressions in equivalent forms to solve problems.

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

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

Understand the relationship between zeros and factors of polynomials.

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

Solve equations and inequalities in one variable.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and solve problems involving addition and subtraction.

Reason about and solve one-variable equations and inequalities.

Solve systems of equations.

Add and subtract within 20.

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

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

Use polynomial identities to solve problems.

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

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

Interpret the structure of expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Generate and analyze patterns.

Use the four operations with whole numbers to solve problems.

Write and interpret numerical expressions.

Represent and solve equations and inequalities graphically.

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

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

Work with radicals and integer exponents.

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

Create equations that describe numbers or relationships

Represent and solve problems involving multiplication and division.

Perform arithmetic operations on polynomials.

Rewrite rational expressions.
problem_activity: Activity
Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name. Refining their language about the different forms prepares students to be more precise in their thinking about the graphs of quadratic functions in later lessons (MP6).
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 1–2 partners to share their response to the final question, “Why do you think that form is called factored form?” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “Can you say more about what each expression means?” or “I understand _____, but can you clarify . . .” Next, provide students with 2–3 minutes to revise their initial draft based on feedback from their peers.
Design Principle(s): Optimize output (for explanation)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer that can be used to brainstorm the activity. Include one column for standard form and one column for factored form. Include a space at the bottom of the organizer for student-generated notes or reminders about standard form and factored form.
Supports accessibility for: Language; Organization
Student Facing
The quadratic expression
\(x^2 + 4x + 3\)
is written in
standard form
.
Here are some other quadratic expressions. The expressions on the left are written in standard form and the expressions on the right are not.
Written in standard form:
\(x^2 – 1\)
\( x^2 + 9x\)
\(\frac12 x^2\)
\(4x^2 – 2x + 5\)
\(\text-3x^2 – x + 6\)
\(1 - x^2\)
Not written in standard form:
\((2x + 3)x\)
\((x+1)(x-1)\)
\(3(x-2)^2 +1\)
\( \text-4(x^2 + x) +7\)
\( (x+8)(\text{-}x+5)\)
What are some characteristics of expressions in standard form?
\((x+1)(x-1)\)
and
\((2x + 3)x\)
in the right column are quadratic expressions written in
factored form
. Why do you think that form is called factored form?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
What quadratic expression can be described as being both standard form and factored form? Explain how you know.
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Display the two columns of expressions for all to see. Solicit students’ ideas on the features of each form. Record their responses for all to see. Invite other students to express agreement or disagreement or to clarify their fellow students’ responses.
Define a quadratic expression in
standard form
explicitly as
\(ax^2 + bx + c\)
. Explain that we refer to
\(a\)
as the coefficient of the squared term
\(x^2\)
,
\(b\)
as the coefficient of the linear term
\(x\)
, and
\(c\)
as the constant term.
Ask students:
“How would you write
\((2x + 3)x\)
in standard form?” (
\(2x^2 + 3x\)
)
“The expression
\(2x^2 + 3x\)
only has two terms. Is it still in standard form?” (Yes, there is no constant term, which means
\(c\)
is 0.)
“How would you write
\(\text {-4}(x^2 + \frac14x) +7\)
in standard form?” (
\(\text {-4}x^2 - x + 7\)
)
“What are the values of the coefficients
\(a\)
and
\(b\)
?” (-4 and -1) “What about the constant term?” (7)
Then, clarify that a quadratic expression in
factored form
is a product of two factors that are each a linear expression. For example,
\((x+1)(x-1)\)
,
\((2x + 3)x\)
, and
\(x(4x)\)
all have two linear expressions for their factors. An expression with two factors that are linear expressions and a third factor that is a constant, for example:
\(2(x+2)(x-1)\)
, is also in factored form.
Tell students that each form gives us interesting or useful information about the function that the expression represents, and they will learn more about this in future lessons.


correct_option_index: 3

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