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.
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fl_problem_000674

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

Add and subtract within 20.

Analyze patterns and relationships.

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

Reason about and solve one-variable equations and inequalities.

Work with radicals and integer exponents.

Generate and analyze patterns.

Rewrite rational expressions.

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

Understand the relationship between zeros and factors of polynomials.

Interpret the structure of expressions.

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

Represent and solve problems involving addition and subtraction.

Write expressions in equivalent forms to solve problems.

Represent and solve problems involving multiplication and division.

Create equations that describe numbers or relationships

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Represent and solve equations and inequalities graphically.

Solve equations and inequalities in one variable.

Perform arithmetic operations on polynomials.

Solve systems of equations.

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

Write and interpret numerical expressions.

Use properties of operations to generate equivalent expressions.

Use the four operations with whole numbers to solve problems.

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

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

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

Use polynomial identities to solve problems.

Multiply and divide within 100.
problem_activity: Activity
Students learn that we can still organize our work with the distributive property in a familiar way, even with negative numbers where thinking in terms of area breaks down.
Launch
Display the image and ask students to write an expression for the area of the big rectangle in at least 3 different ways.
[IMG]
Collect responses. If students simply say “16,” ask them to explain how they calculated 16 and record these processes for all to see. Remind students that thinking about area gives us a way to understand the distributive property. This diagram can be used to show that
\(2\boldcdot 5+2\boldcdot 3=2(5+3)\)
. Be sure that students see you write the partial products in the diagram, and that they see every piece of the associated identity
\(2\boldcdot 5+2\boldcdot 3=2(5+3)\)
.
[IMG]
Tell students that when we are working with negative numbers, thinking about area doesn’t work so well, but the distributive property still holds when there are negative numbers. The expressions involved still have the same structure, and we can still organize our work the same way.
Representation: Internalize Comprehension.
Differentiate the degree of difficulty or complexity by beginning with an example with more accessible values, such as the one given. Highlight connections between representations by recording the calculations in the boxes. In addition, consider creating a display showing a general example using only variables and keeping it as a reference throughout the remainder of the unit.
Supports accessibility for: Conceptual processing
Student Facing
Write two expressions for the area of the big rectangle.
[IMG]
Use the distributive property to write an expression that is equivalent to
\(\frac12(8y + \text-x + \text-12)\)
. The boxes can help you organize your work.
[IMG]
Use the distributive property to write an expression that is equivalent to
\(\frac12(8y - x - 12)\)
.
Student Response
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Student Facing
Are you ready for more?
Here is a calendar for April 2017.
[IMG]
Let's choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11.
Average these four numbers. What do you notice?
Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice?
Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Solicit responses to the second question and demonstrate thinking about one product at a time:
[IMG]
Then ask students to share how they approached the last question. Highlight responses where students noticed that
\(\frac12(8y - x - 12)\)
can be rewritten like
\(\frac12(8y + -x + -12)\)
(because of what they talked about in the warm-up). So the two questions have the same answer.
Speaking, Representing: MLR7 Compare and Connect.
Use this routine when students present their expressions. Ask students “What is the same and what is different?” about the approaches and representations involving subtraction with the distributive property. Help students connect how the expressions that have a subtraction operation are equivalent to expressions that add its additive inverse. These exchanges strengthen students’ mathematical language use and reasoning with the distributive property and the subtraction operation.
Design Principle(s): Maximize meta-awareness


correct_option_index: 26