few_shots: Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Your response:
Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001392

8
options: Gain familiarity with factors and multiples.

Represent and analyze quantitative relationships between dependent and independent variables.

Analyze patterns and relationships.

Multiply and divide within 100.

Represent and solve problems involving multiplication and division.

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

Represent and solve problems involving addition and subtraction.

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

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

Work with radicals and integer exponents.

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

Work with addition and subtraction equations.

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

Use the four operations with whole numbers to solve problems.

Understand the relationship between zeros and factors of polynomials.

Solve equations and inequalities in one variable.

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

Interpret the structure of expressions.

Solve systems of equations.

Write expressions in equivalent forms to solve problems.

Add and subtract within 20.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

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

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Create equations that describe numbers or relationships

Perform arithmetic operations on polynomials.

Understand solving equations as a process of reasoning and explain the reasoning.
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
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
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
Teachers with a valid work email address can
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: 25