few_shots: Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
Answer: B

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

Example 3:
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.


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_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Understand the connections between proportional relationships, lines, and linear equations.

Work with radicals and integer exponents.

Generate and analyze patterns.

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

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

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

Use properties of operations to generate equivalent expressions.

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

Add and subtract within 20.

Represent and solve problems involving multiplication and division.

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

Reason about and solve one-variable equations and inequalities.

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

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Rewrite rational expressions.

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

Interpret the structure of expressions.

Solve systems of equations.

Understand the relationship between zeros and factors of polynomials.

Represent and solve equations and inequalities graphically.

Multiply and divide within 100.

Write and interpret numerical expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Perform arithmetic operations on polynomials.

Use the four operations with whole numbers to solve problems.

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

Analyze patterns and relationships.

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

Gain familiarity with factors and multiples.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use polynomial identities to solve problems.

Solve equations and inequalities in one variable.
problem_activity: Activity
The purpose of this activity is for students to solve two related quadratic equations by completing the square. The process of solving will be nearly identical, with the difference being that one will involve square roots of a negative number and the other will not. It is not necessary for students to prove a general statement about all monic quadratic equations, but it is important for students to compare and understand why some have real solutions and others have non-real solutions.
Launch
Representation: Internalize Comprehension.
Activate or supply background knowledge about completing the square. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Solve these equations by completing the square.
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 19 = 0\)
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
For which values of
\(a\)
does the equation
\(x^2 - 8x + a = 0\)
have two real solutions? One real solution? No real solutions? Explain your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share their solutions. Record and display their thinking for all to see. If not brought up during the discussion, ask students to clarify why one equation has real solutions while the other has complex solutions.
Display these three equations for all to see:
\(x^2+14x+44=0\)
\(x^2+14x+49=0\)
\(x^2+14x+54=0\)
Ask students, “How many solutions do these equations have? Which, if any, of these equations has solutions that involve imaginary numbers?” (
\(x^2+14x+44\)
has two real solutions,
\(x^2+14x+49\)
has one real solution, and
\(x^2+14x+54\)
has two non-real solutions. Compare each left hand side to the perfect square
\((x+7)^2\)
, which is equal to
\(x^2+14x+49\)
. When completing the square, the first equation will have
\((x+7)^2 = 5\)
, the second equation will have
\((x+7)^2 = 0\)
, and the third equation will have
\((x+7)^2 = \text-5\)
.)
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to compare characteristics of equations that have real and complex solutions. Call students’ attention to phrases such as “the right hand side is negative”, “factored form”, or “replace with the imaginary number
\(i\)
.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language to explain solutions to quadratic equations.
Design Principle(s): Maximize meta-awareness


correct_option_index: 33