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 ratio concepts and use ratio reasoning to solve problems.

Analyze proportional relationships and use them to solve real-world and mathematical problems.
problem_activity: Activity
This activity is intended to build confidence and facility in writing an equation for a proportional relationship. Students build on their understanding that measurement conversions can be represented by proportional relationships, which they studied in an earlier lesson and will revisit in future lessons. Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and use repeated reasoning (MP8) to arrive at an equation. In addition, they are expected to identify the relationship between the constant of proportionality when going from centimeters to meters and vice versa (reciprocals). Since students have already explored a similar relationship (centimeters and millimeters) in a previous lesson, this activity may go very quickly.
As students work, look for students writing an equation like
\(100k = 1\)
(for table 2) as a step to finding the constant of proportionality. Encourage them to say how they would solve the equation. Ask students to say why using this equation makes sense in the scenario.
Launch
Introduce the task: “In a previous lesson, you examined the relationship between millimeters and centimeters. Today we will examine the relationship between centimeters and meters.”
Student Facing
There are 100 centimeters (cm) in every meter (m).
| length (m) | length (cm) |
|------------|-------------|
|     1      |     100     |
|    0.94    |             |
|    1.67    |             |
|   57.24    |             |
|   \(x\)    |             |
| length (cm) | length (m) |
|-------------|------------|
|     100     |     1      |
|     250     |            |
|    78.2     |            |
|    123.9    |            |
|    \(y\)    |            |
Complete each of the tables.
For each table, find the constant of proportionality.
What is the relationship between these constants of proportionality?
For each table, write an equation for the proportional relationship. Let
\(x\)
represent a length measured in meters and
\(y\)
represent the same length measured in centimeters.
Student Response
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Student Facing
Are you ready for more?
How many cubic centimeters are there in a cubic meter?
How do you convert cubic centimeters to cubic meters?
How do you convert the other way?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Invite students to share their answers. These questions can guide the discussion:
"How can we find an equation for each table?"
"Where does the constant of proportionality occur in the table and equation?"
"What is the relationship between the two constants of proportionality? How can you use the equations to see why this should be true?"
The equations can help students see why the constants of proportionality are reciprocals:
\(\displaystyle y = 100x\)
\(\displaystyle \left(\frac{1}{100}\right)y = \frac{1}{100} \boldcdot (100x)\)
\(\displaystyle \left(\frac{1}{100}\right)y = x\)
\(\displaystyle x = \left(\frac{1}{100}\right)y\)
This line of reasoning illustrated above should be accessible to students, because it builds on grade 6 work with expressions and equations.
Ask students to interpret the meaning of the equations in the context: "What do the equations tell us about the conversion from meters to centimeters and back?"
Given the length in meters, to find the length in centimeters, multiply the number of centimeters by 100.
Given the length in centimeters, to find the the length in meters, multiply the number of meters by
\(\frac{1}{100}\)
.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because….,” “I noticed _____ so I….,” “Why did you . . .?,” and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion supports.
Provide students with sentence frames such as “I found the constant of proportionality by _____” to help explain their reasoning. Revoice student ideas to model mathematical language use by restating a response as a question in order to clarify, and apply appropriate language.
Design Principle(s): Maximize linguistic and cognitive awareness


correct_option_index: 1